THE CLASSIFICATION OF THE SIMPLE LIE SUPERALGEBRAS OF VECTOR FIELDS

DIMITRY LEITES^{1}, IRINA SHCHEPOCHKINA^{2}

Abstract. Infinite dimensional Lie superalgebras of vector fields with polynomial or formal coefficients together with particular gradings, called Weisfeiler gradings, will be briefly called W-graded vectorial Lie superalgebras. The importance of W-gradings and the name of these algebras are due to the fact that they are determined by a maximal subalgebra L_{0} ⊂ L of finite codimension, so every vectorial L is realized by means of vector fields on (L/L_{0})^{∗}. We classify simple complex W-graded vectorial Lie superalgebras, and discuss their deformations (the classification will be given elsewhere).

We show that a straightforward extension of the original É. Cartan’s problem to superalgebras, i.e., “classification of primitive Lie superalgebras”, is a wild problem and distill a tame subproblem solved here.

Observe that (1) unlike Lie algebras, the Lie superalgebra of vector fields may have several (finitely many) non-isomorphic W-gradings all of which we list; (2) the parameters of deformations of Lie superalgebras can run over a supervariety.

To the memory of A. N. Tyurin.

Introduction

This is a continuation of [Sh14]. The term “exceptional”, first used in [Sh3], implies that a classification is ready or “almost”. In [LSh1], a preliminary result was described together with main steps of classification. In 1996-97 we have explicitly announced the classification of W-graded simple complex vectorial Lie superalgebras at various conferences, e.g., [LSh2]. Here is the proof and examples of deformations (the classification will be given elsewhere).

Summary: there are 4 series of simple W-graded Lie algebras (Cartan); passing to Lie superalgebras we find 34 series and 15 exceptional algebras.

Several of these superalgebra admit deformations. These deformations may have odd parameters; e.g., the quantization of the Poisson bracket is a deformation with even parameter, the quantization of the antibracket is a deformation with odd parameter.

As abstract ones, W-gradings disregarded, these algebras can be collected into 13 series (one of which is exceptional) and 5 exceptional families. We would like to draw the attention of the reader to the fact that our economical way to describe vectorial Lie superalgebras as regradings of several particular ones, chosen as “points of reference”, hides somewhat the peculiarity of Lie superalgebras of having several totally distinct models as vectorial algebras, i.e., as regards to the structures they preserve or, in inner terms, as (generalized) Cartan

1991 Mathematics Subject Classification. 17A70 (Primary) 17B01, 17B70 (Secondary). Key words and phrases. Lie superalgebras, vector fields, deformations of Lie superalgebras. We are thankful to D. Alekseevsky, J. Bernstein, P. Grozman, I. Kantor, Yu. Kochetkov, G. Post and

A. Rudakov for help. Discussions with I. Kantor, D. Alekseevsky and A. Rudakov in 1978 were especially crucial; Kantor was teh first to observe several of the regradings. Financial support of D.L. by NFR throughout the work, and of I.Shch. by INTAS grant 96-0538 at the last stage is thankfully acknowledged. D.L. also thanks Mittag-Leffler Institute, Harvard University; IHES; Hebrew University of Jerusalem, Tel-Aviv University and IAS, Princeton, for hospitality at the earlier stages of this research and where main results and ideas were delivered (1986 – 1991) and especially MPIM, Bonn, where the first draft ([LSh1]) was typed and where the current Introduction was finally moulded.

2 DIMITRY LEITES^{1}, IRINA SHCHEPOCHKINA^{2}

prolongs. Passage to real forms makes it manifest how vital it is to distinguish the filtration (realization): real forms do not, as a rule, survive regradings.

I.0. On setting of the problem. Selection of Lie algebras with reasonably nice properties is a matter of taste and is influenced by the initial “big” problem considered. One of the usual choices is the class of simple algebras: they have a richer structure and, therefore, are easier to study than other types of Lie algebras; they also illuminate important symmetries. Even representatives of the “complementary” types — nilpotent and solvable algebras — are often of real interest only when subalgebras of simple algebras.

Of simple Lie algebras, finite dimensional ones are naturally the first to study. Over ℂ, they constitute three “classical” series (𝔰𝔩, 𝔬 and 𝔰𝔭) and five exceptional algebras; all are neatly encoded by very simple graphs — Dynkin diagrams — or Cartan matrices.

Next on the agenda are ℤ-graded Lie algebras of polynomial growth (let us call them ZGLAPGs for short). Some of these algebras have Cartan matrix; they resemble finite dimensional simple Lie algebras more than others, and have proved very useful in various branches of mathematics and theoretical physics.

Let us recall the meaning of the terms entering ZGLAPG. A ℤ-graded algebra A = ⊕A_{i} is said to be simple graded if it has no homogeneous or better say graded ideals (i.e., no ideals I = ⊕I_{i} such that I_{i} = I ∩ A_{i}; a simple graded algebra can be nonsimple as an abstract algebra: e.g., loop algebras with values in a simple Lie algebra are simple graded but nonsimple). If dim A_{i} < ∞ for all i, one can define the growth of A to be gth(A) =

\overline{lim_{n→∞}} \frac{ln dim ⊕_{|i|≤n}A_{i}}{ln n}. We say that A is of polynomial growth if gth(A) < ∞.

Around 1966 V. Kac and Weisfeiler started to study simple filtered Lie algebras of polynomial growth, i.e., Lie algebras such that the associated graded algebra is of polynomial growth. The first step in the study was, of course, the simple graded algebras. Kac classified simple ZGLAPGs provided they are generated by elements of degree ±1. Among simple ZGLAPGs only 𝔳𝔢𝔠𝔱(1) = 𝔡𝔢𝔯 ℂ[x] and 𝔳𝔢𝔠𝔱^{L}(1) = 𝔡𝔢𝔯 ℂ[x^{−1}, x] are not generated by elements of degree ±1 with respect to any grading. Kac conjectured that these examples plus his list exhaust all the simple ZGLAPGs. Twenty years after O. Mathieu proved this conjecture in a string of rather complicated papers culminating in [M].

It goes without saying that we look at the simple Lie superalgebras as a preliminary (though vital) material. The central extensions, deformations and algebras of outer derivations of simple Lie superalgebras are often more interesting in applications (like affine Kac– Moody algebras are “more useful” than loop algebras; likewise, Poisson algebras and their deformations are often no less appealing to the heart of the physicist than even the Lie algebras of Hamiltonian vector fields, vital in the classical mechanics). So the real object of interest is the answer to the following Problem 0:

0. Problem. Starting with a simple Lie algebra of class ZGLAPG, list the results of iterated combinations of the following operations:

a) 𝔤 ⟶ 𝔢(𝔤), the nontrivial central extension of 𝔤 (example: central extensions of loop algebras; the Poisson algebra);

b) 𝔤 ⟶ 𝔡𝔢𝔯(𝔤), the derivation algebra of 𝔤 (example: affine Kac–Moody algebras);

c) deformations of 𝔤 (though they sometimes lead out of the class ZGLAPG, we should not disregard them since such deformed Lie algebras are often no less important in applications than the original algebras; examples: the result of the quantization of either the Poisson algebra or of the algebra of Hamiltonian vector fields; P.Golod’s and Krichever-Novikov’s examples);

d) all filtered completions of 𝔤 and various ℤ-gradings; e) forms of these algebras over non-closed fields.

We will loosely refer to the results of the iterated procedures described in Problem 0 as classical Lie algebras and call the algebras obtained from a simple one, 𝔤, by the above procedures the relatives of 𝔤 and each other. To list them, their counterparts in the supersetting and/or in prime characteristic, their various (not only ℤ-) gradings is the main strategic goal of our study. Remark . The reader should not think, as was customary 30 years ago, that “bigger” algebras are useless or too difficult to study. For example, various versions of 𝔤𝔩(∞) and algebras studied by Borcherds and Gritsenko-Nikulin are both, though huge, of huge interest and their representations describable.

I.0.1. Types of the classical Lie algebras. Let us qualitatively describe the simple Lie algebras of polynomial growth to better visualize them. They break into the disjoint union of the following types:

1) finite dimensional algebras (growth 0).

2) loop algebras, perhaps, twisted (growth 1); more important in applications are their non-simple “relatives” called affine Kac–Moody algebras, cf. [K3].

3) vectorial algebras, i.e., Lie algebras of vector fields^{1} with polynomial coefficients (growth is equal to the number of indeterminates) or their completions with formal power series as coefficients.

4) the stringy^{2} algebra 𝔳𝔢𝔠𝔱^{L}(1) = 𝔡𝔢𝔯 ℂ[t^{−1}, t] (the superscript stands for Laurent). This class consists of one algebra. It is often called Witt algebra and denoted 𝔴𝔦𝔱𝔱 in honor of Witt who considered its analog over fields of prime characteristic, and by physicists the centerless Virasoro algebra because its nontrivial central extension (discovered by Fuchs and Gelfand), 𝔳𝔦𝔯, is called the Virasoro algebra in honor of Virasoro who rediscovered the corresponding central extension and indicated its importance in physical models.

Strictly speaking, stringy algebras are also vectorial, but we retain the generic term vectorial for algebras with polynomial or formal coefficients.

The above are graded algebras. Filtered algebras are not classified (see examples to Problem 0). Among the known examples of filtered algebras, we distinguish various deformations of the above and

5) the Lie algebra of matrices of complex size and its generalizations, cf. [GL2, LSe, DGS]. These algebras are simple filtered Lie algebras of polynomial growth whose associated graded algebras are not simple. (Though they are much more numerous than simple graded ones, we still hope that it is possible to distill a tame subproblem: up to some twists, the examples known to us (modulo current algebras and algebras of vector fields on affine varieties, cf. [L7]) are multiparameter deformations of the Poisson algebras of functions on the orbits in the coadjoint representation of simple finite dimensional or Lie algebras of matrices over the rings of differential operators. Super version of this class is even wider due to S. Montgomery’s and Konstein’s constructions ([GL2, KV]).)

1These algebras are sometimes known under the recently introduced clumsy name “algebras of Cartan type”: just imagine “the Cartan subalgebra in the Lie superalgebra of Cartan type with Cartan matrix”.

2The term induced by the lingo of imaginative physicists who now play (and share with us sensation of beauty of stringy superalgebras [GSW] and urge to investigate them, or take a calmer attitude [WL]) with the idea that an elementary particle is not a point but rather a slinky springy string; the term “stringy algebra”

means “pertaining to string theory” but also mirrors their structure as a collection of several strings — the 𝔴𝔦𝔱𝔱-modules. In our papers “stringy” means “simple vectorial superalgebra on a supermanifold whose base is a circle or a relative of such algebra (see Problem 0)”.

The algebras of the above classes 1) – 5) can be characterized by other properties that are sometimes taken for their definition. For example, algebras 𝔤 of classes 1), 2) and 4) are “symmetric”, i.e. with every root space 𝔤_{α} they also have 𝔤_{−α} of the same dimension as 𝔤_{α}. Additionally, algebras of classes 1) and 2) have Cartan matrix. This matrix is always symmetrizable and, moreover, not only integer but of so simple form that it can be encoded by a simple graph (Dynkin diagram). Algebras with a symmetric Cartan matrix and (most of the) algebras of class 5) possess an invariant symmetric nondegenerate bilinear form.

I.0.2. Superization. After Wess and Zumino made importance of supersymmetries manifest, cf. [D, WZ], it was natural to classify simple “classical” Lie superalgebras. It immediately turned out that the above types of simple Lie algebras and their properties are intermixed under the passage to superalgebras.

— Finite dimensional superalgebras. They split into two types:

“symmetric” ones, subdivided into

sy^{CM} the ones with (symmetrizable) Cartan matrix (since the matrix is symmetrizable, the algebras possess a nondegenerate even supersymmetric bilinear form) and

sy^{W CM} the ones without Cartan matrix in the conventional sense but with a nondegenerate supersymmetric bilinear form that can be either even or odd;

“skew-symmetric” or vectorial types: with more positive roots than it has negative ones.

— (Twisted) loop superalgebras. It is obvious that loop superalgebras with values in simple Lie superalgebras are also simple (as graded algebras). Unlike Lie algebras, some of the loop superalgebras have no Cartan matrix, some have a non-symmetrizable ones, etc., so classification of (twisted) loop superalgebras splits into several very different cases. The very first examples showed that even if a finite dimensional 𝔤 possesses a Cartan matrix, the twisted loop superalgebra 𝔤_{φ}^{(k)} (for notations and theory in non-supercase see [K3]) may possess no Cartan matrix and vice versa.

An intrinsic characterization of loop superalgebras without apeal to Cartan matrix (deduced from Mathieu [M]) is given in [GLS1] together with same for stringy superalgebras: both types of algebras are ℤ-graded 𝔤 = ⊕_{i=−d}^{∞} 𝔤_{i} of infinite depth d but in the adjoint repre-

sentation they act differently:

for the loop algebras every root vector corresponding to any real root

acts locally nilpotently in the adjoint representation,

for the stringy algebras this is not so.

(∗)

In the absence of the above intrinsic definition, one of us (DL) tried to approach the classification “from the end” and, in order to get a supply of examples, conjectured that, as for Lie algebras, simple twisted loop superalgebras correspond to outer automorphisms of φ ∈ Aut𝔤/Int𝔤. Serganova listed these automorphisms and amended the conjecture: to get distinct algebras, one should factorize Aut 𝔤 modulo the connected component of the unity (which is somewhat greater than Int 𝔤, the group of inner automorphisms of 𝔤), see [Se3], [FLS]. An a priori classification of Lie superalgebras of polynomial growth with symmetrizable Cartan matrix is due to J. van de Leur [vdL]; his classification and Serganova’s unpublished classification of Lie superalgebras of polynomial growth with nonsymmetrizable Cartan matrix cited in [LS1], [GLS1] support Leites-Serganova’s conjecture on the completeness of the list from [FLS].

— Stringy superalgebras. For the conjectural list of simple ones see [GLS1]. Partly (with several extra conditions) this conjecture is proved in [KvdL], [K8]. Observe that some of the stringy superalgebras possess Cartan matrix, though non-symmetrizable ones [GLS1], and have deformations whereas their vectorial namesakes are rigid.

Remark . In the literature there are several definitions of what we call stringy superalgebras, mostly self-contradicting ones. For example, in all physical papers and books stringy superalgebras are called “superconformal” ones (meaning conformal superalgebras). In reality, only 𝔴𝔦𝔱𝔱, 𝔨^{L}(1|1) and 𝔨^{M}(1|1) are conformal in the original sense of the term ([GLS1]); but even if other superalgebras had been conformal, their central extensions are not, though the term “superconformal” is magnanimously applied to all. In some papers the “superconformal” superalgebras are defined as “simple ... such that considered as 𝔴𝔦𝔱𝔱-module ...”, whereas in physical papers everybody calls Virasoro, Neveu-Schwarz and Ramond superalgebras “superconformal” ones, though they are neither simple nor conformal. Therefore, we suggest the term stringy for the general class of algebras inside of which some are conformal, some are simple, etc.

Some of the simple stringy superalgebras ([GLS1]) are distinguished: they admit a nontrivial central extensions. In [GLS1], there are also indicated exceptional stringy superalgebras and occasional isomorphisms unnoticed in previous papers and ignored, so far, in later ones.

In this paper we consider the remaining type of simple graded Lie superalgebras of polynomial growth:

Vectorial Lie superalgebras. Main examples to keep in the mind’s eye are the Lie algebra L = 𝔡𝔢𝔯 ℂ[x] of polynomial vector fields with grading and filtration given by setting deg x_{i} = 1 for all i, and its (x)-adic completion, the filtered Lie algebra L = 𝔡𝔢𝔯 ℂ[[x]] of formal vector fields.

Cartan came to these algebras from geometrical problems in which “primitive” Lie algebras naturally arise. Infinite dimensional primitive Lie algebras turned out to be isomorphic to the algebras of derivations of simple ones (one of the items in Problem 0), and since dim 𝔡𝔢𝔯 𝔤/𝔤 ≤ 1, to classify infinite dimensional primitive algebras is practically the same as to classify the simple vectorial ones (the list of finite dimensional primitive algebras is much longer than that of simple ones but still reasonable [O]). Contrariwise, the classification problem of primitive Lie superalgebras is wild, as shown in [ALSh] and Appendix. So, having distinguished a reasonable problem, we will study simple Lie superalgebras.

I.1. Classificational problems and Main Theorem. Consider infinite dimensional complex filtered Lie superalgebras L̃ with decreasing filtration of the form

L̃ = L̃_{−d̃} ⊃ L̃_{−d̃+1} ⊃ ⋯ ⊃ L̃_{0} ⊃ L̃_{1} ⊃ …  (F)

where depth d̃ is finite and where 1) L̃_{0} is a maximal subalgebra of finite codimension;

2) L̃_{0} does not contain ideals of the whole L̃. The algebra L̃ with such a subalgebra L̃_{0} is called a primitive Lie superalgebra.

We assume that these Lie superalgebras L̃ are complete with respect to a natural topology whose basis of neighborhoods of zero is formed by the spaces of finite codimension, e.g., the L̃_{i}. (In the absence of odd indeterminates this topology is the most natural one: we consider two vector fields k-close if their coefficients coincide up to terms of degree > k.) This topology is naturally (see. §1) called, “briefly”, projective limit topology but even more unfortunate term “linearly compact topology” is also used. No other topology will be encountered, so all topological terms (complete, open, closed) refer to this topology.

Observe that the very term “filtered algebra” implies that [L̃_{i}, L̃_{j}] ⊂ L̃_{i+j}. Set L̃_{i} = L̃_{i}/L̃_{i+1}. Conditions 1) and 2) manifestly imply that dim L̃_{i} < ∞ for all k and the ℤ-graded Lie superalgebra L̃ = ⊕_{k≥−d̃}L̃_{k} associated with L̃ grows polynomially, i.e., dim ⊕_{k≤n} L̃_{k} grows

as a polynomial in n.

Weisfeiler endowed every such filtered Lie algebra L̃ with another, refined, filtration L̃ = L:

L = L_{−d} ⊃ L_{−d+1} ⊃ ⋯ ⊃ L_{0} ⊃ L_{1} ⊃ …  (W F 1)

by setting L_{0} = L̃_{0} and letting L_{−1} to be a minimal L_{0}-invariant subspace strictly containing L_{0}, different from L_{0} and L_{0}-invariant, the other terms being defined by the formula (for i ≥ 1):

L_{−i−1} = [L_{−1}, L_{−i}] + L_{−i} and L_{i} = {D ∈ L_{i−1} | [D, L_{−1}] ⊂ L_{i−1}}.  (W F 2)

The d in (WF1) is called the depth of L and of the associated with L̃ graded Lie superalgebra L.

An advantage of the Weisfeiler filtrations is that for the corresponding regraded Lie superalgebra L the L_{0}-action on L_{−1} is irreducible (see sec. I.0.2). These refined filtrations are called, after [W], Weisfeiler filtrations even for Lie superalgebras, where Weisfeiler’s construction is literally applied; we will shortly write W-filtrations and call the gradings associated with W-filtrations W-gradings.

Now, observe that condition 2) on filtrations considered is equivalent to the following condition which is sometimes more convenient to use:

2′) for any non-zero x ∈ L̃_{k} for k ≥ 0, where L̃_{k} = L̃_{k}/L̃_{k+1}, there exists y ∈ L̃_{−1} such that [x, y] ≠ 0.

When the L_{0}-module L_{−1} is faithful, as is always the case for simple Lie superalgebras L, such filtered Lie superalgebras L (and the associated with them graded ones, L) can be realized by vector fields on the supermanifold corresponding to the linear superspace (L/L_{0})^{∗} with formal (resp. polynomial) coefficients. So, being primarily interested in simple Lie superalgebras, we assume that L_{0}-module L_{−1} is faithful. We refer to the Lie superalgebras of vector fields as vectorial ones for short.

The following problems arise:

A′) Classify simple W-graded vectorial Lie superalgebras as abstract ones, i.e., distinction in grading disregarded.

Immediately observe that this problem is very unnatural: quite distinct algebras become equivalent. Indeed, as early as in [ALSh], we observed that 𝔳𝔢𝔠𝔱(1|1), the Lie algebra of all vector fields on (1|1)-dimensional superspace, is isomorphic as an abstract algebra to 𝔨(1|2), the Lie algebra of contact vector fields on (1|2)-dimensional superspace (and even to 𝔪(1), another, “odd” type of contact Lie superalgebra). A more natural formulation is, therefore, the following one:

A) Classify simple W-graded vectorial Lie superalgebras.

In some applications it suffices to confine ourselves to graded algebras ([L8]), in other applications, e.g., in the study of representations ([GLS4]), complete algebras are sometimes more important and natural than the associated with them graded ones. So the following problem might be a reasonable goal:

B) Classify simple W-filtered complete vectorial Lie superalgebras.

Observe natural (“trivial”) examples of complete algebras: algebras L̄ obtained from the answers L to Problem A by taking formal series, rather than polynomials, as coefficients. In addition to these examples there might occur nontrivial filtered deforms (as Gerstenhaber calls the result of a deformation), i.e., complete algebras L̃ such that the associated graded of L̄ and L̃ is the same, L. Examples will be given later.

On the road to solution of Problem B the following problem seems to be a natural step: B′) Classify simple W-filtered complete vectorial Lie superalgebras as abstract

ones, i.e., distinction in filtrations disregarded.

Observe that whereas the difference between Problems A′ and B′ is as large as that between Problems A and B, the difference between Problems B and B′ is negligible. Actually, it is impossible to solve Problem B′ and not solve Problem B. Indeed, various filtered deformations are not necessarily isomorphic as abstract algebras, so in order to solve Problem B′ we have to know first all the W-filtrations or the associated W-gradings and then describe filtered deformations for every W-grading.

However, even Problem B is not the most natural one: there exist deforms of algebras from class B which do not lie in class B: e.g., the result of the factorization of the quantized Poisson algebra modulo center. So the true problem one should solve is

C) Describe all the deformations (not only filtered ones) of the simple Wfiltered vectorial Lie superalgebras.

I.1.1. Notations. Comments. We will describe all the Lie superalgebras from the tables in our Main Theorem in detail in due course, but to simplify grasping the general picture from the displayed formulas of the following theorem, let us immediately inform the reader that the prime example, the general vectorial algebra, 𝔳𝔢𝔠𝔱(m|n; r), is the Lie superalgebra of vector fields whose coefficients are polynomials (or formal power series, depending on the actual problem) in m commuting and n anticommuting indeterminates with the filtration (and grading) determined by equating the degrees of r (0 ≤ r ≤ m) of odd indeterminates to 0, the degrees of the remaining indeterminates being equal to 1. The regradings of other series are determined similarly, but in a more complicated way, see I.2.2. Usually, we omit indicating parameter r if r = 0.

On vectorial superalgebras, there are two analogs of trace. The divergence is one of them, the linear functional that vanishes on the brackets is another one. Accordingly, the special (divergence free) subalgebra of 𝔤 is denoted by 𝔰𝔤, e.g., 𝔰𝔳𝔢𝔠𝔱(n|m); the superscript ^{o} singles out the traceless ideal of codimension 1 (or 0|1). Observe that the usual notation is a ′, rather that ^{o}, but we are not considering the derived series and ^{o} is more graphic.

𝔨 and 𝔥 are the straightforward analogs of the contact and hamiltonian series, respectively; 𝔭𝔬(2n|m), the central extension of 𝔥(2n|m), is the Poisson Lie superalgebra.

𝔪 and 𝔩𝔢 are the “odd” analogs of the contact and hamiltonian series, respectively. The bracket in 𝔟, the central extension of 𝔩𝔢, is the classical Schouten bracket, more popular now under the name antibracket. We say “odd” in quotation marks: the series 𝔪 preserves the Pfaff equation with an even contact form, and all these “odd” analogs have both even and odd parts.

The antibracket can be deformed, the deformed algebra is 𝔟_{λ}.

The algebras 𝔰𝔪, 𝔰𝔩𝔢, 𝔥^{o} and 𝔰𝔟 are divergence free subalgebras in respective algebras. 𝔰̃𝔟 is a simple algebra, it is a deformation of a non-simple 𝔰𝔟(n|n + 1; n). Observe an outstanding property of the bracket in 𝔥_{λ}(2|2) ≃ 𝔟_{λ}(2|2): it can be considered

as a deformation of an even bracket as well as a deformation of an odd one.

In the sequel for any 𝔤 we write 𝔠𝔤 = 𝔤⊕ℂ·z or 𝔠(𝔤) to denote the trivial central extension with the 1-dimensional even center generated by z. Any nontrivial central extension is shorthanded as 𝔢(𝔤).

Let 𝔞 ⊂ + 𝔟 or 𝔟 ⊃ + 𝔞 denote the semidirect sum in which 𝔞 is the ideal.

In proofs we use notations of roots in the simple finite dimensional Lie algebras from [OV], Table 5. The standard (identity) representation of a matrix Lie superalgebra 𝔤, i.e., a subalgebra of 𝔤𝔩(V), in V is denoted by id or, for clarity, id_{𝔤}. The trivial representation of 𝔤_{0} is denoted by ℂ (over a simple algebra) whereas ℂ[k] denotes a representation of 𝔤_{0} trivial on its semi-simple part and such that k is the value of the central element z from 𝔤_{0} chosen so that z|_{𝔤_{i}} = i · id_{𝔤_{i}}.

Further elucidations. In formulas (S) and (E) below: in parentheses after the “family name” of the algebra there stands the superdimension of the superspace of indeterminates and — after semicolon — a shorthand description of the regrading r ≠ 0.

Passing from one regrading to another one, we take a “minimal” realization (i.e., with a minimal dim L/L_{0}) as the point of reference. For the exceptional Lie superalgebras another point of reference is often more convenient, the consistent grading, K.

The regradings of the series are governed by a parameter r described in sec. I.2. All regradings are given in (hopefully) suggestive notations, e.g., 𝔨𝔞𝔰^{ξ}(1|6; 3η) means that having taken 𝔨𝔞𝔰^{ξ} as the point of reference we set deg η = 0 for each of the three η’s (certainly, this imposes some conditions on the degrees of the other indeterminates).

The exceptional grading E_{H} of 𝔥_{λ}(2|2) is described in passing in the list of occasional isomorphisms (OI); it is described in detail for another incarnation of this algebra (I.2.1.1).

The finite dimensional Lie superalgebra 𝔥(0|n) of hamiltonian vector fields is not simple: it contains a simple ideal 𝔥^{o}(0|n).

On drop-outs. Several algebras are “drop-outs” from the series. Examples: 𝔰𝔳𝔢𝔠𝔱^{∘}(1|n; r) are “drop-outs” from the series 𝔰𝔳𝔢𝔠𝔱(m|n; r) since the latter are not simple for m = 1 but contain the simple ideal 𝔰𝔳𝔢𝔠𝔱^{∘}(1|n; r). Similarly, 𝔩𝔢(n|n; r), 𝔟_{1}^{∘}(n|n+1; r) and 𝔟_{∞}^{∘}(n|n+1; r) are “drop-outs” from the series 𝔟_{λ}(n|n + 1; r) corresponding to λ = 0, 1 and ∞, respectively, having either a simple ideal of codimension 1 or a center the quotient modulo which is simple. Though 𝔰𝔪 is not a drop-out due to the above reason, it is singled out by its divergence free property, hence, deserves a separate line.

One should not treat the drop-outs lightly, be it the case of characteristic p, or the supercase. People justly consider the Lie algebras of Hamiltonian vector fields and its central extension, Poisson algebra (or loop algebra and its affine Kac-Moody relative) as totally distinct algebras. The difference between 𝔰𝔩 and 𝔭𝔰𝔩 in characteristic p and in the super-case is enourmous: e.g., the irreducible representation of least dimension of 𝔭𝔰𝔩 is the adjoint one.

Thus, all drop-outs, without exception, are quite separate items on the list.

Theorem . (Solution of Problem A) The simple W-graded vectorial Lie superalgebras 𝔤 constitute the following series (S) and five exceptional families of fifteen individual algebras (E). They are pair-wise non-isomorphic, as graded and filtered superalgebras, bar for occasional isomorphisms (OI).

For completeness, our tables include finite dimensional “degenerations” labelled by “FD” instead of the number.

All these algebras are the results of the (generalized) Cartan prolongation (described in sec 0.5) and, therefore, are determined by the terms 𝔤_{i} with i ≤ 0 (or i ≤ 1 in some cases). These terms are listed in sec. I.2 (and the form of these terms might drastically vary with n and r though they can be united in “families”).

N the family and conditions for its simplicity
1 𝔳𝔢𝔠𝔱(m|n; r) for m ≥ 1 and 0 ≤ r ≤ n
FD 𝔳𝔢𝔠𝔱(0|n; r) for n > 1 and 0 ≤ r ≤ n
2 𝔰𝔳𝔢𝔠𝔱(m|n; r) for m > 1, 0 ≤ r ≤ n
FD 𝔰𝔳𝔢𝔠𝔱(0|n; r) for n > 2 and 0 ≤ r ≤ n
3 𝔰𝔳𝔢𝔠𝔱^{∘}(1|n; r) for n > 1, 0 ≤ r ≤ n
FD { \atop 𝔰𝔳𝔢𝔠𝔱}(0|n) for n > 2

(S_{vect})

4 𝔨(2m + 1|n; r) for 0 ≤ r ≤ [\frac{n}{2}] unless (m|n) = (0|2k)

𝔨(1|2k; r) for 0 ≤ r ≤ k except r = k − 1

5 𝔥(2m|n; r) for m > 0 and 0 ≤ r ≤ [\frac{n}{2}]

𝔥_{λ}(2|2; r) for λ ≠ 0, 1, ∞, and r = 0, 1 and E_{H} (see (OI) and sec. I.2.1.1)

FD 𝔥^{o}(0|n) for n > 3

(S_{k})

6 𝔪(n|n + 1; r) for 0 ≤ r ≤ n except r = n − 1
7 𝔰𝔪(n|n + 1; r) for n > 1 but n ≠ 3 and 0 ≤ r ≤ n except r = n − 1
8 𝔟_{λ}(n|n + 1; r) for n > 1, where λ ≠ 0, 1, ∞ and 0 ≤ r ≤ n except r = n − 1

𝔟_{λ}(2|3; r), where λ ≠ 0, 1, ∞ for r = 0, 2 and E (see sec. I.2.1.1)

9 𝔟_{1}^{∘}(n|n + 1; r) for n > 1 and 0 ≤ r ≤ n except r = n − 1
10 𝔟_{∞}^{∘}(n|n + 1; r) for n > 1 and 0 ≤ r ≤ n except r = n − 1
11 𝔩𝔢(n|n; r) for n > 1 and 0 ≤ r ≤ n except r = n − 1
12 𝔰𝔩𝔢^{∘}(n|n; r) for n > 2 and 0 ≤ r ≤ n except r = n − 1
13 𝔰̃𝔟_{μ}(2^{n−1} − 1|2^{n−1}) for μ ≠ 0 and n > 2

(S_{m})

Hereafter we inconsistently abbreviate 𝔪(n|n + 1; r), 𝔟_{λ}(n|n + 1; r), 𝔩𝔢(n|n; r), etc., i.e., algebras from (S_{m}), to 𝔪(n; r), 𝔟_{λ}(n; r), 𝔩𝔢(n; r), etc., respectively.

For clarity, instead of 𝔟_{λ}(n; r), where λ = \frac{2a}{n(a−b)} ∈ ℂ ∪ \{∞\}, we sometimes write 𝔟_{a,b}(n; r). The parameters a, b, even more natural than λ, are described in sec. 0.7.

Similarly looking notations 𝔰𝔳𝔢𝔠𝔱_{a,b}(0|n) and 𝔰𝔭𝔢_{a,b}(n) are shorthand for 𝔤 ⊂ + ℂ(az + bd), where 𝔤 is 𝔰𝔳𝔢𝔠𝔱(0|n) or 𝔰𝔭𝔢(n), d is the operator that determines the standard ℤ-grading of 𝔤, and z the trivial center.

(OI): Occasional isomorphisms:

𝔳𝔢𝔠𝔱(1|1) ≅ 𝔳𝔢𝔠𝔱(1|1; 1);

𝔰𝔳𝔢𝔠𝔱(2|1) ≅ 𝔩𝔢(2; 2); 𝔰𝔳𝔢𝔠𝔱(2|1; 1) ≅ 𝔩𝔢(2)

𝔰𝔪(n) ≅ 𝔟_{2/(n−1)}(n); in particular, 𝔰𝔪(2) ≅ 𝔟_{2}(2), and

𝔰𝔪(3) ≅ 𝔟_{1}(3), hence, 𝔰𝔪(3) is not simple;

𝔰𝔩𝔢^{∘}(3) ≅ 𝔰𝔩𝔢^{∘}(3; 3);

𝔟_{1/2}(2; 2) ≅ 𝔥_{1/2}(2|2) = 𝔥(2|2); 𝔥_{λ}(2|2) ≅ 𝔟_{λ}(2; 2); 𝔥_{λ}(2|2; 1) ≅ 𝔟_{λ}(2);

𝔟_{a,b}(2; E) ≅ 𝔟_{−b,−a}(2) ≅ 𝔟_{b,a}(2) for a ≠ b and b ≠ 0;

𝔟_{a,0}(2; E) ≅ 𝔩𝔢(2) and 𝔟_{a,a}(2; E) ≅ 𝔟_{∞}^{∘}(2);

𝔥_{λ}(2|2) ≅ 𝔥_{−1−λ}(2|2), in particular, 𝔟_{0}(n) ≃ 𝔟_{−1}(n) and 𝔟_{1}(n) ≃ 𝔟_{−2}(n),

hence, the fundamental domain is Reλ ≥ −\frac{1}{2} or Reλ ≤ −\frac{1}{2};

𝔰̃𝔟_{μ}(2^{n−1} − 1|2^{n−1}) ≅ 𝔰̃𝔟_{ν}(2^{n−1} − 1|2^{n−1}) for μν ≠ 0.

(OI)

Observe that the exceptional regrading E of 𝔟_{λ}(2) and isomorphisms 𝔥_{λ}(2|2) ≅ 𝔟_{λ}(2; 2) determine the exceptional grading E_{H} of 𝔥_{λ}(2|2). These regradings (E and E_{H}) are not, however, of any interest from the classificational point of view, thanks to the isomorphisms (OI). On the other hand, they contribute to the automorphisms of 𝔥_{λ}(2|2) ≅ 𝔟_{λ}(2; 2).

Our original notations of exceptional simple vectorial superalgebras, though reflect the way they are constructed and the geometry preserved (𝔨𝔰𝔩𝔢 reflects that it is a subalgebra of 𝔨 related to 𝔰𝔩𝔢, and so on), are rather long. But to write just 𝔢(dim) is to create confusion: the superdimensions of the superspaces (L/L)^{∗} on which the algebra L is realized coincide sometimes for different regradings. So, in accordance with table (E ⊂ S)in sec. I.2, we set (by K we denote consistent gradings, CK is an exceptional grading found by Cheng and Kac):

Lie superalgebra  its regradings (shorthand)
𝔳𝔩𝔢(4|3; r), r = 0, 1, K  𝔳𝔩𝔢(4|3), 𝔳𝔩𝔢(5|4), and 𝔳𝔩𝔢(3|6)
𝔳𝔞𝔰(4|4)  𝔳𝔞𝔰(4|4)
𝔨𝔞𝔰(1|6; r), r = 0, 1ξ, 3ξ, 3η 𝔨𝔞𝔰(1|6), 𝔨𝔞𝔰(5|5), 𝔨𝔞𝔰(4|4), and 𝔨𝔞𝔰(4|3)
𝔪𝔟(4|5; r), r = 0, 1, K  𝔪𝔟(4|5), 𝔪𝔟(5|6), and 𝔪𝔟(3|8)
𝔨𝔰𝔩𝔢(9|6; r), r = 0, 2, K, CK 𝔨𝔰𝔩𝔢(9|6), 𝔨𝔰𝔩𝔢(11|9), 𝔨𝔰𝔩𝔢(5|10), and 𝔨𝔰𝔩𝔢(9|11)

(E)

Remark . 1) Though 𝔟_{λ}(2; r) and 𝔥_{λ}(2|2; r) are isomorphic, we consider them separately because they are deformations of very distinct structures: the odd and the even bracket, respectively. In I.2.4.2 we show wherefrom comes the mysterious at this point isomorphism 𝔥_{λ}(2|2) ≅ 𝔥_{−1−λ}(2|2).

Actually, 𝔥_{λ}(2|2) ≅ 𝔟_{λ}(2; 2) should be considered as an exceptional family.

2) Warning. Isomorphic abstract Lie superalgebras might be quite distinct as filtered or graded: e.g., regradings provide us with isomorphisms ([ALSh])

𝔨(1|2) ≅ 𝔳𝔢𝔠𝔱(1|1) ≅ 𝔪(1) as abstract algebras.  (IAA)

Observe that of the above three nonisomorphic graded algebras only one is W-graded.

3) The excluded regrading 𝔨(1|2) is often considered in physical papers, despite the fact that this grading (or filtration) is not a Weisfeiler one: for it the 𝔤_{0}-module 𝔤_{−1} is reducible. Over ℝ, however, 𝔨(1|2k; k − 1) is a W-grading, at least, for some real forms, and it is these forms that are considered.

4) { \atop 𝔰𝔳𝔢𝔠𝔱}(0|n), as well as 𝔰̃𝔟_{μ}(2^{n−1} − 1|2^{n−1}), depend on an odd parameter if n is odd.

The above Lie superalgebras sometimes admit deformations that do not possess Weisfeiler filtrations. These deformations are considered in detail in [LS5]; far partial results see [L4], [Ko1], [LSh3].

The five families of exceptional Lie superalgebras are given below in their minimal realizations as Cartan’s prolongs (𝔤_{−1}, 𝔤_{0})_{∗} or generalized (see sect. 0.5) Cartan’s prolongs (𝔤_{−}, 𝔤_{0})_{∗}^{mk}.

For 𝔤_{−} = ⊕_{−2≤i≤−1} 𝔤_{i}, instead of (𝔤_{−}, 𝔤_{0})_{∗}^{mk} we write (𝔤_{−2}, 𝔤_{−1}, 𝔤_{0})_{∗}^{mk}. Indicated is also one of

the Lie superalgebras from the list of series (S) as an ambient which contains the exceptional one as a maximal subalgebra. The superalgebras of depth 3 appear as regradings of the listed ones at certain value of r; the corresponding terms will be given below.

𝔳𝔩𝔢(4|3; r) = (Π(Λ(3))/ℂ · 1, 𝔠𝔳𝔢𝔠𝔱(0|3))_{∗} ⊂ 𝔳𝔢𝔠𝔱(4|3; R), r = 0, 1, K
𝔳𝔞𝔰(4|4) = (spin, 𝔞𝔰)_{∗} ⊂ 𝔳𝔢𝔠𝔱(4|4)
𝔨𝔞𝔰^{ξ}(1|6; r) ⊂ 𝔨(1|6; r), r = 0, 1ξ, 3ξ
𝔨𝔞𝔰^{ξ}(1|6; 3η) = (Vol_{0}(0|3), 𝔠(𝔳𝔢𝔠𝔱(0|3)))_{∗} ⊂ 𝔰𝔳𝔢𝔠𝔱(4|3)
𝔪𝔟(4|5; r) = (𝔞𝔟(4), 𝔠𝔳𝔢𝔠𝔱(0|3))_{∗}^{m} ⊂ 𝔪(4|5; R), r = 0, 1, K
𝔨𝔰𝔩𝔢(9|6; r) = (𝔥𝔢𝔦(8|6), 𝔰𝔳𝔢𝔠𝔱(0|4)_{3,4})_{∗}^{k} ⊂ 𝔨(9|6; r), r = 0, 2, CK
𝔨𝔰𝔩𝔢(9|6; K) = (id_{𝔰𝔩(5)}, Λ^{2}(id_{𝔰𝔩(5)}^{∗}), 𝔰𝔩(5))_{∗}^{k} ⊂ 𝔰𝔳𝔢𝔠𝔱(5|10; R)

(E ⊂ S)

In (E ⊂ S), most of the regradings R of the ambients are the same as that of the embedded algebra, i.e., R(r) = r. Certain regradings R(r) of the ambients are so highly nonstandard that even the homogeneous fibers are of infinite dimension. The following table describes these regradings (for 𝔨𝔞𝔰 notations r = 0, 1ξ, 3ξ are clear):

\begin{matrix}𝔳𝔩𝔢(4|3) & R(0) = (1111|111), R(1) = (2110|011), R(K) = (2220|111) \\ 𝔪𝔟(4|5) & R(0) = (1111|1111; 2), R(1) = (0211|2011; 2), R(K) = (0222|3111; 3) \\ 𝔨𝔰𝔩𝔢(9|6) & R(K) = (22222|1 … 1) \end{matrix} (R)

The description of the degrees of the generating functions is given in Theorem I.2.1.2.

Summary. 1) In [ALSh] we have singled out a tame subproblem: classify simple vectorial Lie superalgebras, from its straightforward but wild classical analog: classify primitive Lie superalgebras (cf. problem formulation in [K2], [K7], [K9]).

2) Solution of Problem A for depth 1. (The most part of this solution was preprinted in [LSh2].)

3) In several of our earlier papers, we also gave examples of simple vectorial Lie superalgebras of greater depth. The union of these examples ([ALSh], [LSh1], [Sh5]) was our conjectural list solving Problem A announced in [LSh2]. And the list is, indeed, the complete one. We make use of the proof from [K7, CK1, CK2], where the case of consistent grading and what we call Kac’s grading for depth > 1 are considered, to solve Problem A in full generality.

Related results. 1) For the list of simple “stringy” Lie superalgebras, their explicit representation by generating functions, and for related structures see [GLS1]. Observe a mysterious difference in presentations for 𝔨𝔞𝔰 generated by polynomials and 𝔨𝔞𝔰^{L} generated by Laurent polynomials, see [GLS4]: we suspect that there are infinitely many relations between generators of 𝔨𝔞𝔰 whereas 𝔨𝔞𝔰^{L} is finitely presented.

2) The description of the irreducible representations of simple vectorial Lie superalgebras was started in [BL1] for the general series 𝔳𝔢𝔠𝔱(m|n) in its standard realization following the pioneering works of A. Rudakov. For a review of results on representations for other series of vectorial Lie superalgebras by 1980 see [L2]. For a description of the irreducible representations of two or three exceptional vectorial Lie superalgebras see [KR]; for an updated review and description of irreducible representations for 11 of the 15 exceptions in a wider setting see [GLS3].

3) For the classification of simple vectorial Lie superalgebras over ℝ see [LSh4].

4) For the classification of deformations of simple vectorial Lie superalgebras see [LS5].

I.2.1. Nonstandard realizations. As follows from Step 7 of our proof (see I.3), the following nonstandard gradings exhaust all the W-gradings of all the simple vectorial Lie superalgebras. In particular, the gradings in the series 𝔳𝔢𝔠𝔱 induce the gradings in the series 𝔰𝔳𝔢𝔠𝔱, 𝔰𝔳𝔢𝔠𝔱^{∘} and the exceptional families 𝔳𝔩𝔢(4|3) and 𝔳𝔞𝔰(4|4); the gradings in 𝔪 induce the gradings in 𝔟_{λ}, 𝔩𝔢, 𝔰𝔩𝔢, 𝔰𝔩𝔢^{∘}, 𝔟, 𝔰𝔟, 𝔰𝔟^{∘} and the exceptional family 𝔪𝔟; the gradings in 𝔨 induce the gradings in 𝔭𝔬, 𝔥 and the exceptional families 𝔨𝔞𝔰 and 𝔨𝔰𝔩𝔢.

The indeterminates are introduced in sec. 0.2 below. In what follows we consider 𝔨(2n + 1|m) as preserving the Pfaff equation α̃ = 0, where (see (0.2.4))

α̃ = dt + ∑_{i≤n}(p_{i}dq_{i} − q_{i}dp_{i}) + ∑_{j≤r}(ξ_{j}dη_{j} + η_{j}dξ_{j}) + ∑_{k≥m−2r} θ_{k}dθ_{k}.

The standard realizations correspond to r = 0, they are marked by an (∗). Observe that the codimension of L_{0} attains its minimum in the standard realization.

