SuperLie and problems (to be) solved with it

Pavel Grozman, Dimitry Leites

Department of Mathematics, University of Stockholm, Roslagsv. 101, Kräftriket hus 6, S-106 91, Stockholm, Sweden; e-mail: mleites@math.su.se; grozman@equa.se or pavel@rixtele.com

Key words and phrases. Lie superalgebras, cohomology, differential equations

PG and DL gratefully acknowledge financial support of the Swedish Institute (1992-94) and NFR (1994-00), and TBSS (2001/02), and MPIM, Bonn (2002).

Abstract. A Mathematica-based package SuperLie for the study of Lie (super)algebras and their cohomology is offered. Among applications we find:

(1) calculation of the Riemannian tensors needed to write Supergravity Equations for any N-extended Minkowski superspace and to classify possible models for these superspaces;

(2) the possibility to study stability of nonholonomic systems (among which we find: ballbearings, gyroscopes, electro-mechanical devices like rotor collector with a gliding contact; ideal gas and market economy, etc.);

(3) description of the analogue of the curvature tensor for nonlinear nonholonomic constraints and the fields of solids or their surfaces, e.g., cones, as in optimal control;

(4) a new method for the study of integrability of differential equations (numerical methods can provide with individual solutions but are unable, generally, to study qualitative behavior, e.g., stability);

(5) lists of “natural” operators, i.e., the operators between sections of tensor fields (or jets) invariant with respect to the group of diffeomorphism or its subgroup;

(6) list central extensions and deformations.

All the above problems can be expressed in terms of Lie algebra cohomology.

SuperLie makes it possible to

(1) determine Lie algebras as subalgebras, via defining relations, from Cartan matrix, realize via vector fields, as polynomials with Poisson or contact (Legendre) bracket, etc.,

(2) determine various modules over these Lie algebras (tensors; modules with a vacuum vector; submodules, quotients, duals, etc.), and calculate the bracket as well as the action of the Lie algebra on its module,

(3) solve systems of equations with scalar and (linear) vector unknowns; find ker and Im of a linear operator;

(4) calculates exterior differential in various complexes.

A partial list of open problems, currently under study, is offered: the students in search of a topic for a MS or a PhD thesis, as well as other interested, are invited to join forces. Please direct your questions/suggestions to PG if SuperLie/Matehmatica related, to DL if mathematics related.

Preface

This is an expanded version of Leites’s Special Invited Lecture delivered at at NATO ASI Special Functions-2000, Tempe, Arizona, June 2000 and a version of the transcript of Leites’ lectures at Stockholm University addressed to students from diferent departments who attended an informal “math circle” once a week for about a year. The aim of the circle was to provide with a background necessary to calculate homology and cohomology of Lie algebras and Lie superalgebras needed in various, sometimes unexpected, applications: from supergravity to stability of ballbearings and in market economy, to formal solvability of differential equations.

These calculations can, sometimes, be performed analytically, but most often than not they are too complicated to be performed by bare hands. Besides, there are few general theorems to calculate (co)homology analytically.

SuperLie is a package designated to compute certain Lie algebra cohomology as an initial goal but it soon evolved into a package that makes it possible to

(1) determine Lie algebras and Lie superalgebras via defining relations, or via Cartan matrix, or realize them via vector fields, or as polynomials (or Laurent polynomials) with Poisson or contact bracket, or antibracket, etc.

(2) construct various modules over these Lie algebras and superalgebras, e.g., tensors, with a vacuum vector, etc.,

(3) construct central extensions and deformations.

In the lectures we also tried not just illustrate some calculations with motivations, but explain how to perform such calculations. The task was not easy since students from departments of computer science (not only at Stockholm University; even from the former Soviet Union, where math background was supposed to be solid) vaguely remember what a vector space is and do not know what the Lie algebras, tensor products and quotient spaces are.

So to be accessible to students and engineers, this review was appended with a brief introduction into representation theory of Lie algebras and finite groups [BKL].