Lie superalgebra  its ℤ-grading
𝔳𝔢𝔠𝔱(n|m; r), deg u_{i} = deg ξ_{j} = 1 for any i, j (∗)
0 ≤ r ≤ m  deg ξ_{j} = 0 for 1 ≤ j ≤ r;
deg u_{i} = deg ξ_{r+s} = 1 for any i, s
𝔪(n; r),  deg τ = 2, deg q_{i} = deg ξ_{i} = 1 for any i (∗)
0 ≤ r ≤ n  deg τ = deg q_{i} = 2, deg ξ_{i} = 0 for 1 ≤ i ≤ r < n;
r ≠ n − 1  deg q_{r+j} = deg ξ_{r+j} = 1 for any j
𝔪(1|n; n)  deg τ = deg q_{i} = 1, deg ξ_{i} = 0 for 1 ≤ i ≤ n
𝔨(2n + 1|m; r), deg t = 2, (∗)

deg p_{i} = deg q_{i} = deg ξ_{j} = deg η_{j} = deg θ_{k} = 1 for any i, j, k

0 ≤ r ≤ [\frac{m}{2}]  deg t = deg ξ_{i} = 2, deg η_{i} = 0 for 1 ≤ i ≤ r ≤ [\frac{m}{2}];
r ≠ k − 1 for m = 2k and n = 0  deg p_{i} = deg q_{i} = deg θ_{j} = 1 for j ≥ 1 and all i
𝔨(1|2m; m)  deg t = deg ξ_{i} = 1, deg η_{i} = 0 for 1 ≤ i ≤ m

I.2.1.1. The exceptional nonstandard regrading E. This is a regrading of 𝔟_{a,b}(2) given by the formulas:

deg τ = 0; deg ξ_{1} = deg ξ_{2} = −1; deg q_{1} = deg q_{2} = 1.  (r = E)

Further

b = 0 or a = b: 𝔟_{a,0}(2; E) ≅ 𝔩𝔢(2) and 𝔟_{a,a}(2; E) ≅ 𝔟_{∞}^{∘}(2), in part. 𝔤_{−2} = 0;

a and b generic: 𝔤_{−2} = Span{Le_{ξ_{1}ξ_{2}}} and 𝔤_{−1} = Span{Le_{ξ_{1}}, Le_{ξ_{2}}, Le_{Q_{1}}, Le_{Q_{2}}},

(I.2.1.1)

where Q_{1} = Aξ_{1}ξ_{2}q_{1} + Bτξ_{2}, Q_{2} = Aξ_{1}ξ_{2}q_{2} − Bτξ_{1} and where A and B are some coefficients determined by a and b. The bracket on 𝔤_{−1} is determined by the odd form ω = c ∑ dQ_{i}dξ_{i}, so 𝔤_{0} must be contained in 𝔪(2)_{0}. The direct calculations show that dim 𝔤_{0} = 4|4 and

𝔤_{0} = 𝔰𝔭𝔢(2) ⊕ ℂX, where X = Le_{aτ+b ∑ q_{i}ξ_{i}}.

Indeed,

𝔰𝔭𝔢(2)_{0} ≅ 𝔰𝔩(2) = Span{Le_{q_{1}ξ_{2}}, Le_{q_{2}ξ_{1}}, Le_{q_{1}ξ_{1}−q_{2}ξ_{2}}}, 𝔰𝔭𝔢(2)_{−1} = ℂ · Le_{1},

and

𝔰𝔭𝔢(2)_{1} = ℂLe_{αξ_{1}ξ_{2}P (q)+βτΔ(ξ_{1}ξ_{2}P (q))},

where P (q) is a monomial of degree 2 and α, β are some constants.

The eigenvalues of X on 𝔤_{−1} are: −a + b on the even part and a + b an the odd part. So

𝔟_{a,b}(2; E) ≅ 𝔟_{−b,−a}(2) ≅ 𝔟_{b,a}(2).

For n > 2, as well as for 𝔪(n) with n > 1, similar regradings are not Weisfeiler ones. The exceptional grading E of 𝔟_{λ}(2) induces the exceptional grading E_{H} of the isomorphic

algebra 𝔥_{λ}(2|2), see (OI).

Thus, the exceptional regradings E or E_{H} do not prowide with new W-graded vectorial algebras. Still, they are important for description of automorphisms.

I.2.1.2. The W-regradings of exceptional algebras.

Theorem . ([Sh14], [CK2]) The W-regradings of the exceptional simple vectorial Lie superalgebras are given by the following regradings of their “standard” ambients listed in Introduction (E ⊂ S):

1) 𝔳𝔩𝔢(4|3; r) = (Π(Λ(3)/ℂ · 1), 𝔠𝔳𝔢𝔠𝔱(0|3))_{∗} ⊂ 𝔳𝔢𝔠𝔱(4|3), r = 0, 1, K; \underline{r = 0}: deg y = deg u_{i} = deg ξ_{i} = 1; \underline{r = 1}: deg y = deg ξ_{1} = 0, deg u_{2} = deg u_{3} = deg ξ_{2} = deg ξ_{3} = 1, deg u_{1} = 2; \underline{r = K}: deg y = 0, deg u_{i} = 2; deg ξ_{i} = 1.

2) 𝔳𝔞𝔰(4|4) = (spin, 𝔞𝔰)_{∗} ⊂ 𝔳𝔢𝔠𝔱(4|4);

3) 𝔨𝔞𝔰 ⊂ 𝔨(1|6; r), r = 0, 1, 3ξ; \underline{r = 0}: deg t = 2, deg η_{i} = 1; deg ξ_{i} = 1; \underline{r = 1}: deg ξ_{1} = 0, deg η_{1} = deg t = 2, deg ξ_{2} = deg ξ_{3} = deg η_{2} = deg η_{3} = 1; \underline{r = 3ξ}: deg ξ_{i} = 0, deg η_{i} = deg t = 1; \underline{r = 3η}: deg η_{i} = 0, deg ξ_{i} = deg t = 1 𝔨𝔞𝔰(1|6; 3η) ⊂ 𝔰𝔳𝔢𝔠𝔱(4|3).

4) 𝔪𝔟(4|5; r) = (𝔞𝔟(4), 𝔠𝔳𝔢𝔠𝔱(0|3))_{∗}^{m} ⊂ 𝔪(4), r = 0, 1, K; \underline{r = 0}: deg τ = 2, deg u_{i} = deg ξ_{i} = 1 for i = 0, 1, 2, 3; \underline{r = 1}: deg τ = deg ξ_{0} = deg u_{1} = 2, deg u_{2} = deg u_{3} = deg ξ_{2} = deg ξ_{3} = 1; deg ξ_{1} =

deg u_{0} = 0; \underline{r = K}: deg τ = deg ξ_{0} = 3, deg u_{0} = 0, deg u_{i} = 2; deg ξ_{i} = 1 for i > 0.

5) 𝔨𝔰𝔩𝔢(9|6; r) = (𝔥𝔢𝔦(8|6), 𝔰𝔳𝔢𝔠𝔱_{3,4}(4))_{∗}^{k} ⊂ 𝔨(9|6), r = 0, 2, CK; \underline{r = 0}: deg t = 2, deg p_{i} = deg q_{i} = deg ξ_{i} = deg η_{i} = 1; \underline{r = 2}: deg t = deg q_{3} = deg q_{4} = deg η_{1} = 2, deg q_{1} = deg q_{2} = deg p_{1} = deg p_{2} = deg η_{2} =

deg η_{3} = deg ζ_{2} = deg ζ_{3} = 1; deg p_{3} = deg p_{4} = deg ζ_{1} = 0; \underline{r = K}: deg t = deg q_{i} = 2, deg p_{i} = 0; deg ζ_{i} = deg η_{i} = 1; 𝔨𝔰𝔩𝔢(9|6; K) ⊂ 𝔰𝔳𝔢𝔠𝔱(5|10; R); \underline{r = CK}: deg t = deg q_{1} = 3, deg p_{1} = 0; deg q_{2} = deg q_{3} = deg q_{4} = deg ζ_{1} = deg ζ_{2} =

deg ζ_{3} = 2; deg p_{2} = deg p_{3} = deg p_{4} = deg η_{1} = deg η_{2} = deg η_{3} = 1 .

Thus, from the point of view of classification of the W-filtered Lie superalgebras, there are five families of exceptional algebras consisting of 15 individual algebras.

I.2.2. Several first terms that determine the Cartan and mk-prolongations. (To recall the notion of Cartan prolongation and for definition of its generalizations see sec. 0.5.) To facilitate the comparison of various vectorial superalgebras, we offer the following Table. The most interesting phenomena occur for extremal values of parameter r and small values of superdimension m|n.

The central element z ∈ 𝔤_{0} is supposed to be chosen so that it acts on 𝔤_{k} as k · id.

Let Λ(r) = ℂ[ξ_{1}, …, ξ_{r}] be the Grassmann superalgebra generated by the ξ_{i} of degree 0 for all i. We set Λ(0) = ℂ, the 𝔳𝔢𝔠𝔱(0|n)-module Λ(n)/ℂ · 1 is denoted by T_{0}(0); set

Vol_{0}(0|m) = {v ∈ Vol(0|m) | ∫ v = 0};

the 𝔰𝔳𝔢𝔠𝔱-module T_{0}^{0}(0) is defined as Vol_{0}(0|m)/ℂ·1 (for more elucidations see sec. 0.6).Over 𝔰𝔳𝔢𝔠𝔱 it is convenient to consider Vol_{0} as a submodule of Λ.

Recall that the range of the parameter r (see I.2.1 and sec. 0.11) is the set of integer from [0, m], where m is the number of odd indeterminates, for the series 𝔨 and 𝔥 the range is [0, [\frac{m}{2}]], and for 𝔪(n), 𝔟_{λ}(n), etc. the range is [0, n] with value r = n − 1 excluded.

Recall that we exclude certain values of r, namely, r = k − 1 for 𝔨(1|2k) and r = n − 1 for 𝔪(n) and their subalgebras because for these values of r the corresponding grading is not a W-grading: the 𝔤_{0}-module 𝔤_{−1} is reducible.

N  𝔤  𝔤_{−2}  𝔤_{−1}  𝔤_{0}
1  𝔳𝔢𝔠𝔱(n|m; r)  −  id ⊗ Λ(r) 𝔤𝔩(n|m − r) ⊗ Λ(r) ⊂ + 𝔳𝔢𝔠𝔱(0|r)
2  𝔳𝔢𝔠𝔱(1|m; m)  −  Λ(m)  Λ(m) ⊂ + 𝔳𝔢𝔠𝔱(0|m)
3 𝔰𝔳𝔢𝔠𝔱(n|m; r), n ≠ 1 −  id ⊗ Λ(r) 𝔰𝔩(n|m − r) ⊗ Λ(r) ⊂ + 𝔳𝔢𝔠𝔱(0|r)
4 𝔰𝔳𝔢𝔠𝔱^{∘}(1|m; r), r ≠ m − id ⊗ Vol_{0}(0|r) 𝔰𝔩(n|m − r) ⊗ Λ(r) ⊂ + 𝔳𝔢𝔠𝔱(0|r)
5  𝔰𝔳𝔢𝔠𝔱^{∘}(1|m; m)  − Vol_{0}(0|m)  Λ(m) ⊂ + 𝔰𝔳𝔢𝔠𝔱(0|m)
𝔰𝔳𝔢𝔠𝔱^{∘}(1|2)  −  T_{0}(0)  𝔳𝔢𝔠𝔱(0|2) ≅ 𝔰𝔩(1|2)
𝔰𝔳𝔢𝔠𝔱(2|1)  −  Π(T_{0}(0))  𝔳𝔢𝔠𝔱(0|2) ≅ 𝔰𝔩(2|1)

6  𝔥(2n|m)  −  id  𝔬𝔰𝔭(m|2n)
7  𝔥(2n|m; r) T_{0}(0)  id ⊗ Λ(r)  𝔬𝔰𝔭(m − 2r|2n) ⊗ Λ(r) ⊂ + 𝔳𝔢𝔠𝔱(0|r)
8 𝔥(2n|2r; r)  −  id ⊗ Λ(r)  𝔰𝔭(2n) ⊗ Λ(r) ⊂ + 𝔳𝔢𝔠𝔱(0|r)
9  𝔥_{λ}(2|2)  −  Π(Vol^{λ}(0|2))  𝔬𝔰𝔭(2|2) ≅ 𝔳𝔢𝔠𝔱(0|2)
10 𝔥_{λ}(2|2; 1)  − id_{𝔰𝔭(2)} ⊗ Vol^{λ}(0|1)  𝔰𝔭(2) ⊗ Λ(1) ⊂ + 𝔳𝔢𝔠𝔱(0|1)
11 𝔨(2n + 1|m; r) Λ(r)  id ⊗ Λ(r)  𝔠𝔬𝔰𝔭(m − 2r|2n) ⊗ Λ(r) ⊂ + 𝔳𝔢𝔠𝔱(0|r)
12 𝔨(1|2m; m)  −  Λ(m)  Λ(m) ⊂ + 𝔳𝔢𝔠𝔱(0|m)
13 𝔨(1|2m + 1; m) Λ(m)  Π(Λ(m))  Λ(m) ⊂ + 𝔳𝔢𝔠𝔱(0|m)

We set p(μ) ≡ n (mod 2), so μ can be odd indeterminate. The Lie superalgebras { \atop 𝔰𝔳𝔢𝔠𝔱}_{μ}(0|n) are isomorphic for nonzero μ’s; hence, so are the algebras 𝔰̃𝔟_{μ}(2^{n−1} − 1|2^{n−1}). So for n even we can set μ = 1, while if μ is odd we should consider it as an additional indeterminate on which the coefficients depend.

N  𝔤  𝔤_{−2}  𝔤_{−1}  𝔤_{0}
14  𝔪(n; r)  Π(Λ(r))  id ⊗ Λ(r)  𝔠𝔭𝔢(n − r) ⊗ Λ(r) ⊂ + 𝔳𝔢𝔠𝔱(0|r)
15  𝔪(n; n)  −  Π(Λ(n))  Λ(n) ⊂ + 𝔳𝔢𝔠𝔱(0|n)
16 𝔰̃𝔟_{μ}(2^{n−1} − 1|2^{n−1})  −  \frac{Π(Vol(0|n))}{ℂ(1+μξ_{1}...ξ_{n})vol(ξ)}  { \atop 𝔰𝔳𝔢𝔠𝔱}_{μ}(0|n)

In what follows λ = \frac{2a}{n(a−b)} ≠ 0, 1, ∞; the three exceptional cases (corresponding to the “drop-outs” 𝔩𝔢(n), 𝔟_{1}^{∘}(n) and 𝔟_{∞}^{∘}(n), respectively) are considered separately. The irreducibility condition of the 𝔤_{0}-module 𝔤_{−1} for 𝔤 = 𝔟_{∞}^{∘} excludes r = n − 1. The case r = n − 2 is extra exceptional, so in the following tables, unless specified,

0 < r < n − 2; additionally a ≠ b and (a, b) ≠ α(n, n − 2) for any α ∈ ℂ.

To further clarify the following tables, denote the superspace of the identity k|k-dimensional representation of 𝔰𝔭𝔢(k) by V; let d = diag(1_{k}, −1_{k}) ∈ 𝔭𝔢(k). Let W = V ⊗ Λ(r) and D ∈ 𝔳𝔢𝔠𝔱(0|r). Let Ξ = ξ_{1} ⋯ ξ_{n} ∈ Λ(ξ_{1}, …, ξ_{n}). Denote by T^{r} the representation of 𝔳𝔢𝔠𝔱(0|r) in 𝔰𝔭𝔢(n − r) ⊗ Λ(r) given by the formula

T^{r}(D) = 1 ⊗ D + d ⊗ \frac{1}{n − r}divD.  (T^{r})

\underline{In 𝔰𝔩𝔢^{∘}(n; r)_{0} for r ≠ n − 2}:

𝔳𝔢𝔠𝔱(0|r) acts on the ideal 𝔰𝔭𝔢(n − r) ⊗ Λ(r) via T^{r};
any X ⊗ f ∈ 𝔰𝔭𝔢(n − r) ⊗ Λ(r) acts in 𝔤_{−1} as id ⊗ f and in 𝔤_{−2} as 0;
any D ∈ 𝔳𝔢𝔠𝔱(0|r) acts in 𝔤_{−1} via T^{r} and in 𝔤_{−2} as D.

(I.2.1.1)

\underline{In 𝔰𝔩𝔢^{∘}(n; n − 2)_{0}}, observe that 𝔰𝔭𝔢(2) ≅ ℂ(Le_{q_{1}ξ_{1}−q_{2}ξ_{2}}) ⊂ + ℂLe_{ξ_{1}ξ_{2}}, whereas 𝔤_{−2} and 𝔤_{−1} are as above, for r < n − 2. Set 𝔥 = ℂ(Le_{q_{1}ξ_{1}−q_{2}ξ_{2}}). In this case

𝔤_{0} ≅ \underline{(𝔥 ⊗ Λ(n − 2) ⊂ + ℂLe_{ξ_{1}ξ_{2}} ⊗ (Λ(n − 2) \ ℂξ_{3} ⋯ ξ_{n}))} ⊂ + T^{1}(𝔳𝔢𝔠𝔱(0|n − 2)). (I.2.1.2)

The action of 𝔳𝔢𝔠𝔱(0|n−2), the quotient of 𝔤_{0} modulo the underlined ideal, is via (I.2.1.1). In the subspace ξ_{1}ξ_{2} ⊗Λ(n−2) ⊂ 𝔤_{0} this action is the same as in the space of volume forms. So we can throw away Ξ, or, speaking more correctly, take the irreducible submodule of forms with integral 0.

In the following table the terms “𝔤_{−i}” denote the superspace isomorphic to the one in quotation marks but with the action given by formulas (I.2.1.1) and (I.2.1.2).

N  𝔤  𝔤_{−2}  𝔤_{−1}  𝔤_{0}
17  𝔩𝔢(n)  −  id  𝔭𝔢(n)
18  𝔩𝔢(n; r)  Π(T_{0}(0)) id ⊗ Λ(r)  𝔭𝔢(n − r) ⊗ Λ(r) ⊂ + 𝔳𝔢𝔠𝔱(0|r)
19  𝔩𝔢(n; n)  −  Π(T_{0}(0))  𝔳𝔢𝔠𝔱(0|n)
20  𝔰𝔩𝔢^{∘}(n)  −  id  𝔰𝔭𝔢(n)
21 𝔰𝔩𝔢^{∘}(n; r) “Π(T_{0}(0))” “id ⊗ Λ(r)” 𝔰𝔭𝔢(n − r) ⊗ “Λ(r)′′ ⊂ + T^{1}(𝔳𝔢𝔠𝔱(0|r))
22 𝔰𝔩𝔢^{∘}(n; n − 2) “Π(T_{0}(0))” “id ⊗ Λ(r)”  see (I.2.1.2)
23 𝔰𝔩𝔢^{∘}(n; n)  −  Π(T_{0}^{0}(0))  𝔰𝔳𝔢𝔠𝔱(0|n)

Next, we consider 𝔟_{a,b}(n; r) for 0 < r < n − 2 and ar − bn ≠ 0; in particular, this excludes 𝔟_{∞}^{∘}(n; n) = 𝔟_{a,a}^{∘}(n; n) and 𝔟_{1}^{∘}(n; n − 2) = 𝔟_{n,n−2}^{∘}(n; n − 2).

If z is the central element of 𝔠𝔰𝔭𝔢(n − r) that acts on 𝔤_{−1} as −id, then

z ⊗ ψ acts on 𝔤_{−1} as −id ⊗ ψ, and on 𝔤_{−2} as −2id ⊗ ψ.  (I.2.1.3)

Set

c = \frac{a}{ar − bn}.

Let a · str ⊗ id be the representation of 𝔭𝔢(n − r) = 𝔰𝔭𝔢(n − r) ⊂ + ℂd which is id_{𝔰𝔭𝔢(n−r)} on 𝔰𝔭𝔢(n − r) and sends d to 2aid. Recall that ℂ[k] is the 1-dimensional representation of 𝔤_{0} where k is the value of the central element z from 𝔤_{0} chosen so that z|_{𝔤_{i}} = i · id_{𝔤_{i}}.

N  𝔤  𝔤_{−2}  𝔤_{−1}  𝔤_{0}
24 𝔟_{λ}(n)  Π(ℂ[−2])  id  𝔰𝔭𝔢(n) ⊂ + ℂ(az + bd)
25 𝔟_{λ}(n; r) Π((−c)str) ⊗ (Vol(0|r))^{2c} ((−\frac{c}{2})str) ⊗ id) ⊗ (Vol(0|r))^{c} (𝔭𝔢(n − r) ⊗ Λ(r)) ⊂ + 𝔳𝔢𝔠𝔱(0|r)
26 𝔟_{λ}(n; n)  −  Π(Vol^{λ}(0|n))  𝔳𝔢𝔠𝔱(0|n)

In the following tables the terms “𝔤_{i}” denote the superspace isomorphic to the one in quotation marks but with the action given by formula (I.2.1.3).

In the exceptional case ar = bn, i.e., λ = \frac{2}{n − r}, the 𝔳𝔢𝔠𝔱(0|r)-action on the ideal 𝔠𝔰𝔭𝔢(n −

r) ⊗ Λ(r) ⊂ + 𝔳𝔢𝔠𝔱(0|r) of 𝔤_{0} and on 𝔤_{−} is the same as for 𝔰𝔩𝔢^{∘}, see (I.2.1.1).

N  𝔤  𝔤_{−2}  𝔤_{−1}  𝔤_{0}
27 𝔟_{2/(n−r)}(n; r) Π(ℂ) ⊗ Λ(r) id ⊗ “Λ(r)′′  𝔠𝔭𝔢(n − r) ⊗ “Λ(r)′′ ⊂ + T^{r}(𝔳𝔢𝔠𝔱(0|r))
28  𝔟_{∞}^{∘}(n)  Π(ℂ)  id  𝔰𝔭𝔢(n)_{a,a}
29  𝔟_{∞}^{∘}(n; r)  Π(ℂ) ⊗ Λ(r)  id ⊗ Λ(r)  ((𝔰𝔭𝔢(n − r)_{a,a}) ⊗ Λ(r)) ⊂ + 𝔳𝔢𝔠𝔱(0|r)
30 𝔟_{∞}^{∘}(n; n), n > 2  −  Π(Λ(n))  (Λ(n) \ ℂΞ) ⊂ + 𝔰𝔳𝔢𝔠𝔱(0|n)
31  𝔟_{1}^{∘}(n)  Π(ℂ)  id  𝔰𝔭𝔢(n)_{n,n−2}
32  𝔟_{1}^{∘}(n; r)  “Π(Vol_{0}(0|r))” id ⊗ “Λ(r)” ((𝔰𝔭𝔢(n − r)_{n,n−2}) ⊗ “Λ(r)′′) ⊂ + T^{r}(𝔳𝔢𝔠𝔱(0|r))
33 𝔟_{1}^{∘}(n; n − 2)  “Π(T_{0}(0))”  id ⊗ “Λ(r)”  (I.2.1.2) for the above line
with 𝔠𝔰𝔭𝔢(2) instead of 𝔰𝔭𝔢(2)
34  𝔟_{1}^{∘}(n; n)  −  Π(Vol_{0}(0|n))  𝔳𝔢𝔠𝔱(0|n)

I.2.3. The exceptional Lie subsuperalgebras. Here are the terms 𝔤_{i} for −2 ≤ i ≤ 0 of the 15 exceptional W-graded algebras (cf. sec. I.2.3):

𝔤  𝔤_{−2}  𝔤_{−1}  𝔤_{0}  dim 𝔤_{−}
𝔳𝔩𝔢(4|3)  −  Π(Λ(3)/ℂ1)  𝔠(𝔳𝔢𝔠𝔱(0|3))  4|3
𝔳𝔩𝔢(4|3; 1)  ℂ[−2]  id ⊗ Λ(2)  𝔠(𝔰𝔩(2) ⊗ Λ(2) ⊂ + T^{1/2}(𝔳𝔢𝔠𝔱(0|2))  5|4
𝔳𝔩𝔢(4|3; K) id_{𝔰𝔩(3)} ⊗ ℂ[−2] id_{𝔰𝔩(3)}^{∗} ⊗ id_{𝔰𝔩(2)} ⊗ ℂ[−1]  𝔰𝔩(3) ⊕ 𝔰𝔩(2) ⊕ ℂz  3|6
𝔳𝔞𝔰(4|4)  −  spin  𝔞𝔰  4|4
𝔨𝔞𝔰  ℂ[−2]  Π(id)  𝔠𝔬(6)  1|6
𝔨𝔞𝔰(; 1ξ)  Λ(1)  id_{𝔰𝔩(2)} ⊗ id_{𝔤𝔩(2)} ⊗ Λ(1)  (𝔰𝔩(2) ⊕ 𝔤𝔩(2) ⊗ Λ(1)) ⊂ + 𝔳𝔢𝔠𝔱(0|1) 5|5
𝔨𝔞𝔰(; 3ξ)  −  Λ(3)  Λ(3) ⊕ 𝔰𝔩(1|3)  4|4
𝔨𝔞𝔰(; 3η)  −  Vol_{0}(0|3)  𝔠(𝔳𝔢𝔠𝔱(0|3))  4|3
𝔪𝔟(4|5)  Π(ℂ[−2])  Vol(0|3)  𝔠(𝔳𝔢𝔠𝔱(0|3))  4|5
𝔪𝔟(4|5; 1)  Λ(2)/ℂ1  id_{𝔰𝔩(2)} ⊗ Λ(2)  𝔠(𝔰𝔩(2) ⊗ Λ(2) ⊂ + T^{1/2}(𝔳𝔢𝔠𝔱(0|2))  5|6
𝔪𝔟(4|5; K) id_{𝔰𝔩(3)} ⊗ ℂ[−2] Π(id_{𝔰𝔩(3)}^{∗} ⊗ id_{𝔰𝔩(2)} ⊗ ℂ[−1])  𝔰𝔩(3) ⊕ 𝔰𝔩(2) ⊕ ℂz  3|8
𝔨𝔰𝔩𝔢(9|6)  ℂ[−2]  Π(T_{0}^{0}(0))  𝔰𝔳𝔢𝔠𝔱(0|4)_{3,4}  9|6
𝔨𝔰𝔩𝔢(9|6; 2)  Π(id_{𝔰𝔩(1|3)})  id_{𝔰𝔩(2)} ⊗ Λ(3)  (𝔰𝔩(2) ⊗ Λ(3)) ⊂ + 𝔰𝔩(1|3)  11|9
𝔨𝔰𝔩𝔢(9|6; K)  id  Π(Λ^{2}(id^{∗}))  𝔰𝔩(5)  5|10
𝔨𝔰𝔩𝔢(9|6; CK) id_{𝔰𝔩(3)}^{∗} ⊗ Λ(1) id_{𝔰𝔩(2)} ⊗ (id_{𝔰𝔩(3)} ⊗ Λ(1)) 𝔰𝔩(2) ⊕ (𝔰𝔩(3) ⊗ Λ(1) ⊂ + 𝔳𝔢𝔠𝔱(0|1)) 9|11

Observe that none of the simple W-graded vectorial Lie superalgebras is of depth > 3 and only two algebras are of depth 3: 𝔪𝔟(4|5; K), for which we have

𝔪𝔟(4|5; K)_{−3} ≅ Π(id_{𝔰𝔩(2)}),

and another one, 𝔨𝔰𝔩𝔢(9|6; CK) = 𝔨𝔰𝔩𝔢(9|11), for which we have :

𝔨𝔰𝔩𝔢(9|11)_{−3} ≃ Π(id_{𝔰𝔩(2)} ⊗ ℂ[−3]).

I.2.3.1 The exceptional Lie subsuperalgebra 𝔨𝔞𝔰 of 𝔨(1|6). Like 𝔳𝔢𝔠𝔱(1|m; m), this Lie superalgebra is not determined by its nonpositive part and requires a closer study. The Lie superalgebra 𝔤 = 𝔨(1|2n) is generated by the functions from ℂ[t, ξ_{1}, …, ξ_{n}, η_{1}, …, η_{n}]. The standard ℤ-grading of 𝔤 is induced by the ℤ-grading of ℂ[t, ξ, η] given by

deg K_{f} = deg f − 2, where deg t = 2, deg ξ_{i} = deg η_{i} = 1.

Clearly, in this grading 𝔤 is of depth 2. Let us consider the functions that generate several first homogeneous components of 𝔤 = ⊕_{i≥−2} 𝔤_{i}:

component 𝔤_{−2} 𝔤_{−1}  𝔤_{0}  𝔤_{1}
its generators 1 Λ^{1}(ξ, η) Λ^{2}(ξ, η) ⊕ ℂ · t Λ^{3}(ξ, η) ⊕ tΛ^{1}(ξ, η)

As one can prove directly, the component 𝔤_{1} generates the whole subalgebra 𝔤_{+} of elements of positive degree. The component 𝔤_{1} splits into two 𝔤_{0}-modules 𝔤_{11} = Λ^{3} and 𝔤_{12} = tΛ^{1}. It is obvious that 𝔤_{12} is always irreducible and the component 𝔤_{11} is trivial for n = 1.

The partial Cartan prolongs of 𝔤_{11} and 𝔤_{12} are well-known:

\begin{matrix}(𝔤_{−} ⊕ 𝔤_{0}, 𝔤_{11})_{∗}^{mk} ≅ 𝔭𝔬(0|2n) ⊕ ℂ · K_{t} ≅ 𝔡(𝔭𝔬(0|2n)); \\ (𝔤_{−} ⊕ 𝔤_{0}, 𝔤_{12})_{∗}^{mk} = 𝔤_{−2} ⊕ 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{12} ⊕ ℂ · K_{t^{2}} ≅ 𝔬𝔰𝔭(2n|2). \end{matrix}

Observe a remarkable property of 𝔨(1|6): only for it does the component 𝔤_{11} split into 2 irreducible modules that we will denote 𝔤_{11}^{ξ} and 𝔤_{11}^{η}: one is contains ξ_{1}ξ_{2}ξ_{3}, the other one η_{1}η_{2}η_{3}.

Observe further, that 𝔤_{0} = 𝔠𝔬(6) ≅ 𝔤𝔩(4). As 𝔤𝔩(4)-modules, 𝔤_{11}^{ξ} and 𝔤_{11}^{η} are the symmetric squares S^{2}(id) and S^{2}(id^{∗}) of the identity 4-dimensional representation and its dual, respectively. Theorem . ([Sh5], [Sh14]) The partial Cartan prolong 𝔨𝔞𝔰^{ξ} = (𝔤_{−}⊕𝔤_{0}, 𝔤_{11}^{ξ}⊕𝔤_{12})_{∗}^{mk} is infinite dimensional and simple. It is isomorphic to 𝔨𝔞𝔰^{η} = (𝔤_{−} ⊕ 𝔤_{0}, 𝔤_{11}^{η} ⊕ 𝔤_{12})_{∗}^{mk}.

When it does not matter which of isomorphic algebras 𝔨𝔞𝔰^{ξ} ≃ 𝔨𝔞𝔰^{η} to take, we will simply write 𝔨𝔞𝔰.

I.2.4. Grozman’s theorem and a description of 𝔤 as 𝔤¯0 and 𝔤¯1. It is convenient to describe the Lie superalgebra 𝔤 of twisted polyvector fields as 𝔳𝔢𝔠𝔱-module. Similarly, in [CK2] the exceptional algebras are described as 𝔤 = 𝔤¯0 ⊕ 𝔤¯1. For most of the series such description is of little value because each homogeneous component 𝔤¯0 and 𝔤¯1 has a complicated structure. For the exceptions the situation is totally different! Observe that apart from being beautiful, such a description is useful for the construction of simple Volichenko algebras, i.e., inhomogeneous with respect to parity subalgebras of simple Lie superalgebras, cf. [LS].

Recall in relation to this a theorem of Grozman. He completely described bilinear differential operators acting in tensor fields and invariant under all changes of coordinates. Miraculously, almost all of the first order operators determine a Lie superalgebra on its domain. Some of these superalgebras turn out to be simple or close to simple. In the constructions below we use some of these invariant operators.

Let ρ be an irreducible representation of the group GL(n) in a finite dimensional vector space V and λ = (λ_{1}, …, λ_{n}) its lowest weight (see [OV]). The tensor field of type ρ or V on an n-dimensional connected manifold M is any section t of the locally trivial vector bundle over M with fiber V such that under the change of coordinates

t(y(x)) = ρ(\frac{∂y}{∂x})t(x).

The space of tensor field of type ρ or V will be denoted by T (ρ) or T (V) or even T (λ).

Denote by T (μ, …, μ) = Vol^{μ} the space of μ-densities. Other important examples: Ω^{r}, the space of differential r-forms, is T (0, …, 0, 1, …, 1) with r-many 1’s; in particular, T (0) = Ω^{0} is the space of functions; 𝔳𝔢𝔠𝔱(n) = T (−1, 0, …, 0) and L^{r} = E_{Ω^{0}}^{r}(𝔳𝔢𝔠𝔱(n)) = T (−1, …, −1, 0, …, 0) with r-many −1’s is the space of r-vector fields.

The spaces of twisted r-forms and twisted r-vector fields with twist μ are defind to be Ω_{μ}^{r} = Ω^{r} ⊗_{Ω^{0}} Vol^{μ} and L_{μ}^{r} = L^{r} ⊗_{Ω^{0}} Vol^{μ}. Obviously, L_{μ}^{r} ≃ Ω_{μ−1}^{n−r} and Vol^{1} = Ω^{n}.

The following statements are excerpts from Grozman’s difficult result [G]. To describe one of the operators, P_{4}, the Nijenhuis bracket, originally defined to be

ω^{k} ⊗ ξ, ω^{l} ⊗ η ↦ (ω^{k} ⊗ ω^{l}) ⊗ [ξ, η]+
(ω^{k} ∧ L_{ξ}(ω^{l}) + (−1)^{k}dω^{k} ∧ ι(ξ)(ω^{l})) ⊗ η+
(−L_{η}(ω^{k}) ∧ ω^{l} + (−1)^{l}ι(η)(ω^{k}) ∧ dω^{l})) ⊗ ξ,

where ι is the inner product. The Nijenhuis bracket has the following interpretation which implies its invariance: the invariant operator D : (Ω^{k} ⊗ 𝔳𝔢𝔠𝔱(M), Ω.) ⟶ Ω. given by the formula

\begin{matrix}D(ω^{k} ⊗ ξ, ω) = \\ d (ω^{k} ∧ ι(ξ)(ω) + (−1)^{k}ω^{k} ∧ ι(ξ)(dω)) = dω^{k} ∧ ι(ξ)(ω) + (−1)^{k}ω^{k} ∧ L_{ξ}(ω) \end{matrix}

is, for a fixed ω^{k} ⊗ ξ, a superdifferentiation of the supercommutative superalgebra Ω. and he Nijenhuis bracket is just the supercommutator of these superdifferentiations. So we can identify Ω. ⊗ 𝔳𝔢𝔠𝔱(M), with the Lie subsuperalgebra of 𝔳𝔢𝔠𝔱(M̂) (recall that M̂ is the supermanifold (M, Ω.(M)) and x̂_{i} = dx_{i}):

C(d) = {D ∈ 𝔳𝔢𝔠𝔱(M̂) | [D, d] = 0}, the centralizer of the exterior differential.

I.2.4.1. Statement . ([G]) The following first order differential operators are, up to dualization and permutation of arguments, all first order bilinear operators D : T (ρ_{1})⊗T (ρ_{2}) ⟶ T (ρ_{3}) (hereafer tensoring over the algebra of functions) invariant under arbitrary changes of variables:

P_{1} : Ω^{r} ⊗ T (ρ_{2}) ⟶ T (ρ_{3}), (w, t) ↦ Z(dw, t),

where Z is the zero-th order operator, the extension of the projection ρ_{1} ⊗ ρ_{2} ⟶ ρ_{3} onto the irreducible component;

P_{2} : V ect ⊗ T (ρ) ⟶ T (ρ), the Lie derivative;
P_{3} : T (S^{p}(id^{∗})) ⊗ T (S^{q}(id^{∗})) ⟶ T (S^{p+q−1}(id^{∗})), the Poisson bracket;

P_{4}: On manifolds, the bracket in C(d) is called the Nijenhuis bracket. This bracket is a linear combination of operators P_{1}, P_{1}^{∗1}, their composition with the permutation operator T (V) ⊗ T (W) ⟶ T (W) ⊗ T (V), and an “irreducible” operator P_{4};

P_{5} : Ω^{p} ⊗ Ω^{q} ⟶ Ω^{p+q+1}; ω_{1}, ω_{2} ↦ (−1)^{p(ω_{1})}a(dω_{1}ω_{2}) + b(ω_{1}dω_{2}), where a, b ∈ ℂ;

P_{6} : Ω_{μ}^{p} ⊗ Ω_{ν}^{q} ⟶ Ω_{μ+ν}^{p+q+1}, where |μ|^{2} + |ν|^{2} ≠ 0 and p + q < n is given by the formula

ω_{1}vol^{μ}, ω_{2}vol^{ν} ↦ (ν(−1)^{p(ω_{1})}dω_{1}ω_{2} − μω_{1}dω_{2}) vol^{μ+ν};
P_{7} : L^{p} ⊗ L^{q} ⟶ L^{p+q−1} the Schouten bracket;

P_{8} : L_{μ}^{p} ⊗ L_{ν}^{q} ⟶ L_{μ+ν}^{p+q−1}, a generalization of the Schouten bracket (on manifolds, for p + q ≤ n; on supermanifolds of dimension n|1 for p, q ∈ ℂ) given by the formula

Xvol^{μ}, Y vol^{ν} ↦ ((ν − 1)(μ + ν − 1)divX · Y + (−1)^{p(X)}(μ − 1)(μ + ν − 1)XdivY −
(μ − 1)(ν − 1)div(XY)) vol^{μ+ν},

where the divergence of a polyvector field is best described in local coordinates (x, x̌) on the supermanifold M̌ associated to any manifold M (see Remark after formula (0.3.7)). I.2.4.2. Statement . The following natural invariant operators determine Lie superalgebras on their domains. Some of these Lie superalgebras are close to simple ones:

P_{3} and P_{4} (by definition). The vectorial Lie superalgebra C(d) is not, however, transitive. P_{5}: for ab = 0 it determines an associative superalgebra structure; for a = b determines the

structure of a nilpotent Lie superalgebra on Π(Ω.). The operator dω_{1}ω_{2} − ω_{1}dω_{2} determines a Lie algebra structure on the space Ω..

P_{6} multiplied by \frac{μ−ν}{μν} determines structures of nilpotent Lie superalgebras on

Π( ⊕_{λ∈ℂ}Ω. ⊗ Vol^{λ})/dΩ., Ω._{+} = Π( ⊕_{λ>0; λ∈ℝ}Ω. ⊗ Vol^{λ}), and Ω._{−} = Π( ⊕_{λ<0; λ∈ℝ}Ω. ⊗ Vol^{λ}).

P_{6} multiplied by \frac{μ−ν}{μ+ν} determines a nilpotent Lie superalgebra structure on

Π(dΩ. ⊕ ⊕_{λ≠0; λ∈ℝ}Ω. ⊗ Vol^{λ}).

P_{8} multiplied by \frac{1}{μν} determines nilpotent Lie superalgebra structures on Ω._{+} and Ω._{−}.

I.2.4.3. In addition to the W -gradings it is sometimes convenient to consider gradings which lead our vectorial Lie superalgebras out of the polynomial growth class but still preserve the finiteness of depth. Namely, we set deg x_{i} = 0 for all even indeterminates x; we denote this grading n̄. Then the exceptional Lie superalgebras and 𝔟_{λ}(n; n̄) look as follows (with the unspecified brackets being given by P_{6} or P_{7}): \underline{𝔤 = 𝔢(5|10)}: 𝔤¯0 = 𝔰𝔳𝔢𝔠𝔱(5|0) ≃ dΩ^{4}, 𝔤¯1 = Π(dΩ^{1}) with the natural 𝔤¯0-action on 𝔤¯1 and the bracketing of odd elements being their product; we identify

dx_{i} ∧ dx_{j} ∧ dx_{k} ∧ dx_{l} ⊗ vol^{−1} = sign(ijklm)\frac{∂}{∂x_{m}} for any permutation (ijklm) of (12345).

\underline{𝔤 = 𝔳𝔞𝔰(4|4)}: 𝔤¯0 = 𝔳𝔢𝔠𝔱(4|0), and 𝔤¯1 = Ω^{1} ⊗ Vol^{−1/2} with the natural 𝔤¯0-action on 𝔤¯1 and the bracketing of odd elements being

[\frac{ω_{1}}{\sqrt{vol}}, \frac{ω_{2}}{\sqrt{vol}}] = \frac{dω_{1} ∧ ω_{2} + ω_{1} ∧ dω_{2}}{vol} ,

where we identify

\frac{dx_{i} ∧ dx_{j} ∧ dx_{k}}{vol} = sign(ijkl)\frac{∂}{∂x_{l}} for any permutation (ijkl) of (1234).

\underline{𝔤 = 𝔳𝔩𝔢(3|6)}: 𝔤¯0 = 𝔳𝔢𝔠𝔱(3|0) ⊕ 𝔰𝔩(2)_{≥0}^{(1)}, where 𝔤_{≥0}^{(1)} = 𝔤 ⊗ ℂ[x_{1}, x_{2}, x_{3}] , and

𝔤¯1 = (Ω^{1} ⊗ Vol^{−1/2}) ⊗ id_{𝔰𝔩(2)_{≥0}^{(1)}} with the natural 𝔤¯0-action on 𝔤¯1.

Recall that id_{𝔰𝔩(2)} is the irreducible 𝔰𝔩(2)-module L^{1} with highset weight 1; its tensor square splits into L^{2} ≃ 𝔰𝔩(2) and the trivial module L^{0}; accordingly, denote by v_{1} ∧ v_{2} and v_{1} • v_{2} the projections of v_{1} ⊗ v_{2} ∈ L^{1} ⊗ L^{1} onto the skew-symmetric and symmetric components, respectively. For f_{1}, f_{2} ∈ Ω^{0}, ω_{1}, ω_{2} ∈ Ω^{1} and v_{1}, v_{2} ∈ L^{1} we set

[\frac{ω_{1} ⊗ v_{1}}{\sqrt{vol}} , \frac{ω_{2} ⊗ v_{2}}{\sqrt{vol}}] = \frac{(ω_{1} ∧ ω_{2}) ⊗ (v_{1} ∧ v_{2}) + (dω_{1} ∧ ω_{2} + ω_{1} ∧ dω_{2}) ⊗ (v_{1} • v_{2})}{vol},

where we identify Ω^{0} with Ω^{3} ⊗_{Ω^{0}} Vol^{−1} and Ω^{2} ⊗_{Ω^{0}} Vol^{−1} with 𝔳𝔢𝔠𝔱(3|0) by setting

\frac{dx_{i} ∧ dx_{j}}{vol} = sign(ijk)\frac{∂}{∂x_{k}} for any permutation (ijk) of (123).

\underline{𝔤 = 𝔪𝔟(3|8)}: 𝔤¯0 = 𝔳𝔢𝔠𝔱(3|0) ⊕ 𝔰𝔩(2)_{≥0}^{(1)}, as for 𝔳𝔩𝔢(3|6), while 𝔤¯1 = 𝔤_{11} ⊕ 𝔤_{12}, where

𝔤_{11} = (Vol^{−1/2}) ⊗ id_{𝔰𝔩(2)_{≥0}^{(1)}} and 𝔤_{12} = (Ω^{2} ⊗ Vol^{−1/2}) ⊗ id_{𝔰𝔩(2)_{≥0}^{(1)}}.

Multiplication is similar to that of 𝔤 = 𝔳𝔩𝔢(3|6). For f_{1}, f_{2} ∈ Ω^{0}, ω_{1}, ω_{2} ∈ Ω^{1} and v_{1}, v_{2} ∈ L^{1} we set

[\frac{ω_{1} ⊗ v_{1}}{\sqrt{vol}} , \frac{ω_{2} ⊗ v_{2}}{\sqrt{vol}}] = 0,

[\frac{f_{1} ⊗ v_{1}}{\sqrt{vol}} , \frac{f_{2} ⊗ v_{2}}{\sqrt{vol}}] = \frac{(df_{1} ∧ df_{2}) ⊗ (v_{1} ∧ v_{2})}{vol},

[\frac{f_{1} ⊗ v_{1}}{\sqrt{vol}} , \frac{ω_{1} ⊗ v_{2}}{\sqrt{vol}}] = \frac{f_{1}ω_{1} ⊗ (v_{1} ∧ v_{2}) + (df_{1}ω_{1} + f_{1}dω_{1}) ⊗ (v_{1} • v_{2})}{vol}.

\underline{𝔤 = 𝔨𝔞𝔰}: 𝔤¯0 = 𝔳𝔢𝔠𝔱(1|0) ⊕ 𝔰𝔩(4)_{≥0}^{(1)}, where 𝔰𝔩(4)_{≥0}^{(1)} = 𝔰𝔩(4) ⊗ ℂ[x], and 𝔤¯1 = 𝔤_{−1} ⊕ 𝔤_{1}, where 𝔤_{−1} = Λ^{2}(id_{𝔰𝔩(4)_{≥0}^{(1)}}) and 𝔤_{1} = S^{2}(id_{𝔰𝔩(4)_{≥0}^{(1)}}^{∗}); clearly, one can interchange 𝔤_{±1}.

The multiplication is natural: the bracket of 𝔤_{1} and 𝔤_{−1} (or the other way round) is the product of the skew matrix by symmetric one (or vice versa); the action of 𝔰𝔩(4)_{≥0}^{(1)} on 𝔤_{±1} is the natural action in the space of bilinear forms (or its dual), 𝔳𝔢𝔠𝔱(1|0) acts on functionscoefficients. \underline{𝔤 = 𝔟_{λ}(n; n̄)}: here n̄ denotes the non-Weisfeiler grading given by the formulas deg q_{i} = 0, deg ξ_{i} = 0 for i = 1, …, n. Then we have 𝔤_{i} = (Π(Λ^{i−1}(𝔳𝔢𝔠𝔱(n|0))) ⊗ Vol^{−(i−1)λ} for i = −1, 0, …, n − 1. The multiplication is given by P_{8}.

Consider n = 2 more attentively. Clearly, one can interchange 𝔤_{±1}; this possibility explains the mysterious isomorphism 𝔟_{λ}(2; ¯2) ≃ 𝔟_{−1−λ}(2; ¯2) mentioned in (OI). In particular, we have additional outer automorphisms of 𝔟_{−1/2}(2; ¯2), whereas for λ = \frac{1}{2} (and λ = −\frac{3}{2}) there is a nontrivial central extension missed in [Ko1] and studied in [LSh3].

I.3. The steps of our classification. (From [LSh1], 1988, pp.235–289; compare with [K9], §3.)

Step 1. Formulation of the main ideas of our proof. Recapitulations and comparison with the Lie algebra case.

(1) Following Weisfeiler, we make use of the fact that every simple filtered vectorial Lie superalgebra has a W-filtration such that the associated graded Lie superalgebra is simple (see [W]);

(2) We show that any simple ℤ-graded vectorial Lie superalgebra ⊕_{i≥−1} 𝔤_{i} is infinite dimen-

sional if and only if 𝔤_{0} acts on 𝔤_{−1} so that the action has a rank 1 operator with an even covector.

We have listed all simple ℤ-graded vectorial Lie superalgebras of depth 1 in the union of [ALSh], [Sh3], [Sh-M], [Sh14]. This union also contains all the examples of simple vectorial superalgebras of depth d > 1 considered as abstract ones and examples of various W-regradings.

(3) The cases when 𝔤_{−1} is a purely odd superspace (consistent grading) or d > 1. We left them to consider at the end. Meanwhile Cheng and Kac investigated them ([K7], [CK2]) and found new (as compared with the known to us at that time) regradings of exceptional superalgebras, but no new abstract examples.

Step 2. Criterion for infinite dimension of a Cartan prolongation: existence of rank 1 operators with an even covector.

Step 3. Main reduction: the three cases. For 𝔤_{0} and an irreducible faithful representation ρ of 𝔤_{0} on 𝔤_{−1} with an operator of rank 1 the following cases are possible:

I) ρ is induced (from a representation of a Lie subsuperalgebra of 𝔤_{0} of codimension 0|k). II) 𝔤_{0} is almost simple or 𝔤_{0} is a central extension of an almost simple Lie superalgebra; III) dim 𝔤_{−1} = (2, 2) and 𝔤_{0} is contained in a particular Lie superalgebra. Step 4. Description of prolongations of induced representations (case I of step 3). Step 5. Description of prolongations of almost simple Lie superalgebras and their central

extensions (case II of step 3).

Step 6. Case III of step 3. (The unique simple infinite dimensional Lie superalgebra arising at this step is a regrading of 𝔟_{∞}^{∘}(2).)

Step 7. Description of W-gradings in the above-found simple superalgebras.

Step 8. The case of consistent grading and depth > 1. See [K7] and [K9].

Step 9. Description of deformations. This is a separate problem, see [LS5].

I.4. On history. Victor Kac was the first to start classification of simple vectorial Lie superalgebras. This was even in a pre-super era: in [K0] three of the four series of simple finite dimensional vectorial Lie superalgebras are described. The realization of supersymmetric nature of the Universe encouraged several researchers to begin classification of simple Lie superalgebras, of finite dimension, to start with. Interesting partial results were obtained by Scheunert, Nahm, Rittenberg [SNR], I. Kaplansky ([FK], [Kap]) and Kac ([K1], [K1C]). The union of these results added up to a complete classification (where various cases required absolutely different approach); Kac summarized the results in [K1.5].

None of these researchers, however, considered deformations with odd parameters; these deformations in the natural category of supermanifolds, i.e., with odd parameters, were first classified by J. Bernstein and Leites (for proof see [LS5]) and simultaneously and independently by M. Duflo. Observe that to throw away odd parameters is contrary to the very idea of supersymmetry.^{3}

Scheunert, Nahm, Rittenberg [SNR] classified all of simple finite dimensional Lie superalgebras except vectorial ones and Kaplansky independently classified the ones with Cartan matrix. Kac considered all finite dimensional algebras but missed some of them ([K1], [K1C]). Kaplansky’s preprint-1975 (a gist of it appeared with some extensions as [FK], and finally as [Kap]) helped Kac to repair gaps in his independent proof of the classification ([K1], [K1C]). Since various cases have to be treated separately, anyway, it is instructive to compare different techniques of [SNR], and [FK], [Kap], with [K2], where earlier partial results of Pais and Rittenberg and Djokovič are cited.

The paper [K2] — a review of the state of the art at that moment — consists of the classification, skillfully rounded up, and a “continuation of the theory” — elements of the representation theory, further classification theorems, etc. Though every original major statement (Theorems 7–10, description of systems of simple roots, even the highest weight theorem) in the “continuation” is somewhat wrong (for some corrections see [S], [Se1]–[Se3], [L2], and this paper), the paper [K2] contains not only a classification, it is the first review of Lie superalgebra theory, partly developed by the author only. The paper was and still is impressive, and it became an unjust habit to write instead of “for classification see [K2]” (as the best sourse) “Kac’s classification [K2]”, thus attributing to Kac the whole classification rather than the summary of (dis)joint efforts.

3It is interesting why for some people it is easier, psychologically, to accept odd central extensions than odd deformations ... Study of real forms (also a cohomological problem) also causes similar problems.

Next on the agenda were infinite dimensional Lie superalgebras: to start with, say, ℤgraded of polynomial growth with the aim to obtain their complete classification similar to that of Kac–Mathieu [M] or to go even further: describe their completions and deformations (the latter task is not completed even for loop algebras and though we hope a version of it is tame, it might be wild as stated, cf. [GL2]).

Simple vectorial Lie algebras were classified by É. Cartan who considered them in the framework of the class of primitive Lie algebras, almost identical with the class of simple Lie algebras.

Kac’s original idea was to follow Cartan and classify primitive Lie superalgebras; he conjectured [K2] (Th. 10 and Conjecture 1) that infinite dimensional primitive Lie superalgebras of vector fields with polynomial or formal coefficients are only straightforward analogs of the four well-known Cartan series of simple algebras 𝔳𝔢𝔠𝔱(n), 𝔰𝔳𝔢𝔠𝔱(n), 𝔥(2n) and 𝔨(2n + 1) and the Lie superalgebras of their derivations.

Since superdimension is a pair of numbers, Kac’s examples of simple vectorial Lie superalgebras “double” Cartan’s list of simple vectorial Lie algebras. It soon became clear that the actual list of simple vectorial Lie superalgebras “doubles” that of Cartan at least twice, not once: the “odd” counterparts 𝔪 of 𝔨 as well as 𝔩𝔢 — the counterpart of 𝔥 — were discovered [L1], in [ALSh] their divergence-free subalgebras, deformations 𝔟_{λ}(n; r) (misattributed to [Ko1] in [CK2], [K7], [K9]) and 𝔰̃𝔟(2^{n} − 1|2^{n}) were added.

At about the same time Grozman found 𝔟_{λ}(n) in a different setting, cf. I.2.4 After [L1] the number of Lie superalgebras serving as the linear part of a simple vectorial

Lie superalgebras grew so rapidly and their shapes were so diverse that at one moment we feared that the classification of simple vectorial superalgebras was a wild problem. Then we had understood that even the Lie superalgebras from the four well-known series (𝔳𝔢𝔠𝔱, 𝔰𝔳𝔢𝔠𝔱, 𝔥 and 𝔨), have, in addition to the dimension of the superspace on which they are usually realized, one more discrete parameter governing their other, nonstandard, realizations ([ALSh]).

In other words, one Lie superalgebra of vector fields has several (but finitely many) distinct nonisomorphic realizations as a filtered or graded Lie superalgebra. This phenomenon is not only totally missed in [K2] but even in Kac’ latest papers on the subject [K7], [K9] it is stated that to describe all filtered vectorial superalgebras (not just as abstract ones) is “probably impossible” (see [K7], the penultimate paragraph on p. 5 addresses an equivalent problem). In [K9] this problem (classify all finite dimensional irreducible faithful representations of Lie superalgebras) is unequivocally qualified as impossible. Perhaps it is, but it is not equivalent to our problem. The equivalent problem is, contrarywise, tame and in this paper we solve it.

Furthermore, several of these simple vectorial Lie superalgebras have deformations (see [L2], [Ko1], [L3], [LSh2]). Some of these deformed Lie superalgebras are very interesting due to their applications to physics. Observe, that it is natural not to confine ourselves to the classification problem of simple Lie superalgebras of certain class (say, filtered deformations) but to investigate where the deformations lead us; in this way we get, among other examples, deforms similar to the result of quantization of the Poisson Lie algebra.

Next, three exceptional vectorial algebras were discovered [Sh3], followed by a fourth exception [Sh-M] and, finally, fifth and the last exception [Sh5]. Regrettably, together with any other of our results this somehow escaped even such a detailed featured review as the one S. Sternberg wrote for Math. Reviews (99m:17006) heralding (somewhat prematurely concidering the proof, as we will see) solution of Problem B′ in [K7].

According to our method (which is a rather straightforward superization of Guillemin, Quillen and Sternberg’s approach [GQS]) — to seek the irreducible modules over finite dimensional Lie superalgebras with a rank 1 even covector — cf. [K7], [K9] — we had to consider various cases and we leisurely passed from case to case. The end became visible after the exceptional cases and a difficult regular case (see §3 and [Sh-M]) were completely done with: we have completely classified infinite dimensional simple vectorial Lie superalgebras of depth 1. Having added center to L_{0} considered, we got a number of examples of depth > 1 and were sure that there are no more examples though.

So, we have announced the complete classification in [LSh2] (as an aside remark to another story), as well as during several talks, in particular, at a seminar of E. Ivanov, JINR, Dubna (July, 1996) and Voronezh Winter School Jan. 12–18, 1997; conference in honor of D. Buchsbaum, Northeastern Univ., Boston, November 1997, see [LSh2].

Observe that, as early as in [ALSh], we wrote that the notion of “primitive Lie superalgebras” is not as “useful”, so to say, as in the Lie algebra setting and resembles that of “semisimple” Lie superalgebras (or even Lie algebras: infinite dimensional or over a field of prime characteristic). Indeed, while for finite dimensional and vectorial complex Lie algebras these notions (primitive and semisimple) can be reduced to a reasonable notion of simple algebras, there are “too many primitive Lie superalgebras”. For this simple but important fact, announced in [ALSh], see Appendix. So, having established that the problem of classification of primitive Lie superalgebras is a wild one ([ALSh]), we only classify simple Lie superalgebras.

Kac’s talk [K6], where he considered consistent gradings of simple exceptional vectorial Lie superalgebras which proved to be of (WF) type, i.e., gave their new, as compared with known to us, realizations was an unexpected contribution to our proof after we had confirmed that we did had announced the classification. Though brief, the talk left the impression that Kac has a new powerful method which proves our classification of simple vectorial Lie superalgebras (Problem A or even B) quicker than we do.

It was, however, unclear why the list of the series of simple vectorial superalgebras Kac gave in [K6] looked shorter than ours, and that of exceptions longer.

While we were eagerly waiting for the details of [K6] we tidied up our proof, and submitted [Sh5], a short version of [Sh14], which by that time was being refereed for almost two years.

When the details of [K6] were published in [K7] we saw, to our amazement, that Kac’s method is very similar to ours, the one we lectured and discussed. Actually, Kac applied the same approach (superization of [GQS]) but more ingeniously and considered the cases we left out so far. While we were busy considering various filtrations, he distinguished one particular filtration for each of complete simple vectorial Lie superalgebras.

Since it does not matter which base of neighborhoods of zero one takes to determine the topology, Kac gave examples of complete simple Lie superalgebras considered as abstract ones and isomorphic to the completions of our graded ones.

The totally new and vital items in [K7], as compared with our results published by that time, were (1) the study of the case of consistent grading (which, as we conjectured, did not give any new examples to our final list if we consider algebras as abstract ones, not as graded or filtered) and (2) recovery, following Weisfeiler, of algebras of depth > 1 from the depth 1 ones.

In [K7], [CK1], [CK2], Cheng and Kac considered a part of our problem only, namely, the classification of complete vectorial Lie superalgebras considered as abstract ones (presumably under a wrong impression that there are “too many” W-filtrations). Still, their result doublechecked the completeness of our classification if we consider our algebras as abstract ones.

Meanwhile we had listed all the distinct W-gradings of these Lie superalgebras and were finishing the proof of the completeness of our list. We considered the following tasks:

(1) to list the examples of simple algebras of depth 1;

(1′) to prove completeness of the similar list for depth > 1;

(2) for a given Lie superalgebra list its W-filtrations;

(3) classify the deformations.

Task (3) leads us outside the realm of the original problem, but applications of its solution make it worth the trouble.

In [K7] Kac skillfully simplified problems (1) and (1′) and ignored problem (2) (dubbing it as “perhaps, impossible”) and in [CK1] together with Cheng he slackened the formulation of problem (3): they only considered filtered deformations, moreover, disregarding odd parameters and throwing them away when they popped up.

Since [CK1] only considers one particular filtration, the problem of complete classification of filtered deformations is still open.

Comparison of notations. Strangely enough, not all of the results from [ALSh] or [Ko4] are reproduced in [K7]. Cheng and Kac considered the third parameter r for exceptional Lie superalgebras only. Cheng and Kac totally ignored subtleties described in I.2.2; so far we had published only (IAA) and some of the isomorphisms listed in (OI) and only these were mentioned in [K7]. Here is a comparison of series.

our notations notations from [K7]
𝔳𝔢𝔠𝔱(m|n; r) W (m|n)
𝔰𝔳𝔢𝔠𝔱(m|n; r) S(m|n)
𝔰𝔳𝔢𝔠𝔱^{∘}(1|n; r) S(1|n)
{ \atop 𝔰𝔳𝔢𝔠𝔱}(0|n)  S̃(0|n)

(S_{vect})

𝔨(2m + 1|n; r) K(2m + 1|n)
𝔥(2m|n; r)  H(2m|n)
𝔥_{λ}(2|2; r)  not singled out
𝔥^{o}(0|n)  H(0|n)

(S_{k})

𝔪(n|n + 1; r)  KO(n|n + 1)
𝔰𝔪(n|n + 1; r)  SKO(n|n + 1)
𝔟_{λ}(n|n + 1; r)  SKO(n|n + 1; β), where β = \frac{b}{a} = 1 − \frac{2}{nλ} ∈ ℂ
𝔟_{1}^{∘}(n|n + 1; r)  SKO(n|n + 1; \frac{n−2}{n})
𝔟_{∞}^{∘}(n|n + 1; r)  SKO(n|n + 1; 1)
𝔩𝔢(n|n; r)  HO(n|n)
𝔰𝔩𝔢^{∘}(n|n; r)  SHO(n|n)
𝔰̃𝔟_{μ}(2^{n−1} − 1|2^{n−1}) not recognized as vectorial; appears as a filtered deformation

(S_{m})

The parameter λ of 𝔟_{λ}(n|n + 1; r) is a projective one: λ ∈ ℂP^{1}. At λ = 0 the Lie superalgebra 𝔟_{0}(n|n + 1) is the Buttin superalgebra — a central extension of 𝔩𝔢(n|n); it is an “odd” analog of the Poisson algebra 𝔭𝔬(2n). Therefore, the whole family 𝔟_{λ}(n|n + 1) can be considered as a deformation of the Buttin superalgebra. In Kac’s notations the parameter β is an affine one: β ∈ ℂ. However, as is clear from the translation formulas, λ = 0 corresponds to β = ∞.

Deformations: discussion. Let us forget at the moment that one should not disregard odd deformations, leave alone general deformations, and consider the following result of computations of filtered deformations [CK1] (considered as final, e.g., in [K9]):

There are only three series L̃ of filtered deformations of Lie superalgebras L and all of them can be described as follows. Let L ⊂ 𝔳𝔢𝔠𝔱(n|m; r) be a subalgebra with the same negative part, let Θ be the product of all m odd indeterminates and t a parameter, p(t) = p(m). Then

L̃ = (1 + tΘ)L  (d.1)

is a subalgebra of 𝔳𝔢𝔠𝔱(n|m; r). It so happens that usually L̃ ≃ L but there are three series of exceptions: 𝔰𝔳𝔢𝔠𝔱(0|n), 𝔰𝔩𝔢(n) and 𝔟_{1}(n).

Now, observe that

(1) all these deformations were known: the first one appeared in [K2], the other two are due to Kochetkov. Kochetkov’s description of deformations is partly buried in inaccessible VINITI depositions [Ko4] (reproduced in [L6]) and partly unpublished (a letter to Leites, at that time Kochetkov’s Ph.D. advisor).

(2) Kochetkov [Ko4] had also found several odd cocycles, see [L6]. In [CK1] they are rediscovered but deliberately thrown away.

(3) In 1985 Kochetkov further observed (unpublished) that nontrivial deformations of the form (d.1) correspond to the Lie superalgebras with a trace, i.e., with an ideal of codimension 1. Likewise, the existence of an integral on the space of values of the divergence map gives rise to similar deformations.

(4) In [CK1] it is shown that filtered deformations of the graded Lie superalgebra 𝔤 of nonzero depth and with a decreasing filtration correspond to 𝔤_{0}-invariants of the superspace H^{2}(𝔤_{−}; 𝔤) (of which Cheng and Kac only take the even part). A problem arises: let 𝔤 ≃ 𝔥 be two ℤ-gradings of the same abstract algebra. Is the formula

H^{2}(𝔤_{−}; 𝔤)^{𝔤_{0}} ≃ H^{2}(𝔥_{−}; 𝔥)^{𝔥_{0}}  (d.2)

true?

If not, [CK1] contains not the description of filtered deformations of all simple W-graded vectorial Lie superalgebras, as claimed^{4}, but \frac{1}{m}th of all possibilities for each n|m-dimensional supermanifold.

If yes, it is remarkable: why should this be true?! A proof is desirable. (For example, the whole spaces H^{2}(𝔤_{−}; 𝔤) and H^{2}(𝔥_{−}; 𝔥) are always distinct in the cases considered in [GLS2].) Dzhumadildaev told us that (d.2) is not true for some simple Lie algebras in characteristic p.

For other examples of deformations see [L6] and [LSh3]. For the answer to the above question (d.2) and the complete description of deformations see [LS5], where we give a simplified proof of the results of [Ko4], and cases left, and so close the problem.

§0. Background and basic examples

0.1. Linear algebra in superspaces. Generalities. A superspace is a ℤ/2-graded space; for any superspace V = V¯0 ⊕ V¯1 denote by Π(V) another copy of the same superspace: with the shifted parity, i.e., (Π(V))ī = V_{ī+¯1}. The superdimension of V is dim V = p + qε, where ε^{2} = 1 and p = dim V¯0, q = dim V¯1. (Usually, dim V is expressed as a pair (p, q) or p|q; this obscures the fact that dim V ⊗ W = dim V · dim W; this fact is clear with the use of ε.)

A superspace structure in V induces the superspace structure in the space End(V). A basis of a superspace is always a basis consisting of homogeneous vectors; let Par = (p_{1}, …, p_{dim V}) be an ordered collection of their parities. We call Par the format of (the basis of) V. A square supermatrix of format (size) Par is a dim V × dim V matrix whose ith row and ith column are of the same parity p_{i} ∈ Par.

One usually considers one of the simplest formats Par, e.g., the format Par_{st} of the form (¯0, …, ¯0; ¯1, …, ¯1) is called standard. In this paper we can do without nonstandard formats. But they are vital in the classification of systems of simple roots of simple finite dimensional simple Lie superalgebras. In particular, the reader might be interested in them in connection with applications to q-quantization or integrable systems. Systems of simple roots corresponding to distinct nonstandard formats are related by so-called odd reflections — analogs of our nonstandard regradings.

The matrix unit E_{ij} is supposed to be of parity p_{i} + p_{j} and the bracket of supermatrices (of the same format) is defined via Sign Rule:

if something of parity p moves past something of parity q the sign (−1)^{pq} accrues; the formulas defined on homogeneous elements are extended to arbitrary ones via linearity.

Examples of application of the Sign Rule: setting [X, Y] = XY − (−1)^{p(X)p(Y)}Y X we get the notion of the supercommutator and the ensuing notions of the supercommutative superalgebra; Lie superalgebra is the one which in addition to superskew-commutativity satisfies the super Jacobi identity, i.e., the Jacobi identity amended with the Sign Rule); the superderivation of a superalgebra A is a linear map D : A ⟶ A that satisfies the super Leibniz rule

D(ab) = D(a)b + (−1)^{p(D)p(a)}aD(b).

In particular, let A = ℂ[x] be the free supercommutative polynomial superalgebra in x = (x_{1}, …, x_{n}), where the superstructure is determined by the parities of the indeterminates: p(x_{i}) = p_{i}. Partial derivatives are defined (with the help of super Leibniz Rule) by the formulas \frac{∂x_{i}}{∂x_{j}} = δ_{i,j}. Clearly, the collection 𝔡𝔢𝔯 ℂ[x] of all superdifferentiations of A is a Lie superalgebra whose elements are of the form ∑ f_{i}(x)\frac{∂}{∂x_{i}} with f_{i}(x) ∈ ℂ[x] for all i.

Observe that sometimes applying Sign Rule requires some dexterity (we have to distinguish between two versions both of which turn in non-supercase into one, called either skew- or anti-commutativity):

4Actually, the claim of [CK1] is more vague and subject to interpretations: “we determine all simple filtered deformations of certain ℤ-graded Lie superalgebras classified in [K3]”, and since in [K3] the algebras are only classified as abstract ones, one might argue that (d.2) is not claimed there.

ba = (−1)^{p(b)p(a)}ab  (symmetry)
ba = −(−1)^{p(b)p(a)}ab  (anti-symmetry)
ba = (−1)^{(p(b)+1)(p(a)+1)}ab (skew-symmetry)
ba = −(−1)^{(p(b)+1)(p(a)+1)}ab (antiskew-symmetry)

In what follows the superskew-symmetric bilinear forms are named according to the above definitions.

Given the supercommutative superalgebra F of “functions” in indeterminates x, define the supercommutative superalgebra Ω of differential forms as polynomial algebra over F in dx, where p(d) = ¯1. Since dx is even for an odd x, we can consider not only polynomials in dx. Smooth or analytic functions in the differentials of the x are called pseudodifferential forms on the supermanifold with coordinates x, see [BL]. We will need them to interpret 𝔟_{λ}(n). The exterior differential is defined on the space of (pseudo) differential forms by the formulas (mind super Leibniz rule):

d(x_{i}) = dx_{i} and d^{2} = 0.

The Lie derivative is defined (minding same Rules) by the formula

L_{D}(df) = (−1)^{p(D)}d(D(f)).

In particular,

L_{D} ((df)^{λ}) = λ(−1)^{p(D)}d(D(f))(df)^{λ−1} for any λ ∈ ℂ and odd f.

The general linear Lie superalgebra of all supermatrices of size Par is denoted by 𝔤𝔩(Par), where Par = (p_{1}, …) is the ordered collection of parities of the rows identical to that of the columns; usually, for the standard (simplest) format 𝔤𝔩(¯0, …, ¯0, ¯1, …, ¯1) is abbreviated to 𝔤𝔩(dim V¯0| dim V¯1). Any nonzero supermatrix from 𝔤𝔩(Par) can be uniquely expressed as the sum of its even and odd parts; in the standard format this is the following block expression:

(\begin{matrix}A & B \\ C & D \end{matrix}) = (\begin{matrix}A & 0 \\ 0 & D \end{matrix}) + (\begin{matrix}0 & B \\ C & 0 \end{matrix}) , p ((\begin{matrix}A & 0 \\ 0 & D \end{matrix})) = ¯0, p ((\begin{matrix}0 & B \\ C & 0 \end{matrix})) = ¯1.

The supertrace is the map 𝔤𝔩(Par) ⟶ ℂ, (A_{ij}) ↦ ∑(−1)^{p_{i}}A_{ii}, where Par = (p_{1}, …). Since str[x, y] = 0, the subsuperspace of supertraceless matrices constitutes the special linear Lie subsuperalgebra 𝔰𝔩(Par).

There are, however, at least two super versions of 𝔤𝔩(n), not one. The other version is called the queer Lie superalgebra and is defined as the one that preserves the complex structure given by an odd operator J, i.e., is the centralizer C(J) of J:

𝔮(n) = C(J) = {X ∈ 𝔤𝔩(n|n) : [X, J] = 0}, where J^{2} = −id.

It is clear that by a change of basis we can reduce J to the form J_{2n} = (\begin{matrix}0 & 1_{n} \\ −1 & 0 \end{matrix}) in the

standard format and then 𝔮(n) takes the form

𝔮(n) = \{(\begin{matrix}A & B \\ B & A \end{matrix})\} .

On 𝔮(n), the queertrace is defined: qtr : (\begin{matrix}A & B \\ B & A \end{matrix}) ↦ trB. Denote by 𝔰𝔮(n) the Lie superal-

gebra of queertraceless matrices.

Observe that the identity representations of 𝔮(V) and 𝔰𝔮(V) in V, though irreducible in supersetting, are not irreducible in the nongraded sense: take homogeneous (with respect to parity) and linearly independent vectors v_{1}, …, v_{n} from V; then Span(v_{1} + J(v_{1}), …, v_{n} + J(v_{n})) is an invariant subspace of V which is not a subsuperspace.

A representation is irreducible of general type or just of G-type if there is no invariant subspace, otherwise it is called irreducible of Q-type (Q is after the general queer Lie superalgebra — a specifically “superish” analog of 𝔤𝔩); an irreducible representation of Q-type has no invariant subsuperspace but has an invariant subspace.

Lie superalgebras that preserve bilinear forms: two types. To the linear map F : V ⟶ W of superspaces there corresponds the dual map F^{∗} : W^{∗} ⟶ V^{∗} between the dual superspaces. In a basis consisting of the vectors v_{i} of format Par, the formula F (v_{j}) = ∑_{i} v_{i}A_{ij} assigns to F the supermatrix A. In the dual bases, to F^{∗} the supertransposed

matrix A^{st} corresponds:

(A^{st})_{ij} = (−1)^{(p_{i}+p_{j})(p_{i}+p(A))}A_{ji}.

The supermatrices X ∈ 𝔤𝔩(Par) such that

X^{st}B + (−1)^{p(X)p(B)}BX = 0 for an homogeneous matrix B ∈ 𝔤𝔩(Par)

constitute the Lie superalgebra 𝔞𝔲𝔱(B) that preserves the bilinear form B^{f} on V whose matrix B is given by the formula B_{ij} = (−1)^{p(B^{f})p(v_{i})}B^{f}(v_{i}, v_{j}) for the basis vectors v_{i}.

Recall that the supersymmetry of the homogeneous form B^{f} means that its matrix B

satisfies the condition B^{u} = B, where B^{u} = (\begin{matrix}R^{t} & (−1)^{p(B)}T^{t} \\ (−1)^{p(B)}S^{t} & −U^{t} \end{matrix}) for the matrix

B = (\begin{matrix}R & S \\ T & U \end{matrix}). Similarly, skew-supersymmetry of B means that B^{u} = −B. Thus, we

see that the upsetting of bilinear forms u : Bil(V, W) ⟶ Bil(W, V), which for the spaces and when V = W is expressed on matrices in terms of the transposition, is a new operation.

Most popular canonical forms of the nondegenerate supersymmetric form are the ones whose supermatrices in the standard format are the following canonical ones, B_{ev} or B′_{ev}:

B′_{ev}(m|2n) = (\begin{matrix}1_{m} & 0 \\ 0 & J_{2n} \end{matrix}) , where J_{2n} = (\begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix}),

or

B_{ev}(m|2n) = (\begin{matrix}antidiag(1, …, 1) & 0 \\ 0 & J_{2n} \end{matrix}) .

The usual notation for 𝔞𝔲𝔱(B_{ev}(m|2n)) is 𝔬𝔰𝔭(m|2n) or, more precisely, 𝔬𝔰𝔭^{sy}(m|2n). Observe that the passage from V to Π(V) sends the supersymmetric forms to superskew-symmetric ones, preserved by the “symplectico-orthogonal” Lie superalgebra, 𝔰𝔭𝔬(2n|m) or, more prudently, 𝔬𝔰𝔭^{sk}(m|2n), which is isomorphic to 𝔬𝔰𝔭^{sy}(m|2n) but has a different matrix realization.

In the standard format the matrix realizations of these algebras are:

\begin{matrix}𝔬𝔰𝔭(m|2n) = \{(\begin{matrix}E & Y & X^{t} \\ X & A & B \\ −Y^{t} & C & −A^{t} \end{matrix})\}; 𝔬𝔰𝔭^{sk}(m|2n) = \{(\begin{matrix}A & B & X \\ C & −A^{t} & Y^{t} \\ Y & −X^{t} & E \end{matrix})\} , \\ where (\begin{matrix}A & B \\ C & −A^{t} \end{matrix}) ∈ 𝔰𝔭(2n), E ∈ 𝔬(m). \end{matrix}

A nondegenerate supersymmetric odd bilinear form B_{odd}(n|n) can be reduced to a canonical form whose matrix in the standard format is J_{2n}. A canonical form of the superskew odd nondegenerate form in the standard format is Π_{2n} = (\begin{matrix}0 & 1_{n} \\ 1_{n} & 0 \end{matrix}). The usual notation

for 𝔞𝔲𝔱(B_{odd}(Par)) is 𝔭𝔢(Par). The passage from V to Π(V) establishes an isomorphism 𝔭𝔢^{sy}(Par) ≅ 𝔭𝔢^{sk}(Par). This Lie superalgebra is called, as A. Weil suggested, periplectic. The matrix realizations in the standard format of these superalgebras is shorthanded to:

𝔭𝔢^{sy} (n) = \{(\begin{matrix}A & B \\ C & −A^{t} \end{matrix}) , where B = −B^{t}, C = C^{t}\};

𝔭𝔢^{sk}(n) = \{(\begin{matrix}A & B \\ C & −A^{t} \end{matrix}) , where B = B^{t}, C = −C^{t}\} .

Observe that though the Lie superalgebras 𝔬𝔰𝔭^{sy}(m|2n) and 𝔬𝔰𝔭^{sk}(m|2n), as well as 𝔭𝔢^{sy}(n) and 𝔭𝔢^{sk}(n), are isomorphic, the difference between them is sometimes crucial, see [Sh5].

The special periplectic superalgebra is

𝔰𝔭𝔢(n) = {X ∈ 𝔭𝔢(n) | strX = 0}.

Of particular interest to us will be also

𝔰𝔭𝔢(n)_{a,b} = 𝔰𝔭𝔢(n) ⊂ + ℂ(az + bd), where z = 1_{2n}, d = diag(1_{n}, −1_{n}).

0.1.1. What a Lie superalgebra is. Dealing with superalgebras it sometimes becomes useful to know their definition. Lie superalgebras were distinguished in topology in 1930’s or earlier. So when somebody offers a “better than usual” definition of a notion which seemed to have been established about 70 year ago this might look strange, to say the least. Nevertheless, the answer to the question “what is a Lie superalgebra?” is still not a common knowledge.

So far we operated as if we know. However, the naive definition suggested above (“apply the Sign Rule to the definition of the Lie algebra”) is manifestly inadequate for considering the (singular) supervarieties of deformations and to apply representation theory to mathematical physics, for example, in the study of the coadjoint representation of the Lie supergroup which can act on a supermanifold but never on a superspace (an object from another category). So, to deform Lie superalgebras and apply group-theoretical methods in “super” setting, we must be able to recover a supermanifold from a superspace, and vice versa.

A proper definition of Lie superalgebras is as follows, cf. [L3]–[L5]. The Lie superalgebra in the category of supermanifolds corresponding to the “naive” Lie superalgebra L = L¯0 ⊕L¯1 is a linear supermanifold L = (L¯0, O), where the sheaf of functions O consists of functions on L¯0 with values in the Grassmann superalgebra on L∗ ; ¯1; this supermanifold should be such that for “any” (say, finitely generated, or from some other appropriate category) supercommutative superalgebra C, the space L(C) = Hom(SpecC, L), called the space of C-points of L, is a Lie algebra and the correspondence C ⟶ L(C) is a functor in C. (A. Weil introduced this approach in algebraic geometry in 1953; in super setting it is called the language of points or families, see [L4], [D].) This definition might look terribly complicated, but fortunately one can show that the correspondence L ⟷ L is one-to-one and the Lie algebra L(C), also denoted L(C), admits a very simple description: L(C) = (L ⊗ C)¯0.

A Lie superalgebra homomorphism ρ : L_{1} ⟶ L_{2} in these terms is a functor morphism, i.e., a collection of Lie algebra homomorphisms ρ_{C} : L_{1}(C) ⟶ L_{2}(C) compatible with morphisms of supercommutative superalgebras C ⟶ C′. In particular, a representation of a Lie superalgebra L in a superspace V is a homomorphism ρ : L ⟶ 𝔤𝔩(V), i.e., a collection of Lie algebra homomorphisms ρ_{C} : L(C) ⟶ (𝔤𝔩(V) ⊗ C)¯0.

Example . Consider a representation ρ : 𝔤 ⟶ 𝔤𝔩(V). The tangent space of the moduli superspace of deformations of ρ is isomorphic to H^{1}(𝔤; V ⊗ V^{∗}). For example, if 𝔤 is the 0|ndimensional (i.e., purely odd) Lie superalgebra (with the only bracket possible: identically equal to zero), its only irreducible representations are the trivial one, 1, and Π(1). Clearly, 1 ⊗ 1^{∗} ≃ Π(1) ⊗ Π(1)^{∗} ≃ 1, and because the Lie superalgebra 𝔤 is commutative, the differential in the cochain complex is trivial. Therefore, H^{1}(𝔤; 1) = E^{1}(𝔤^{∗}) ≃ 𝔤^{∗}, so there are dim 𝔤 odd parameters of deformations of the trivial representation. If we consider 𝔤 “naively” all of the odd parameters will be lost.

Which of these infinitesimal deformations can be extended to a global one is a separate much tougher question, usually solved ad hoc.

Examples that lucidly illustrate why one should always remember that a Lie superalgebra is not a mere linear superspace but a linear supermanifold are, e.g., deforms { \atop 𝔰𝔳𝔢𝔠𝔱}(0|2n + 1) and 𝔰̃𝔟_{μ}(2^{2n}−1|2^{2n}) with odd parameters. In the category of supermanifolds these are simple Lie superalgebras.

0.1.2. Projectivization. If 𝔰 is a Lie algebra of scalar matrices, and 𝔤 ⊂ 𝔤𝔩(n|n) is a Lie subsuperalgebra containing 𝔰, then the projective Lie superalgebra of type 𝔤 is 𝔭𝔤 = 𝔤/𝔰. Lie superalgebras 𝔤_{1} ⨀ 𝔤_{2} described in sect. 3.1 are also projective.

Projectivization sometimes leads to new Lie superalgebras, for example: 𝔭𝔤𝔩(n|n), 𝔭𝔰𝔩(n|n), 𝔭𝔮(n), 𝔭𝔰𝔮(n); whereas 𝔭𝔤𝔩(p|q) ≅ 𝔰𝔩(p|q) if p ≠ q.

0.1.3. What a semisimple Lie superalgebra is. Recall that the Lie superalgebra 𝔤 without proper ideals and of dimension > 1 is called simple. Examples: 𝔰𝔩(m|n) for m > n ≥ 1, 𝔭𝔰𝔩(n|n) for n > 1, 𝔭𝔰𝔮(n) for n > 2, 𝔬𝔰𝔭(m|2n) for mn ≠ 0 and 𝔰𝔭𝔢(n) for n > 2.

In §5 we will need the remaining simple finite dimensional Lie superalgebras of non-vectorial type. These superalgebras, discovered by I. Kaplansky (an unpublished preprint, see [FK]) are 𝔬𝔰𝔭_{α}(4|2), the deforms of 𝔬𝔰𝔭(4|2), and the two exceptions that we denote by 𝔞𝔤_{2} and 𝔞𝔟_{3}. These Lie superalgebras do not contribute to our classification; for their description we refer to [FK], [K2] completed in [vdL], [Sud] and [GL1].

We say that 𝔥 is almost simple if it can be sandwiched (non-strictly) between a simple Lie superalgebra 𝔰 and the Lie superalgebra 𝔡𝔢𝔯 𝔰 of derivations of 𝔰, i.e., 𝔰 ⊂ 𝔥 ⊂ 𝔡𝔢𝔯 𝔰.

By definition, 𝔤 is semisimple if its radical is zero. Literally following the description of semisimple Lie algebras over the fields of prime characteristic, V. Kac [K2] gave the following description of semisimple Lie superalgebras. Let 𝔰_{1}, …, 𝔰_{k} be simple Lie superalgebras, let n_{1}, …, n_{k} be pairs of non-negative integers n_{j} = (n¯0 ; j, n¯1 ; j), let F(n_{j}) be the supercommutative superalgebra of polynomials in n¯0 ; j even and n¯1 ; j odd indeterminates, and 𝔰 = ⊕_{j} (𝔰_{j} ⊗ F(n_{j})).

Then

𝔡𝔢𝔯 𝔰 = ⊕_{j} ((𝔡𝔢𝔯 𝔰_{j}) ⊗ F(n_{j}) ⊂ + id_{𝔰_{j}} ⊗ 𝔳𝔢𝔠𝔱(n_{j})) .  (0.1.3)

Let 𝔤 be a subalgebra of 𝔡𝔢𝔯𝔰 containing 𝔰. If the projection of 𝔤 on id_{𝔰_{j}} ⊗ 𝔳𝔢𝔠𝔱(n_{j})_{−1} is onto for each j, then 𝔤 is semisimple and all semisimple Lie superalgebras arise in the manner indicated, i.e., as sums of almost simple superalgebras corresponding to the summands of (0.1.3).

0.1.4. A. Sergeev’s central extension. In 70’s A. Sergeev proved that there is just one nontrivial central extension of 𝔰𝔭𝔢(n) for n > 2. It exists only for n = 4 and we denote it by 𝔞𝔰. Let us represent an arbitrary element A ∈ 𝔞𝔰 as a pair A = x + d · z, where x ∈ 𝔰𝔭𝔢(4), d ∈ ℂ and z is the central element. The bracket in 𝔞𝔰 is

[(\begin{matrix}a & b \\ c & −a^{t} \end{matrix}) + d · z, (\begin{matrix}a′ & b′ \\ c′ & −a′^{t} \end{matrix}) + d′ · z] = [(\begin{matrix}a & b \\ c & −a^{t} \end{matrix}) , (\begin{matrix}a′ & b′ \\ c′ & −a′^{t} \end{matrix})] + tr cc̃′ · z, (0.1.4.1)

where ˜ is extended via linearity from matrices c_{ij} = E_{ij} − E_{ji} on which c̃_{ij} = c_{kl} for any even permutation (1234) ↦ (ijkl).

The Lie superalgebra 𝔞𝔰 can also be described with the help of the spinor representation. For this we need several vectorial superalgebras defined in sect. 0.3. Consider 𝔭𝔬(0|6), the Lie superalgebra whose superspace is the Grassmann superalgebra Λ(ξ, η) generated by ξ_{1}, ξ_{2}, ξ_{3}, η_{1}, η_{2}, η_{3} and the bracket is the Poisson bracket (0.3.6). Recall that 𝔥(0|6) = Span(H_{f} : f ∈ Λ(ξ, η)).

Now, observe that 𝔰𝔭𝔢(4) can be embedded into 𝔥(0|6). Indeed, setting deg ξ_{i} = deg η_{i} = 1 for all i we introduce a ℤ-grading on Λ(ξ, η) which, in turn, induces a ℤ-grading on 𝔥(0|6) of the form 𝔥(0|6) = ⊕_{i≥−1} 𝔥(0|6)_{i}. Since 𝔰𝔩(4) ≅ 𝔬(6), we can identify 𝔰𝔭𝔢(4)_{0} with 𝔥(0|6)_{0}.

It is not difficult to see that the elements of degree −1 in the standard gradings of 𝔰𝔭𝔢(4) and 𝔥(0|6) constitute isomorphic 𝔰𝔩(4) ≅ 𝔬(6)-modules. It is subject to a direct verification that it is possible to embed 𝔰𝔭𝔢(4)_{1} into 𝔥(0|6)_{1}.

Sergeev’s extension 𝔞𝔰 is the result of the restriction to 𝔰𝔭𝔢(4) ⊂ 𝔥(0|6) of the cocycle that turns 𝔥(0|6) into 𝔭𝔬(0|6). The quantization deforms 𝔭𝔬(0|6) into 𝔤𝔩(Λ(ξ)); the through maps T_{λ} : 𝔞𝔰 ⟶ 𝔭𝔬(0|6) ⟶ 𝔤𝔩(Λ(ξ)) are representations of 𝔞𝔰 in the 4|4-dimensional modules spin_{λ} isomorphic to each other for all λ ≠ 0. The explicit form of T_{λ} is as follows:

T_{λ} : (\begin{matrix}a & b \\ c & −a^{t} \end{matrix}) + d · z ↦ (\begin{matrix}a & b − λc̃ \\ c & −a^{t} \end{matrix}) + λd · 1_{4|4},  (0.1.4.2)

where 1_{4|4} is the unit matrix and c̃ is defined in formula (0.1.4.1). Clearly, T_{λ} is an irreducible representation for any λ.

0.2. Vectorial Lie superalgebras. The standard realization. The elements of the Lie algebra L = 𝔡𝔢𝔯 ℂ[[u]] are considered as vector fields. The Lie algebra L has only one maximal subalgebra L_{0} of finite codimension (consisting of the fields that vanish at the origin). The subalgebra L_{0} determines a filtration of L: set

L_{−1} = L and L_{i} = {D ∈ L_{i−1} | [D, L] ⊂ L_{i−1}} for i ≥ 1.  (0.2.1)

The associated graded Lie algebra L = ⊕_{i≥−1} L_{i}, where L_{i} = L_{i}/L_{i+1}, consists of the vector

fields with polynomial coefficients.