Rival or not so rival codes. Several packages for computation of Lie algebra and Lie superalgebra homology and cohomology are known. None of them is, so far, sufficiently efficient to overcome D. Kalinin’s results [GKF] of early 1970’s.

After D. Kalinin’s untimely death, no trace was left as to how he computed so wonderfully. This is indeed a challenge: compared with modern machines, the computer Kalinin used does not differ much from abacus, still his result was never surpassed!

The known to us other packages that compute Lie algebra (co)homology or, at least, the modules over them, distinguish an irreducible submodule in a module, etc., are as follows.

Their documetation (and some additional manuals) are also available at the homepage of MPIM, Bonn, see

http://www.math.usu.edu/f̃g_{−}mp/Pages/SymbolicsPage/Symbolics.html There are at least two codes that complement Grozman’s SuperLie. (Ideally, it would be

great to join forces, or, at least, make links, and smoothen the seams. But this is a long and expencive job.) These codes are:

1) The package http://www.mpim-bonn.mpg.de/html/computer/software.html can calculate cohomology of finite dimensional Lie algebras with trivial coefficients, and relative cohomology (with respect to a subalgebra).

Moreover, this code can compute many things SuperLie can not (the derived series, verification of nilpotency, etc.), and not only for Lie algebras (it can consider tensors, manifolds, differential equations and Spencer cohomology; given time and money, we should have written a comprehensive toc with comments as to pros and cons).

It can not, however, compute anything in supersetting nor deal directly with infinite dimensional case, whereas SuperLie can.

The package is based on Maple. Documentation is on approximately same level as currently that for SuperLie (in blunt nordic terms: rather pitiful) but in better English.

We did not compare the speeds of performances.

2) http://www.mpim-bonn.mpg.de/html/computer/software This is the link to LiE, A software package for Lie group computations. The package is designed for work with reductive Lie groups and Lie algebras. LiE returns properties of Lie groups, Lie algebras and modules over them in the framework of a reference book, but — unlike the reference book — for any given reductive Lie group or Lie algebra.

LiE can effectively compute, among other things, dimensions of modules, decomposition of the tensor product, symmetric and alternating powers of irreducible representations into irreducible components, systems of weights, characters, orbits, and Dynkin diagrams.

The elements of Lie algebras and Lie groups are not available in LiE, so there is no (clear) way to compute anything with them. Moreover, LiE does not see any difference between Lie groups and Lie algebras, since it only deals with common properties of both.

These properties are expressed as integers, vectors, matrices or integer Laurent polynomials. All vectors and matrices are only numerical ones, and LiE can perform various calculations with them. Calculations with the same objects but with rational coefficients are not realized.

LiE can only work with finite dimensional modules. (An attempt to compute the dimension of the highest weight 𝔰𝔩(2)-module with highest weight −1 returned 2.)

LiE is an independent package, written in C for Unix. The initial code is available. Moreover, certain computations are available at

http://young.sp2mi.univ-poitiers.fr/˜ marc/LiE/form.html.

Summary: LiE has few features in common with SuperLie. LiE and SuperLie nicely complement each other. It would be useful if somebody would have taught LiE deal with superalgebras. In future we should teach SuperLie how to appeal to LiE when needed.

Other codes for dealing with certain problems of Lie superalgebras.

1) http://www-gap.dsc.st-and.ac.uk/ gap for computations related with Groups, Algorithms and Programming

2) Macauley2 is also dealing with various aspects of Linear superalgebra. It is a favourite of D. Eisenbud and his colleagues with impressive results. These results and approaches are, so far, transversal to ours, though also deal with superalgebras.

Miscellanea. SuperLie is Mathematica-based, hence, rather slow. Moreover, dealing with more voluminous problems we have realized that Mathematica has certain inbuilt (but not publicized) restrictions: in can not process data if its volume is greater than 2GB or so. So the motto “if your code is slow, push the button a day earlier” does not help: the machine just aborts. As a result we were unable, regrettably, to use facilities of medics.polytechnique.fr run extremely user-friendly personell directed by Marc Giusti.