Superization. For a simple Lie superalgebra L (for example, take L = 𝔡𝔢𝔯 ℂ[u, ξ]), suppose L_{0} ⊂ L is a maximal subalgebra of finite codimension. Let L_{−1} be a minimal subspace of L containing L_{0}, different from L_{0} and L_{0}-invariant. A Weisfeiler filtration of L is determined by setting for i ≥ 1:

L_{−i−1} = [L_{−1}, L_{−i}] + L_{−i} and L_{i} = {D ∈ L_{i−1} | [D, L_{−1}] ⊂ L_{i−1}}.  (0.2.2)

Since the codimension of L_{0} is finite, the filtration takes the form

L = L_{−d} ⊃ ⋯ ⊃ L_{0} ⊃ …  (0.2.3)

for some d. This d is called the depth of L and of the associated graded Lie superalgebra L.

Considering the subspaces (0.2.1) as the basis of a topology, we can complete the graded or filtered Lie superalgebras L or L; the elements of the completion are the vector fields with formal power series as coefficients. Though the structure of the graded algebras is easier to describe, in applications the completed Lie superalgebras are usually needed. Remarks . 1) Not all filtered or graded Lie superalgebras of finite depth are vectorial, i.e., realizable with vector fields on a supermanifold of the same dimension as that of L/L_{0}; only those with faithful L_{0}-action on L_{−} are.

2) Unlike Lie algebras, simple vectorial superalgebras possess several nonisomorphic maximal subalgebras of finite codimension, see I.2.2.

1) General algebras. Let x = (u_{1}, …, u_{n}, θ_{1}, …, θ_{m}), where the u_{i} are even indeterminates and the θ_{j} are odd ones. Set 𝔳𝔢𝔠𝔱(n|m) = 𝔡𝔢𝔯 ℂ[x]; it is called the general vectorial Lie superalgebra.

2) Special algebras. The divergence of the field D = ∑_{i} f_{i} \frac{∂}{∂u_{i}} + ∑_{j} g_{j} \frac{∂}{∂θ_{j}} is the function

(in our case: a polynomial, or a series)

divD = ∑_{i} \frac{∂f_{i}}{∂u_{i}} + ∑_{j}(−1)^{p(g_{j})} \frac{∂g_{i}}{∂θ_{j}} .  (0.2.4)

• The Lie superalgebra 𝔰𝔳𝔢𝔠𝔱(n|m) = {D ∈ 𝔳𝔢𝔠𝔱(n|m) : divD = 0} is called the special (or divergence-free) vectorial superalgebra.

It is clear that it is also possible to describe 𝔰𝔳𝔢𝔠𝔱(n|m) as {D ∈ 𝔳𝔢𝔠𝔱(n|m) | L_{D}vol_{x} = 0}, where vol_{x} is the volume form with constant coefficients in coordinates x (see 0.6) and L_{D} the Lie derivative with respect to D.

• The Lie superalgebra 𝔰𝔳𝔢𝔠𝔱_{λ}(0|m) = {D ∈ 𝔳𝔢𝔠𝔱(0|m) | div(1 + λθ_{1} · · · · · θ_{m})D = 0}, where p(λ) ≡ m (mod 2), — the deform of 𝔰𝔳𝔢𝔠𝔱(0|m) — is called the deformed special (or divergence-free) vectorial superalgebra. Clearly, 𝔰𝔳𝔢𝔠𝔱_{λ}(0|m) ≅ 𝔰𝔳𝔢𝔠𝔱_{μ}(0|m) for λμ ≠ 0. So we briefly denote these deforms by { \atop 𝔰𝔳𝔢𝔠𝔱}(0|m).

Observe that for odd m the parameter of deformation, λ, is odd. Remark . As is customary in differential geometry, where meaningful notations prevail, we sometimes write 𝔳𝔢𝔠𝔱(x) or 𝔳𝔢𝔠𝔱(V) if V = Span(x) and use similar notations for the subalgebras of 𝔳𝔢𝔠𝔱 introduced below. Some algebraists sometimes abbreviate 𝔳𝔢𝔠𝔱(n) and 𝔰𝔳𝔢𝔠𝔱(n) to W_{n} (in honor of Witt) and S_{n}, respectively.

3) The algebras that preserve Pfaff equations and differential 2-forms.

• Set u = (t, p_{1}, …, p_{n}, q_{1}, …, q_{n}); let

α̃_{1} = dt + ∑_{1≤i≤n}(p_{i}dq_{i} − q_{i}dp_{i}) + ∑_{1≤j≤m} θ_{j}dθ_{j} and ω̃_{0} = dα̃_{1} .  (0.2.5)

The form α̃_{1} is called contact, the form ω̃_{0} is called symplectic. Sometimes it is more convenient to redenote the θ’s and set Θ = (ξ, η), or, perhaps, Θ = (ξ, η, θ), where

ξ_{j} = \frac{1}{\sqrt{2}}(θ_{j} − iθ_{r+j}); η_{j} = \frac{1}{\sqrt{2}}(θ_{j} + iθ_{r+j}) for j ≤ r = [m/2] (here i^{2} = −1), θ = θ_{2r+1}

and in place of ω̃_{0} or α̃_{1} take α_{1} and ω_{0} = dα_{1}, respectively, where

α_{1} = dt + ∑_{1≤i≤n}(p_{i}dq_{i} − q_{i}dp_{i}) + ∑_{1≤j≤r}(ξ_{j}dη_{j} + η_{j}dξ_{j}) \begin{cases}& if m = 2r \\ +θdθ & if m = 2r + 1. \end{cases} (0.2.6)

The Lie superalgebra that preserves the Pfaff equation α_{1} = 0, i.e., the superalgebra

𝔨(2n + 1|m) = {D ∈ 𝔳𝔢𝔠𝔱(2n + 1|m) | L_{D}α_{1} = f_{D}α_{1} for some f_{D} ∈ ℂ[t, p, q, θ]},

is called the contact superalgebra. The Lie superalgebra

𝔭𝔬(2n|m) = {D ∈ 𝔨(2n + 1|m) | L_{D}α_{1} = 0}

is called the Poisson superalgebra. (A geometric interpretation of the Poisson superalgebra: it is the Lie superalgebra that preserves the connection with form α in the line bundle over a symplectic supermanifold with the symplectic form dα, cf. [Kos].)

• Similarly, set u = q = (q_{1}, …, q_{n}), let θ = (ξ_{1}, …, ξ_{n}; τ) be odd. Set

α_{0} = dτ + ∑_{i}(ξ_{i}dq_{i} + q_{i}dξ_{i}), ω_{1} = dα_{0}

and call these forms the odd-contact (actually it is even, though “odd”) and periplectic, respectively. The periplectic form is odd.

The Lie superalgebra that preserves the Pfaff equation α_{0} = 0, i.e., the superalgebra

𝔪(n) = {D ∈ 𝔳𝔢𝔠𝔱(n|n + 1) | L_{D}α_{0} = f_{D} · α_{0} for some f_{D} ∈ ℂ[q, ξ, τ]}

is called the odd-contact superalgebra.^{5}

The Lie superalgebra

𝔟(n) = {D ∈ 𝔪(n) | L_{D}α_{0} = 0}

is called the Buttin superalgebra ([L4]). (A geometric interpretation of the Buttin superalgebra: it is the Lie superalgebra that preserves the connection with form α_{1} in the line bundle of rank ε = (0|1) over a periplectic supermanifold, i.e., over a supermanifold with the periplectic form dα_{0}.)

The Lie superalgebras

𝔰𝔪(n) = {D ∈ 𝔪(n) | div D = 0} , 𝔰𝔟(n) = {D ∈ 𝔟(n) | div D = 0}

are called the divergence-free (or special) odd-contact and special Buttin superalgebras, respectively. Remark . A relation with finite dimensional geometry is as follows. Clearly, ker α_{1} = ker α̃_{1}. The restriction of ω̃_{0} to ker α_{1} is the orthosymplectic form B_{ev}(m|2n); the restriction of ω̃_{0} to ker α̃_{1} is B′_{ev}(m|2n). Similarly, the restriction of ω_{1} to ker α_{0} is B_{odd}(n|n).

0.3. Generating functions. A laconic way to describe 𝔨, 𝔪 and their subalgebras is via generating functions.

• Odd form α_{1}. For any f ∈ ℂ[t, p, q, θ] set :

K_{f} = (2 − E)(f)\frac{∂}{∂t} − H_{f} + \frac{∂f}{∂t} E,  (0.3.1)

where E = ∑_{i} y_{i} \frac{∂}{∂y_{i}} (here the y_{i} are all the coordinates except t) is the Euler operator (which

counts the degree with respect to the y_{i}), and H_{f} is the hamiltonian field with Hamiltonian f that preserves dα̃_{1}:

H_{f} = ∑_{i≤n}(\frac{∂f}{∂p_{i}} \frac{∂}{∂q_{i}} − \frac{∂f}{∂q_{i}} \frac{∂}{∂p_{i}}) − (−1)^{p(f)} (∑_{j≤m} \frac{∂f}{∂θ_{j}} \frac{∂}{∂θ_{j}} ) .  (0.3.2)

The choice of the form α_{1} instead of α̃_{1} only affects the shape of H_{f} that we give for m = 2k + 1:

H_{f} = ∑_{i≤n}(\frac{∂f}{∂p_{i}} \frac{∂}{∂q_{i}} − \frac{∂f}{∂q_{i}} \frac{∂}{∂p_{i}}) − (−1)^{p(f)} ∑_{j≤k}(\frac{∂f}{∂ξ_{j}} \frac{∂}{∂η_{j}} + \frac{∂f}{∂η_{j}} \frac{∂}{∂ξ_{j}} + \frac{∂f}{∂θ} \frac{∂}{∂θ}).

• Even form α_{0}. For any f ∈ ℂ[q, ξ, τ] set:

M_{f} = (2 − E)(f)\frac{∂}{∂τ} − Le_{f} − (−1)^{p(f)} \frac{∂f}{∂τ} E,  (0.3.3)

5To call 𝔪 “odd contact superalgebra” is misleading since the even part of 𝔪 is nonzero. A really good name is desired.

where E = ∑_{i} y_{i} \frac{∂}{∂y_{i}} (here the y_{i} are all the coordinates except τ), and

Le_{f} = ∑_{i≤n}(\frac{∂f}{∂q_{i}} \frac{∂}{∂ξ_{i}} + (−1)^{p(f)} \frac{∂f}{∂ξ_{i}} \frac{∂}{∂q_{i}}).  (0.3.4)

Since

\begin{matrix}L_{K_{f}} (α_{1}) = 2\frac{∂f}{∂t} α_{1} = K_{1}(f)α_{1}, \\ L_{M_{f}} (α_{0}) = −(−1)^{p(f)}2\frac{∂f}{∂τ} α_{0} = −(−1)^{p(f)}M_{1}(f)α_{0}, \end{matrix}  (0.3.5)

it follows that K_{f} ∈ 𝔨(2n + 1|m) and M_{f} ∈ 𝔪(n). Observe that

p(Le_{f}) = p(M_{f}) = p(f) + ¯1.

• To the (super)commutators [K_{f}, K_{g}] or [M_{f}, M_{g}] there correspond contact brackets of the generating functions:

[K_{f}, K_{g}] = K_{{f, g}_{k.b.}}; [M_{f}, M_{g}] = M_{{f, g}_{m.b.}}

The explicit formulas for the contact brackets are as follows. Let us first define the brackets on functions that do not depend on t (resp. τ).

The Poisson bracket {·, ·}_{P.b.} (in the realization with the form ω̃_{0}) is given by the formula

\{f, g\}_{P.b.} = ∑_{i≤n} (\frac{∂f}{∂p_{i}} \frac{∂g}{∂q_{i}} − \frac{∂f}{∂q_{i}} \frac{∂g}{∂p_{i}} ) − (−1)^{p(f)} ∑_{j≤m} \frac{∂f}{∂θ_{j}} \frac{∂g}{∂θ_{j}} for any f, g ∈ ℂ[p, q, θ]

(0.3.6)

and in the realization with the form ω_{0} for m = 2k + 1 it is given by the formula

\begin{matrix}\{f, g\}_{P.b.} = ∑_{i≤n} (\frac{∂f}{∂p_{i}} \frac{∂g}{∂q_{i}} − \frac{∂f}{∂q_{i}} \frac{∂g}{∂p_{i}} )− \\ (−1)^{p(f)}[ ∑_{j≤m}(\frac{∂f}{∂ξ_{j}} \frac{∂g}{∂η_{j}} + \frac{∂f}{∂η_{j}} \frac{∂g}{∂ξ_{j}}) + \frac{∂f}{∂θ} \frac{∂g}{∂θ}] for f, g ∈ ℂ[p, q, ξ, η, θ]. \end{matrix}

The Buttin bracket {·, ·}_{B.b.} is given by the formula

\{f, g\}_{B.b.} = ∑_{i≤n} (\frac{∂f}{∂q_{i}} \frac{∂g}{∂ξ_{i}} + (−1)^{p(f)} \frac{∂f}{∂ξ_{i}} \frac{∂g}{∂q_{i}} ) for any f, g ∈ ℂ[q, ξ].  (0.3.7)

Remark . The what we call here “Buttin bracket” was discovered in pre-super era by Schouten; Buttin was the first to prove that this bracket establishes a Lie superalgebra structure. The interpretations of the Buttin superalgebra similar to that of the Poisson algebra and of the elements of 𝔩𝔢 as analogs of Hamiltonian vector fields was given in [L1]. The Buttin bracket and “odd mechanics” introduced in [L1] was rediscovered by Batalin with Vilkovisky (and, even earlier, by Zinn-Justin, but his papers went mainly unnoticed, see [FLSf]); it gained a great deal of currency under the name antibracket, cf. [GPS]. The Schouten bracket was originally defined on the superspace of polyvector fields on a manifold, i.e., on the superspace of sections of the exterior algebra (over the algebra F of functions) of the tangent bundle, Γ(Λ.(T (M))) ≅ Λ._{F}(V ect(M)). The explicit formula of the Schouten bracket (in which the hatted slot should be ignored, as usual) is

\begin{matrix}[X_{1} ∧ ⋯ ∧ ⋯ ∧ X_{k}, Y_{1} ∧ ⋯ ∧ Y_{l}] = \\ ∑_{i,j}(−1)^{i+j}[X_{i}, Y_{j}] ∧ X_{1} ∧ ⋯ ∧ X̂_{i} ∧ ⋯ ∧ X_{k} ∧ Y_{1} ∧ ⋯ ∧ Ŷ_{j} ∧ ⋯ ∧ Y_{l}. \end{matrix}  (∗)

With the help of Sign Rule we easily superize formula (∗), i.e., replace M with a supermanifold M. Let x and ξ be the even and odd coordinates on M. By setting

θ_{i} = Π(\frac{∂}{∂x_{i}}) = x̌_{i}, q_{j} = Π(\frac{∂}{∂ξ_{j}}) = ξ̌_{j}  (∗∗)

we get an identification of the Schouten bracket of polyvector fields on M with the Buttin bracket of functions on the supermanifold M̌ the coordinates on it being x, ξ and x̌, ξ̌, where the transformation rule of the checked variables is induced by that of unchecked ones via (∗∗).

In terms of the Poisson and Buttin brackets, respectively, the contact brackets are

\{f, g\}_{k.b.} = (2 − E)(f)\frac{∂g}{∂t} − \frac{∂f}{∂t} (2 − E)(g) − \{f, g\}_{P.b.}  (0.3.8)

and

\{f, g\}_{m.b.} = (2 − E)(f)\frac{∂g}{∂τ} + (−1)^{p(f)} \frac{∂f}{∂τ} (2 − E)(g) − \{f, g\}_{B.b.}.  (0.3.9)

The Lie superalgebras of Hamiltonian fields (or Hamiltonian superalgebra) and its special subalgebra (defined only if n = 0) are

𝔥(2n|m) = {D ∈ 𝔳𝔢𝔠𝔱(2n|m) | L_{D}ω_{0} = 0} and 𝔥^{o}(m) = {H_{f} ∈ 𝔥(0|m) | ∫ fvol_{θ} = 0}.

The “odd” analogues of the Lie superalgebra of Hamiltonian fields are the Lie superalgebra of vector fields Le_{f} introduced in [L1] and its special subalgebra:

𝔩𝔢(n) = {D ∈ 𝔳𝔢𝔠𝔱(n|n) | L_{D}ω_{1} = 0} and 𝔰𝔩𝔢(n) = {D ∈ 𝔩𝔢(n) | divD = 0}.

It is not difficult to prove the following isomorphisms (as superspaces):

\begin{aligned}𝔨(2n + 1|m) & ≅ & Span(K_{f} | f ∈ ℂ[t, p, q, ξ]); & 𝔩𝔢(n) & ≅ & Span(Le_{f} | f ∈ ℂ[q, ξ]); \\ 𝔪(n) & ≅ & Span(M_{f} | f ∈ ℂ[τ, q, ξ]); & 𝔥(2n|m) & ≅ & Span(H_{f} | f ∈ ℂ[p, q, ξ]). \end{aligned}

Set 𝔭𝔬^{o}(m) = {K_{f} ∈ 𝔭𝔬(0|m) | ∫ fvol_{ξ} = 0}; then 𝔥^{o}(m) = 𝔭𝔬^{o}(m)/ℂ · K_{1}.

0.3.1. Deduction of the explicit form of the odd-contact bracket. In what follows F_{x} is a shorthand for \frac{∂}{∂x}F.

On the one hand, since M_{F} α_{0} = (−1)^{p(F)+1}2F_{τ}α_{0}, it follows that

\begin{matrix}(M_{F} M_{G} − (−1)^{(p(F)+1)(p(G)+1)}M_{G}M_{F})α_{0} = \\ \{(−1)^{p(G)+1}2[(ξF_{τ} + F_{q})G_{ξτ} + (−1)^{p(F)}(F_{ξ} − qF_{τ})G_{qτ} + (−1)^{p(G)(p(F)+1)}2G_{τ} F_{τ}] \\ −(−1)^{(p(F)+1)p(G)}2[(ξG_{τ} + G_{q})F_{ξτ} + (−1)^{p(G)}(G_{ξ} − qG_{τ})F_{qτ} + (−1)^{p(F)(p(G)+1)}2F_{τ} G_{τ}]\}α_{0}. \end{matrix} (0.3.10)

On the other hand, we have

((2 − E)(F)G_{τ})_{τ} = 2F_{τ}G_{τ} + ξF_{τξ}G_{τ} − qF_{τq}G_{τ};  (0.3.11)
(F_{τ}(2 − E)(G))_{τ} = (−1)^{p(F)+1}(2F_{τ}G_{τ} + F_{τ}ξG_{τξ} − F_{τ}qG_{τq});  (0.3.12)
(F_{q}G_{ξ} + (−1)^{p(F)}F_{ξ}G_{q})_{τ} = F_{τq}G_{q} + (−1)^{p(F)}F_{q}G_{τξ} − F_{ξ}G_{τq}.  (0.3.13)

Clearly,

(−1)^{p(F)p(G)}(0.3.11) + (−1)^{p(G)}(0.3.12) + (−1)^{p(F)+p(G)}(0.3.13) = {… },

where {… } is the coefficient of α_{0} in (0.3.10). Therefore,

{F, G}_{m.b.} = (−1)^{p(F)+p(G)}[(2 − E)(F)G_{τ} + (−1)^{p(F)}F_{τ}(2 − E)(G) + {F, G}_{B.b.}].

The arguments for the even contact bracket are similar with α_{0} replaced with α_{1} or α̃_{1} and fields M_{∗} with K_{∗}, respectively.

0.4. Divergence-free subalgebras. Since, as is easy to calculate,

divK_{f} = (2n + 2 − m)K_{1}(f),

it follows that the divergence-free subalgebra of the contact Lie superalgebra either coincides with it (for m = 2n + 2) or is the Poisson superalgebra. For the “odd” contact series the situation is more interesting: the divergence free subalgebra is simple and new (as compared with the above list).

Since

divM_{f} = (−1)^{p(f)}2 ((1 − E)\frac{∂f}{∂τ} − ∑_{i≤n} \frac{∂^{2}f}{∂q_{i}∂ξ_{i}} ) ,  (0.4.1)

it follows that the divergence-free subalgebra of the odd-contact superalgebra is

𝔰𝔪(n) = Span (M_{f} ∈ 𝔪(n) | (1 − E)\frac{∂f}{∂τ} = ∑_{i≤n} \frac{∂^{2}f}{∂q_{i}∂ξ_{i}} ) .

In particular,

divLe_{f} = (−1)^{p(f)}2 ∑_{i≤n} \frac{∂^{2}f}{∂q_{i}∂ξ_{i}} .  (0.4.2)

The odd analog of the Laplacian, namely, the operator

Δ = ∑_{i≤n} \frac{∂^{2}}{∂q_{i}∂ξ_{i}}  (0.4.3)

on a periplectic supermanifold appeared in physics under the name of BRST operator, cf. [GPS], or Batalin-Vilkovysky operator. Observe that it is just the Fourier transform (with respect to the ghost indeterminates (odd ones, if considered on a manifold) of the exterior oerator.

The divergence-free vector fields from 𝔰𝔩𝔢(n) are generated by harmonic functions, i.e., such that Δ(f) = 0.

Lie superalgebras 𝔰𝔩𝔢(n), 𝔰𝔟(n) and 𝔰𝔳𝔢𝔠𝔱(1|n) have traceless ideals 𝔰𝔩𝔢^{∘}(n), 𝔰𝔟^{∘}(n) and 𝔰𝔳𝔢𝔠𝔱^{∘}(n) of codimension 1 defined from the exact sequences

0 ⟶ 𝔰𝔩𝔢^{∘}(n) ⟶ 𝔰𝔩𝔢(n) ⟶ ℂ · Le_{ξ_{1}...ξ_{n}} ⟶ 0,
0 ⟶ 𝔰𝔟^{∘}(n) ⟶ 𝔰𝔟(n) ⟶ ℂ · M_{ξ_{1}...ξ_{n}} ⟶ 0,

0 ⟶ 𝔰𝔳𝔢𝔠𝔱^{∘}(n) ⟶ 𝔰𝔳𝔢𝔠𝔱(1|n) ⟶ ℂ · ξ_{1} … ξ_{n} \frac{∂}{∂t} ⟶ 0.

0.5. The Cartan prolongs. We will repeatedly use the Cartan prolongation. So let us recall the definition and generalize it somewhat. Let 𝔤 be a Lie algebra, V a 𝔤-module, S^{i} the operator of the i-th symmetric power. Set 𝔤_{−1} = V, 𝔤_{0} = 𝔤 and for k > 0 define the k-th Cartan prolong (the result of Cartan’s prolongation) of the pair (𝔤_{−1}, 𝔤_{0}) as

𝔤_{k} = {X ∈ Hom(𝔤_{−1}, 𝔤_{k−1}) | X(v_{0})(v_{1}, v_{2}, … v_{k}) = X(v_{1})(v_{0}, v_{2}, …, v_{k})
for any v_{0}, v_{1}, … v_{k} ∈ 𝔤_{−1}}.

Equivalently, let

i : S^{k+1}(𝔤_{−1})^{∗} ⊗ 𝔤_{−1} ⟶ S^{k}(𝔤_{−1})^{∗} ⊗ 𝔤_{−1}^{∗} ⊗ 𝔤_{−1}  (0.5.1)

be the natural embedding and

j : S^{k}(𝔤_{−1})^{∗} ⊗ 𝔤_{0} ⟶ S^{k}(𝔤_{−1})^{∗} ⊗ 𝔤_{−1}^{∗} ⊗ 𝔤_{−1}  (0.5.2)

the natural map. Then 𝔤_{k} = i(S^{k+1}(𝔤_{−1})^{∗} ⊗ 𝔤_{−1}) ∩ j(S^{k}(𝔤_{−1})^{∗} ⊗ 𝔤_{0}).

The Cartan prolong of the pair (V, 𝔤) is (𝔤_{−1}, 𝔤_{0})_{∗} = ⊕_{k≥−1} 𝔤_{k}.

(In what follows . in superscript denotes, as is now customary, the collection of all degrees, while ∗ is reserved for dualization; in the subscripts we retain the old-fashioned ∗ instead of . to avoid too close a contact with the punctuation marks.)

Suppose that the 𝔤_{0}-module 𝔤_{−1} is faithful. Then, clearly,

(𝔤_{−1}, 𝔤_{0})_{∗} ⊂ 𝔳𝔢𝔠𝔱(n) = 𝔡𝔢𝔯 ℂ[x_{1}, ..., x_{n}], where n = dim 𝔤_{−1} and 𝔤_{i} = {D ∈ 𝔳𝔢𝔠𝔱(n) | deg D = i, [D, X] ∈ 𝔤_{i−1} for any X ∈ 𝔤_{−1}}.

It is subject to an easy verification that the Lie algebra structure on 𝔳𝔢𝔠𝔱(n) induces same on (𝔤_{−1}, 𝔤_{0})_{∗}.

Of the four simple vectorial Lie algebras, three are Cartan prolongs:

𝔳𝔢𝔠𝔱(n) = (id, 𝔤𝔩(n))_{∗}, 𝔰𝔳𝔢𝔠𝔱(n) = (id, 𝔰𝔩(n))_{∗}, 𝔥(2n) = (id, 𝔰𝔭(n))_{∗}.

The fourth one — 𝔨(2n + 1) — is the result of a trifle more general construction described as follows.

0.5.1. A generalization of the Cartan prolong. Let 𝔤_{−} = ⊕_{−d≤i≤−1} 𝔤_{i} be a nilpotent ℤ-

graded Lie algebra and 𝔤_{0} ⊂ 𝔡𝔢𝔯_{0}𝔤 a Lie subalgebra of the ℤ-grading-preserving derivations. Let

i : S^{k+1}(𝔤_{−})^{∗} ⊗ 𝔤_{−} ⟶ S^{k}(𝔤_{−})^{∗} ⊗ 𝔤_{−}^{∗} ⊗ 𝔤_{−}  (0.5.1′)

and

j : S^{k}(𝔤_{−})^{∗} ⊗ 𝔤_{0} ⟶ S^{k}(𝔤_{−})^{∗} ⊗ 𝔤_{−}^{∗} ⊗ 𝔤_{−}  (0.5.2′)

be the natural maps similar to (0.5.1) and (0.5.2), respectively. For k > 0 define the k-th prolong of the pair (𝔤_{−}, 𝔤_{0}) to be:

𝔤_{k} = (j(S.(𝔤_{−})^{∗} ⊗ 𝔤_{0}) ∩ i(S.(𝔤_{−})^{∗} ⊗ 𝔤_{−}))_{k} ,

where the subscript k in the right hand side singles out the component of degree k. Set (𝔤_{−}, 𝔤_{0})_{∗} = ⊕_{i≥−d} 𝔤_{i}; then, as is easy to verify, (𝔤_{−}, 𝔤_{0})_{∗} is a Lie algebra.

What is the Lie algebra of contact vector fields in these terms? Denote by 𝔥𝔢𝔦(2n) the Heisenberg Lie algebra: its space is W ⊕ ℂ · z, where W is a 2n-dimensional space endowed with a nondegenerate skew-symmetric bilinear form B and the bracket in 𝔥𝔢𝔦(2n) is given by the following relations:

z is in the center and [v, w] = B(v, w) · z for any v, w ∈ W.  (0.5.3)

Clearly, 𝔨(2n + 1) ≅ (𝔥𝔢𝔦(2n), 𝔠𝔰𝔭(2n))_{∗}.

0.5.2. Lie superalgebras of vector fields as Cartan’s prolongs. The superization of the constructions from sec. 0.5 are straightforward: via Sign Rule. We thus get infinite dimensional Lie superalgebras

𝔳𝔢𝔠𝔱(m|n) = (id, 𝔤𝔩(m|n))_{∗}; 𝔰𝔳𝔢𝔠𝔱(m|n) = (id, 𝔰𝔩(m|n))_{∗};
𝔥(2m|n) = (id, 𝔬𝔰𝔭^{sk}(m|2n))_{∗};
𝔩𝔢(n) = (id, 𝔭𝔢^{sk}(n))_{∗}; 𝔰𝔩𝔢(n) = (id, 𝔰𝔭𝔢^{sk}(n))_{∗}.

Remark. Observe that the Cartan prolongs (id, 𝔬𝔰𝔭^{sy}(m|2n))_{∗} and (id, 𝔭𝔢^{sy}(n))_{∗} are finite dimensional.

The generalization of Cartan’s prolongations described in sec. 0.5.1 was first defined in [ALSh] (and repeatedly used later, e.g., in [CK1]). Observe that after superization it has two analogs associated with the contact series 𝔨 and 𝔪, respectively.

• Define the Lie superalgebra 𝔥𝔢𝔦(2n|m) on the direct sum of a (2n, m)-dimensional superspace W endowed with a nondegenerate skew-symmetric bilinear form B and a (1, 0)- dimensional space spanned by z by (0.5.3).

Clearly, we have

𝔨(2n + 1|m) = (𝔥𝔢𝔦(2n|m), 𝔠𝔬𝔰𝔭^{sk}(m|2n))_{∗}

and, given 𝔥𝔢𝔦(2n|m) and a subalgebra 𝔤 of 𝔠𝔬𝔰𝔭^{sk}(m|2n), we call (𝔥𝔢𝔦(2n|m), 𝔤)_{∗} the k-prolong of (W, 𝔤), where W is the identity 𝔬𝔰𝔭^{sk}(m|2n)-module.

• The “odd” analog of 𝔨 is associated with the following “odd” analog of 𝔥𝔢𝔦(2n|m). Denote by 𝔞𝔟(n) the antibracket Lie superalgebra: its space is W ⊕ ℂ · z, where W is an n|ndimensional superspace endowed with a nondegenerate skew-symmetric odd bilinear form B; the bracket in 𝔞𝔟(n) is given by the following relations:

z is odd and lies in the center; [v, w] = B(v, w) · z for any v, w ∈ W.  (0.5.4)

Clearly,

𝔪(n) = (𝔞𝔟(n), 𝔠𝔭𝔢^{sk}(n))_{∗}

and, given 𝔞𝔟(n) and a subalgebra 𝔤 of 𝔠𝔭𝔢^{sk}(n), we call (𝔞𝔟(n), 𝔤)_{∗} the m-prolong of (W, 𝔤), where W is the identity 𝔭𝔢^{sk}(n)-module.

Generally, given a nondegenerate form B on a superspace W and a superalgebra 𝔤 that preserves B, we refer to the above generalized prolongations as to mk-prolongation of the pair (W, 𝔤).

0.5.3. A partial Cartan prolong or the prolong of a positive part. Take a 𝔤_{0}- submodule 𝔥_{1} in 𝔤_{1} such that [𝔤_{−1}, 𝔥_{1}] = 𝔤_{0}. If such 𝔥_{1} exists (usually, [𝔤_{−1}, 𝔥_{1}] ⊂ 𝔤_{0}), define the 2nd prolongation of ( ⊕_{i≤0} 𝔤_{i}, 𝔥_{1}) to be

𝔥_{2} = {D ∈ 𝔤_{2} | [D, 𝔤_{−1}] ∈ 𝔥_{1}}.

The terms 𝔥_{i}, i > 2, are similarly defined. Set 𝔥_{i} = 𝔤_{i} for i < 0 and 𝔥_{∗} = ⊕𝔥_{i}.

Examples: 𝔳𝔢𝔠𝔱(1|n; n) is a subalgebra of 𝔨(1|2n; n). The former is obtained as Cartan’s prolong of the same nonpositive part as 𝔨(1|2n; n) and a submodule of 𝔨(1|2n; n)_{1}, cf. Table I.2.2. The simple exceptional superalgebra 𝔨𝔞𝔰 introduced in I.2.3 ([Sh5, Sh14]) is another example. Remark . In nonsuper setting generalized Cartan prolongation was first introduced by Tanaka [T], see a review [Y]. Different from ours emphasis of these papers delayed our recognition of similarity of constructions. The partial prolong seems to be first discovered in [ALSh].

0.6. The modules of tensor fields. To advance further, we have to recall the definition of the modules of tensor fields over 𝔳𝔢𝔠𝔱(m|n) and its subalgebras, see [BL2], [L2]. For any other ℤ-graded vectorial Lie superalgebra the construction is identical.

Let 𝔤 = 𝔳𝔢𝔠𝔱(m|n) and 𝔤_{≥} = ⊕_{i≥0} 𝔤_{i}. Clearly, 𝔳𝔢𝔠𝔱_{0}(m|n) ≅ 𝔤𝔩(m|n). Let V be the 𝔤𝔩(m|n)-

module with the lowest weight λ = lwt(V). Make V into a 𝔤_{≥}-module setting 𝔤_{+} · V = 0 for 𝔤_{+} = ⊕_{i>0} 𝔤_{i}. Let us realize 𝔤 by vector fields on the m|n-dimensional linear supermanifold

C^{m|n} with coordinates x = (u, ξ). The superspace T (V) = Hom_{U(𝔤_{≥})}(U(𝔤), V) is isomorphic, due to the Poincaré–Birkhoff–Witt theorem, to ℂ[[x]] ⊗ V. Its elements have a natural interpretation as formal tensor fields of type V. When λ = (a, …, a) we will simply write T (a⃗) instead of T (λ). We will usually consider 𝔤-modules induced from irreducible 𝔤_{0}- modules.

Examples: 𝔳𝔢𝔠𝔱(m|n) as 𝔳𝔢𝔠𝔱(m|n)- and 𝔰𝔳𝔢𝔠𝔱(m|n)-modules is T (id). More examples: T (0) is the superspace of functions; Vol(m|n) = T (1, …, 1; −1, …, −1) (the semicolon separates the first m (“even”) coordinates of the weight with respect to the matrix units E_{ii} of 𝔤𝔩(m|n)) is the superspace of densities or volume forms. We denote the generator of Vol(m|n) corresponding to the ordered set of coordinates x by vol(x). The space of λdensities is Vol^{λ}(m|n) = T (λ, …, λ; −λ, …, −λ). In particular, Vol^{λ}(m|0) = T (λ⃗) but Vol^{λ}(0|n) = T (\overrightarrow{−λ}). We set:

Vol_{0}(0|m) = {v ∈ Vol | ∫ v = 0} and T_{0}(0) = Λ(m)/ℂ · 1.

If the generator vol of Vol is fixed, then Vol ≅ T (0), as 𝔰𝔳𝔢𝔠𝔱(m|n)-modules. We set T_{0}^{0}(0) to denote the 𝔰𝔳𝔢𝔠𝔱(m|n)-module Vol_{0}(0|m)/ℂ vol(ξ).

Remark. To view the volume element as “d^{m}ud^{n}ξ” is totally wrong: the superdeterminant can never appear as a factor under the changes of variables. We can try to use the usual notations of differentials provided all the differentials anticommute. Then the linear transformations that do not intermix the even u’s with the odd ξ’s multiply the volume element vol(x), viewed as the fraction \frac{du_{1}·...·du_{m}}{dξ_{1}·...·dξ_{n}}, by the Berezinian of the transformation. But how could we justify this? Let X = (x, ξ). If we consider the usual, exterior, differential forms, then the dX_{i}’s super anti-commute, hence, the dξ_{i} commute; whereas if we consider the symmetric product of the differentials, as in the metrics, then the dX_{i}’s supercommute, hence, the dx_{i} commute. However, the \frac{∂}{∂ξ_{i}} anticommute and, from transformations’ point of view, \frac{∂}{∂ξ_{i}} = \frac{1}{dξ_{i}}. The notation, du_{1} · ... · du_{m} · \frac{∂}{∂ξ_{1}} · . . . · \frac{∂}{∂ξ_{n}}, is, nevertheless, still wrong: almost any transformation A : (u, ξ) ↦ (v, η) sends du_{1} · ... · du_{m} · \frac{∂}{∂ξ_{1}} · ... · \frac{∂}{∂ξ_{n}} to the correct element, ber(A)(du^{m} · \frac{∂}{∂ξ_{1}} · ... · \frac{∂}{∂ξ_{n}}), plus extra terms. Indeed, the fraction du_{1} · ... · du_{m} · \frac{∂}{∂ξ_{1}} · ... · \frac{∂}{∂ξ_{n}} is the highest weight vector of an indecomposable 𝔤𝔩(m|n)-module and vol(x) is the notation of the image of this vector in the 1-dimensional quotient module modulo the invariant submodule that consists precisely of all the extra terms.

0.7. Deformations of the Buttin superalgebra. (After [ALSh].) As is clear from the definition of the Buttin bracket, there is a regrading (namely, 𝔟(n; n) given by deg ξ_{i} = 0, deg q_{i} = 1 for all i) under which 𝔟(n), initially of depth 2, takes the form 𝔤 = ⊕_{i≥−1} 𝔤_{i} with

𝔤_{0} = 𝔳𝔢𝔠𝔱(0|n) and 𝔤_{−1} ≅ Π(ℂ[ξ]). Replace now the 𝔳𝔢𝔠𝔱(0|n)-module 𝔤_{−1} of functions (with inverted parity) with the module of λ-densities, i.e., set 𝔤_{−1} ≅ Π(Vol(0|n)^{λ}), where

L_{D}(vol_{ξ})^{λ} = λdivD · vol_{ξ}^{λ} and p(vol_{ξ})^{λ} = ¯1.

0.7.1. Define 𝔟_{λ}(n; n) as the Cartan’s prolong

(𝔤_{−1}, 𝔤_{0})_{∗} = (Π(Vol(0|n)^{λ}), 𝔳𝔢𝔠𝔱(0|n))_{∗}.

Clearly, it is a deform of 𝔟(n; n). The collection of these 𝔟_{λ}(n; n) for all λ’s is called the main deformation. (Though main, this deformation is not the quantization of the Buttin bracket, cf. [Ko1], [L4], [LSh3].)

The deform 𝔟_{λ}(n) of 𝔟(n) is a regrading of 𝔟_{λ}(n; n) described as follows. Set

𝔟_{λ}(n) = \{M_{f} ∈ 𝔪(n) | a divM_{f} = (−1)^{p(f)}2(a − bn)\frac{∂f}{∂τ} \}.  (0.7)

For future use, we will denote the operator that singles out 𝔟_{λ}(n) in 𝔪(n) as follows:

div_{λ} = (bn − aE)\frac{∂}{∂τ} − aΔ.  (0.8)

Taking into account the explicit form (0.4.1) of the divergence of M_{f} we get

𝔟_{λ}(n) = \{M_{f} ∈ 𝔪(n) | (bn − aE)\frac{∂f}{∂τ} = aΔf\} = \{D ∈ 𝔳𝔢𝔠𝔱(n|n + 1) | L_{D}(vol_{q,ξ,τ}^{a}α_{0}^{a−bn}) = 0\}.

It is subject to a direct check that 𝔟_{a,b}(n) is isomorphic to 𝔟_{λ}(n) for λ = \frac{2a}{n(a−b)}. This isomorphism shows that λ actually runs over ℂP^{1}, not ℂ.

Observe that 𝔟_{nb,b}(n) ≅ 𝔰𝔪(n). Observe also that 𝔟_{a,−a}(2; 2) ≅ 𝔥_{1/2}(2|2), see (OI).

As follows from the description of 𝔳𝔢𝔠𝔱(m|n)-modules ([BL2]) and the criteria for simplicity of ℤ-graded Lie superalgebras ([K2]), the Lie superalgebras 𝔟_{λ}(n) are simple for n > 1 and λ ≠ 0, 1, ∞. It is also clear that the 𝔟_{λ}(n) are nonisomorphic for distinct λ’s.

The Lie superalgebra 𝔟(n) = 𝔟_{0}(n) is not simple: it has an ε-dimensional, i.e., (0|1)- dimensional, center. At λ = 1 and ∞ the Lie superalgebras 𝔟_{λ}(n) are not simple either: they has an ideal of codimension ε^{n} and ε^{n+1}, respectively. The corresponding exact sequences are

0 ⟶ ℂM_{1} ⟶ 𝔟(n) ⟶ 𝔩𝔢(n) ⟶ 0,
0 ⟶ 𝔟_{1}^{∘}(n) ⟶ 𝔟_{1}(n) ⟶ ℂ · M_{ξ_{1}...ξ_{n}} ⟶ 0,
0 ⟶ 𝔟_{∞}^{∘}(n) ⟶ 𝔟_{∞}(n) ⟶ ℂ · M_{τξ_{1}...ξ_{n}} ⟶ 0.

0.8. The exceptional simple vectorial Lie superalgebras as Cartan’s prolongs. For the description of the exceptions as Cartan prolongs see Introduction. For a more detailed description of the “standard” realizations of the exceptions see [Sh14], [ShP].

0.9. The structures preserved. It is always desirable to find the structure preserved by the Lie superalgebra under the study. To see what do the vectorial superalgebras in nonstandard realizations preserve, we have to say, first of all, what is the structure that 𝔤_{0} = 𝔳𝔢𝔠𝔱(0|n) preserves on 𝔤_{−1} = Λ(n).

Let 𝔤 = 𝔳𝔢𝔠𝔱(0|n), set further

W = Λ(n), V = Λ(n)/ℂ · 1, V_{0} = {φ ∈ Λ(n) | φ(0) = 0}.

The projection p : W ⟶ V establishes a natural isomorphism between V and V_{0}. Let i : V_{0} ⟶ W be the “inverse” embedding.

Denote by mult : W ⊗ W ⟶ W the multiplication on W given by the tensor mult of valency (2, 1). Since V_{0} is an ideal in the associative supercommutative superalgebra W, the image mult|_{V_{0}⊗V_{0}} is contained in V_{0}. Denote by mult^{∘} the tensor which coincides with mult on V_{0} ⊗ V_{0} and vanishes on ℂ · 1 ⊗ W ⊕ W ⊗ ℂ · 1. By means of the projection p and the embedding i we can 𝔤-invariantly transport mult^{∘} to V. The tensor obtained will be also denoted by mult^{∘}.

For any monomial φ ∈ W denote by φ^{∗} the dual functional (let B(W) denote the monomial basis). Then

mult = mult^{∘} + ∑_{φ∈B(W)} 1^{∗} ⊗ φ^{∗} ⊗ φ = mult^{∘} + 1^{∗} ⊗ ∑_{φ∈B(W)} φ^{∗} ⊗ φ.

By definition of 𝔤, it preserves mult, i.e., L_{D}(mult) = 0 for any D ∈ 𝔤. Hence,

L_{D}(mult^{∘}) = −L_{D}(1^{∗}) ⊗ ∑_{φ∈B(W)} φ^{∗} ⊗ φ − (1^{∗}) ⊗ L_{D} ( ∑_{φ∈B(W)} φ^{∗} ⊗ φ) .

Under the restriction onto V_{0} ⊗ V_{0} the second summand vanishes. Observe that

∑_{φ∈B(W)} φ^{∗} ⊗ φ = id|_{W} .

Thus,

L_{D}(mult^{∘}|_{V_{0}⊗V_{0}}) = −L_{D}(1^{∗}) ⊗ id|_{V_{0}}.

The lift of this identity operator to W reads as follows:

L_{D}(mult) = α(D) ⊗ id|_{W} for a 1-form α on W .

Thus, all the structures preserved by 𝔤_{0} on 𝔤_{−1} are clear, except for those preserved by several of the exceptional algebras. Namely, these structures are:

(1) the tensor products B ⊗ mult of a bilinear or a volume form B preserved (perhaps, conformally, up to multiplication by a scalar) in the fiber of a vector bundle over a 0|rdimensional supermanifold on which the structure governed by mult is preserved,

(2) mult^{∘}, or mult twisted by divergence with factor λ. Observe that the volume element B may be not just vol(ξ) but (1 + αξ_{1} … ξ_{n})vol(ξ) as well.

The structures of another type, namely, certain pseudodifferential forms preserved by 𝔟_{λ}(n), are already described.

0.9.1. On Shmelev’s interpretations of 𝔥_{λ}(2|2) ≅ 𝔟_{λ}(2; 2). When n = 2 the algebras 𝔟_{λ}(n) seemed to have one more realization. G. Shmelev interpreted 𝔥_{λ}(2|2) ≅ 𝔟_{λ}(2; 2) as a deformation 𝔥_{λ}(2|2) of the Lie superalgebra 𝔥(2|2) of Hamiltonian vector fields as preserving either the pseudodifferential form

dη^{\frac{2λ−1}{1−λ}} (λdqdp + (1 − λ)dξdη)

or, equivalently, as preserving the pseudodifferential form

dη^{\frac{1}{λ} −2} (dqdp + dξdη) .

We trustingly reproduced this in [ALSh] and other places, but verification shows that Shmelev made a mistake calculating Lie derivatives and these interpretations are wrong.

§1. Step 1. Main ideas of our proof

For Lie algebras the following facts and ideas (sec. 1.0) are known, though, perhaps, not as well as they deserve. For completeness sake we recall them without (or only with a sketch of a) proof. We learned them from a preprint by A. Joseph [J] that we saw in Moscow ca. 1974, where Cartan’s list of primitive Lie algebras is obtained as in [GQS].

1.0. Lie algebras. Given a primitive Lie algebra L, consider its Weisfeiler filtration (WF), see Introduction, and the associated graded Lie algebra 𝔤 := ⊕_{i≥−d} 𝔤_{i}, where 𝔤_{i} = L_{i}/L_{i+1}.

1.0.1. Proposition . If 𝔤 is any primitive graded Lie algebra, then 1) 𝔤_{−1} generates 𝔤_{−} := ⊕_{i<0} 𝔤_{i};

2) Let 𝔤^{0} := ⊕_{i≥0} 𝔤_{i}. If a ∈ 𝔤^{0} and [𝔤_{−1}, a] = 0, then a = 0 (transitivity condition);

3) the closure \overline{𝔤} = ∏ 𝔤_{i} is an algebra with projective limit topology, \overline{𝔤}^{0} is its maximal subalgebra, and gr\overline{𝔤} ≅ 𝔤.

The notion of projective limit topology is useful in theoretical questions. Recall that the vector space endowed with one of the following topologies is isomorphic and homeomorphic to a space endowed with any of the other two topologies:

a) the dual to a space with discrete topology;

b) the direct product of finite dimensional spaces;

c) the projective limit of finite dimensional spaces.

The topological vector space whose topology is given by any of the above ways will be referred to as a space with a projective limit topology. Notice that this topology is highly disconnected: every subspace W containing one space V from the base of neighborhoods of zero is open (since W is the union of shifts of V) and closed (since the complement to W is also the union of shifted base spaces). The “topology” here is a means to impose finiteness type conditions on subspaces and maps.

The algebra with a projective limit topology is a one whose space is endowed with such a topology and the multiplication is a continuous map with respect to it. In applications, one usually considers continuous representations of algebras with a projective limit topology in either projective limit or discrete spaces, cf. [R1], [BL2].

Example. Consider O_{n} = ℂ[[x_{1}, …, x_{n}]] with the (x)-adic topology; the (x)-adic topology is equivalent to the one determined (for a given c ∈ ℕ, c > 1) by the norm |f| = c^{− deg f} for any f ∈ O_{n}.

Then O_{n} ≃ ℂ^{ℕ} with the product topology (the right hand side is the set of coefficients of monomials). The continuity of the multiplication O_{n} × O_{n} ⟶ O_{n} follows from the fact that O_{n} · (x)^{p} ≃ (x)^{p} · O_{n} ⊂ (x)^{p} for any p.

1.0.2. Lemma . Let 𝔤 = ⊕_{i≥−d} 𝔤_{i} be a primitive graded algebra. Then

a) [a, 𝔤_{s}] = 𝔤_{r+s} for any nonzero a ∈ 𝔤_{r}, where r, s < 0 and r + s ≥ −d;

b) for any nonzero a ∈ 𝔤_{r}, −d ≤ r < 0 we have either 𝔤_{r} = Span(a) or 𝔤_{r} = [a, 𝔤_{0}].

An action of a Lie algebra 𝔤 in a space V is called transitive if 𝔤v = V for any nonzero v ∈ V.

Thus, Lemma 1.0.2 shows that 𝔤_{0} : 𝔤_{−1} is a faithful transitive representation for every primitive Lie algebra 𝔤. There are very few such pairs: 1.0.3. Proposition . Let 𝔤 : V be a faithful transitive representation. Then the following cases are possible:

1) 𝔤 = 𝔰, where 𝔰 is simple and, moreover, 𝔰 is either 𝔰𝔩(n) or 𝔰𝔭(2n) and V is either the identity representation id of 𝔰 or its dual, id^{∗};

2) 𝔤 = ℂz, the 1-dimensional commutative Lie algebra,

3) 𝔤 = 𝔰 ⊕ ℂz, the trivial central extension of the case 1) and the 𝔤-action on V is the natural extension of the 𝔰-action: z acts as multiplication by a scalar.

Proof. 1) Let 𝔯 ⊂ 𝔤 be a radical. By Lie’s theorem, there is an eigenspace

V_{χ} = {v ∈ V | rv = χ(r)v for any r ∈ 𝔯 and some character χ : 𝔯 ⟶ ℂ}.

To prove that V_{χ} = V, it suffices to show that V_{χ} is a submodule. Indeed,

r(gv) = [r, g]v + g(rv) = χ([r, g])v + χ(r)gv for any r ∈ 𝔯, g ∈ 𝔤, v ∈ V_{χ}.

Let us show that χ([r, g]) = 0. Indeed, in the space W spanned by v, gv, g^{2}v, … (W is an eigenspace of r and g) the elements of the radical 𝔯 are represented in the above-indicated basis by matrices with χ(r) on the diagonal, i.e., χ(r) dim W = tr_{W} r. But

0 = tr_{W} [r, g] = χ([r, g]) dim W,

hence, V_{χ} is a submodule. So we deduce:

Transitivity implies irreducibility.  (1.0.3.1)

Hence, V_{χ} = V, and the radical is at most the center of 𝔤 due to the faithfulness of the 𝔤-action on V.

2) If 𝔤 = ℂz, then dim V = 1.

3) Let 𝔰 ≠ 0. In 𝔰, select the maximal torus 𝔥; let R^{+} be the set of positive roots of 𝔰 (with respect to an ordering) and 𝔟 the Borel subalgebra corresponding to R^{+}. Let ν^{+} and ν^{−} be the highest and the lowest weights of V, w^{±} the corresponding weight vectors. By transitivity, gw^{−} = w^{+} for some g ∈ 𝔤. Let g = ∑_{α∈R∪{0}} c_{α}X_{α} + λz. Then

w^{+} = ∑ c_{α}X_{α}w^{−} + λw^{−}.

Since the vectors of distinct weights are linearly independent, only one summand in the right-hand side is nonzero for α = ν^{+} − ν^{−}. Since −ν^{−} is the highest weight of V^{∗}, we have a representation of α as the sum of two positive (even dominant) weights.

This implies simplicity of 𝔰. Indeed, if 𝔰 = 𝔰_{1} ⊕ ⋯ ⊕ 𝔰_{k}, where the summands are simple, then V = V_{1} ⊗ ⋯ ⊗ V_{k}, where V_{i} is an irreducible 𝔰_{i}-module. Now,

ν^{+} = ν_{1}^{+} + ⋯ + ν_{k}^{+}, −ν^{−} = −ν_{1}^{−} − ⋯ − ν_{k}^{−}

with nonnegative summands ν_{i}^{+}, −ν_{i}^{−} ∈ 𝔥_{i}^{∗}. But since α is a root, ν^{+} − ν^{−} = α ∈ 𝔥_{i}^{∗} for some i and the other summands are all zero. Hence, k = 1 by faithfulness of the module V and, moreover,

A root of 𝔤_{0} is the sum of dominant weights.  (1.0.3.2)

From representation theory (see, e.g., [OV]) we know that this can only happen if 𝔰 = 𝔰𝔩(n) or 𝔰𝔭(2n) and these dominant weights are:

\underline{for 𝔰𝔩(n)}: ω_{1} + ω_{n−1} = ε_{1} − ε_{n} is a root for the fundamental weights ω_{1} = ε_{1} and ω_{n−1} = ε_{1} + ⋯ + ε_{n−1} = −ε_{n}.

\underline{for 𝔰𝔭(2n)} : ω_{1} + ω_{1} = 2ε_{1} is a root. The representation corresponding to ω_{1} is the standard one; the case of 𝔰𝔩(n)-module id^{∗} with highest weight ω_{n−1} reduces to the standard module id with the help of the outer automorphism given by X ↦ −X^{t}. □

1.0.4. Proposition . Let 𝔤 be an infinite dimensional primitive graded Lie algebra. Then 1) 𝔤_{0} is 𝔰 or 𝔰 ⊕ ℂz with 𝔰 = 𝔰𝔩(n) or 𝔰𝔭(2n) and 𝔤_{−1} is the standard 𝔤_{0}-module;

2) dim 𝔤_{−2} = 0 or 1;

3) 𝔤_{−3} = 0.

Proof. 1) is already proved.

2) Since 𝔤_{−2} = [a, 𝔤_{−1}] for any nonzero a ∈ 𝔤_{−1} and [a, a] = 0, we have dim 𝔤_{−2} ≤ dim 𝔤_{−1}−1. The action of 𝔰 on 𝔤_{−2} is transitive and, since dim 𝔤_{−2} ≠ dim 𝔤_{−1}, the description of transitive 𝔰-actions implies that the 𝔰-action on 𝔤_{−2} is zero. Hence, dim 𝔤_{−2} ≤ 1.

3) By the same arguments as in 2) dim 𝔤_{−3} ≤ 1. If 𝔤_{−3} ≠ 0, then there exists a unique a ∈ 𝔤_{−2} which can organize an epimorphism ada : 𝔤_{−1} ⟶ 𝔤_{−3} and ker(ada)|_{𝔤_{−1}} ≠ 0 (otherwise dim 𝔤_{−1} = 1 and 𝔤_{−2} = [𝔤_{−1}, 𝔤_{−1}] = 0).

By Lemma 1.0.2 [g, 𝔤_{−2}] = 𝔤_{−3} for any g ∈ 𝔤_{−1}; now, take g ∈ ker ada to get 𝔤_{−3} = 0. □

1.0.5. Theorem . The following table lists all primitive infinite dimensional Lie algebras 𝔤.

𝔤_{0}  𝔰𝔩(n) 𝔰𝔩(n) ⊕ ℂz 𝔰𝔭(2n) 𝔰𝔭(2n) ⊕ ℂz
𝔤 with 𝔤_{−2} = 0 𝔰𝔳𝔢𝔠𝔱(n) 𝔳𝔢𝔠𝔱(n) 𝔥(2n) 𝔡𝔢𝔯 𝔥(2n)
𝔡𝔢𝔯(𝔰𝔳𝔢𝔠𝔱(n))
𝔤 with 𝔤_{−2} ≠ 0  −  −  −  𝔨(2n + 1)

Proof is based on the following facts:

1) if 𝔤_{−2} = 0, then 𝔤 ⊂ 𝔳𝔢𝔠𝔱(n), where n = dim 𝔤_{−1}, by the uniquness of Cartan’s prolongation;

2) the semisimple part of 𝔤_{0} is either 𝔰𝔩 or 𝔰𝔭;

3) every 𝔤_{i} is a completely reducible 𝔤_{0}-module;

4) 𝔤_{i+1} ⊂ 𝔤_{i}^{(1)}, the Cartan prolong of 𝔤_{i};

5) the first Cartan prolong of a 𝔤_{0}-submodule is a 𝔤_{0}-submodule (indeed, since [v, [g, h]] = [[v, g], h] + [g, [v, h]] for any v ∈ V, g ∈ 𝔤_{0}, h ∈ 𝔤_{i}^{(1)}, it follows that [g, h] ∈ 𝔤_{i}^{(1)}).

It remains to decompose 𝔤_{i} into 𝔤_{0}-submodules and collect from them the Lie algebras from the table. Let us decompose S^{i}(V^{∗}) ⊗ V into irreducible 𝔰-modules:

S^{i}(V^{∗}) ⊗ V = \begin{cases}L^{(i−1)ω_{1}} ⊕ L^{iω_{1}+ω_{n−1}} & for 𝔰 = 𝔰𝔩(n) \\ L^{(i−1)ω_{1}} ⊕ L^{(i+1)ω_{1}} ⊕ L^{(i−1)ω_{1}+ω_{2}} & for 𝔰 = 𝔰𝔭(2n). \end{cases}

Here are all the possibilities (𝔤_{1} ∉ 𝔤_{0}^{(1)} means that this case is impossible):

𝔤_{0} = 𝔰𝔩(n)  𝔤_{1} = L^{ω_{1}}  ⟹ 𝔤_{1} ∉ 𝔤_{0}^{(1)}
𝔤_{1} = L^{2ω_{1}+ω_{n−1}} ⟹ 𝔤 = 𝔰𝔳𝔢𝔠𝔱(n)
𝔤_{0} = 𝔤𝔩(n) 𝔤_{1} = L^{ω_{1}} ⟹ 𝔤 = 𝔰𝔩(n + 1)
𝔤_{1} = L^{2ω_{1}+ω_{n−1}} ⟹ 𝔤 = 𝔡𝔢𝔯(𝔰𝔳𝔢𝔠𝔱(n))
𝔤_{0} = 𝔰𝔭(2n), 𝔤_{1} = L^{2ω_{1}} ⟹ 𝔤 = 𝔥(2n)
𝔤_{1} = L^{ω_{1}}  ⟹ 𝔤_{1} ∉ 𝔤_{0}^{(1)}
𝔤_{1} = L^{ω_{2}+ω_{1}}  ⟹ 𝔤_{1} ∉ 𝔤_{0}^{(1)}
𝔤_{0} = 𝔰𝔭(2n) ⊕ ℂz, 𝔤_{1} = L^{2ω_{1}}  ⟹ 𝔤 = 𝔡𝔢𝔯 𝔥(2n)
𝔤_{1} = L^{ω_{1}}  ⟹ 𝔤_{1} ∉ 𝔤_{0}^{(1)}
𝔤_{1} = L^{ω_{2}+ω_{1}}  ⟹ 𝔤_{1} ∉ 𝔤_{0}^{(1)}

1.1. Lie superalgebras. Given a Lie superalgebra L with Weisfeiler filtration (WF), see Introduction, consider the associated graded Lie superalgebra 𝔤 := ⊕_{i≥−d} 𝔤_{i}, where 𝔤_{i} =

L_{i}/L_{i+1}. There are two distinct cases: that of a consistent ℤ-grading, i.e., when 𝔤¯0 = ⊕_{i≡0 (mod 2)} 𝔤_{i} and 𝔤¯1 = ⊕_{i≡1 (mod 2)} 𝔤_{i}, and when the grading is inconsistent.

To the \underline{inconsistent grading} certain arguments of sec. 1.0 are applicable: in particular, the proof of the facts that

1) 𝔤_{−1} generates 𝔤_{−} := ⊕_{i<0} 𝔤_{i};

2) If a ∈ 𝔤^{0} and [𝔤_{−1}, a] = 0, then a = 0 (transitivity condition);

3) the 𝔤_{0}-module 𝔤_{−1} is transitive, moreover, it is irreducible.

The proof of the fact that

a root of 𝔤_{0} is the sum of positive weights of 𝔤_{−1}  (1.1)

is given in §2, where (1.1) reads as follows: the Cartan prolong of the pair 𝔤_{0} : 𝔤_{−1} is of infinite dimension if and only if the action of 𝔤_{0} on 𝔤_{−1} has a rank 1 operator (the coroot corresponding to (1.1)) with the even covector.

After that we “only” have to find the pairs 𝔤_{0} : 𝔤_{−1} for which (1.1) holds. The search of such pairs and the proof of (mutatis mutandis) the analog of Theorem 1.0.5 occupy §§3–6.

Observe that analogs of headings 2) and 3) of Prop. 1.0.4 fail to remain true as stated for Lie superalgebras but the fact that 𝔤_{0} has only two or three simple subquotients and possibility to apply Prop. 1.0.4 to a nonzero even a ∈ 𝔤_{−1} demonstrates that the only cases of depth > 1 correspond, same as for Lie algebras, to a nondegenerate bilinear form (·, ·) on 𝔤_{−1}, perhaps, with values in a multidimensional vector superspace in contradistinction from the one dimensional target space for Lie algebras.

Finally, it is only a posteriori that we see the restrictions on depth of W-filtered simple vectorial Lie superalgebras given in Introduction.

1.2. Lie superalgebras: consistent gradings.

Theorem . ([K7]) The only simple vectorial Lie superalgebras with consistent Weisfeiler filtration are: 𝔨(1|m) and the exceptions 𝔨𝔞𝔰, 𝔳𝔩𝔢(4|3; K), 𝔨𝔰𝔩𝔢(9|6; K) and 𝔪𝔟(4|5; K), see sec. I.2.2.

§2. Step 2. Criterion for infinite dimension of the Cartan prolong

Here we generalize to Lie superalgebras the following Guillemin–Quillen–Sternberg’s theorem proved à la Wilson. 2.1. Theorem . Let 𝔤 be an infinite dimensional transitive Lie algebra of the form 𝔤 = ⊕_{i≥−1} 𝔤_{i}. Then there exists g ∈ 𝔤_{0} such that rk(adg)|_{𝔤_{−1}} = 1.

The converse statement is obvious: if such g exists, then, clearly, Cartan’s prolong of (𝔤_{−1}, 𝔤_{0}) is infinite dimensional. Indeed, in some basis, g takes the form x\frac{∂}{∂y} and all the elements x^{n} \frac{∂}{∂y} belong to the Cartan prolong.

2.2. A reformulation of Theorem. This theorem is, in fact, a special case of a more general statement. Let A = ⊕_{i≤0} A_{i} be the commutative ℤ-graded algebra with unity gener-

ated by A_{−1} = A and M = ⊕_{i≥k} M_{i} a ℤ-graded A-module. (In the case we are interested in

A = S.(𝔤_{−1}) and M = 𝔤.) We will say that M is transitive if

Am ≠ 0 for any nonzero m ∈ M_{i} and i ≥ k + 1.

Notice that each element m ∈ M determines a linear map

m : A ⟶ M, a ↦ am.

Clearly, M is transitive if and only if there exists i ≥ k + 1 and a nonzero m ∈ M_{i} such that ker m ≠ A.

Now it is clear that Theorem 2.1 can be reformulated in the following equivalent form which is more convenient for us. Theorem . Let M = ⊕_{i≥k} M_{i} be a ℤ-graded infinite dimensional transitive A-module. Then

there exists m ∈ M_{k+1} such that codim ker m = 1; hence, rk m = 1.

Proof. Without loss of generality we may assume that k = −1, i.e., M = ⊕_{i≥−1} M_{i} and

m ∈ M_{0}.

1^{∘}. Due to the commutativity of A, ker am ⊃ ker m for any a ∈ A, m ∈ M.

2^{∘}. Set r_{i} = max_{m∈M_{i}}{dim ker m}. By the above, r_{i+1} ≤ r_{i} ; hence, the sequence r_{i} stabilizes:

r_{N} = r_{N+1} = ⋯:= r for some N. Due to transitivity, r < dim A, so it remains to prove that r = dim A − 1.

3^{∘}. We will say that m is maximally degenerate if dim ker m = r.

By item 1^{∘}, if a ∈ A and m ∈ M_{n} for n ≥ N + 1 is maximally degenerate, so am is maximally degenerate too and ker am = ker m.

4^{∘}. Let m_{0} ∈ M_{N+l} for l ≥ 0 be a maximally degenerate element. For any n ≥ N set

V_{n}(m_{0}) = {m ∈ M_{n} | ker m ⊃ ker m_{0}}.

It is clear that any nonzero element m ∈ V_{n}(m_{0}) is maximally degenerate and ker m = ker m_{0}.

5^{∘}. The following Lemma is crucial in the next item. Actually, this Lemma is the key point of the whole proof of Theorem 2.2.

Lemma . Let U, V, W be finite dimensional vector spaces, U ⊂ Hom(V, W) a subspace such that any nonzero homomorphism u ∈ U is a monomorphism. Then dim U ≤ dim W − dim V + 1.

Proof (This elegant proof due to J. Bernstein replaces an original elementary but ugly proof.) Let P (X) be the projective space corresponding to the vector space X. Since U⊗V → W is a monomorphism for any nonzero u ∈ U, there exists a map i : P (U)×P (V) ⟶ P (W) which induces embeddings i : a × P (V) ⟶ P (W) and i : P (U) × b ⟶ P (W) for each fixed a ∈ P (U) and b ∈ P (V).

Let h ∈ H^{2}(P (W)), h_{1} ∈ H^{2}(P (U)) and h_{2} ∈ H^{2}(P (V)) be the generators of the respective cohomology rings. Since i : a×P (V) ⟶ P (W) and i : P (U)×b ⟶ P (W) are embeddings, it follows that i^{∗}(h) = h_{1} + h_{2}.

On the other hand, the only relations h, h_{1} and h_{2} satisfy are

h^{dim W +1} = 0, h_{1}^{dim U+1} = 0, h_{2}^{dim V +1} = 0.

Hence, replacing h with i^{∗}(h) = h_{1} + h_{2} we see that the first identity is only possible if

dim U + dim V ≤ dim W + 1. □

6^{∘}. Each m ∈ V_{n}(m_{0}) determines a monomorphism A/ ker m_{0} ⟶ V_{n−1}(m_{0}). By Lemma 5^{∘} we have

dim V_{n}(m_{0}) ≤ dim V_{n−1}(m_{0}) − (dim A − r) + 1 ⟹
dim V_{n−1}(m_{0}) ≥ dim V_{n}(m_{0}) + (dim A − r) − 1.

7^{∘}. Suppose r < dim A − 1. Then by item 6^{∘}, dim V_{n−1}(m_{0}) ≥ dim V_{n}(m_{0}) + 1. Let us take m_{0} ∈ M_{N+s}, where s > 0 is arbitrary. Then V_{N+s}(m_{0}) ≠ 0; hence,

dim M_{n} ≥ dim V_{N}(m_{0}) ≥ V_{N+1}(m_{0}) + 1 ≥ ⋯ ≥ dim V_{N+s}(m_{0}) + s ≥ 1 + s.

Since dim M_{N} < ∞, we get a contradiction. Thus, r = dim A − 1.

8^{∘}. So for some s ≥ N we get an element m_{0} ∈ M_{s} with codim ker m_{0} = 1. By transitivity there are a_{1}, ⋯, a_{s} ∈ A such that 0 ≠ m = a_{1} ⋯ a_{s}m_{0} ∈ M_{0}. Due to item 1^{∘}, ker m ⊃ ker m_{0} and by transitivity ker m ≠ A. Then ker m = ker m_{0} and m is the element we need. □

Remark . The proof of the reformulated theorem is outlined in this way by D. Alekseevsky and elaborated by A. Rudakov (1979). An advantage of this proof is that it is short and superizable as shown in sec 2.3.

2.3. Let A = ⊕_{i≤−1}A_{i} be the supercommutative superalgebra generated by A_{−1} = A = A¯0 ⊕ A¯1. Let M ⊕_{i≥−1} M_{i} be an A-module. As above, we will say that M is transitive if ker m ≠ 0 for any nonzero m ∈ M_{i}, where i ≥ 0. Theorem . In any infinite dimensional transitive graded A-module M there exists an element m ∈ M_{0} such that codim ker m = 1|0.

Proof. Let dim A¯1 = 0|n.

1^{∘}. Due to supercommutativity of A we have

a^{2}m = 0 for any a ∈ A¯1, m ∈ M;  (2.3.1)
a_{1} ⋯ a_{i}m = 0 for any m ∈ M and a_{1}, ⋯, a_{i} ∈ A¯1 if i > n.  (2.3.2)

2^{∘}. Due to (2.3.2) and transitivity of M we see that A¯0m ≠ 0 for any m ∈ M_{i} with i ≥ n. 3^{∘}. Let A_{0} ⊂ A be the subalgebra generated by A¯0. Let M̃ = M/ ⊕_{i=−1}^{n−1} M_{i}. Let us

consider M̃ as an A_{0}-module and apply Theorem 2.2 to M̃. We get an element m ∈ M_{n+1} such that dim ker m ∩ A¯0 = dim A¯0 − 1. If ker m ∩ A¯1 ≠ A¯1, there exists a_{1} ∈ A¯1 such that a_{1} ∉ ker m. Then a_{1}m ≠ 0 and by (2.3.1), ker a_{1}m ⊃ ker m ⊕ ℂa_{1}.

Similarly, for i ≤ n, there exist a_{1}, ⋯, a_{i} such that m_{1} = a_{1} ⋯ a_{i}m ≠ 0 and ker m_{1} ⊃ ker m + A¯1.

Notice that m_{1} ∈ M_{s}, where s = n + 1 − i > 0. Hence, ker m_{1} ≠ A; thus, codim ker m_{1} = 1|0.

Now we can complete the proof in the same way as in item 8^{∘} of the proof of Theorem 2.2. □

Corollary . Let 𝔤 = ⊕_{i≥−1}𝔤_{i} be a transitive infinite dimensional ℤ-graded Lie superalgebra with dim 𝔤_{i} < ∞ for all i. Then there exists v^{∗} ⊗ w ∈ 𝔤_{0}, where v ∈ (𝔤_{−1})¯0 and w ∈ 𝔤_{−1}.

§3. Step 3. Main reduction: the three cases

3.0. Lemma . If 𝔤_{∗} = ⊕_{i≥−1} 𝔤_{i} is simple, then

1) [𝔤_{−1}, 𝔤_{1}] = 𝔤_{0};

2) the representation of 𝔤_{0} in 𝔤_{−1} is irreducible, i.e., 𝔤_{≥} = ⊕_{i≥0} 𝔤_{i} is a maximal Lie subsu-

peralgebra; 3) the representation of 𝔤_{0} in 𝔤_{−1} is faithful.

Proof. When one of the conditions 1) − 3) is violated it is easy to find an ideal in 𝔤_{∗}. If 1) fails, take 𝔦 = 𝔤_{−1} ⊕ [𝔤_{−1}, 𝔤_{1}] ⊕ (⊕_{i>0}𝔤_{i}). In the other cases take 𝔦 = ⊕_{n≥0} 𝔦_{n}, where 𝔦_{n} are recursively defined (here ρ is the 𝔤_{0}-action

in 𝔤_{−1}):

𝔦_{−1} = \begin{cases}a 𝔤_{0}-invariant subsuperspace & in case 2) \\ 0 & in case 3) \end{cases}

𝔦_{0} = ker ρ in case 3)

𝔦_{n} = \{g ∈ 𝔤_{n} | [𝔤_{−1}, g] ⊂ 𝔦_{n−1}\} for n ≥ \begin{cases}0 & in case 2) \\ 1 & in case 3); \end{cases}

The Jacobi identity provides the inclusion

[𝔤_{−1}, [𝔤_{k}, 𝔦_{n}]] ⊂ [𝔤_{k−1}, 𝔦_{n}] + [𝔤_{k}, [𝔤_{−1}, 𝔦_{n}]];

the double induction on k and n shows that 𝔦 is an ideal in 𝔤_{∗}. □

The conditions 2) and 3) of Lemma allow us to think of 𝔤_{0} as of an irreducible linear Lie superalgebra, i.e., as of a subalgebra of 𝔤𝔩(V), where V = 𝔤_{−1}, and the identity representaiton in V is irreducible.

3.1. Some notations. The four cases.

1) Let V_{1} and V_{2} be linear superspaces, V = V_{1} ⊗ V_{2}. The tensor product of 𝔤𝔩(V_{i})-actions in V_{1} ⊗ V_{2} gives us a representation

ρ(A, B) = A ⊗ 1_{V_{2}} + 1_{V_{1}} ⊗ B for any A ∈ 𝔤𝔩(V_{1}), B ∈ 𝔤𝔩(V_{2}).

It is clear that ρ has a one-dimensional kernel ℂ(1_{V_{1}}−1_{V_{2}}). Denote the image ρ(𝔤𝔩(V_{1})⊕𝔤𝔩(V_{2})) by

𝔤𝔩(V_{1}) ⨀ 𝔤𝔩(V_{2}).

2) The space of all operators in a superspace V may be endowed with two structures: that of the Lie superalgebra 𝔤𝔩(V) and of the associative superalgebra Mat(V). We will carefully distinguish them. (Recall that we endow with index _{L} the Lie superalgebra constructed on the space of an associative superalgebra with the dot product replaced with the bracket. In particular, Mat(V)_{L} = 𝔤𝔩(V).)

Let V_{0} be a (1, 1)-dimensional superspace and A ⊂ Mat(1|1) = Mat(V_{0}) the associative

subsuperalgebra of supermatrices of the form (\begin{matrix}a & b \\ b & a \end{matrix}), where a, b ∈ ℂ and C(A) ⊂ Mat(V_{0})

is the associative subsuperalgebra of supermatrices of the form (\begin{matrix}c & d \\ −d & c \end{matrix}), where c, d ∈ ℂ.

Set J = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) and Π = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}); clearly 1 and J generate A while 1 and Π generate

C(A) as associative superalgebras.

Observe that

J^{2} = −1, Π^{2} = 1; JΠ = ΠJ = 0

and, therefore, A ≅ C(A) over ℂ and [A, C(A)] = 0. (This justifies our notations: C(A) is the centralizer of A in Mat(1|1).)

If V_{1} and V_{2} are linear superspaces of dimension mε and n, respectively, then the Lie superalgebras (Mat(V_{1}) ⊗ A)_{L} and (C(A) ⊗ Mat(V_{2}))_{L} of the operators in V_{1} ⊗ V_{0} and V_{0} ⊗ V_{2}, respectively, can be identified with 𝔮(m) and 𝔮(n), respectively.

Now, let V = V_{1} ⊗ V_{0} ⊗ V_{2}. Then a natural 𝔮(n_{1}) ⊕ 𝔮(n_{2})-action ρ in V is determined:

for any (\begin{matrix}A & B \\ B & A \end{matrix}) ∈ 𝔮(m), and (\begin{matrix}C & D \\ D & C \end{matrix}) ∈ 𝔮(n) set ρ((\begin{matrix}A & B \\ B & A \end{matrix}) , (\begin{matrix}C & D \\ D & C \end{matrix})) =
(A ⊗ 1_{V_{0}} + B ⊗ J) ⊗ 1_{V_{2}} + 1_{V_{1}} ⊗ (1_{V_{0}} ⊗ C + Π ⊗ D).

This representation has a one-dimensional kernel. Denote by 𝔮(n_{1}) ⨀ 𝔮(n_{2}) the image of the Lie superalgebra 𝔮(n_{1}) ⊕ 𝔮(n_{2}) under this representation.

3) Let V_{1} be a linear superspace of dimension (r, s), 𝔫 = 𝔤𝔩(V_{1}) ⊗ Λ(n), 𝔨 = 𝔳𝔢𝔠𝔱(0|n) and 𝔤 = 𝔫 ⊂ + 𝔨 with the natural action of 𝔨 on the ideal 𝔫. The Lie superalgebra 𝔤 has a natural faithful representation ρ in the linear superspace V = V_{1} ⊗ Λ(n) defined by the formulas

\begin{aligned}ρ(A ⊗ φ)(v ⊗ ψ) & = & (−1)^{p(φ)p(v)}Av ⊗ φψ \\ ρ(D)(v ⊗ ψ) & = & −(−1)^{p(D)p(v)}v ⊗ D(ψ) \end{aligned} for any A ⊗ φ ∈ 𝔫, D ∈ 𝔨, v ⊗ ψ ∈ V.

In the sequel, we will always identify the elements of 𝔤 with their images under ρ. Therefore, we will consider 𝔤 embedded in 𝔤𝔩(V).

4) Let 𝔥𝔢𝔦(0|2n) = Span(ξ_{1}, …, ξ_{n}; η_{1}, …, η_{n}; z) be the Heisenberg superalgebra, see 0.5.2, with the nonzero relations [ξ_{i}, η_{j}] = δ_{ij}z. The nonzero irreducible finite dimensional representation of 𝔥𝔢𝔦(0|2n) that maps z to the operator of multiplication by 1 is realized in Λ(ξ) via ξ_{i} ↦ ξ_{i}, η_{i} ↦ ∂_{i}. The normalizer of 𝔥𝔢𝔦(0|2n) in 𝔤𝔩(Λ(n)) is 𝔤 = 𝔥𝔢𝔦(0|2n) ⊂ + 𝔬(2n) or, in terms of differential operators:

𝔥𝔢𝔦(0|2n) ⊂ + 𝔬(2n) = Span(1, ξ_{i}, ∂_{i}, ξ_{i}∂_{j}, ∂_{i}∂_{j}, ξ_{i}ξ_{j})_{i,j=1}^{n}.

3.2. Theorem . If 𝔤 ⊂ 𝔤𝔩(V) is an irreducible linear Lie superalgebra neither almost simple nor a central extension of an almost simple, then 𝔤 is contained in one of the following four Lie superalgebras:

1) 𝔤 ⊂ 𝔤𝔩(V_{1}) ⨀ 𝔤𝔩(V_{2}) ⊂ 𝔤𝔩(V) for any V_{1}, V_{2} such that V = V_{1} ⊗ V_{2} and dim V_{i} > 1 for i = 1, 2;

2) 𝔤 ⊂ 𝔮(m) ⨀ 𝔮(n) ⊂ 𝔤𝔩(V), where V = V_{1} ⊗ V_{0} ⊗ V_{2} for any V_{0}, V_{1}, V_{2} such that dim V_{0} = (1, 1), dim V_{1} = mε, dim V_{2} = n and mn > 1;

3) 𝔤 ⊂ 𝔤𝔩(V_{1}) ⊗ Λ(n) ⊂ + 𝔳𝔢𝔠𝔱(0|n) ⊂ 𝔤𝔩(V) for any V_{1} and n such that V = V_{1} ⊗ Λ(n); the natural representation of 𝔤 in V is induced;

4) 𝔤 ⊂ 𝔥𝔢𝔦(0|2n) ⊂ + 𝔬(2n) ⊂ 𝔤𝔩(V) for n > 1 if V ≅ Λ(n) or Π(Λ(n)).

For proof see [Sh-M] or sec. 3.3.

Corollary . If 𝔤𝔩(V) is an irreducible linear superalgebra with a rank 1 operator in V, then the following 3 cases are possible:

1) the natural representation of 𝔤 in V is induced;

2) 𝔤 is almost simple or a central extension of an almost simple Lie superalgebra;

3) dim V = (2, 2) and 𝔤 ⊂ 𝔥𝔢𝔦(0|4) ⊂ + 𝔬(4).

Indeed, neither the matrix Lie superalgebras 𝔤𝔩(V_{1}) ⨀ 𝔤𝔩(V_{2}) with dim V_{i} > 1 nor 𝔮(n_{1}) ⨀ 𝔮(n_{2}) with n_{1}n_{2} > 1 have operators of rank 1; the only possibility for the Lie superalgebra 𝔤 = 𝔥𝔢𝔦(0|2n) ⊂ + 𝔬(2n) to have an operator of rank 1 occures for n = 2.

3.3. Proof of Theorem 3.2. In this section 𝔤 is an irreducible linear Lie superalgebra, ρ its standard representation in superspace V. We will assume that dim V > 1 (because the case dim V = 1 is trivial). By hypothesis 𝔤 contains an ideal 𝔨.

Our proof of Theorem 3.2 is largely based on the following Theorem 3.3.1 — the superization of a statement well-known to the reader from Dixmier’s book [Di], sec.5.4.1. 3.3.1. Theorem . Let 𝔤 be an irreducible linear Lie superalgebra, ρ its standard representation (in particular, ρ is faithful). Let 𝔨 be an ideal of 𝔤 and dim 𝔨 > 1. Then only one of the following 3 cases is possible:

A) ρ|𝔨 is irreducible;

B) ρ|𝔨 is a multiple of an irreducible 𝔨-module τ and the multiplicity of τ is > 1; C) there exists a subalgebra 𝔥 ⊂ 𝔤 such that 𝔨 ⊂ 𝔥 and ρ ≡ ind_{𝔥}^{𝔤}σ for an irreducible

𝔨-module σ.

Proof largely follows the lines of [Di] with one novel case: irreducible modules of Q-type might occure. Fortunately, their treatment is rather straightforward and we would rather save paper by ommiting the verification.

We will say that the representation ρ is of type A, B, or C with respect to the ideal 𝔨 if the corresponding case holds. Lemmas 3.3.5, 3.3.2 and 3.3.3 deal with types A, B and C, respectively. 3.3.2. Lemma . B) If ρ is of type B with respect to the ideal 𝔨, then either 𝔤 ⊂ 𝔤𝔩(V_{1}) ⨀ 𝔤𝔩(V_{2}) or 𝔤 ⊂ 𝔮(V_{1}) ⨀ 𝔮(V_{2}) for some V_{1} and V_{2}.

Proof. Let V be the superspace of representation τ. Then V = V_{1} ⊗ U for some U and

ρ(h)(v ⊗ u) = τ(h)v ⊗ u for any h ∈ 𝔨, v ∈ V_{1}, u ∈ U

i.e.,

the operators ρ(h) are of the form τ(h) ⊗ 1.  (3.3.2.1)

On the other hand, 𝔤 is the stabilizer of τ. So for any g ∈ 𝔤 there exists s(g) ∈ EndV_{1} such that τ([g, h]) = [s(g), τ(h)] for any h ∈ 𝔨.

Let us consider the operator T (g) = ρ(g) − s(g) ⊗ 1 in V. For h ∈ 𝔨 we have

[T (g), ρ(h)] = [ρ(g) − s(g) ⊗ 1, ρ(h)] = ρ([g, h]) − [s(g), τ(h)] ⊗ 1 = 0 ,

i.e., T (g) commutes with all the operators of the form ρ(h), h ∈ 𝔥, and, since τ is irreducible, T (g) is of the form

T (g) = \begin{cases}1 ⊗ A(g) & if τ is of G-type \\ 1 ⊗ A(g) + J ⊗ B(g) & if τ is of Q-type , \end{cases}

where J is an odd generator of C(τ), the centralizer of τ.

Therefore, ρ(g) is either of the form

ρ(g) = s(g) ⊗ 1 + 1 ⊗ A(g), if τ is of G-type  (3.3.2.2)

or

ρ(g) = s(g) ⊗ 1 + 1 ⊗ A(g) + J ⊗ B(g) if τ is of Q-type .  (3.3.2.3)

Formulas (3.3.2.1)−(3.3.2.3) show that if τ is of G-type, then 𝔤 is contained in 𝔤𝔩(V_{1}) ⨀ 𝔤𝔩(V_{2}) ⊂ 𝔤𝔩(V), where V_{2} = U, and if τ is of Q-type, then 𝔤 is contained in 𝔮(V_{1}) ⨀ 𝔮(V_{2}) ⊂ 𝔤𝔩(V), where V_{2} = Span(1, J) ⊗ U. □

3.3.3. Lemma . C) If ρ is of type C with respect to the ideal 𝔨, then 𝔤 ⊂ 𝔤𝔩(V) ⊗ Λ(n) ⊂ + 𝔳𝔢𝔠𝔱(0|n) for some V and n.

Lemma C follows from the definition of the induced representaiton.

3.34. To complete the proof of Theorem 3.2 it suffices to consider the case when ρ possesses the following property: for any ideal 𝔨 ⊂ 𝔤 either ρ is of type A with respect to 𝔨 or ρ|_{𝔨} is the multiple of a character.

Let 𝔯 be the radical of the linear Lie superalgebras 𝔤. Lemma . If 𝔯 is commutative and ρ belongs to type A or B with respect to 𝔯, then dim 𝔯 = 0 or 1.

Proof follows from the fact that any irreducible representation of a commutative Lie superalgebra is 1-dimensional (up to the change of parity) and the faithfulness of ρ. □

3.3.5. In the case when 𝔯 is not commutative, consider the derived series of 𝔯:

𝔯 ⊃ 𝔯_{1} ⊃ ⋯ ⊃ 𝔯_{k} ⊃ 𝔯_{k+1} = 0,

where 𝔯_{i+1} = [𝔯_{i}, 𝔯_{i}]. Clearly, each 𝔯_{i} is an ideal in 𝔤 and the last nonzero ideal, 𝔯_{k}, is commutative. Lemma . A) If ρ is of type A with respect to 𝔯_{k−1} and ρ|_{𝔯_{k}} is scalar, then either 𝔯_{k−1} ≅ 𝔥𝔢𝔦(0|2n) or 𝔯_{k−1} ≅ 𝔥𝔢𝔦(0|2n − 1) and V ≅ Λ(n) or Π(Λ(n)) and 𝔤 ⊂ 𝔥𝔢𝔦(0|2n) ⊂ + 𝔬(2n).

Proof. We see that if the restrictions ρ|_{𝔯_{i}} are irreducible for all i, then 1) dim 𝔯_{k} = 1 because ρ is faithful;

2) 𝔯_{k−1} = 𝔥𝔢𝔦(0|m) for some m;

3) ρ|_{𝔯_{k−1}} is irreducible and faithful, so it can be realized in the superspace of functions, Λ(2n), or in Π(Λ(2n)), where n = [\frac{m}{2}] + 1;

4) 𝔤 is contained in the normalizer of 𝔥𝔢𝔦(0|2n) in its spinor representation, i.e., 𝔤 ⊂ 𝔥𝔢𝔦(0|2n) ⊂ + 𝔬(2n). □

3.3.6. Lemma . Let dim 𝔯 ≤ 1, i.e., 𝔤 is either semisimple or a central extension of a semisimple Lie superalgebra but NOT an almost simple or a central extension of an almost simple Lie superalgebra. Then we can always choose an ideal 𝔨 such that ρ is of type B or C with respect to 𝔨.

Proof. Let dim 𝔯 = 0, i.e., let 𝔤 be semisimple. As 𝔤 is not almost simple, then due to [K2] the alternative arises: either

𝔤 contains an ideal 𝔨 of the form 𝔨 = 𝔥 ⊗ Λ(n) with simple 𝔥 and n > 0  (3.3.6)

or

𝔤 = ⊕_{j≤k} 𝔰_{j}, where each 𝔰_{j} is almost simple and k > 1.

3.3.6.1. Lemma . If 𝔤 = ⊕_{j≤k} 𝔰_{j} and k ≥ 2, then any irreducible faithful representation of 𝔤

is of type B with respect to any its ideal 𝔰_{j}.

Proof. Since the stabilizer of any irreducible representation of 𝔰_{j} is the whole 𝔤, the type of any irreducible representation of 𝔤 with respect to 𝔰_{j} can be either A or B. Due to faithfulness case A is ruled out. □

3.3.6.2. Lemma . Let 𝔥 be a simple Lie superalgebra and 𝔨 = 𝔥 ⊗ Λ(n). Then 𝔨 has no faithful irreducible finite dimensional representations.

For proof see 3.3.6.6.

3.3.6.3. Corollary . If 𝔤 contains an ideal 𝔨 of the form (3.3.6), then 𝔤 can not have any faithful irreducible finite dimensional representation of type A with respect to the ideal 𝔨.

3.3.6.4. Corollary . Lemma 3.3.6.1 and Corollary 3.3.6.2 prove Lemma 3.3.6 for semisimple Lie superalgebras.

3.3.6.5. Lemma . If 𝔥 is a simple Lie superalgebra, then

[𝔥¯1, 𝔥¯1] = 𝔥¯0 and [𝔥¯0, 𝔥¯1] = 𝔥¯1 .

Proof is left to the reader. (Hint: see [K2].) □

3.3.6.6. Proof of Lemma 3.3.6.2. For n > 0 the Lie superalgerba 𝔨 contains supercommutative ideal

𝔫_{1} = 𝔥 ⊗ ξ_{1} … ξ_{n}.

Moreover, if n > 2, then 𝔤 contains a supercommutative ideal

𝔫_{2} = 𝔥 ⊗ Λ^{n−1}(ξ) ⊕ 𝔫_{1}.

Thanks to Theorem 3.3.1, any irreducible representation ρ of 𝔨 is obtained as follows: take an irreducible representation (character χ) of an ideal 𝔫_{i}, the subsuperalgebra 𝔰𝔱_{𝔨}(χ), and an irreducible representation ρ̃ of 𝔰𝔱_{𝔨}(χ) whose restriction onto 𝔫_{i} is a multiple of χ. Then ρ = ind_{𝔰𝔱_{𝔨}(χ)}^{𝔨}(ρ̃) for some ρ̃.

Let us show that if dim ρ < ∞, then χ|_{𝔥⊗ξ_{1}···ξ_{n}} = 0. Since 𝔥 ⊗ ξ_{1} ⋯ ξ_{n} is an ideal in 𝔨, this would contradict the faithfulness of ρ. Indeed, dim ρ < ∞ if and only if 𝔰𝔱_{𝔨}(χ) ⊃ 𝔨¯0.

1) n = 2k + 1. For any f ∈ 𝔥 and g = h ⊗ 1 ∈ 𝔥_{0} ⊗ 1 we have 0 = χ([g, f ⊗ ξ_{1} ⋯ ξ_{n}]) = χ([h, f] ⊗ ξ_{1} ⋯ ξ_{n}) implying that

χ|_{[𝔥¯0,𝔥¯1]⊗ξ_{1}···ξ_{n}} =^{(Lemma 3.3.6.5)} χ|_{(𝔥¯1⊗ξ_{1}···ξ_{n})} = 0.

Since χ is a character, then χ|_{𝔥¯0⊗ξ_{1}···ξ_{n}} = 0, i.e., χ|_{𝔥⊗ξ_{1}···ξ_{n}} = 0.

2) n = 2k > 2. For any

\begin{matrix}g = h ⊗ l_{1}(ξ) ∈ 𝔥¯1 ⊗ Λ^{1}(ξ) ⊂ 𝔨¯0 \\ g_{1} = h_{1} ⊗ l_{n−1}(ξ) ∈ 𝔥¯1 ⊗ Λ^{n−1}(ξ) ⊂ (𝔫_{2})¯0 \end{matrix}

we have

0 = χ([g, g_{1}]) = ±χ([h, h_{1}]) ⊗ l_{1}(ξ)l_{n−1}(ξ),

i.e.,

χ|_{[𝔥¯1,𝔥¯1]⊗ξ_{1}···ξ_{n}} = χ|_{𝔥¯0⊗ξ_{1}···ξ_{n}} = 0 .

But χ is a character; therefore,

χ|_{𝔥¯1⊗ξ_{1}···ξ_{n}} = 0

and, finally,

χ|_{𝔥⊗ξ_{1}···ξ_{n}=0} .

3) n = 2. Then 𝔨 contains a commutative ideal 𝔫 = 𝔥 ⊗ ξ_{1}ξ_{2}; let χ be a character of 𝔫.

If χ generates a finite dimensional representation of 𝔨, then 𝔰𝔱_{𝔨}(χ) ⊃ 𝔨¯0, i.e.,

χ([f, g ⊗ ξ_{1}ξ_{2}]) = 0 for any f ∈ 𝔥¯0, g ∈ 𝔥.

If g ∈ 𝔥¯1, then formula (3.3.6) holds automatically by the parity considerations. Let g ∈ 𝔥¯0. Then, nevertheless,

χ([f, g ⊗ ξ_{1}ξ_{2}]) = χ([f, g] ⊗ ξ_{1}ξ_{2}) = 0

and, therefore, χ|_{[𝔥¯0,𝔥¯0]⊗ξ_{1}ξ_{2}} = 0.

Set

𝔪 = 𝔥 ⊗ (Λ^{1}(ξ) ⊕ Λ^{2}(ξ)).

Clearly, 𝔪 is an ideal of 𝔨 and 𝔰𝔱_{𝔫}(χ) = 𝔪. Therefore, the irreducible representation ρ of 𝔪 given by χ is such that ρ|_{𝔫} is a multiple of χ. Let

𝔫_{1} = 𝔥¯0 ⊗ Λ^{1}(ξ) ⊕ ([𝔥¯0, 𝔥¯0] ⊕ 𝔥¯1) ⊗ Λ^{2}(ξ).

Clearly, 𝔫_{1} is an ideal in 𝔪 and

[𝔫_{1}, 𝔪] ⊂ ([𝔥¯0, 𝔥¯0] ⊕ 𝔥¯1) ⊗ Λ^{2}(ξ) ⊂ ker χ.

Thus, 𝔫_{1} ⊂ ker ρ since the subspace V^{ρ(𝔫_{1})} = {v ∈ V | ρ(𝔫_{1})v = 0} is ρ(𝔪_{1})-invariant and non-zero. But 𝔪/𝔫_{1} is a solvable Lie algebra and by the Lie theorem any its irreducible finite dimensional representation is of the same form as χ. Let it be χ, for definiteness sake.

This means that χ|_{[𝔥¯1,𝔥¯1]⊗Λ^{2}(ξ)} = 0, i.e., χ|_{𝔥⊗Λ^{2}(ξ)} = 0 and since 𝔥 ⊗ ξ_{1}ξ_{2} is an ideal in 𝔨, then the representation of 𝔨 given by χ cannot be faithful. Lemma 3.3.6.2 is proved. □

3.3.6.7. Central extensions of semisimple Lie superalgebras. Let 𝔤 be a Lie superalgebra, 𝔯 its radical of dimension 1. Then 𝔯 is the center of 𝔤 and 𝔤̃ = 𝔤/𝔯 is semisimple.

First, consider the case when 𝔤̃ = ⊕_{i≤k} 𝔰_{i}, where the 𝔰_{i} are almost simple and k > 1. Let

π : 𝔤 → 𝔤̃ be the natural projection. The Lie superalgebra 𝔨 := π^{−1}(𝔰_{1}) is an ideal in 𝔤 and dim 𝔨 > 1.

3.3.6.8. Lemma . If k > 1, then ρ can not be irreducible of type A with respect to 𝔨.

Proof. Set 𝔤_{+} = ⊕_{i≥1}(𝔰_{i})¯0. Clearly, [𝔰_{1}, 𝔤_{+}] = 0. Since ρ(𝔯) acts by scalar operators and ρ is a

finite dimensional representation, it follows that [π^{−1}(𝔤_{+}), 𝔨] = 0, i.e., ρ(π^{−1}(𝔤_{+})) consists of intertwining operators of ρ|_{𝔫}, i.e., ρ(π^{−1}(𝔤_{+})) = ℂ·1 and, since ρ is faithful, then π^{−1}(𝔤_{+}) = 𝔯 implying 𝔤_{+} = 0. □

3.3.6.9. Now, let 𝔤̃ contain an ideal 𝔨 of the form (3.3.6). The central extension is defined by a cocycle c : 𝔤̃ × 𝔤̃ ⟶ ℂ. The cocycle condition is

c(f, [g, h]) = c([f, g], h) + (−1)^{p(f)p(g)}c(g, [f, h]) for anyf, g, h ∈ 𝔤̃.

As earlier, we assume that 𝔤 has a faithful finite dimensional representation; so the restriction of c to 𝔤̃¯0 × 𝔤̃¯0 is trivial. Besides, c|_{𝔤̃¯0×𝔤̃¯1} = 0 by parity considerations. Therefore,

c|_{𝔤̃¯0×𝔤̃} = 0.  (3.5)

Lemma . Set 𝔏^{m} = 𝔥 ⊗ Λ^{m}(ξ_{1}, …, ξ_{n}). We have c|_{𝔏^{n}×𝔏^{n}} = 0.

Proof. If n = 2k + 1 the condition (3.5) means that

c|_{𝔥¯1⊗Λ^{n}(ξ)×𝔥¯1⊗Λ^{n}(ξ)} = 0.

Let g¯1, h¯1 ∈ 𝔥¯1 and h¯0 ∈ 𝔥¯0. Then

\begin{matrix}c(f¯0 ⊗ ξ_{1} · · · · · ξ_{n}, [g¯1, h¯1] ⊗ ξ_{1} · · · · · ξ_{n}) = −c(f¯0 ⊗ ξ_{1} · · · · · ξ_{n}, [g¯1 ⊗ ξ_{1} · · · · · ξ_{n}, h¯1]) = \\ −c([f¯0 ⊗ ξ_{1} · · · · · ξ_{n}, g¯1 ⊗ ξ_{1} · · · · · ξ_{n}], h¯1) − c(g¯1 ⊗ ξ_{1} · · · · · ξ_{n}, [f¯0 ⊗ ξ_{1} · · · · · ξ_{n}, h¯1]) = 0. \end{matrix}

Since 𝔥¯0 = [𝔥¯1, 𝔥¯1] (by Lemma 3.3.6.5), it follows that c|_{𝔏^{n}×𝔏^{n}} = 0.

If n = 2k the condition (3.5) means that

c|_{𝔥¯0⊗Λ^{n}(ξ)×𝔥¯0⊗Λ^{n}(ξ)} = 0.

Let f¯1, h¯1 ∈ 𝔥¯1 and g¯0 ∈ 𝔥¯0. Then

\begin{matrix}c(f¯1 ⊗ ξ_{1} · · · · · ξ_{n}, [g¯0, h¯1] ⊗ ξ_{1} · · · · · ξ_{n}) = −c(f¯1 ⊗ ξ_{1} · · · · · ξ_{n}, [g¯0 ⊗ ξ_{1} · · · · · ξ_{n}, h¯1]) = \\ −c([f¯1 ⊗ ξ_{1} · · · · · ξ_{n}, g¯0 ⊗ ξ_{1} · · · · · ξ_{n}], h¯1) − c(g¯0 ⊗ ξ_{1} · · · · · ξ_{n}, [f¯1 ⊗ ξ_{1} · · · · · ξ_{n}, h¯1]) = 0. \end{matrix}

Since 𝔥¯1 = [𝔥¯0, 𝔥¯1] (by Lemma 3.3.6.5), it follows that c|_{𝔏^{n}×𝔏^{n}} = 0. □

3.3.6.10. Lemma . c|_{𝔏^{k}×𝔏^{m}} = 0 for any m = 0, 1, …, n and k = n − m, n − m + 1, …, n.

Proof. We will perform the inverse double induction on k and m. For k = m = n Lemma 3.3.6.9 will do.

Let the statement be true for all k > k_{0} and k_{0} + m ≥ n. Let us show that it is true for k = k_{0} as well. Observe that due to [K2], 𝔤̃ contains n elements η_{i} such that adη_{i}|_{𝔥} = ∂_{ξ_{i}} +D_{i}, where D_{i} ∈ 𝔳𝔢𝔠𝔱(0|n) and deg D_{i} > 0 in the standard grading of 𝔳𝔢𝔠𝔱(0|n).

Let φ ∈ Λ^{k_{0}+1}(ξ_{1}, …, ξ_{n}), ψ ∈ Λ^{m}(ξ_{1}, …, ξ_{n}); let g, h ∈ 𝔥 be such that p(g ⊗ ψ) = ¯0; p(h ⊗ ψ) = ¯1.

Then we have

\begin{matrix}c([η_{i}, g ⊗ φ], h ⊗ ψ) = (−1)^{p(g)}c(g ⊗ η_{i}(φ), h ⊗ ψ) \\ = (−1)^{p(g)} (c(g ⊗ \frac{∂φ}{∂ξ_{i}} , h ⊗ ψ) + c(g ⊗ D_{i}φ, h ⊗ ψ)) . \end{matrix}

As to D_{i}φ ∈ ( ⊕_{s≥k_{0}+1} Λ^{s}(ξ)) from the last summand, it vanishes by the inductive hypothesis.

On the other hand,

c([η_{i}, g ⊗ φ], h ⊗ ψ) = c(η_{i}, [g ⊗ φ, h ⊗ ψ]) + c([η_{i}, h ⊗ ψ], g ⊗ φ).

As deg φ+deg ψ = k_{0} +1+m > n, the bracket in the first summand above is equal to 0. The second summand is equal to 0 by (3.5). So c(g ⊗ \frac{∂φ}{∂ξ_{i}} , h ⊗ ψ) = 0, for arbitrary g ⊗ \frac{∂φ}{∂ξ_{i}} ∈ 𝔏k_{0} ; ¯1

and h ⊗ ψ ∈ 𝔏m ; ¯1. □

3.3.6.11. Lemma . 𝔨 has no faithful irreducible finite dimensional representations.

Proof. Word-for-word reproduction of the proof of Lemma 3.3.6.2 with the help of Lemmas 3.3.6.9 and 3.3.6.10. □

Corollary . ρ can not be of type A with respect to the ideal 𝔨.

□

Lemmas 3.3.2 (Lemma B), 3.3.3 (Lemma C), 3.3.4, 3.3.5 (Lemma A) and Lemma 3.3.6 put together prove Theorem 3.2. □

§4. Step 4. Prolongation of the induced representations

In this step we describe all the simple infinite dimensional ℤ-graded Lie superalgebras of depth 1 for which the irreducible representation of 𝔤_{0} on 𝔤_{−1} is induced from a representation of a subsuperalgebra 𝔥 ⊂ 𝔤_{0}.

But, first of all, we describe certain universal Lie superalgebras.

4.1. Universality of 𝔳𝔢𝔠𝔱, 𝔨 and 𝔪. Let 𝔤 be a ℤ-graded transitive Lie superalgebra.

1. If 𝔤 = ⊕_{i=−1}^{∞} 𝔤_{i} and dim 𝔤_{−1} = (m|n), then 𝔤 ⊂ 𝔳𝔢𝔠𝔱(m|n).

2. If 𝔤 = ⊕_{i=−2}^{∞} 𝔤_{i} and 𝔤_{<0} = 𝔥𝔢𝔦(2m|n), then 𝔤 ⊂ 𝔨(2m + 1|n).

3. If 𝔤 = ⊕_{i=−2}^{∞} 𝔤_{i} and 𝔤_{<0} = 𝔞𝔟(n) (see 0.5.2), then 𝔤 ⊂ 𝔪(n).

The following are all the Lie superalgebras with “big” nonpositive part:

1′. Let 𝔤 = ⊕_{i=−1}^{∞} 𝔤_{i} and dim 𝔤_{−1} = (m|n). Let, besides, 𝔤_{0} ⊃ 𝔰𝔩(m|n) and dim 𝔤 = ∞. If

m > 1, then, 𝔤 = 𝔳𝔢𝔠𝔱(m|n), or 𝔰𝔳𝔢𝔠𝔱(m|n), or 𝔡𝔰𝔳𝔢𝔠𝔱(m|n). If m = 1, then there are two more possibilities: 𝔤 = 𝔰𝔳𝔢𝔠𝔱^{∘}(1|n) and 𝔡𝔰𝔳𝔢𝔠𝔱^{∘}(1|n).

2′. Let 𝔤 = ⊕_{i=−2}^{∞} 𝔤_{i} ⊂ 𝔨(2m + 1|n) be such that 𝔤_{≤0} = 𝔨(2m + 1|n)_{≤0} and dim 𝔤 = ∞. Then

𝔤 = 𝔨(2m + 1|n) or 𝔤 = 𝔡𝔭𝔬(2m|n) or, if m = 0 and n = 6, then 𝔤 = 𝔨𝔞𝔰.

3′. If 𝔤 = ⊕_{i=−2}^{∞} 𝔤_{i} ⊂ 𝔪(n) is such that 𝔤_{<0} = 𝔞𝔟(n), dim 𝔤 = ∞ and 𝔤 is simple, then

𝔤 = 𝔪(n) or 𝔤 = 𝔟_{∞}(n) or 𝔤 = 𝔟_{λ}(n) for λ ≠ 0.

The described universal properties are immediate corollaries of the properties of Cartan prolongs.

4.2. Lemma . Let 𝔤 = ⊕_{i=−1}^{∞} 𝔤_{i} be a ℤ-graded transitive Lie superalgebra and W ⊂ 𝔤_{−1} a

linear subsuperspace. Let F ∈ 𝔤_{k}, k > 0, be such that [w, F] = 0 for any w ∈ W. Then for any u ∈ 𝔤_{−1} the commutator [u, F] also commutes with W.

Proof. By the Jacoby identity

[w, [u, F]] = (−1)^{p(w)p(u)}[u, [w, F]] + [[w, u], F] = 0. □

Actually, Lemma 4.2 says that if the expression of F does not involve certain indeterminates, then [∂_{i}, F] does not involve these indeterminates for any i, either. 4.3. Theorem . Let ξ = (ξ_{1}, …, ξ_{k}) and V a linear superspace of dimension m|n, where (m|n) ≠ (1|0) or (0|1).

a) (Λ(ξ), Λ(ξ) ⊂ + 𝔳𝔢𝔠𝔱(ξ))_{∗} = 𝔨(1|2k; k).

b) (Π(Λ(ξ)), Λ(ξ) ⊂ + 𝔳𝔢𝔠𝔱(ξ))_{∗} = 𝔪(k; k).

c) (V ⊗ Λ(ξ), 𝔤𝔩(V) ⊗ Λ(ξ) ⊂ + 𝔳𝔢𝔠𝔱(ξ))_{∗} = 𝔳𝔢𝔠𝔱(m|n + k; k).

Proof. It is clear that the Lie superalgebras in the rhs of a) – c) are subalgebras of the respective Lie superalgebras in the lhs. Let 𝔤 = ⊕_{l=−1}^{∞} 𝔤_{l} be a superalgebra from the lhs.

Let y = −id_{V} ⊗ 1 ∈ 𝔤𝔩(V) ⊗ Λ(ξ). Then ady determines a natural ℤ-grading in 𝔤.

Let x = Σξ_{i}∂_{ξ_{i}} ∈ 𝔤_{0}. Then adx determines a ℤ-grading in each component 𝔤_{l}.

Let 𝔤_{l,i} = {g ∈ 𝔤_{l} | adx(g) = i · g}. Then 𝔤 = ⊕_{l,i} 𝔤_{l,i}.

Observe that 𝔤_{−1} = ⊕_{i≥0} 𝔤_{−1,i} and 𝔤_{0} = ⊕_{i≥−1} 𝔤_{0,i}. Set

W = 𝔤_{−1,0} ⊕ 𝔤_{−1,1} = (V ⊗ 1) ⊕ (V ⊗ Λ^{1}(ξ)).

We see that 𝔤_{0} does not contain nonzero elements that commute with W. Hence, by Lemma 4.2 none of the 𝔤_{l} for l > 0 contain nonzero elements that commute with W, either.

But if 𝔤_{l} = ⊕_{i≥−s} 𝔤_{1,i}, then the components 𝔤_{l+1,t} with t < −s − 1 should commute with W.

Hence, 𝔤_{l+1} = ⊕_{i≥−s−1} 𝔤_{1+1,i} and 𝔤_{l} = ⊕_{i≥−l−1} 𝔤_{1,i}.

\underline{Cases a) and b)}. Consider the operator T = ad(2y + x). Its eigenvalues are integers and in each 𝔤_{l} they are not less then l − 1 of linearly independent eigenvectors. Set 𝔤_{s}^{T} = {g ∈

𝔤 | T (g) = s · g}. Clearly, 𝔤_{s}^{T} ⊂ ⊕_{l=−1}^{s+1} 𝔤_{l} and dim 𝔤_{s}^{T} < ∞. Hence, in 𝔤, we obtain a new

ℤ-grading: 𝔤^{T} = ⊕_{s=−2}^{∞} 𝔤_{s}^{T}. The negative part of this ℤ-grading is as follows:

𝔤_{−2}^{T} = 𝔤_{−1,0} = ℂ · 1 (case a), or ℂ · π(1) (case b) ;
𝔤_{−1}^{T} = 𝔤_{−1,1} ⊕ 𝔤_{0,−1} = Span{ξ_{1}, …, ξ_{k}, ∂_{ξ_{1}}, …, ∂_{ξ_{k}}} (case a), or
Span{π(ξ_{1}), …, π(ξ_{k}), ∂_{ξ_{1}}, …, ∂_{ξ_{k}}} (case b).

Let us check that 𝔤^{T} is transitive, i.e., if g ∈ 𝔤_{l}^{T} with l ≥ 0 and [𝔤_{−1}^{T}, g] = 0, then g = 0. Indeed, for such g ∈ 𝔤 we have [W, g] = 0; hence, g ∈ 𝔤_{−1}. But only the subspace 𝔤_{−2}^{T} = 𝔤_{−1,0} ⊂ 𝔤_{−1} commutes with the ∂_{ξ_{i}} for all i.

Due to the universality of the Lie superalgebras of series 𝔨 and 𝔪 (see sec. 4.1) this means that

in case a) 𝔤^{T} ⊂ 𝔨(1|k) and thus 𝔤^{T} = 𝔨(1|k);

in case b) 𝔤^{T} ⊂ 𝔪(k) and thus 𝔤^{T} = 𝔪(k).

\underline{Case c)}. Set W_{0} = 𝔤_{−1,0} = V ⊗ 1.

4.3.1. Lemma . If dim V ≠ (1|0) or (0|1) and F ∈ 𝔤_{1} is such that [F, W_{0}] = 0, then F = 0. In particular, 𝔤_{1,−2} = 0.

Proof. Due to Lemma 4.2 the element [F, v ⊗ φ] ∈ 𝔤_{0} commutes with W_{0} for any v ∈ V and φ ∈ Λ(ξ). Hence, [F, v ⊗ φ] belongs to 𝔳𝔢𝔠𝔱(0|k) ⊂ 𝔤_{0}.

Denote [F, v ⊗ φ] by D_{v,φ}. By transitivity, D_{v,φ}ψ ≠ 0 for some v ∈ V and φ, ψ ∈ Λ(ξ). Take u ∈ V linearly independent with v. Then

\begin{matrix}[[F, v ⊗ φ], u ⊗ ψ] ± [v ⊗ φ, [F, u ⊗ ψ]] = \\ ±u ⊗ D_{v,φ}ψ ± v ⊗ D_{u,ψ}φ ≠ 0. \end{matrix}  (4.1)

(The symbol ± stands for a sign; this sign is not essential to us.) By the Jacobi identity (4.1) should vanish. This contradiction completes the proof. □

Lemmas 4.2 and 4.3.1 together mean that none of the 𝔤_{l} with l > 0 contains elements that commute with W_{0}. Hence, the least degree of ℤ-grading wrt ξ in 𝔤_{l+1} cannot be less than same in 𝔤_{l} for l ≥ 0. Thus, 𝔤_{l} = ⊕_{i≥−1} 𝔤_{l,i}.

Consider operator T = ad(y + x). Its eigenvalues are integers and in each 𝔤_{l} they are not less then l − 1 of linearly independent eigenvectors. So, as in cases a) and b), dim 𝔤_{s}^{T} = dim{g ∈ 𝔤 | T (g) = s · g} < ∞. Hence, in 𝔤, we get a new ℤ-grading:

𝔤^{T} = ⊕_{s=−1}^{∞} 𝔤_{s}^{T} .

The negative part of this ℤ-grading is

𝔤_{−1}^{T} = 𝔤_{−1,0} ⊕ 𝔤_{0,−1} = V ⊗ 1 ⊕ Span(∂_{ξ_{1}}, …, ∂_{ξ_{k}}).

Moreover, by Lemma 4.3.1 the elements of 𝔤 that commute with W_{0} = V ⊗1 ⊂ 𝔤_{−1}^{T} form the subspace 𝔤_{−1}⊕𝔳𝔢𝔠𝔱(0|k). We can directly see that the elements of this subspace that commute with all ∂_{ξ_{i}} span 𝔤_{−1}^{T}. Hence, 𝔤^{T} is transitive. Due to the universal property 4.1 of the Lie superalgebra 𝔳𝔢𝔠𝔱(m|n+k) this means that 𝔤^{T} ⊂ 𝔳𝔢𝔠𝔱(m|n+k); thus, 𝔤^{T} = 𝔳𝔢𝔠𝔱(m|n+k). □

4.3.2. Corollary . 1) If 𝔤 ⊂ 𝔨(1|2k; k) with 𝔤_{−1} = 𝔨(1|2k; k)_{−1} = Λ(ξ_{1}, ⋯, ξ_{k}) is such that 𝔤_{1} does not contain nonzero elements that commute with the subspace W_{0} = ℂ · 1 ⊂ 𝔤_{−1}, then 𝔤 ⊂ 𝔳𝔢𝔠𝔱(1|k; k).

2) If 𝔤 ⊂ 𝔪(k; k) with 𝔤_{−1} = 𝔪(k; k)_{−1} = Π(Λ(ξ_{1}, ⋯, ξ_{k})) is such that 𝔤_{1} does not contain nonzero elements that commute with the subspace W_{0} = ℂ · Π(1) ⊂ 𝔤_{−1}, then 𝔤 ⊂ 𝔳𝔢𝔠𝔱(0|k + 1; k); hence, dim 𝔤 < ∞.

The proof is the same as for case c) of Theorem 4.3. □ Now we are able to return to the classification.

4.4. Let, as above, ξ = (ξ_{1}, …, ξ_{k}) and V an (m|n)-dimensional linear superspace. We denote the coordinates in V by u = (u_{1}, …, u_{n+m}).

Theorem . Let 𝔤 = ⊕_{i=−1}^{∞} 𝔤_{i} be a simple infinite dimensional transitive ℤ-graded Lie superal-

gebra with 𝔤_{−1} = V ⊗ Λ(ξ) and 𝔤_{0} ⊂ 𝔤𝔩(V) ⊗ Λ(ξ) ⊂ + 𝔳𝔢𝔠𝔱(ξ).

1) If dim V > 1, then 𝔤 is either 𝔳𝔢𝔠𝔱(1|n + k; k) or 𝔰𝔳𝔢𝔠𝔱(m|n + k; k) (for m > 1) or 𝔰𝔳𝔢𝔠𝔱^{∘}(m|n + k; k);

2) if dim V = (1, 0) then 𝔤 is either 𝔳𝔢𝔠𝔱(1|k; k) or 𝔰𝔳𝔢𝔠𝔱^{∘}(1|k; k) or 𝔨(1|2k; k) or 𝔨𝔞𝔰^{ξ}(; ξ).

3) if dim V = (0, 1), then 𝔤 is either 𝔪(k; k) or 𝔟_{λ}(k; k) for λ ∈ ℂP^{1} \ {0} .

Proof. \underline{Case 1)}. By Theorem 4.3 𝔤 ⊂ 𝔳𝔢𝔠𝔱(m|n + k; k). Thus, we can use the realization of elements of 𝔤 by vector fields in u, ξ. The ℤ-grading wrt ξ in each component 𝔳𝔢𝔠𝔱(m|n+k; k)_{l} determines a filtration in 𝔤_{l} ⊂ 𝔤.

Let gr(𝔤) = ⊕_{l=−1}^{∞} ⊕_{i} gr(𝔤)_{l,i} be the corresponding bigraded Lie superalgebra, i.e.,

gr(𝔤)_{l,i} = \frac{𝔤_{l} ∩ ⊕_{j≥i} 𝔳𝔢𝔠𝔱(m|n + k; k)_{l,j}}{𝔤_{l} ∩ ⊕_{j>i} 𝔳𝔢𝔠𝔱(m|n + k; k)_{l,j}} .

By Lemma 3.1 the 𝔤_{0}-module 𝔤_{−1} is irreducible. This gives us the following conditions on

gr(𝔤)_{0,∗} := ⊕_{i} gr(𝔤)_{0,i}:

gr(𝔤)_{0,−1} = Span{∂_{ξ_{1}}, …, ∂_{ξ_{k}}} = 𝔳𝔢𝔠𝔱(m|n + k; k)_{0,−1};  (4.2)
gr(𝔤)_{0,0} ⊂ 𝔳𝔢𝔠𝔱(m|n + k; k)_{0,0} = 𝔤𝔩(V) ⊕ 𝔤𝔩(ξ) = 𝔤𝔩(m|n) ⊕ 𝔤𝔩(0|k)
acts irreducibly in V ⊗ 1 ⊂ 𝔤_{−1}.

(4.3)

4.4.1. Without loss of generality we may assume that the gr(𝔤)_{0,0}-module gr(𝔤)_{0,−1} = Span(∂_{ξ_{i}} | i = 1, …, k) is irreducible. Indeed, if it is reducible and, e.g., Span(∂_{ξ_{1}}, …, ∂_{ξ_{s}}) is a submodule, then

𝔤 ⊂ ( (𝔤𝔩(V) ⊗ Λ(ξ_{1}, …, ξ_{s}) ⊂ + 𝔳𝔢𝔠𝔱(ξ_{1}, …, ξ_{s})) ⊗ Λ(ξ_{s+1}, …, ξ_{k})) ⊂ + 𝔳𝔢𝔠𝔱(ξ_{s+1}, …, ξ_{k}).

So we set Ṽ = V ⊗ Λ(ξ_{1}, …, ξ_{s}) and ξ̃ = (ξ_{s+1}, ⋯, ξ_{k}). Now let us use the condition [𝔤_{−1}, 𝔤_{1}] = 𝔤_{0}. As we have seen in the proof of Theorem 4.3,

𝔳𝔢𝔠𝔱(m|n + k; k)_{1} = ⊕_{i≥−1} 𝔳𝔢𝔠𝔱(m|n + k; k)_{1,i}

and the component gr(𝔤)_{0,−1} can be only obtained by commuting gr(𝔤)_{−1,0} = V ⊗ 1 = Span(∂_{u_{1}}, …, ∂_{u_{m+n}}) with 𝔳𝔢𝔠𝔱(m|n + k; k)_{1,−1} = Span(u_{i}∂_{ξ_{j}} | all possible i, j).

By (4.2), for each j = 1, ⋯, k, there exists u_{α}∂_{ξ_{j}} ∈ gr(𝔤)_{1,−1}. But, due to (4.3), gr(𝔤)_{0,0} acts irreducibly on V^{∗}. Thus, gr(𝔤)_{1,−1} contains the elements of the form u_{α}∂_{ξ_{j}} for any α = 1, …, m + n and j = 1, …, k. Hence, gr(𝔤)_{1,−1} = 𝔳𝔢𝔠𝔱(m|n + k; k)_{1,−1}.

Thus, for any α, β = 1, …, m + n and i, j = 1, …, k we have:

[ξ_{i}∂_{u_{β}}, u_{α}∂_{ξ_{j}}] = δ_{αβ}ξ_{i}∂_{ξ_{j}} − (−1)^{(p(u_{α})+1)(p(u_{β})+1)}δ_{ij}u_{α}∂_{u_{β}} ∈ gr(𝔤)_{0,0}.

Consequently, gr(𝔤)_{0,0} ⊃ (𝔤𝔩(V) ⊕ 𝔤𝔩(ξ)) ∩ 𝔰𝔳𝔢𝔠𝔱(m|n + k).

Now, consider the regrading 𝔤^{T} of gr(𝔤) into the standard ℤ-grading of 𝔳𝔢𝔠𝔱(m|n + k). Then

𝔤_{−1}^{T} = gr(𝔤)_{−1,0} ⊕ gr(𝔤)_{0,−1} = 𝔳𝔢𝔠𝔱(m|n + k; k)_{−1};
𝔤_{0}^{T} = gr(𝔤)_{−1,1} ⊕ gr(𝔤)_{0,0} ⊕ gr(𝔤)_{1,−1} ⊃ 𝔰𝔩(m|n + k).

Next, by property 1′, sect. 4.1, we see that gr(𝔤) = 𝔳𝔢𝔠𝔱(m|n), or 𝔰𝔳𝔢𝔠𝔱(m|n), or 𝔡𝔰𝔳𝔢𝔠𝔱(m|n) for m > 1.

If m = 1, then there are two more possibilities: gr(𝔤) = 𝔰𝔳𝔢𝔠𝔱^{∘}(1|n) and gr(𝔤) = 𝔡𝔰𝔳𝔢𝔠𝔱^{∘}(1|n). \underline{Case 2)}. If dim V = (1|0), then 𝔤_{−1} = Λ(ξ) and 𝔤_{0} ⊂ Λ(ξ) ⊂ + 𝔳𝔢𝔠𝔱(0|k). By Corollary 4.3.2, if the component 𝔤_{1} does not contain nonzero elements that com-

mute with 𝔤_{−1,0} = ℂ · 1, then 𝔤 ⊂ 𝔳𝔢𝔠𝔱(1|k; k) and by above arguments 𝔤 = 𝔳𝔢𝔠𝔱(1|k; k) or 𝔰𝔳𝔢𝔠𝔱^{∘}(1|k; k).

Let the component 𝔤_{1} contain nonzero elements that commute with 𝔤_{−1,0} = ℂ · 1. Then 𝔤 ⊂ 𝔨 := 𝔨(1|2k; k).

In terms of the standard indeterminates t, ξ_{1}, …, ξ_{k}, η_{1}, …, η_{k} for 𝔨 we have

𝔨_{−1} = 𝔤_{−1} = Λ(ξ);
Λ(ξ) ⊗ (t − ∑ ξ_{i}η_{i}) ≅ Λ(k) ⊂ 𝔨_{0};
Λ(ξ) ⊗ Λ^{1}(η) ≅ 𝔳𝔢𝔠𝔱(0|k) ⊂ 𝔨_{0};
𝔤_{1} = t^{1} ⊗ Λ(ξ) ⨁ t ⊗ Λ(ξ) ⊗ Λ^{1}(η) ⨁ Λ(ξ) ⊗ Λ^{2}(η).

The subspace W_{1} ⊂ 𝔨_{1} of elements that commute with 𝔤_{−1,0} coincides with Λ(ξ) ⊗ Λ^{2}(η). Consider gr(𝔤). By (4.2), gr(𝔤)_{0,−1} = Span(η_{1}, …, η_{k}) and η_{i} acts in W_{1} as ∂_{ξ_{i}}. Since 𝔤_{1} ∩ W_{1} ≠ 0, then 𝔤_{1} ∩ 𝔨_{1,−2} ≠ 0, where 𝔨_{1,−2} = Λ^{2}(η) ⊗ 1. Hence, gr(𝔤)_{1,−2} ≠ 0. Set

𝔥 = ⊕_{i=−1}^{k−1} 𝔥_{i} = gr(𝔤)_{0,∗} ∩ Λ^{1}(η) ⊗ Λ(ξ).

Clearly, 𝔥 ⊃ [𝔤_{−1}, gr(𝔤)_{1,−2}].

4.4.2. Lemma . 𝔥_{k−2} is not contained in 𝔰𝔳𝔢𝔠𝔱(0|k)_{k−2}. In particular, 𝔥_{k−2} ≠ 0.

Proof. Let X = ξ_{1} ⋯ ξ_{k} ∈ 𝔤_{−1}. Then

\begin{matrix}[X, ∑_{i<j} α_{ij}η_{i}η_{j}] = (−1)^{k+1} ∑_{i<j} α_{ij}(\frac{∂X}{∂ξ_{i}} η_{j} − \frac{∂X}{∂ξ_{j}} η_{i}) ↦ \\ (−1)^{k+1} ∑_{i<j} α_{ij}(\frac{∂X}{∂ξ_{i}} ∂_{ξ_{j}} − \frac{∂X}{∂ξ_{j}} ∂_{ξ_{i}}) = D. \end{matrix}

But divD = 2(−1)^{k+1} ∑_{i<j} α_{ij} \frac{∂^{2}X}{∂ξ_{i}∂ξ_{j}}. So divD = 0 if and only if α_{ij} = 0 for all i, j. Since

gr(𝔤)_{1,−2} ≠ 0, this proves Lemma. □

We have established that:

1) 𝔥 ⊂ 𝔳𝔢𝔠𝔱(0|k);

2) 𝔥_{−1} = 𝔳𝔢𝔠𝔱(0|k)_{−1} and 𝔥_{−1} is an irreducible 𝔥_{0}-module;

3) 𝔥_{k−2} is not contained in 𝔰𝔳𝔢𝔠𝔱(0|k)_{k−2}. Comparing these conditions with the list in [K2] (p.71, Th. 4) we conclude that 𝔥 = 𝔳𝔢𝔠𝔱(0|k) for n > 3. If n = 3, then, additionally, 𝔥 can be equal to 𝔰𝔩(1|3) ⊂ 𝔳𝔢𝔠𝔱(0|3). Anyway, gr(𝔤)_{0,∗} ⊃ Λ^{1}(η) ⊗ Λ^{1}(ξ) ≅ 𝔤𝔩(k). Thus, by the irreducibility of the exterior square of the standard 𝔤𝔩(k)-module, gr(𝔤)_{1} ⊃ Λ^{2}(η).

The irreducibility of the 𝔤_{0}-module 𝔤_{−1} means that gr(𝔤)_{0,∗} should contain at least some elements from Λ(ξ) ⊗ (t − ∑ ξ_{i}η_{i}). Hence, t ∈ gr(𝔤)_{0,∗}.

Now we are able to consider the regrading 𝔤^{T} of gr(𝔤) into the standard ℤ-grading of 𝔨(1|2k). By the above, 𝔤_{≤0}^{T} = 𝔨(1|2k)_{≤0}. Hence, 𝔤^{T} is either 𝔨(1|2k) or 𝔨𝔞𝔰^{ξ} ⊂ 𝔨(1|6) and 𝔤 = 𝔨(1|2k; k) or 𝔨𝔞𝔰^{ξ}(; ξ), respectively.

\underline{Case 3)}. If dim V = (0|1), then 𝔤_{−1} = Π(Λ(ξ)), 𝔤_{0} ⊂ Λ(ξ) ⊂ + 𝔳𝔢𝔠𝔱(ξ) and 𝔤 ⊂ 𝔪(k; k). The operators of rank 1 with even covector in Λ(ξ) ⊂ + 𝔳𝔢𝔠𝔱(ξ) are

g_{i} = ξ_{1} ⋯ ξ_{n}∂_{i}, and h_{i} = ξ_{1} … ξ̂_{i} … ξ_{n}∂_{i} for i = 1, …, n.

By Theorem 2.1 𝔤_{0} contains at least one of them. Sect. 4.4.1 implies that 𝔤_{0} contains either all the g_{i} or all the h_{i}.

a) Let 𝔤_{0} ⊃ Span(g_{1}, …, g_{n}). Then from (4.2) we deduce that 𝔤_{0} ⊃ 𝔳𝔢𝔠𝔱(ξ_{1}, … ξ_{n}). Since the 𝔤_{0}-module 𝔤_{−1} is irreducible, 𝔤_{0} = Λ(ξ) ⊂ + 𝔳𝔢𝔠𝔱(ξ).

b) For the same reason, if 𝔤_{0} ⊃ Span(h_{1}, …, h_{n}), then 𝔤_{0} ⊃ 𝔰𝔳𝔢𝔠𝔱(ξ) and the following cases are possible:

1) 𝔤_{0} ≅ 𝔤_{0}^{λ} = {v + λ · div (v) | v ∈ 𝔳𝔢𝔠𝔱(ξ)}, where λ ≠ 0;
2) 𝔤_{0} ≅ Vol_{0}(0|n) ⊂ + 𝔰𝔳𝔢𝔠𝔱(ξ);
3) 𝔤_{0} ≅ Λ(ξ) ⊂ + 𝔰𝔳𝔢𝔠𝔱(ξ).

In all the cases 𝔤_{0} is ℤ-graded by powers of ξ and 𝔤_{0,0} ⊃ 𝔰𝔩(0|k). Due to Corollary 4.3.1 𝔤_{1,−2} ≠ 0; hence, 𝔤_{1,−2} = S^{2}(id).

Consider the regrading 𝔤^{T} of 𝔤 into the standard ℤ-grading of 𝔪(k). We get 𝔤_{<0}^{T} = 𝔪_{<0} and 𝔤_{0} ⊃ 𝔰𝔭𝔢(k). Then by universal property 3′ of sec. 4.1 𝔤 = 𝔪(k) or 𝔟_{λ}(k), where λ ∈ ℂP^{1} \ {0}. □

4.5. Corollary . For the simple vectorial Lie superalgebra 𝔤 with W-filtration the Lie superalgebra 𝔤_{0} is solvable in the following cases only: 𝔤 = 𝔳𝔢𝔠𝔱(1|1) or 𝔰𝔳𝔢𝔠𝔱^{∘}(1|2; 1) or 𝔳𝔢𝔠𝔱(1|2; 1) and the corresponding 𝔤_{0} is isomorphic to 𝔤𝔩(1|1) or 𝔰𝔩(1|1) ⊗ Λ(1) ⊂ + 𝔳𝔢𝔠𝔱(0|1) or 𝔤𝔩(1|1) ⊗ Λ(1) ⊂ + 𝔳𝔢𝔠𝔱(0|1), respectively.

§5. Step 5. Prolongations of almost simple Lie superalgebras and their central extensions

In this step there are two essentially different cases: vectorial and non-vectorial Lie superalgebras (Steps 5B and 5A, respectively).

Let 𝔤 be an almost simple Lie superalgebra or a central extension of an almost simple classical Lie superalgebra (all such Lie superalgebras are listed in Tables 8.1 and 8.2). Let T be a faithful irreducible representation of 𝔤 in the finite dimensional superspace V. Our goal is to describe all the collections (𝔤, T, V) for which there exists a rank 1 operator T (g), g ∈ 𝔤, with an even covector.

Irreducible representations of simple vectorial Lie superalgebras and their “relatives” — central extensions of almost simple Lie superalgebras related to them — are completely described, and to make exposition selfcontained we give an account in Appendix. From this description we easily deduce the list of admissible collections (𝔤, T, V).

For non-vectorial Lie superalgebras, there is a complete description of the typical (i.e., generic) representations (Berezin [B1], [B2] and Kac [K5]). But (bar a couple of exceptions) the rank 1 operators occure, if ever, in atypical representations whose description is only conjectural.

The established results concerning atypical representations are, so far, only partial (for reviews covering various aspects of the problem see [L2], [KW], [Se1]).

Though we share the belief in the validity of the remarkable general formula of Penkov and Serganova, we should better rely on something else: it is conjectural at the moment. Therefore, our proof in Step 5A does not appeal to the description of irreducible representations. Inevitably then, the proof is rather lengthy.

Our method of description and of search after such collections (𝔤, T, V) is based on pretty straightforward considerations which, nevertheless, provide with essential restrictions on the weight diagram of T.

Step 5A (sec. 5.0–5.6). Non-vectorial Lie superalgebras

To help the reader, we have gathered main results of Step 5A in Proposition 5.0. 5.0. Proposition . For the almost simple non-vectorial Lie superalgebra 𝔤_{0} or any central extension of such, the representations (T, V) with a rank 1 operator T (g), g ∈ 𝔤_{0}, with an even covector are only the following ones (we have normed the action of the generator of the center of 𝔤_{0} to be equal to id):

V = 𝔤_{−1}  𝔤_{0}  (𝔤_{−1}, 𝔤_{0})_{∗}
id or id^{∗}  𝔰𝔩(m|n) and 𝔤𝔩(m|n) 𝔰𝔳𝔢𝔠𝔱(m|n) and 𝔳𝔢𝔠𝔱(m|n)
Π(id) or Π(id^{∗}) 𝔰𝔩(n|m) and 𝔤𝔩(n|m) 𝔰𝔳𝔢𝔠𝔱(n|m) and 𝔳𝔢𝔠𝔱(n|m)
id 𝔬𝔰𝔭^{sk}(m|2n) 𝔥(2n|m)
𝔠𝔬𝔰𝔭^{sk}(m|2n)  𝔡(𝔥(2n|m))
id 𝔰𝔭𝔢^{sk}(n) and 𝔭𝔢^{sk}(n) 𝔰𝔩𝔢(n) and 𝔩𝔢(n)
id  𝔠(𝔰𝔭𝔢^{sk}(n)) and 𝔠(𝔭𝔢^{sk}(n)) 𝔡(𝔰𝔩𝔢(n)) and 𝔡(𝔩𝔢(n))
spin_{λ}, λ ≠ 0 𝔞𝔰 𝔳𝔩𝔢(4|4)

Now we only have to test the simplicity of the prolongs (𝔤_{−1}, 𝔤_{0})_{∗}. 5.1. Proposition . If T (𝔤) contains an odd operator of rank 1, then either T (𝔤) contains even operators of rank (1, 0) and (0, 1) or T (𝔤) contains an even operator of rank (1, 1).

Proof. Let ξ ∈ 𝔤¯1 be such that rkT (ξ) = 1, i.e., T (ξ)v = α(v) · v_{ξ}, where α ∈ V^{∗} and v_{ξ} is a fixed nonzero vector. (In other words, T (ξ) = α ⊗ v_{ξ}.) Let η ∈ 𝔤¯1. Then

T ([η, ξ]) = T (η)T (ξ) + T (ξ)T (η).

Consider the two operators:

A_{η} = α ⊗ T (η)v_{ξ} and B_{η} = −T (η)^{∗} · α ⊗ v_{ξ}

i.e., explicitly

\begin{matrix}A_{η}(v) = T (η)T (ξ)v = α(v) · T (η)T (ξ), \\ B_{η}(v) = T (ξ)T (η)v = α(T (η)v_{ξ}. \end{matrix}

Clearly, for each fixed η both A_{η} and B_{η} are of rank not greater than 1; besides, they act in the spaces of opposite parity.

5.2. Lemma . Let W = ∩_{η∈𝔤¯1} ker T (η) and U = T (𝔤¯1)V. Then the subspaces U and W are

𝔤-invariant.

Since T is irreducible, U = V and W = 0.

Proof. 𝔤¯1-invariance is obvious. Let g ∈ 𝔤¯0, ξ ∈ 𝔤¯1, v ∈ V and u = ξv ∈ U. Let us show that gu ∈ U. Indeed,

gu = gξv = ξgv + [g, ξ]v ∈ T (𝔤¯1)V = V

because gv ∈ V and [g, ξ] ∈ 𝔤¯1.

Let w ∈ W, g ∈ 𝔤¯0. Let us show that gw ∈ W. Indeed,

η(gw) = g(ηw) + [η, g]w = 0

because ηw = 0 and [η, g] ∈ 𝔤¯1. Consequently, gw ∈ W. □

Corollary . There exist η_{1}, η_{2} ∈ 𝔤¯1 such that rkA_{η_{1}} = 1 and rkB_{η_{2}} = 1.

To complete the proof of Proposition, it suffices to observe that if rkB_{η_{1}} = 1 (or rkA_{η_{2}} = 1), then

rkT ([η_{1}, ξ]) = rk(A_{η_{1}} + B_{η_{1}}) = (1, 1), resp. rkT ([η_{2}, ξ]) = (1, 1),

whereas if rkB_{η_{1}} = rkA_{η_{2}} = 0, then the operators T ([η_{1}, ξ]) and T ([η_{2}, ξ]) are of rank 1 and act in the spaces of opposite parity. □

Conclusion. If dim 𝔤 = ∞, then there exists an element g_{0} ∈ 𝔤¯0 such that rk(T (g_{0}))|_{V¯0} = 1 and rk(T (g_{0})|_{V¯1} ≤ 1. 5.3. Proposition . Let 𝔤_{j} for j = 1, …, k be simple Lie algebras (not superalgebras), 𝔯 a solvable Lie algebra and 𝔤 their direct sum. Let T be a (perhaps, reducible) representation of 𝔤 in space V with a rank 1 operator. Then either

1) there exists a one-dimensional 𝔤-invariant subspace of V, or

2) up to renumbering, 𝔤_{1} = 𝔰𝔩(n) or 𝔰𝔭(2n) and V = W ⊕ U, where U = ker T (𝔤_{1}) and W is isomorphic, as 𝔤-module, to either id ⊗ χ_{0} ⊗ λ or id^{∗} ⊗ χ_{0} ⊗ λ, where id is the identity representation of 𝔤_{1} and χ_{0} is the zero (one-dimensional) representation of 𝔤_{2} ⊕⋯⊕𝔤_{k}, and λ is a character of 𝔯.

Proof. Since every finite dimensional representation of the semisimple Lie algebra 𝔤_{ss} = 𝔤_{1} ⊕ ⋯ ⊕ 𝔤_{k} is completely reducible and the sum of 𝔤_{ss} with 𝔯 is the direct one, then V can be represented in the form V = ⊕_{i}V_{i}, where T |_{V_{i}} = τ_{1}^{i} ⊗ ⋯ ⊗ τ_{k}^{i} ⊗ τ_{𝔯}^{i} and τ_{j}^{i} is an irreducible representation of 𝔤_{j} and τ_{𝔯}^{i} is an irreducible representation of 𝔯.

Since T contains a rank 1 operator, so does the restriction T |_{V_{i_{0}}} for some i_{0}. Therefore, not more than one of the spaces of the representations τ_{1}^{i_{0}}, … τ_{k}^{i_{0}}, τ_{𝔯}^{i_{0}} can be of dimension > 1.

If dim τ_{1}^{i_{0}} = ⋯ = dim τ_{k}^{i_{0}} = dim τ_{𝔯}^{i_{0}} = 1, then dim V_{i_{0}} = 1 and possibility 1) of Proposition 5.3 is realized.

If dim τ_{1}^{i_{0}} = ⋯ = dim τ_{k}^{i_{0}} = 1, but dim τ_{𝔯}^{i_{0}} > 1, then, due to the fact that 𝔯 is solvable, we deduce that V_{i_{0}} contains a one-dimensional 𝔯-submodule V_{0}. Since 𝔤_{ss}V_{i_{0}} = 0, it follows that V_{0} is a 𝔤-submodule, i.e., once again the same possibility 1) of Prop. 5.3 is realized.

Finally, let dim τ_{j_{0}}^{i_{0}} > 1 for some j_{0}. Let us show that in this case possibility 2) or Prop. 5.2 is realized. Indeed, up to renumbering of the subalgebras 𝔤_{j}, we may assume that j_{0} = 1. Then dim τ_{2}^{i_{0}} = ⋯ = dim τ_{k}^{i_{0}} = dim τ_{𝔯}^{i_{0}} = 1 and the image T (𝔤)|_{V_{i_{0}}} is isomorphic to either 𝔤_{1} or its trivial central extension 𝔠(𝔤_{1}). Moreover, T (𝔤_{2} ⊕ ⋯ ⊕ 𝔤_{k})|_{V_{i_{0}}} = 0 and T (𝔯)|_{V_{i_{0}}} = λ is a character.

Since there are only three series of simple infinite dimensional Lie algebras with Weisfeiler filtration of depth one, it follows that a faithful irreducible representation τ of the Lie algebra 𝔥 contains a rank 1 operator if and only if for a character λ of 𝔥 we have

𝔥 = 𝔰𝔩(n) and τ = id or id^{∗};
𝔥 = 𝔤𝔩(n) and τ = id ⊗ λ or id^{∗} ⊗ λ;
𝔥 = 𝔰𝔭(2n) and τ = id ≅ id^{∗};
𝔥 = 𝔠𝔰𝔭(2n) and τ = id ⊗ λ.

In our case the role of 𝔥 is played by the image T (𝔤_{1})|_{V_{i_{0}}} and we see that 𝔤_{1} is isomorphic to either 𝔰𝔩(n) or 𝔰𝔭(2n); while T (𝔤_{1})|_{V_{i_{0}}} is either id or id^{∗}.

Thus, for the role of W we take V_{i_{0}}. To complete the proof, it suffices to observe that the rank of any operator T (g) is caclulated from the formula

rkT (𝔤) = ∑_{i=1}^{k} rkT (𝔤)|_{V_{i}}

and, therefore, T (𝔤)|_{V_{i}} = 0 for i ≠ i_{0}. □

5.4. The cases of Prop. 5.3: the two regular and two exceptional subcases. Now, return to the situation of interest to us: 𝔤 is an almost simple Lie superalgebra or a central extension thereof, T its faithful representation in a finite dimensional superspace V such that among the operators T (g), g ∈ 𝔤, there is one of rank 1 with an even covector.

Consider the restriction of T onto the even part of 𝔤 — the Lie algebra 𝔤¯0. Thanks to Proposition 5.1 the representation of 𝔤¯0 in the subspace V¯0 must possess a rank 1 operator whereas the representation of 𝔤¯0 in the whole V has either a rank (1, 0) operator or a rank (1, 1) operator. Moreover, from Table 8.1 we see that 𝔤¯0 is always of the form 𝔤_{1}⊕⋯⊕𝔤_{k} ⊕𝔯, where the 𝔤_{i} are simple and 𝔯 is a solvable Lie algebra (𝔯 can be zero and almost always it is commutative). Hence, Prop. 5.3 is applicable to the 𝔤¯0-action in the spaces V¯0 and V¯1.

Let, as in Prop. 5.3, 𝔤_{ss} = 𝔤_{1} ⊕ ⋯ ⊕ 𝔤_{k}. Consider, separately, the two cases corresponding to the possibilities of Prop. 5.3:

1) among the operators T (𝔤_{ss}), there is no operator of rank (1, 0) or (1, 1);

2) among the operators T (𝔤_{ss}), there is such an operator.

Case 1). As is clear from the proof of Prop. 5.3, we are in the setting of possibility 1) and the operator of rank (1, 0) or (1, 1) is in this case of the form T (h), where h ∈ 𝔯. Moreover, due to the irreducibility of T the operator h can not belong to the center of the whole 𝔤.

This is possible in the following regular subcases:

𝔤 = 𝔰𝔩(m|n) for m ≠ n, 𝔤𝔩(m|n), 𝔬𝔰𝔭(2|2n), 𝔠𝔬𝔰𝔭(2|2n)

and in the exceptional subcases to be considered separately:

a) 𝔭𝔰𝔩(2|2) ⊂ 𝔤 ⊂ 𝔡𝔢𝔯 𝔭𝔰𝔩(2|2) and 1 ≤ dim 𝔤/𝔭𝔰𝔩(2|2) ≤ 2;

b) the same as in a) but with 𝔠𝔭𝔰𝔩(2|2) instead of 𝔭𝔰𝔩(2|2).

5.4.1. The regular subcases. In these subcases 𝔤 possesses a ℤ-grading of the form 𝔤 = 𝔤^{−1} ⊕ 𝔤_{0} ⊕ 𝔤^{1}, where 𝔤_{0} = 𝔤¯0 and, therefore, 𝔤¯1 = 𝔤^{−1} ⊕ 𝔤^{1}. Then, under a proper norming, the operator h acts as the grading operator:

[h, g] = ±g for any g ∈ 𝔤_{±1}.

Let W¯0 = ℂw_{0} be a one-dimensional 𝔤¯0-invariant subspace of V¯0 and h · w_{0} = λw_{0}. Then

h(gw_{0}) = g(hw_{0}) + [h, g]w_{0} = λ(gw_{0}) ± (gw_{0}) = (λ ± 1)(g · w_{0}),

i.e., on each of the subspaces W_{−1} = 𝔤^{−1}W and W_{1} = 𝔤^{1}W the operator h acts as multiplication by a scalar.

Consider the action of 𝔤_{ss} on W_{±1}. If g_{0} ∈ 𝔤_{ss}, ξ ∈ 𝔤^{±1}, then

g_{0}(ξw_{0}) = ξ(g_{0}w_{0}) + [g_{0}, ξ] · w_{0} = [g_{0}, ξ]w_{0}.

Hence, each of the subspaces W_{±1} is 𝔤¯0-invariant and the representation of 𝔤_{ss} in W_{±1} is isomorphic to the quotient representation of 𝔤_{ss} in 𝔤^{±1}.

Since in all the cases considered the representation of 𝔤_{ss} in 𝔤^{±1} is irreducible, we see that either W_{±1} = 0 or dim W_{±1} = dim 𝔤^{±1} > 1.

But then, to ensure that rkT (h) = 1 or (1, 1), we should have λ = ±1; moreover, if λ+k ≠ 0, then W_{k} = 0 for k = ±1. Hence, 𝔤¯1W¯0 = W_{−λ} and h|_{W_{−λ}} = 0. Then h|_{𝔤^{±1}·W_{−λ}} = ±id and 𝔤^{±1}W_{−λ} ⊂ V¯0.

But since rkT (h)|_{V¯0} = 1, we get:

𝔤_{λ}W_{−λ} = W¯0, 𝔤_{−λ}W_{−λ} = 0 and 𝔤2 ¯1W¯0 = W¯0.

Therefore, W_{λ} ⊕ W¯0 is invariant with respect to the whole 𝔤; hence, V = W_{λ} ⊕ W¯0 and dim V = (1, n), where n = dim 𝔤^{1} = dim 𝔤^{−1}.

Thus,

𝔤 ⊂ 𝔤𝔩(1|n) and dim 𝔤¯1 = 2n = dim 𝔤𝔩(1|n)¯1.

Therefore, either 𝔤 = 𝔰𝔩(1|n) or 𝔤𝔩(1|n) and either T = id or T = id^{∗} (see the agreement on the action of the center in Prop. 5.0).

5.4.2. The exceptional subcases. These cases are related with the simple Lie superalgebra 𝔭𝔰𝔩(2|2) whose Lie superalgebra of outer derivations is isomorphic to 𝔰𝔩(2).

Subcase a): 𝔭𝔰𝔩(2|2) ⊂ 𝔤 ⊂ 𝔡𝔢𝔯 𝔭𝔰𝔩(2|2) and 1 ≤ dim 𝔤/𝔭𝔰𝔩(2|2) ≤ 2;

Subcase b): 𝔠𝔭𝔰𝔩(2|2) ⊂ 𝔤 ⊂ 𝔡𝔢𝔯 𝔠𝔭𝔰𝔩(2|2) and 1 ≤ dim 𝔤/𝔠𝔭𝔰𝔩(2|2) ≤ 2.

Denote by z the central element of 𝔤 in subcase b).

Let 𝔯_{0} ⊂ 𝔰𝔩(2) be a subalgebra of the Lie algebra of outer derivations of 𝔭𝔰𝔩(2|2) such that 1 ≤ dim 𝔯_{0} ≤ 2. Let

r = \begin{cases}𝔯_{0} & in subcase a) \\ 𝔯_{0} ⊕ ℂz & in subcase b). \end{cases}

Denote by R_{0} = id_{𝔰𝔩(2)}|_{𝔯_{0}} the restriction of the standard representation of 𝔰𝔩(2) onto 𝔯_{0}. Then 𝔤¯0 = 𝔰𝔩(2) ⊕ 𝔰𝔩(2) ⊕ 𝔯_{0} and 𝔤_{ss} = 𝔰𝔩(2) ⊕ 𝔰𝔩(2) ≅ 𝔬(4). Let χ_{0} be the trivial character on ℂz; then, as 𝔤¯0-module, 𝔤¯1 is isomorphic to id ⊗ id ⊗ R, where

R = \begin{cases}R_{0} & in subcase a) \\ R_{0} ⊗ χ_{0} & in subcase b). \end{cases}

Set X = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) ∈ 𝔰𝔩(2) and Y = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) ∈ 𝔰𝔩(2). Clearly, up to the choice of a basis

in 𝔤¯1, there are three essentially distinct possibilities for 𝔯_{0}:

1) 𝔯_{0} = Span(X, Y);

2) 𝔯_{0} = ℂX;

3) 𝔯_{0} = ℂY.

Recall that we are looking for an irreducible representation T of 𝔤 in a superspace V such that T (𝔤¯0) contains an operator of rank (1, 0) or (1, 1) whereas T (𝔤_{ss})|_{V¯0} has no such operators.

Then the operator of rank (1, 0) or (1, 1) is T (r), where r ∈ 𝔯_{0}. Without loss of generality we may assume that either r = X or Y.

If r = X, then the arguments of the regular case are applicable and we deduce that 𝔤 has no such representations.

Suppose that r = Y. Let, as in the general case, W¯0 = ℂw_{0} ⊂ V¯0 be a one-dimensional 𝔤¯0-invariant subspace. Then T (𝔤_{ss})w_{0} = 0, the representation T determines a character χ_{λ,α} on 𝔯 from the formulas

\begin{matrix}T (z)w_{0} = χ_{λ,α}(z)w_{0} = λw_{0} for λ ≠ 0 \\ T (Y)w_{0} = χ_{λ,α}(Y)w_{0} = αw_{0}. \end{matrix}

(Clearly, if dim 𝔯_{0} = 2, then α = 0.)

Consider the 𝔤¯0-module 𝔤¯1W¯0 ⊂ V¯1. It is isomorphic to a quotient of the 𝔤¯0-module 𝔤¯1 ⊗ W_{0} = id ⊗ id ⊗ (R ⊗ χ_{λ,α}). Therefore, the three cases are possible:

1) 𝔤¯1W¯0 = 0. Then W¯0 is a 𝔤-invariant subspace. Hence, V = W¯0 and T is not faithful.

2) 𝔤¯1W¯0 ≅ 𝔤¯1⊗W_{0}. Then rk T (Y)|_{𝔤¯1⊗W_{0}} = \begin{cases}4 & if α = 0 \\ 8 & if α ≠ 0. \end{cases} Both the possibilities contradict

our assumption on the rank.

3) 𝔤¯1W¯0 ≅ id ⊗ id ⊗ χ_{λ,α}. If α ≠ 0, then rk T (Y)|_{𝔤¯1W¯0} = 4. Hence, α = 0, i.e., T (Y)|_{W_{0}} = T (Y)|_{𝔤¯1W¯0} = 0. Set

𝔤_{Y} = Im adY ⊂ 𝔤¯1.  (5.4.2.1)

Then

T (Y)W¯0 = T (Y)𝔤¯1W¯0 − 𝔤¯1T (Y)W¯0 = 0.

Thus, w_{0} is the highest weight vector in T.

In subcase a) this means that T is one-dimensional and, therefore, not faithful.

In subcase b), let c : 𝔤¯1 × 𝔤¯1 be the cocycle that determines the central extension 𝔭𝔰𝔩(2|2) ⟶ 𝔰𝔩(2|2). As is clear from Table 8.2,

c|_{𝔤¯1×𝔤_{Y}} = 0.  (5.4.2.2)

Let 𝔤_{−Y} ⊂ 𝔤¯1 be a 𝔤_{ss}-invariant subspace complementary to 𝔤_{Y}. By (5.4.2.1), 𝔤¯1W¯0 = 𝔤_{−Y} W¯0. Take some nonzero ξ ∈ 𝔤_{Y} and η ∈ 𝔤_{−Y}. Then

ξ · ηw_{0} = −ηξw_{0} = [ξ, η]w_{0} = c(ξ, η) · λw_{0}.

By (5.4.2.2), both Y and all the elements ξ ∈ 𝔤_{Y} annihilate the space 𝔤¯1W¯0. Hence, 𝔤¯1W¯0 generates a 𝔤-invariant subspace in V which does not contain W¯0. This contradicts irreducibility of T.

Conclusion: in none of the exceptional subcases 𝔤 has irreducible representations we are looking for.

5.5. Case 2). Due to Prop. 5.3, 𝔤_{ss} = 𝔤_{1} ⊕ 𝔤_{comp}, where 𝔤_{1} = 𝔰𝔩(n) or 𝔰𝔭(2n) and the complementary Lie algebra 𝔤_{comp} is either semisimple or zero. So V = W ⊕ U for (decomposable) 𝔤_{1}- and 𝔤_{comp}-modules W and U, where T (𝔤_{1})|_{U} = 0, the 𝔤_{ss}-module W¯0 is isomorphic to either id ⊗ χ_{0} or id^{∗} ⊗ χ_{0} and where the 𝔤_{ss}-module W¯1 is either zero or isomorphic to either id ⊗ χ_{0} or id^{∗} ⊗ χ_{0}.

Consider the weight diagrams of V¯0 and V¯1 with respect to 𝔤_{ss}. Denote by ε_{1}, …, ε_{n} the standard vectors that span the space dual to the Cartan subalgebra of 𝔤_{1}, see [OV], and by δ_{1}, …, δ_{N} the similar vectors for 𝔤_{comp}. (Clearly, if 𝔤_{ss} is simple and, therefore, 𝔤_{comp} = 0 these complementary vectors δ_{i} are absent.) Then the collection of weights (wrt 𝔤_{ss}) in each of W¯0 and W¯1 is as follows (with multiplicity one):

for 𝔤_{1} = 𝔰𝔩(n): either ε_{1}, …, ε_{n} or −ε_{1}, …, −ε_{n}.

for 𝔤_{1} = 𝔰𝔭(2n): ε_{1}, …, ε_{n} and −ε_{1}, …, −ε_{n}.

In the spaces U¯0 and U¯1 all the weights are of the form φ(δ_{1}, …, δ_{N}), i.e., depend only on the δ’s. The weight diagram of such a form will be referred to as a desirable one. Let α be an odd root of 𝔤 and ξ_{α} the corresponding root vector; let λ be a weight of V and v_{λ} the corresponding weight vector. Then ξ_{α} · v_{λ} is a weight vector of weight λ + α.

5.5.1. The scheme of our further considerations. 1^{∘}) We start with a 𝔤¯0-invariant subspace W¯0 whose weights we know.

2^{∘}) We apply 𝔤¯1 to this subspace and get a subspace of V¯1. By comparing the desirable weight diagram of V¯1 with weights λ′ of the form λ + α we get an information about possible weight diagrams of 𝔤¯1W¯0.

3^{∘}) By applying heading 2^{∘}) to a 𝔤¯0-invariant subspace of 𝔤¯1W¯0, we deduce, as a result: a) 𝔤2 ¯1W¯0 = W¯0 and, therefore, V = W¯0 ⊕ 𝔤¯1W¯0;

b) the description of the weight diagram of V.

4^{∘}) Since the dimension of V is very low and its homogeneous components are irreducible with respect to 𝔤¯0, the direct check and dimension considerations quickly bring about the result. If they fail, we compare the superspace V obtained, the two of whose highest weights with respect to 𝔤¯0 are known, with the irreducible 𝔤-modules with the same highest weights. Since all such modules are quotients of the induced modules, it suffices to investigate the so-called singular vectors (see Appendix) of two induced modules only. 5.6. Case-by-case study.

5.6.1. 𝔤_{1} = 𝔰𝔭(2n). Then either 𝔤 = 𝔬𝔰𝔭(m|2n) or 𝔬𝔰𝔭(m|2n) ⊕ ℂz. The weights λ of W¯0 are ±ε_{1}, …, ±ε_{n} .

\underline{m > 2}. The odd roots are of the form α = ±ε_{i} ± δ_{j} for 𝔬𝔰𝔭(2m|2n). For 𝔬𝔰𝔭(2m + 1|2n), to the roots of the above form we must add the roots α = ±ε_{i}.

Thus, the sums λ + α which are desirable weights, are only those of the form:

λ′ = \begin{cases}±δ_{j} & for the series 𝔬𝔰𝔭(2m|2n), \\ ±δ_{j} and 0 & for the series 𝔬𝔰𝔭(2m + 1|2n). \end{cases}

By considering the sums λ′′ of the form λ′ +α, we see that among them, the only desirable weights are those of the form ±ε_{i}. Since their multiplicity in V¯0 is equal to one, 𝔤2 ¯1W¯0 = W¯0; hence, V = W¯0 ⊕ 𝔤¯1W¯0.

Consider the 𝔤_{ss}-module 𝔤¯1W¯0 more closely. Clearly, it is isomorphic to the quotient of 𝔤¯1 ⊗ W¯0. As 𝔤_{ss}-module, 𝔤¯1 is id_{𝔰𝔭(2n)} ⊗ id_{𝔬(m)} and W¯0 is id_{𝔰𝔭(2n)} ⊗ χ_{0}. Thus, 𝔤¯1 ⊗ W¯0 is the quotient of (id_{𝔰𝔭(2n)} ⊗ id_{𝔰𝔭(2n)}) ⨂ id_{𝔬(m)} with the zero 𝔰𝔭(2n)-action.

Now, observe that the tensor square of the standard 𝔰𝔭(2n)-module is reducible:

id_{𝔰𝔭(2n)} ⊗ id_{𝔰𝔭(2n)} = S^{2}(id_{𝔰𝔭(2n)}) ⊕ Λ^{2}(id_{𝔰𝔭(2n)}),

moreover, S^{2}(id_{𝔰𝔭(2n)}) ≅ ad_{𝔰𝔭(2n)} (hence, irreducible) and Λ^{2}(id_{𝔰𝔭(2n)}) is the direct sum of the trivial one-dimensional module and the complementary to it irreducible submodule. Therefere, the subspace of invariants in the tensor square id_{𝔰𝔭(2n)}⊗id_{𝔰𝔭(2n)} is one-dimensional. Therefore, the 𝔤_{ss}-module 𝔤¯1 ⊗ W¯0 can be either trivial or isomorphic to χ_{0} ⊗ id_{𝔬(m)}.

Thus, dim V = (2n, 1) or (2n, m). The dimension considerations (thanks to the faithfulness of the 𝔬(m)-action) show that only the second possibility can be realized; the representation T is, actually, Π(id).

\underline{m = 2}. Then 𝔤_{ss} = 𝔰𝔭(2n). The weights λ of W¯0 are ±ε_{1}, …, ±ε_{n} and the odd roots are also ±ε_{1}, …, ±ε_{n}. Each root is of multiplicity 2 and the center of 𝔤¯0 separates the odd roots into the negative and positive ones: 𝔤¯1 = 𝔤^{−1} ⊕ 𝔤^{1}; the 𝔤_{ss}-acton in 𝔤^{−1} is id, same in 𝔤^{1} is id^{∗}.

The desirable weight λ′ of the form α + λ is only λ′ = 0. The subspace 𝔤^{−1}W¯0 is the quotient of 𝔤^{−1} ⊗ W¯0 ≅ id^{∗} ⊗ id and 𝔤^{1}W¯0 is the quotient of 𝔤^{1} ⊗ W¯0 ≅ id ⊗ id. As we have already noted, each of the modules id^{∗} ⊗ id and id ⊗ id only contains a one-dimensional subspace of invariants. Thus,

dim 𝔤¯1W¯0 = 0, 1 or 2.

The desirable weights of the form λ′ + α are equal to ±ε_{i}. Hence, 𝔤2 ¯1W¯0 = W¯0 and V = W¯0 ⊕ 𝔤¯1W¯0.

As above, the dimension considerations imply that T (𝔤¯1) = 𝔤𝔩(2n|1)_{−}. Hence, dim V = (2n|2) and T = Π(id).

\underline{m = 1}. Then 𝔤_{ss} = 𝔰𝔭(2n) and 𝔤¯1 ≅ id, as 𝔤_{ss}-module. The same arguments as above imply that dim 𝔤¯1W¯0 = 1 and V = W¯0 ⊕ 𝔤¯1W¯0. Thus, T = Π(id).

5.6.2. 𝔤_{1} = 𝔰𝔩(n). This case splits into subcases A) – F).

\underline{A) 𝔤 = 𝔰𝔭𝔢(n), 𝔰𝔭𝔢(n) ⊕ ℂz, 𝔞𝔰, 𝔭𝔢(n), 𝔭𝔢(n) ⊕ ℂz}. We will assume that n > 3 (since 𝔰𝔭𝔢(3) ≅ 𝔰𝔳𝔢𝔠𝔱(0|3), this case and all it relatives is considered in the study of vectorial Lie superalgebras).

For these Lie superalgebras 𝔤_{ss} = 𝔰𝔩(n) and the weights of W¯0 are either λ = ε_{1}, …, ε_{n} or λ = −ε_{1}, …, −ε_{n}. The odd roots are α = ±(ε_{i} + ε_{j}) for i ≠ j and 2ε_{i}. Observe that 𝔤¯1 natuarally splits into the positive and negative part: 𝔤¯1 = 𝔤^{−1} ⊕ 𝔤^{1}.

The roots in 𝔤^{−1} are α = −ε_{i} − ε_{j} while those in 𝔤^{1} are α = ε_{i} + ε_{j}, 2ε_{i}.

The weights λ′ of the form λ + α that occure in the desirable diagram are as follows. If λ = ε_{1}, …, ε_{n}, then λ′ = −ε_{1}, …, −ε_{n} and 𝔤^{1}W¯0 = 0. If λ = −ε_{1}, …, −ε_{n}, then λ = ε_{1}, …, ε_{n} and 𝔤^{−1}W¯0 = 0. Recall that the multiplicity of the roots of the form ±ε_{i} in V¯1 can only be equal to 1.

Therefore, dim 𝔤¯1W¯0 = n.

Similarly, the desirable weights λ′′ of the form λ′ + α are as follows.

If λ′ = ε_{1}, …, ε_{n}, then λ′ + α = −ε_{1}, …, −ε_{n} and 𝔤2 ¯1W¯0 = W¯0.

If λ′ = −ε_{1}, …, −ε_{n}, then λ + α = ε_{1}, …, ε_{n} and 𝔤2 ¯1W¯0 = W¯0.

Thus, V = W¯0 ⊕ 𝔤¯1W¯0 and dim V = (n|n). Hence, if 𝔤 = 𝔰𝔭𝔢(n) or 𝔭𝔢(n), then either T = id or T = id^{∗}. Both these representations contain odd operators of rank 1, but for T = id^{∗} the covector is odd.

For Lie superalgebras 𝔤 = 𝔰𝔭𝔢(n)⊕ℂz and 𝔤 = 𝔭𝔢(n)⊕ℂz we see that T = id⊗χ, where χ is an arbitrary character of the center, and for 𝔤 = 𝔞𝔰 there is the one-parametric family T_{λ} of (4|4)-dimensional representations that differ from each other by the value on the center.

\underline{B) 𝔤 = 𝔭𝔰𝔮(n), 𝔰𝔮(n), 𝔭𝔮(n), 𝔮(n)}. In these cases 𝔤_{ss} = 𝔰𝔩(n).

The odd roots are α = ε_{i} − ε_{j}. Let, as always, λ denote a weight of W¯0, let λ′ = λ + α run over the weights of desirable form from V¯1 and λ′′ = λ′ + α the weights of desirable form from V¯0. Then

\begin{matrix}λ & λ′ & λ′′ \\ ε_{1}, …, ε_{n} & ε_{1}, …, ε_{n} & ε_{1}, …, ε_{n} \\ −ε_{1}, …, −ε_{n} & −ε_{1}, …, −ε_{n} & −ε_{1}, …, −ε_{n} \end{matrix}

Thus, 𝔤2 ¯1W¯0 = W¯0 and dim 𝔤¯1W¯0 = n. Hence, dim V = (n|n). But for 𝔤 = 𝔭𝔰𝔮(n) and 𝔭𝔮(n) there are no such representations at all ([K4]), while for 𝔤 = 𝔰𝔮(n) and 𝔮(n) this is the standard representation which has no rank 1 operators.

\underline{C) 𝔤 = 𝔰𝔩(n|m), 𝔤𝔩(n|m), 𝔭𝔰𝔩(n|n).} Observe that for n = m = 1, the Lie superalgebra 𝔤𝔩(1|1) is solvable; hence, all its irreducible representations are induced [S]; the case considered in Step 4. Due to an isomorphism 𝔰𝔩(2|1) ≅ 𝔳𝔢𝔠𝔱(0|2) we can and will consider this case in Step 5B.

\underline{m > 1}. Then 𝔤_{ss} = 𝔰𝔩(n) ⊕ 𝔰𝔩(m) and 𝔤¯1 = 𝔤^{−1} ⊕ 𝔤^{1}, where the roots of 𝔤^{−1} are α_{−1} = −ε_{i} + δ_{j} and the roots of 𝔤^{1} are α_{1} = ε_{i} − δ_{j}.

First, suppose \underline{n > 2}. Denote by λ′_{±1} the desirable weights of the form λ + α_{±1} and by λ′′_{±1} the desirable weights of the form λ′_{±1} + α_{±1}. We get:

\begin{matrix}λ & λ′_{−1} & λ′_{1} & λ′′_{−1} & λ′′_{1} \\ ε_{1}, …, ε_{n} & δ_{1}, …, δ_{n} & ∅ & ∅ & ε_{1}, …, ε_{n} \\ −ε_{1}, …, −ε_{n} & ∅ & δ_{1}, …, δ_{n} & −ε_{1}, …, −ε_{n} & ∅ \end{matrix}  (5.6.2)

Thus, we see that 𝔤2 ¯1W¯0 = W¯0. Let us consider the 𝔤_{ss}-module 𝔤¯1W¯0 in more detail. First, let W¯0, as 𝔤_{ss}-module, be id ⊗ χ_{0} (the weights are ε_{1}, …, ε_{n}). Then, as is clear from (5.6.2), 𝔤^{1}W¯0 = 0 and 𝔤¯1W¯0 = 𝔤^{−1}W¯0; hence, 𝔤¯1W¯0 is isomorphic to a quotient of the tensor product

𝔤^{−1} ⊗ W¯0 ≅ (id ⊗ id^{∗}) ⨂ (id ⊗ χ_{0}) .

But the 𝔰𝔩(n)-module id⊗id^{∗} is the sum of the adjoint module and the trivial one-dimensional one. Since the 𝔰𝔩(n)-action in 𝔤^{−1} ⊗ W¯0 is trivial, we deduce that dim 𝔤^{−1} ⊗ W¯0 = (0|m) and dim V = (n|m).

Therefore,

T = \{ \begin{matrix}id & for 𝔤 = 𝔰𝔩(n|m) and 𝔤𝔩(n|n); \\ id ⊗ χ, where χ is an arbitrary character, & for 𝔤 = 𝔤𝔩(n|m). \end{matrix}

For 𝔤 = 𝔭𝔰𝔩(n|n) there are no such representations: from dimension considerations and the fact that the irreducible 𝔤-module of the least dimension is the adjoint one [K4].

The case W¯0 = id^{∗} ⊗ χ_{0} is similarly considered. We obtain:

T = \{ \begin{matrix}id^{∗} & for 𝔤 = 𝔰𝔩(n|m) and 𝔤𝔩(n|n); \\ id^{∗} ⊗ χ, where χ is an arbitrary character, & for 𝔤 = 𝔤𝔩(n|m). \end{matrix}

\underline{n = 2}. In this case by the study of the weight diagram only we cannot conclude that either 𝔤^{−1}W¯0 = 0 or 𝔤^{1}W¯0 = 0. We will consider this case in sec. 5.6.3.

\underline{m = 1 and n > 2}. Then 𝔤_{ss} = 𝔰𝔩(n) and 𝔤¯1 = 𝔤^{−1} ⊕ 𝔤^{1}, where the roots of 𝔤^{−1} are α_{−1} = −ε_{i} and the roots of 𝔤^{1} are α_{1} = ε_{i}. The list of desirable weights in this case is obtained from (5.6.2) by replacing δ_{1}, …, δ_{n} with 0. The rest of our arguments are the same as in the above subcase. 5.6.3. Lemma . Let the Lie superalgebra 𝔥 = 𝔥¯0 ⊕ 𝔥¯1 be such that 𝔥¯0 = 𝔰𝔩(n) and 𝔥¯1 = id^{∗}, as 𝔰𝔩(n)-module. Let V be an 𝔥-module and W ⊂ V its 𝔰𝔩(n)-invariant subspace isomorphic to id as 𝔰𝔩(n)-module. Let, moreover, 𝔥¯1W be a one-dimensional trivial 𝔰𝔩(n)-module. Then 𝔥2 ¯1W = 0.

Proof. By hypothesis, 𝔥¯1 and W can be considered as dual linear spaces. Let w_{1}, …, w_{n} be a basis of W and h_{1}, …, h_{n} the dual basis of 𝔥¯1. As we have already noticed, the 𝔰𝔩(n)- module 𝔥¯1W is the quotient of the tensor product 𝔥¯1⊗W modulo the adjoint submodule. The adjoint submodule is singled out as the kernel of the natural bilinear form B(h ⊗ w) = h(w) on 𝔥¯1 ⊗ W. Then the generator u of 𝔥¯1W can be expressed as u = h_{1}w_{1} = ⋯ = h_{n}w_{n}. For i ≠ j we get

h_{i}u = h_{i}h_{j}w_{j} = −(−1)^{p(h_{i})p(h_{j})}h_{j}h_{i}w_{j} = 0. □

Corollary . 1) The statement of Lemma holds also if 𝔥¯1 = id and W = id^{∗}.

2) For n = 2 the modules id and id^{∗} are isomorphic, so Lemma 5.6.3 holds also if 𝔥¯1 ≅ W ≅ id and 𝔥¯1 ≅ W ≅ id^{∗}.

Thus, let 𝔤 = 𝔰𝔩(2|m), 𝔤𝔩(2|m) for m > 2 or any almost simple Lie superalgebra related to 𝔭𝔰𝔩(2|2) except 𝔡𝔢𝔯 𝔭𝔰𝔩(2|2) ≅ 𝔬𝔰𝔭(4|2; 0). Then 𝔤_{ss} = 𝔰𝔩(2) ⊕ 𝔰𝔩(m) and it remains to consider the case when 𝔤_{1} = 𝔰𝔩(2), i.e., the subspace W¯0 ⊂ V is isomorphic to id ⊗ χ_{0} as 𝔤_{ss}-module.

Observe that the submodules 𝔤^{−1}W¯0 and 𝔤^{1}W¯0 can not vanish simultaneously. Let, for example, 𝔤^{−1}W¯0 ≠ 0. Then 𝔤^{−1}W¯0 = χ_{0} ⊗id_{𝔰𝔩(m)} and, thanks to Lemma 5.6.3, (𝔤^{−1})^{2}W¯0 = 0. But then 𝔤^{1}(𝔤^{−1}W¯0) ≠ 0 due to irreducibility of the 𝔤-module V and our usual arguments appealing to the weight diagram show that 𝔤^{1}(𝔤^{−1}W¯0) = W¯0. Moreover, W¯0 ≅ id_{𝔰𝔩(2)} ⊗ χ_{0} arises as the quotient module of the tensor product

𝔤^{1} ⊗ (𝔤^{−1}W¯0) ≅ (id_{𝔰𝔩(2)} ⊗ id_{𝔰𝔩(m)}^{∗}) ⊗ (χ_{0} ⊗ id_{𝔰𝔩(m)}) =
id_{𝔰𝔩(2)} ⨂ (id_{𝔰𝔩(m)}^{∗} ⊗ id_{𝔰𝔩(m)}) .

By applying Lemma 5.6.3 once again we deduce that (𝔤^{1})^{2}(𝔤^{−1}W¯0) = 𝔤^{1}W¯0 = 0, i.e., dim V = (2|m) and T (𝔤) = 𝔰𝔩(2|m) or 𝔤𝔩(2|m).

Let 𝔤 = 𝔬𝔰𝔭(4|2n) for n > 1. Then

\begin{matrix}𝔤¯0 = 𝔬(4) ⊕ 𝔰𝔭(2n) ≅ 𝔰𝔩(2) ⊕ 𝔰𝔩(2) ⊕ 𝔰𝔭(2n), \\ 𝔤¯1 ≅ id_{𝔰𝔩(2)} ⊗ id_{𝔰𝔩(2)} ⨂ id_{𝔰𝔭(2n)}. \end{matrix}

Let 𝔤_{1} = 𝔰𝔩(2). By means of our routine arguments we deduce that

𝔤¯1W¯0 ≅ χ_{0}(𝔰𝔩(2)) ⊗ id_{𝔰𝔩(2)} ⨂ id_{𝔰𝔭(2n)} and 𝔤2 ¯1W¯0 = W¯0.

This implies that V = W¯0 ⊕ 𝔤¯1W¯0 and dim V = (2|4n).

Let us show that there are no such 𝔤-modules. Indeed, in the space dual to the Cartan subalgebra of 𝔬(4), select basis elements ε_{1}, ε_{2}; for the roots of 𝔤_{1} take ±2ε_{1} and let the roots of the complementary to 𝔤_{1} subalgebra in 𝔬(4) be equal to ±2ε_{2}; let δ_{1}, …, δ_{n} be the standard basis of the dual space to the Cartan subalgebra of 𝔰𝔭(2n). Then the weight basis of 𝔤¯1W¯0 consists of the vectors v_{λ} of weight λ = ±ε_{2} ± δ_{i}, whereas the weight basis of W¯0 consists of v_{±ε_{1}}. Let ξ_{±ε_{1}±ε_{2}±δ_{i}} be an odd root vector of 𝔤. Then

v_{±ε_{1}} = ξ_{±ε_{1}±ε_{2}±δ_{i}} · v_{∓ε_{2}∓δ_{i}} and ξ_{α} · v_{λ} = 0 if α + λ ≠ ±ε_{1}.

For i ≠ k we get

ξ_{ε_{1}±ε_{2}±δ_{i}}v_{ε_{1}} = 0;
ξ_{−ε_{1}±ε_{2}±δ_{i}}v_{ε_{1}} = ξ_{−ε_{1}±ε_{2}±δ_{i}}(ξ_{ε_{1}±ε_{2}±δ_{k}}v_{∓ε_{2}∓δ_{k}}) =
−ξ_{ε_{1}±ε_{2}±δ_{k}} · ξ_{−ε_{1}±ε_{2}±δ_{i}}v_{∓ε_{2}∓δ_{k}} = 0.

The penultimate equality holds because [ξ_{ε_{1}±ε_{2}±δ_{k}}, ξ_{−ε_{1}±ε_{2}±δ_{i}}] = 0 and the latter is true since ±2ε_{2} ± (δ_{k} + δ_{i}) is not a root of 𝔤. Thus, 𝔤¯1v_{ε_{1}} = 0. Similarly, 𝔤¯1v_{−ε_{1}} = 0, i.e., 𝔤¯1W¯0 = 0 and V = W_{0}. Contradiction.

\underline{D) 𝔤 = 𝔞𝔟_{3}}. Then 𝔤¯0 = 𝔤_{ss} = 𝔰𝔩(2)⊕𝔬(7). For this case we modify our notations somewhat. Instead of two vectors ε_{1} and ε_{2} we will use one vector, ε, normed so that the roots of 𝔰𝔩(2) be equal to ±2ε.

The weights of W¯0 are λ = ±ε; the odd roots are α = ±ε + \frac{1}{2}(±δ_{1} ± δ_{2} ± δ_{3}) and 𝔤¯1 ≅ id_{𝔰𝔩(2)} ⊗ spin_{7} as 𝔤¯0-module.

The desirable weights are

λ′ = λ + α = \frac{1}{2}(±δ_{1} ± δ_{2} ± δ_{3}) and λ′′ = λ′ + α = ±ε.

Therefore, 𝔤2 ¯1W¯0 = W¯0 and V = W¯0 ⊕ 𝔤¯1W¯0. Observe that 𝔤¯1W¯0 is the quotient of the 𝔤¯0-module

𝔤¯1 ⊗ W¯0 ≅ (id_{𝔰𝔩(2)} ⊗ id_{𝔰𝔩(2)}) ⨂ (spin_{7} ⊗ χ_{0}) .

The dimension of the trivial subspace of the 𝔰𝔩(2)-module id⊗id is equal to one and, since the 𝔰𝔩(2)-action in 𝔤¯1W¯0 should be trivial, we deduce that dim 𝔤¯1W¯0 = 1 · dim(spin_{7}) = 8, i.e., dim V = (2|8).

Our routine arguments demonstrate that 𝔤 = 𝔞𝔟_{3} has no such irreducible representations. \underline{E) 𝔤 = 𝔞𝔤_{2}}. Then 𝔤¯0 = 𝔤_{ss} = 𝔰𝔩(2) ⊕ 𝔤_{2} and 𝔤¯1 ≅ id_{𝔰𝔩(2)} ⊗ R(π_{1}), as 𝔤¯0-module, where

R(π_{1}) is the irreducible representation of 𝔤_{2} with the highest weight π_{1}, see [OV]. Recall that dim R(π_{1}) = 7. The arguments similar to those in the above case show that the desirable representation should be of dimension (2, 7) but 𝔞𝔤_{2} has no such irreducible representations, see [K5] and [KW].

\underline{F) 𝔤 = 𝔬𝔰𝔭(4|2; α)}. Then 𝔤¯0 = 𝔤_{ss} = 𝔰𝔩(2) ⊕ 𝔰𝔩(2) ⊕ 𝔰𝔩(2) and arguments similar to those used in the above two subcases yield that dim V = (2|4); there are no such irreducible 𝔤-modules for generic α ≠ 1.

Step 5B. Vectorial Lie superalgebras. The four cases

5.7. Let 𝔥_{0} be a finite dimensional Lie superalgebra. Let z be either its outer differentiation or a trivial center. Set 𝔤_{0} = 𝔥_{0} ⊂ + ℂ · z (if z is central, this sum is direct).

Let 𝔥_{−1} = 𝔤_{−1} be an irreducible finite dimensional 𝔤_{0}-module. Consider two Cartan prolongations:

𝔥 = ⊕_{i≥−l} 𝔥_{i} = (𝔥_{−1}, 𝔥_{0})_{∗} and 𝔤 = ⊕_{i≥−l} 𝔤_{i} = (𝔤_{−1}, 𝔤_{0})_{∗}.

5.7.0. Lemma . Either 𝔤_{1} = 𝔥_{1}, hence, 𝔤 = 𝔡𝔥 or 𝔤_{1} = 𝔥_{1} ⊕ 𝔤_{−1}^{∗} (the direct sum of vector spaces).

Proof. For any F ∈ 𝔤_{1}, v ∈ 𝔤_{−1} the decomposition

[F, v] = α_{F} (v)z + h,

where h ∈ 𝔥_{0}, determines a linear form α_{F} ∈ 𝔤_{−1}^{∗}. By the definition of Cartan prolongation, α_{F} = 0 if and only if F ∈ 𝔥_{1}. Hence, as a 𝔤_{0}-module, the space 𝔤_{1}/𝔥_{1} is a quotient of the 𝔤_{0}-module 𝔤_{−1}^{∗} which is irreducible together with 𝔤_{−1}. This proves Lemma. □

\underline{1) 𝔥_{0} = 𝔳𝔢𝔠𝔱(0|n), where n ≥ 3, and 𝔤_{0} = 𝔠𝔳𝔢𝔠𝔱(0|n) = 𝔥_{0} ⊕ ℂ · z}. (Recall that 𝔥_{0} does not have any outer differentiation or nontrivial central extentions.) The rank 1 operator with even covector exists only for the following 𝔥_{0}-modules:

1a) 𝔥_{−1} = Π(Λ(n)/ℂ · 1) ⟹ 𝔥 = (𝔥_{−1}, 𝔥_{0})_{∗} = 𝔩𝔢(n; n);

1b) 𝔥_{−1} = Vol_{0}(0|n) ⟹ 𝔥 = (𝔥_{−1}, 𝔥_{0})_{∗} = 𝔰𝔳𝔢𝔠𝔱^{∘}(1|n; n);

1c) 𝔥_{−1} = Π(Vol_{0}(0|n)) ⟹ 𝔥 = (𝔥_{−1}, 𝔥_{0})_{∗} = 𝔟_{1}^{∘}(n; n);

1d) 𝔥_{−1} = Π(Vol^{λ}(n)) ⟹ 𝔥 = (𝔥_{−1}, 𝔥_{0})_{∗} = 𝔟_{λ}(n; n) for λ ≠ 0, 1.

The element z acts in 𝔤_{−1} as a nonzero scalar. We normalize z so that adz|_{𝔤_{−1}} = −1 and use for such 𝔠𝔳𝔢𝔠𝔱(0|n)-modules the same notations as for 𝔳𝔢𝔠𝔱(0|n)-modules. 5.7.1. Proposition . Let 𝔤_{0} = 𝔠𝔳𝔢𝔠𝔱(0|n), where n ≥ 3, and 𝔤_{−1} a 𝔤_{0}-module of type 1a) – 1d). Let 𝔤 = (𝔤_{−1}, 𝔤_{0})_{∗} be Cartan prolongation. Then in the respective cases we have:

1a) if 𝔤_{−1} = Π(Λ(n)/ℂ · 1), then 𝔤 = 𝔳𝔩𝔢(4|3) for n = 3 and 𝔤 = 𝔡𝔩𝔢(n; n) for n > 3; 1b) if 𝔤_{−1} = Vol_{0}(0|n), then 𝔤 = 𝔨𝔞𝔰^{ξ}(; 3η) for n = 3 and 𝔤 = 𝔡𝔰𝔳𝔢𝔠𝔱^{∘}(1|n; n) for n > 3; 1c) if 𝔤_{−1} = Π(Vol_{0}(0|n)), then 𝔤 = 𝔡𝔟_{1}^{∘}(n; n); 1d) if 𝔤_{−1} = Π(Vol^{λ}(n)), then 𝔤 = 𝔡𝔟_{λ}(n; n) for λ ≠ 0, 1.

Proof. Observe that 𝔤 has a bigrading: each component 𝔤_{k} has a ℤ-grading by powers of ξ

(call it ξ-grading) such that 𝔤_{0} = ⊕_{i=−1}^{n−1} 𝔤_{0,i} and z ∈ 𝔤_{0,0}; we have 𝔤_{−1} = ⊕_{i=k}^{m} 𝔤_{−1,i}, where k = 0

in cases 1b) – 1d) and k = 1 in case 1a); m = n in cases 1a), 1d) and m = n − 1 in cases 1b), 1c).

By Lemma 5.7.0 if 𝔤 ≠ 𝔡𝔥, then the subspace 𝔤_{−1}^{∗} ⊂ 𝔤_{1} contains the components of degree from −m to −k with respect to ξ-grading. In particular, 𝔤_{1,−m} ≠ 0.

But in each case we have a subspace W = ⊕_{i=k}^{s} 𝔤_{−1,i} ⊂ 𝔤_{−1} such that 𝔤_{0} does not contain

operators commuting with the whole of W. Due to Lemma 4.2 this means that 𝔤_{1,i} = 0 for all i < −1 − s, i.e., the necessary condition for 𝔤 ≠ 𝔡𝔥 is −m ≥ −1 − s, i.e., m ≤ 1 + s. Specifically,

1a) s = 2, m = n ⟹ n ≤ 3;

1b), 1c) s = 1, m = n − 1 ⟹ n ≤ 3;

1c) s = 1, m = n ⟹ n ≤ 2.

In cases 1a), 1b) for n = 3 we get exceptional Lie superalgebras 𝔳𝔩𝔢 and 𝔨𝔞𝔰.

The case 1c) for n = 3 calls for subtler arguments. We have 𝔤_{−1} = ⊕_{i=0}^{2} 𝔤_{−1,i}, where

𝔤_{−1,0} = ℂΠ(1); 𝔤_{−1,1} = Π(Span(ξ_{j} : j = 1, 2, 3)); 𝔤_{−1,2} = Π(Span(ξ_{j}ξ_{l} : j, l = 1, 2, 3)).

Suppose 𝔤 ≠ 𝔡𝔥 and let F = Π(ξ_{1}ξ_{2})^{∗} ⊂ 𝔤_{1}, i.e.,

[F, Π(ξ_{1}ξ_{2})] = z + h, where h ∈ 𝔤_{0,0}, i.e., h = ∑_{i,j} a_{ij}ξ_{i}∂_{ξ_{j}}.

Then: F ∈ 𝔤_{1,−2} and, thus, [F, Π(1)] = 0. By Jacobi identity,

0 = [[F, Π(ξ_{1}ξ_{2}], Π(1)] = (z + h)Π(1) = −Π(1) + h(Π(1)).

Hence, h(Π(1)) = Π(1) and

divh = − ∑_{i=1}^{3} a_{ii} = 1.  (5.7.1.1)

Since p(Π(ξ_{1}ξ_{2})) = ¯1, we have

0 = [[F, Π(ξ_{1}ξ_{2})], Π(ξ_{1}ξ_{2})] = (z + h)Π(ξ_{1}ξ_{2}) = −Π(ξ_{1}ξ_{2}) + h(Π(ξ_{1}ξ_{2})).

Hence, h(Π(ξ_{1}ξ_{2})) = Π(ξ_{1}ξ_{2}) and a_{11} + a_{22} + divh = 1 or

a_{11} + a_{22} = 0.  (5.7.1.2)

Clearly, [F, Π(ξ_{3})] ∈ 𝔤_{0,−1}, i.e., [F, Π(ξ_{3})] = ∑ b_{j}∂_{ξ_{j}}. By Jacobi identity

0 = [[F, Π(ξ_{1}ξ_{2})], Π(ξ_{3})] − [Π(ξ_{1}ξ_{2}), [F, Π(ξ_{3})]] =
−Π(ξ_{3}) + h(Π(ξ_{3})) − (b_{1}Π(ξ_{2}) − b_{2}Π(ξ_{1})).

Hence, a_{33} + divh = 1, i.e., a_{33} = 0. The last equality contradicts (5.7.1.1) and (5.7.1.2). □

\underline{2) 𝔥_{0} = 𝔰𝔳𝔢𝔠𝔱(0|n) for n ≥ 3}. The rank 1 operator with even covector exists only for the 𝔥_{0}-module: 𝔥_{−1} = Π(T_{0}^{0}(0), hence, 𝔥 = (𝔥_{−1}, 𝔥_{0})_{∗} = 𝔰𝔩𝔢^{∘}(n; n).

We know that 𝔤_{0} has no nontrivial central extentions, but it has an outer derivation, D = ∑ ξ_{i}∂_{ξ_{i}} ∈ 𝔳𝔢𝔠𝔱(0|n). So we have to consider the following cases:

2a) 𝔤_{0} = 𝔥_{0} ⊕ ℂz, 𝔤_{−1} = 𝔥_{−1} as a linear space and 𝔤_{−1} = 𝔥_{−1} ⊗ (−1) as 𝔤_{0}-module; 2b) 𝔤_{0} = 𝔥_{0} ⊂ + ℂD, 𝔤_{−1} = 𝔥_{−1} as a linear space, where D acts as ∑ ξ_{i}∂_{ξ_{i}} + λ · 1; 2c) 𝔤_{0} = (𝔥_{0} ⊕ ℂz) ⊂ + ℂD, 𝔤_{−1} = 𝔥_{−1} as a linear space, where z acts as multiplication by

−1 and D as ∑ ξ_{i}∂_{ξ_{i}}.

2d) 𝔤_{0} = 𝔥_{0} ⊂ + ℂ(az + bD) for a, b ∈ ℂ and 𝔤_{−1} = 𝔥_{−1}. 5.7.2. Proposition . Let (𝔤_{−1}, 𝔤_{0}) be one of the pairs 2a) – 2c) and 𝔤 = (𝔤_{−1}, 𝔤_{0})_{∗} its Cartan’s prolong. Then in 2b) for n = 3, λ = 0 we have 𝔤 = 𝔩𝔢(3). In all other cases, 𝔤 = 𝔡𝔥.

Proof. In all cases, 𝔤_{−1} = ⊕_{i=1}^{n−1} 𝔤_{−1,i}. Due to Lemma 5.7.0, if 𝔤 ≠ 𝔡𝔥, then 𝔤_{1,−n+1} ≠ 0. On

the other hand, 𝔤_{0} = ⊕_{i=−1}^{n−2} 𝔤_{−1,i} and 𝔤_{0} does not contain elements that commute with the

subspace W = 𝔤_{−1,1} ⊕ 𝔤_{−1,2}. Hence, by Lemma 4.2, 𝔤_{1,s} = 0 for all s such that s + 2 < −1. Thus, −n + 1 ≥ −3, i.e., n ≤ 4.

(Observe that 𝔰𝔳𝔢𝔠𝔱(0|3) ≅ 𝔰𝔭𝔢(3) and our Proposition is a particular case of 5.6.2.)

Let us consider the case n = 4 more carefully. We have 𝔤_{−1} = ⊕_{i=1}^{3} 𝔤_{−1,i}, where

\begin{matrix}𝔤_{−1,1} = Span(Π(ξ_{i}))_{i=1}^{4}; 𝔤_{−1,2} = Span(Π(ξ_{i}ξ_{j}))_{i=1}^{4}; \\ 𝔤_{−1,3} = Span(Π(ξ_{1}ξ_{2}ξ_{3})\}. \end{matrix}

Suppose 𝔤_{1} ≠ 𝔥_{1}. In all cases 2a) – 2d) there should exist elements F ∈ 𝔤_{1,−3} and αz + βD ∈ 𝔤_{0} such that

g = [F, Π(ξ_{1}ξ_{2}ξ_{3})] = −α · 1 + βD + h,

where α, β ∈ ℂ, αβ ≠ 0 and h ∈ 𝔥_{0,0} = 𝔰𝔩(4).

By grading reasons, [F, 𝔤_{−1,1}] = 0. Due to the Jacobi identity we have

h|_{𝔤_{−1,1}} = α · 1 − βD|𝔤_{−1,1} = (α − β) · 1.

Since h ∈ 𝔰𝔩(4), it follows that α − β = 0 and h = 0. Without loss of generality we may assume that α = β = 1. Then adg|_{𝔤_{−1,2}} = −1 + D|_{𝔤_{−1,2}} = 1. Hence, 𝔤_{0,0} = 𝔤𝔩(4) ≅ 𝔠𝔬(6) and, as 𝔤_{0,0}-modules, 𝔤_{−1,2} ≅ id_{𝔬(6)} ⊗ 1. Note that [𝔤_{−1,2}, 𝔤_{1,−2}] ⊂ 𝔤_{0,0}. Set

V = {F ∈ 𝔤_{1,−2} | [F, 𝔤_{−1,2}] = 0}.

Then the quotient 𝔤_{1,−2}/V naturally embeds into the first component of Cartan prolong (Π(id), 𝔠𝔬(6))_{∗} = 𝔡𝔥(0|6). Due to Lemma 5.7.0 this contradicts the assumption 𝔤_{1} ≠ 𝔥_{1}. □

\underline{3) 𝔥_{0} = { \atop 𝔰𝔳𝔢𝔠𝔱}(0|n) for n ≥ 3}. The rank 1 operator with even covector exists only for the 𝔥_{0}-module 𝔥_{−1} = \frac{Π(Vol(0|n))}{ℂ(b+αξ_{1}...ξ_{n})vol(ξ)}. Hence, 𝔥 = (𝔥_{−1}, 𝔥_{0})_{∗} = 𝔰̃𝔟_{μ}(2^{n−1} − 1|2^{n−1}) with μ = \frac{α}{b}.

We know that 𝔥_{0} does not have any nontrivial central extentions or outer differentiations. Let 𝔤_{0} = 𝔥_{0} ⊕ ℂz, then 𝔤_{−1} = 𝔥_{−1}, as 𝔥_{0}-modules, and let z act on 𝔤_{−1} by multiplication by −1. 5.7.3. Proposition . (𝔤_{−1}, 𝔤_{0})_{∗} = 𝔡𝔥.

Proof. The components 𝔤_{k} have the filtration by powers of ξ. Consider the corresponding bigraded Lie superalgebra gr𝔤. If 𝔤_{1} ≠ 𝔥_{1}, then, due to Lemma 5.7.0 (gr𝔤)_{1} = (gr𝔥)_{1}⊕(gr𝔤)_{−1}^{∗}.

But (gr𝔤)_{−1} = ⊕_{i=0}^{2n−1} 𝔤_{−1,i}. Hence, (gr𝔤)_{1,−2n+1} ≠ 0.

On the other hand, (gr𝔤)_{0} = ⊕_{i=−1}^{2n−2} 𝔤_{0,i} and gr𝔤_{0} does not contain elements that commute

with the subspace W = ⊕_{i=0}^{1} 𝔤_{−1,i}. Hence, (gr𝔤)_{1,s} = 0 for all s such that s+1 < −1. Therefore,

−2n + 1 + 1 ≥ −1 or 2n ≤ 3. This contradicts to the fact that n ≥ 2. □

\underline{4) 𝔥_{0} = 𝔥^{o}(0|n) for n ≥ 5 and related subcases}. As follows from the description of the irreducible representations of these Lie superalgebras, see [Sha1], [Sha2], no induced representation has a rank 1 operator; among the exceptional representations only the representation in the superspace of functions could have qualified, but, it has no rank 1 operators either.

§6. Step 6. A particular case: case 3) of Corollary 3.1

Let 𝔤_{−1} = Λ(ξ_{1}, ξ_{2}) or Π(Λ(ξ_{1}, ξ_{2})); let 𝔤_{0} ⊂ 𝔥𝔢𝔦(0|4) ⊂ + 𝔬(4), where

\begin{aligned}𝔥𝔢𝔦(0|4) & = & Span(1, ξ_{1}, ξ_{2}, ∂_{1}, ∂_{2}); \\ 𝔰𝔩_{1}(2) & = & Span(ξ_{1}∂_{2}, ξ_{2}∂_{1}, ξ_{1}∂_{1} − ξ_{2}∂_{2}); \\ 𝔰𝔩_{2}(2) & = & Span(∂_{1}∂_{2}, ξ_{1}ξ_{2}, ξ_{1}∂_{1} + ξ_{2}∂_{2} − 1). \end{aligned}

We realize the elements of 𝔤_{−1}⊕𝔤_{0} with vector fields, i.e., we embed 𝔤_{−1}⊕𝔤_{0} into 𝔳𝔢𝔠𝔱(2|2). Redenote the respective vectors of the monomial basis of 𝔤_{−1} = Π(Λ(2)):

π(ξ_{1}) = ∂_{x}, π(ξ_{2}) = ∂_{y}; π(1) = ∂_{η}, π(ξ_{1}ξ_{2}) = ∂_{ζ}.

To the basis vectors from 𝔤_{0} we assign the following vector fields of degree 0 that act in the space of functions in x, y, η, ζ:

\begin{aligned}𝔥𝔢𝔦(0|4) : & −1 & ↦ & x∂_{x} + y∂_{y} + η∂_{η} + ζ∂_{ζ} \\ & ξ_{1} & ↦ & η∂_{x} − y∂_{ζ} \\ & ξ_{2} & ↦ & η∂_{y} + x∂_{ζ} \\ & ∂_{1} & ↦ & ζ∂_{y} − x∂_{η} \\ & ∂_{2} & ↦ & −ζ∂_{x} − y∂_{η} \end{aligned} ; \begin{aligned}𝔰𝔩_{1}(2) : & ξ_{1}∂_{2} & ↦ & −y∂_{x} & 𝔰𝔩_{2}(2) : & ∂_{2}∂_{1} & ↦ & −ζ∂_{η} \\ & ξ_{2}∂_{1} & ↦ & −x∂_{y} & & ξ_{1}ξ_{2} & ↦ & −η∂_{ζ} \\ & ξ_{1}∂_{1} − ξ_{2}∂_{2} & ↦ & y∂_{y} − x∂_{x} & & ξ_{1}∂_{1} + ξ_{2}∂_{2} − 1 & ↦ & −η∂_{η} − ζ∂_{ζ} \end{aligned}

We see that this realization of 𝔤_{0} is invariant under the changes of parity in 𝔤_{−1}. So, both 𝔤_{−1} = Λ(2) and 𝔤_{−1} = Π(Λ(2)) bring about the same result (isomorphic algebras (𝔤_{−1}, 𝔤_{0})_{∗}).

Consider the following subalgebra 𝔥_{0} of 𝔤_{0}:

𝔥_{0} = 𝔥𝔢𝔦(0|4) ⊂ + 𝔰𝔩_{1}(2) =

Span(1, ξ_{1}, ξ_{2}) ⊂ + Span(∂_{1}, ∂_{2}, ξ_{1}∂_{2}, ξ_{2}∂_{1}, ξ_{1}∂_{1} − ξ_{2}∂_{2}) ≅
(Λ(2) \ ℂ · ξ_{1}ξ_{2}) ⊂ + 𝔰𝔳𝔢𝔠𝔱(0|2).

We see that the Cartan’s prolong (𝔤_{−1}, 𝔥_{0})_{∗} is isomorphic to 𝔟_{∞}(2; 2).

The direct calculations show that

(𝔤_{−1}, 𝔤_{0})_{∗} ≅ (𝔤_{−1}, 𝔥_{0})_{∗} ⊕ 𝔰𝔩_{2}(2) and (𝔤_{−1}, 𝔥𝔢𝔦(0|4))_{∗} ≅ 𝔤_{−1} ⊕ 𝔥𝔢𝔦(0|4).

So a necessary condition for simplicity, i.e., [𝔤_{−1}, 𝔤_{1}] = 𝔤_{0}, implies that 𝔤_{0} = 𝔥_{0} and 𝔤 ≅ 𝔟_{a,a}(2; 2) ≅ 𝔟_{∞}(2; 2).

§7. Step 7. ℤ-gradings of vectorial Lie superalgebras

Let 𝔤 = ⊕_{i≥−d} 𝔤_{i} be a ℤ-graded vectorial superalgebra. By a Weisfeiler regrading (or just

regrading) of 𝔤 we understand the passage from 𝔤 to the Lie superalgebra 𝔥 = ⊕_{i≥−D} 𝔥_{i}

isomorphic to 𝔤 as abstract Lie superalgebra and such that 𝔥 is transitive; 𝔥^{0} = ⊕_{i≥0} 𝔥_{i} is a

maximal Lie superalgebra of finite codimension; D < ∞, i.e., 𝔥 is of finite depth.

In what follows in this section 𝔥 denotes a Lie superalgebra obtained from 𝔤 by a regrading. Clearly, not every two gradings on the same algebra determine a bigrading; fortunately, it suffices for our purposes to consider only compatible gradings for which (7.1.1) is well-defined by definition.

For the proof of the next three Lemmas see [Sh14]. 7.1.1. Lemma . Let 𝔥 = ⊕_{i≥−D} 𝔥_{i} be a W-regrading of the Lie superalgebra 𝔤 = ⊕_{j≥−d} 𝔤_{j} that determines a bigrading:

𝔥_{i} = ⊕_{j≥m(i)}^{M(i)} 𝔥_{i,j}, where 𝔥_{i,j} = 𝔥_{i} ∩ 𝔤_{j}.  (7.1.1)

Suppose dim(𝔥_{0} ∩ 𝔤_{−}) = 0|k. Then the number M = M(−1) − m(−1) + 1 of the summands 𝔥_{−1,j} in the decomposition (7.1.1) of 𝔥_{−1} does not exceed the number of homogeneous components with respect to the ℤ-grading of Λ(𝔥_{0} ∩ 𝔤_{−}).

In particular, if 𝔥_{0}∩𝔤_{−} is homogeneous with respect to the ℤ-grading in 𝔤, i.e., 𝔥_{0}∩𝔤_{−} ⊂ 𝔤_{j} for some j, then M ≤ k + 1. 7.1.2. Lemma . If 𝔥_{0} ∩ 𝔤_{−} = 0, then 𝔥_{0} ⊂ 𝔤_{0} and 𝔥_{−1} is homogeneous. 7.1.3. Lemma . If 𝔥_{0} ∩ 𝔤_{−} = 0 and there exists a nonzero x ∈ 𝔤_{−1} ∩ 𝔥_{−}, then the gradings of 𝔥 and 𝔤 coincide.

7.2. How to describe all the W-regradings.

1) Determine 𝔥_{0} ∩ 𝔤_{−1}.

2) Construct a “minimal” (better, though informally speaking, “most tightly compressed”) regrading with the given intersection, i.e., such that preserves in 𝔥_{−1} all the elements of 𝔤_{−1} except for those that have to go away in view of the condition on the intersection.

3) If the “minimal” regrading is a Weisfeiler one, then with the help of Lemmas 7.1.2 and 7.1.3 we prove that any other W-regrading with the given intersection coincides with the minimal one.

4) If the “minimal” regrading is not a Weisfeiler one, then with the help of Lemma 7.1.1 we prove that there are no W-regradings with the given intersection. 7.3. Theorem . Any Weisfeiler regrading 𝔥 of the simple vectorial Lie superalgebra is a regrading from Tables I.2.2.

Let 1 ≤ r ≤ m, let φ(s) = {i_{1}, …, i_{r}} be a set of distinct indices from 1 to m. Permuting the indeterminates, if necessary, we may and will assume that φ(r) = {1, …, r}. Set deg ξ_{1} = … = deg ξ_{r} = 0 and define the remaining degrees from Tables I.2.2. In Tables I.2.1 and I.2.2 the terms 𝔥_{−1}, 𝔥_{0} are indicated for the regraded algebras 𝔥 for the series and the exceptions, respectively.

Proof. 1) Suppose 𝔤 = 𝔳𝔢𝔠𝔱(n|m) = 𝔳𝔢𝔠𝔱(x, ξ) or 𝔤 = 𝔰𝔳𝔢𝔠𝔱(n|m) = 𝔰𝔳𝔢𝔠𝔱(x, ξ). Then 𝔤_{−} = 𝔤_{−1} = Span(∂_{x_{1}}, …, ∂_{x_{n}}, ∂_{ξ_{1}}, …, ∂_{ξ_{n}}). Since the depth of 𝔥 if finite, deg x_{i} > 0 for all i. Hence, ∂_{x_{i}} ∈ 𝔥_{−} ∩ 𝔤_{−1}.

If deg ξ_{j} ≠ 0 for all j, then 𝔥_{0} ∩ 𝔤_{−} = 0 and by Lemma 7.1.3 𝔥 = 𝔤.

2) Suppose now that deg ξ_{i} = 0 for i ≤ r and deg ξ_{i} ≠ 0 for i > r. Let us denote the regraded superalgebras by 𝔯 = 𝔳𝔢𝔠𝔱(n|m; r) or 𝔯 = 𝔰𝔳𝔢𝔠𝔱(n|m; r). Then

𝔯_{−1} = Span(∂_{x_{1}}, …, ∂_{x_{n}}; ∂_{ξ_{j}} | j > r) ⊗ Λ(ξ_{i} | i ≤ r).

Let us show that 𝔥 ≅ 𝔯.

Since ∂_{x_{i}} ∈ 𝔯_{−1}, then by Lemma 7.1.3 it suffices to show that 𝔯_{−1} ∩ 𝔥_{0} = 0. By definition of 𝔥 we have

(Span(∂_{x}, ∂_{ξ_{j}}) ⊗ 1) ∩ 𝔥_{0} = 0 for j > r.

But ∂_{ξ_{i}} ∈ 𝔥_{0} for i < r. Since 𝔥_{0} is a subalgebra, the following implication holds:

if any element from 𝔯_{−1} belongs to 𝔥_{0}, then the elements from Span(∂_{x}, ∂_{ξ_{j}})_{j>r} belong to 𝔥_{0}.

Therefore, 𝔯_{−1} ∩ 𝔥_{0} = 0.

3) For the other series take into account that the preserved form (ω_{0}, ω_{1}, α_{0}, α_{1}, α̃_{1}, etc.) should be homogeneous. For the series 𝔥 and 𝔨 it is convenient to split the indeterminates into dual pairs: the even ones, p_{1}, …, p_{m} and q_{1}, …, q_{m} and the odd ones: ξ_{1}, …, ξ_{k} and η_{1}, …, η_{k}; the remaining odd coordinates being the θ’s. Then the Poisson bracket takes the form

\{f, g\}_{P.B.} = ∑_{i≤m}(\frac{∂f}{∂p_{i}} \frac{∂g}{∂q_{i}} − \frac{∂f}{∂q_{i}} \frac{∂g}{∂p_{i}} )

−(−1)^{p(f)}[∑_{i≤k}(\frac{∂f}{∂ξ_{i}} \frac{∂g}{∂η_{i}} + \frac{∂f}{∂η_{i}} \frac{∂g}{∂ξ_{i}} ) + ∑_{j≤n−2k} \frac{∂f}{∂θ_{i}} \frac{∂g}{∂θ_{i}} )].

Appropriate modifications for contact series are obvious.

Observe that the cases 𝔨(1|2n; n) and 𝔪(n; n) are obtained from the general considerations given above by “compressing” the grading: it is clear that for a non-compressed grading 𝔥_{2a+1} = 0 for all a.

The case of the exceptional grading of 𝔟_{λ}(2) is considered in sec. I.2.2. □

W-regradings of 𝔨𝔰𝔩𝔢(9|6). Though description of all possible regradings of 𝔨𝔰𝔩𝔢(9|6) in [CK2] is correct, there is only given an outline of the idea (any regrading is determined by the degrees of the indeterminates the vectorial Lie algebra is expressed with) and the final result. Since some of the regradings of 𝔨𝔞𝔰 were overlooked in [CK2], we give a detailed proof of the completeness of the list of regradings of 𝔨𝔰𝔩𝔢(9|6); unfortunately, this was the place in the proof in [Sh14] written by analogy and we missed 𝔨𝔰𝔩𝔢(9|11).

There are two “least codimension realizations” for 𝔨𝔰𝔩𝔢(9|6) and to describe regradings is more convenient starting with Kac’s realization: 𝔤¯0 = 𝔰𝔳𝔢𝔠𝔱(5); 𝔤¯1 = Π(dΩ^{1}), cf. sect. .

Set 𝔤_{−2} = Span(∂_{1}, …, ∂_{5}) and 𝔤_{−1} = Span(dx_{i}dx_{j} | 1 ≤ i < j ≤ 5). Let 𝔥 be a regrading of 𝔤. Then, clearly, 𝔤_{−2} ⊂ 𝔥_{−} and 𝔥_{0} ∩ 𝔤_{−} ⊂ 𝔤_{−1}. Since for any permutation σ ∈ S_{5} we have

[dx_{σ(1)}dx_{σ(2)}, dx_{σ(3)}dx_{σ(4)}] = (−1)^{sign(σ)} \frac{∂}{∂x_{σ(5)}} ,

it follows that dx_{σ(1)}dx_{σ(2)} and dx_{σ(3)}dx_{σ(4)} can not simultaneously appear in 𝔥_{0} after the regrading. Hence, dim 𝔥_{0} ∩ 𝔤_{−} ≤ 4.

So we have to consider the following cases:

1) dim 𝔥_{0} ∩ 𝔤_{−} = 1. In this case 𝔥_{0} ∩ 𝔤_{−} ≃ Span(dx_{1}dx_{2}).

2) dim 𝔥_{0} ∩ 𝔤_{−} = 2. In this case 𝔥_{0} ∩ 𝔤_{−} ≃ Span(dx_{1}dx_{2}, dx_{1}dx_{3}).

3) dim 𝔥_{0} ∩ 𝔤_{−} = 3. In this case either (a) 𝔥_{0} ∩ 𝔤_{−} ≃ Span(dx_{1}dx_{2}, dx_{1}dx_{3}, dx_{1}dx_{4}) or (b) 𝔥_{0} ∩ 𝔤_{−} ≃ Span(dx_{1}dx_{2}, dx_{1}dx_{3}, dx_{2}dx_{3}).

4) dim 𝔥_{0} ∩ 𝔤_{−} = 4. In this case 𝔥_{0} ∩ 𝔤_{−} ≃ Span(dx_{1}dx_{2}, dx_{1}dx_{3}, dx_{1}dx_{4}, dx_{1}dx_{5}). The standard arguments (sec. 7.2 or based on Lemma 7.1.3 in [Sh14]) show that in the

cases 2) and 3a) 𝔥 is not a W-regrading, whereas in case 1) we obtain 𝔨𝔰𝔩𝔢(9|11), in case 3b) we obtain 𝔨𝔰𝔩𝔢(11|9) and in case 4) we obtain 𝔨𝔰𝔩𝔢(9|6). The corresponding W-regradings listed in [CK2] are:

1) deg x_{1} = deg x_{2} = 3, deg x_{3} = deg x_{4} = deg x_{5} = 2; deg d = −3; 3b) deg x_{1} = deg x_{2} = 1, deg x_{3} = deg x_{4} = deg x_{5} = 2; deg d = −2;

4) deg x_{i} = 1 for i = 1, …, 4, deg x_{5} = 2; deg d = −\frac{3}{2}.

§8. Tables
𝔤  𝔤¯0  R¯1  𝔤¯0 : 𝔤¯1
𝔰𝔩(n|1)  𝔰𝔩(n) ⊕ ℂh  ±(ε_{i} − δ)  (id ⊗ χ_{0}) ⊕ (id ⊗ χ_{0})^{∗}
𝔰𝔩(n|m)  𝔰𝔩(n) ⊕ 𝔰𝔩(m) ⊕ ℂh  ±(ε_{i} − δ_{j})  (id_{𝔰𝔩(n)} ⊗ id_{𝔰𝔩(m)}^{∗} ⊗ χ_{0}) ⊕ (id_{𝔰𝔩(n)} ⊗ id_{𝔰𝔩(m)}^{∗} ⊗ χ_{0})^{∗}

n ≠ m

𝔭𝔰𝔩(n|n) 𝔰𝔩(n) ⊕ 𝔰𝔩(n) ±(ε_{i} − δ_{j}) (id_{𝔰𝔩(n)} ⊗ id_{𝔰𝔩(n)}^{∗}) ⊕ (id_{𝔰𝔩(n)} ⊗ id_{𝔰𝔩(n)}^{∗})^{∗}

n > 2

𝔬𝔰𝔭(1|2n)  𝔰𝔭(2n)  ±δ_{j}  id_{𝔰𝔭(2n)}
𝔬𝔰𝔭(2|2n)  𝔰𝔭(2n) ⊕ ℂh  ±ε ± δ_{j}  (id_{𝔰𝔭(2n)} ⊗ χ_{1}) ⊕ (id_{𝔰𝔭(n)}^{∗} ⊗ χ_{−1})
𝔬𝔰𝔭(m|2n)  𝔰𝔭(2n) ⊕ 𝔬(m)  ±ε_{i} ± δ_{j} if m = 2k  id_{𝔰𝔭(2n)} ⊗ id_{𝔬(m)}
m > 2  ±ε_{i} ± δ_{j} and ±δ_{j} if m = 2k + 1
𝔬𝔰𝔭_{α}(4|2) 𝔰𝔩_{1}(2) ⊕ 𝔰𝔩_{2}(2) ⊕ 𝔰𝔩_{3}(2)  ±ε_{1} ± ε_{2} ± ε_{3}  id_{1} ⊗ id_{2} ⊗ id_{3}
𝔭𝔰𝔮(n)  𝔰𝔩(n)  ±(ε_{i} − ε_{j}) and 0  ad
𝔞𝔤_{2}  𝔰𝔩(2) ⊕ 𝔤_{2}  ±ε_{i} ± δ; ±δ  id_{𝔰𝔩(2)} ⊗ R(π_{1})
i = 1, 2, 3; ∑ ε_{i} = 0
𝔞𝔟_{3}  𝔰𝔩(2) ⊕ 𝔬_{7}  \frac{1}{2} (±ε_{1} ± ε_{2} ± ε_{3} ± δ)  id_{𝔰𝔩(2)} ⊗ spin_{7}

Table 8.1. Odd roots of simple Lie superalgebras of non-vectorial type and their relatives.

\begin{matrix}𝔤 & 𝔭𝔰𝔩(n|n), n > 2 & 𝔭𝔰𝔩(2|2) & 𝔭𝔰𝔮(n) \\ 𝔢𝔤 & 𝔰𝔩(n|n) & \begin{cases}𝔰𝔩(2|2) \\ 𝔰𝔭𝔬(0|4) \end{cases} & 𝔰𝔮(n) \\ 𝔡𝔢𝔯 𝔢𝔤 & 𝔤𝔩(n|n) & \begin{cases}𝔤𝔩(2|2) \\ 𝔭𝔬(0|4) ⊕ ℂE \end{cases} & 𝔮(n) \end{matrix}

Table 8.2. Central extensions and outer derivations of simple Lie superalgebras of non-vectorial type and their relatives.

§9. Appendix. Irreducible representations of simple finite dimensional Lie superalgebras of vector fields and their relatives

9.1. Induced representations. Consider L = 𝔳𝔢𝔠𝔱(0|m) and its subsuperalgebras 𝔰𝔳𝔢𝔠𝔱(0|m) and 𝔤 = 𝔥^{o}(0|m) in their standard filtration.

Let us prepare a supply of finite dimensional irreducible L-modules. Take an irreducible L_{0}-module L^{χ} with highest weight χ; assume that the highest weight vector (as well as the whole L^{χ}) is even. Set L_{1}L^{χ} = 0; this allows one to define the L-module by setting

I(χ) = ind_{L_{0}}^{L}(L^{χ}) = U(L) ⊗_{L_{0}} L^{χ}.

It turns out that for the generic χ the modules I(χ) are irreducible; the exceptional — reducible — modules line up into a sequence which is exact for 𝔳𝔢𝔠𝔱(0|m) and { \atop 𝔰𝔳𝔢𝔠𝔱}(0|m) and “almost exact” for 𝔰𝔳𝔢𝔠𝔱(0|m), see [BL1].

To formulate the answer graphically, let us pass from the induced modules I(χ) to their duals, the coinduced modules T (χ) = (I(χ ⊗ (−tr)))^{∗}, where tr is the one-dimensional 𝔤𝔩 = L_{0}-module determined by the trace, see sec. 0.6 and [BL1].

Theorem . Let L = 𝔳𝔢𝔠𝔱(0|m), 𝔰𝔳𝔢𝔠𝔱(0|m) or { \atop 𝔰𝔳𝔢𝔠𝔱}(0|m). Then the L-modules T (χ) are irreducible except for the cases when L^{χ} is isomorphic to one of the following: S^{k}(id) or S^{k}(id^{∗}) ⊗ (−tr), where id is the identity representation of L_{0}. These exceptional modules, denoted by Ω^{k} and Σ_{−k}, respectively, are interpreted as the spaces of k-th degree differential or, respectively, (−k)-th degree integral forms on the (0|m)-dimensional supermanifold. The exterior differential and the Berezin integral unite the exceptional modules in one (infinite in both directions) exact sequence:

… ⟶^{d} Σ_{−k} ⟶^{d} … ⟶^{d} Σ_{−1} ⟶^{d} Σ_{0} \underrightarrow{∫} Ω^{0} ⟶^{d} Ω^{1} ⟶^{d} … ⟶^{d} Ω^{k} ⟶^{d} …

and, therefore, the kernels (or the images) of the avove maps are the only irreducible submodules of the exceptional modules if L is of series 𝔳𝔢𝔠𝔱 or { \atop 𝔰𝔳𝔢𝔠𝔱}.

For the L = 𝔰𝔳𝔢𝔠𝔱(0|m), since Σ_{0} ≅ Ω^{0}, the module of functions, Ω^{0}, has an additional submodule T_{0}^{0}(0) of the functions with integral zero.

We see that except for m = 2 (which was already considered since 𝔳𝔢𝔠𝔱(0|2) ≃ 𝔰𝔩(1|2) ≃ 𝔬𝔰𝔭(2|2)) an operator of rank 1 can only occure in the modules T (χ) coinduced from the 1-dimensional module, or in a quotient of such. Or in the modules obtained from such by applying the change of parity. The corresponding cases are listed in tables of sec. I.2.1. For example, the corresponding rank 1 operators from 𝔳𝔢𝔠𝔱(0|m) and { \atop 𝔰𝔳𝔢𝔠𝔱}(0|m) are

ξ_{1} … ξ_{m} \frac{∂}{∂ξ_{i}} realized in Λ(ξ) and even in Vol(ξ)^{λ} for any i if m is even or
in Π(Λ(ξ)) or Π(Vol(ξ)^{λ}) for any i if m is odd.

In the divergence-free case we take ξ_{1} … ξ_{m−1} \frac{∂}{∂ξ_{m}} and analogous operators.

Several of our exceptional vectorial superalgebra exist due to the fact that adding the trivial center to the above-considered L we get in certain small dimensions new nontrivial Cartan prolongs or mk-prolongs. For the details see [Sh14].

If L = 𝔥^{o}(0|m) realised preserving the 2-form ω, then the exceptional modules are still realized in the spaces of the differential forms (since Σ_{−i} ≃ Ω^{i} thanks to nondegeneracy of ω). The complete description of irreducible modules over L = 𝔥^{o}(0|m) and all its relatives — central extensions (i.e., finite dimensional Poisson Lie superalgebras) and superalgebras of derivations — is performed by Shapovalov [Sha1], [Sha2]. Except for the cases of small dimension when an occasional isomorphism occurs (as 𝔥^{o}(0|4) ≃ 𝔭𝔰𝔩(2|2)) and, therefore, are already considered, there are no operators of rank 1.

§10. Appendix. On primitive Lie superalgebras

Recall that a filtered Lie (super)algebra 𝔤 (or an associated graded algebra) is called primitive if it possesses a maximal subalgebra of finite codimension which does not contain an homogeneous ideal of 𝔤.

Let 𝔯 ⊂ 𝔳𝔢𝔠𝔱(ξ) be a subalgebra such that pr_{−1}(𝔯) = 𝔳𝔢𝔠𝔱_{−1}(ξ), where pr_{−1} is the projection of 𝔳𝔢𝔠𝔱(ξ) onto 𝔳𝔢𝔠𝔱_{−1}(ξ) = Span(\frac{∂}{∂ξ_{1}} , …, \frac{∂}{∂ξ_{n}}).

Let 𝔥 = ⊕_{i≥−d} 𝔥_{i} be a primitive Lie superalgebra (with ⊕_{i≥0} 𝔥_{i} being its maximal subalgebra)

such that 𝔥_{−1} is an irreducible 𝔥_{0}-module and 𝔥_{−} is generated by 𝔥_{−1}. Consider the following two Lie superalgebras:

𝔤_{min} = 𝔥 ⊗ Λ(n) ⊂ + 𝔯 and 𝔤_{max} = (𝔥_{−1} ⊗ Λ(n), 𝔥_{0} ⊗ Λ(n) ⊂ + 𝔯)_{∗},  (∗)

where (V, 𝔤)_{∗} denote the Cartan prolong of (V, 𝔤). It is not difficult to see that any Lie superalgebra 𝔭 sandwiched between 𝔤_{min} and 𝔤_{max}, i.e., such that 𝔤_{min} ⊂ 𝔭 ⊂ 𝔤_{max}, is primitive; 𝔤_{min} and 𝔤_{max} are also primitive.

We conjecture that every primitive 𝔭 is of the form 𝔤_{min} for a suitable 𝔥. Be it so or not, there are indescribably many primitive Lie superalgebras.

§11. Appendix. On Lie superalgebra E(2, 2) from [K7]

While this paper was being written and [Sh14] refereed, there appeared a proof of a part of our classification. In it, V. Kac [K7] states, among other things, that there exist six infinite dimensional exceptional simple ℤ-graded Lie superalgebras. In addition to the five (considered as abstract Lie superalgebras) exceptions described already in [Sh5] Kac distinguished one more “exception” — the Lie superalgebra E(2, 2) in his notations.

Let us prove that, E(2, 2) is, actually 𝔟_{∞}^{∘}(2; 2) or, in Kac’s notations, “SKO(2, 3; 1) with subprincipal gradation”. So it is not an exception at all. Indeed, by definition, E(2, 2) is the Cartan prolong 𝔤 = (𝔤_{−1}, 𝔤_{0})_{∗}, where “the 𝔤_{0}-module 𝔤_{−1} gives rise to the linear Lie superalgebra spin_{4}^{∘}.” This means that 𝔤_{0} is the semidect sum 𝔤_{0} = 𝔥𝔢𝔦(0|4) ⊂ + 𝔰𝔩(2) and 𝔤_{−1} is the space of unique irreducible representation of 𝔥𝔢𝔦(0|4) for which the central element corresponds to the identity operator. Hence, we may identify 𝔤_{−1} either with Λ(2) = Λ(ξ_{1}, ξ_{2}) or (due to uniqueness) with Π(Λ(2)) = Π(Λ(ξ_{1}, ξ_{2})). The second variant is more convenient for our purposes.

The action of 𝔥𝔢𝔦(0|4) is given by operators (1, ξ_{1}, ξ_{2}, ∂_{ξ_{1}}, ∂_{ξ_{2}}), while 𝔰𝔩(2) acts via the standard representation on the even component of 𝔤_{−1} and via the trivial representation on the odd component of 𝔤_{−1}.

Now what is 𝔟_{∞}^{∘}(2; 2)? It is the Cartan prolong 𝔥 = (𝔥_{−1}, 𝔥_{0})_{∗}, where 𝔥_{−1} = Π(Λ(2)) = Π(Λ(η_{1}, η_{2})) and

𝔥_{0} = Span(1, η_{1}, η_{2}) ⊂ + 𝔰𝔳𝔢𝔠𝔱(0|2) ≅ (Span(1, η_{1}, η_{2}) ⊂ + Span(∂_{η_{1}}, ∂_{η_{2}})) ⊂ + 𝔰𝔩(2) ≅ 𝔤_{0}.

The subalgebra 𝔰𝔩(2) acts via the standard representation on the subspace Span(Π(η_{1}), Π(η_{2})), i.e., on the even component of 𝔥_{−1} and via the trivial representation on the subspace Span(Π(1), Π(η_{1}η_{2})), i.e., on the odd component of 𝔤_{−1}.

We get the isomorphism 𝔤_{−1} ⊕ 𝔤_{0} ≅ 𝔥_{−1} ⊕ 𝔥_{0}, hence, the isomorphism 𝔤 ≅ 𝔥.

In a recent paper [CK2] the result of this section is briefly stated but the description of 𝔤_{0} in [CK2] should still be corrected.

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1Department of Mathematics, University of Stockholm, Kräftriket hus 6, SE-106 91, Stockholm, Sweden; mleites@matematik.su.se and ^{2}Independent University of Moscow, Bolshoj Vlasievsky per, dom 11, RU-121 002 Moscow, Russia; Ira@Paramonova.mccme.ru