So a natural way out is to rewrite everything in a protho-language, say, in C.

PROJECTS: A ROUGH DESCRIPTION OF SELECTED INTERESTING PROBLEMS SOLVABLE WITH SUPERLIE OR ITS FUTURE GENERALIZATION 5

• V. Kornyak undertook the task but, so far, his code can not produce any result not obtained several years earlier by SuperLie, or easily obtainable by SuperLie.

• Certain aspects of Lie superalgebras are considered (see MR 97j:17006) in Mikhalev, A A.; Zolotykh, A. A. Combinatorial aspects of Lie superalgebras. With 1

IBM-PC floppy disk (3.5 inch; HD). CRC Press, Boca Raton, FL, 1995. viii+260 pp.

• There are several other groups working (pitifuly independently in view of scanty funding) on various aspects of representation theory. We would like to mention one: http://www.mathematik.uni-bielefeld.de/birep

To be done. We did not (for the lack of resources) try several natural ways to improve the performance of SuperLie:

1) the matrix of the differential for the Lie (super)algebra (co)homology for any algebra and any module is very sparce. We never used this.

2) When the structure constants are integers, one can try to compute cohomology mod p. We did this recently but the going is so sloooooooow that clearly this is not THE way.

3) We never tried to parallelize the computations.

PROJECTS: a rough description of selected interesting problems solvable with SuperLie or its future generalization

The following are projects for computer-assisted study of certain tempting mathematical problems. It was not so long ago that some of the physical problems considered in what follows were formulated in purely mathematical terms and, therefore, became solvable, at least theoretically.

Many relations with other branches of mathematics (like calculations of local invariants of nonholonomic manifolds ranging from supergravity to stability of rolling bodies to stability of markets) are far from being clear yet. But even the parts we have understood, lead to a possibility to solve old problems. One “only” has to be able to use computers and interprete the results ALREADY obtainable by means of SuperLie.

The advantage of these problems is that one can start at virtually any level. The only formally needed knowledge is that of how to take partial derivative and definitions of tensor (or symmetric, or exterior) products of linear spaces.

In reality, of course, one has to learn representation theory (the more the better) of Lie algebras (and, to understand it, of finite and compact groups).

Direction 1. Selected applications of Lie superalgebra cohomology.

Background. Invariants of various types help us to describe the physical picture. The knowledge of how to use invariants responsible for stability can be rather vital (Chernobyl or misguided missiles illustrate what we mean). It is known that the sign of the curvature of the space of positions governs stability: if it is positive, as on the sphere, then geodesics (the admissible trajectories of the particle) converge (as meridians do) and if the curvature is negative, then geodesics diverge.

It was only recently understood that the curvature can be interpreted as certain Lie algebra cohomology. Similar cohomology appear as obstructions to formal integrability of differential equations. These obstructions were previously expressed in other terms (Spencer cohomology), difficult to compute, and so far were never used by engineers.

To solve differential equations, people usually use various numerical methods. Here one more, totally new, approach to several types of problems is suggested: in terms of Lie algebra cohomology one can

(1) define criteria of formal integrability of differential equations

(2) calculate the curvature tensor not only for holonomic systems, but — for the first time since Hertz raised the problem — for nonholonomic dynamical systems (which seem to be omnipresent, unlike the holonomic ones, with which the modern textbooks on analytical mechanics tend to confine themselves). One can even consider nonlinear constraints on velocities (example: the car with cruise control switched ON).

This project (to compute (co)homology with a computer) was launched in the talk [LPo]. In order to describe the main ingredient of the new approach, recall the main definitions.

1) Having fixed a basis in a space V of dimension n, we identify End(V), the space of linear operators acting in V, with the space Mat(n) of n × n matrices. The subspace 𝔤 of Mat(n) is called a Lie algebra if it is closed with respect to the bracket multiplication [x, y] = xy − yx.

2) Suppose that d : W ⟶ W is a linear operator whose square vanishes. This implies that Im d is a subspace of Ker d. The quotient space H(d, W) = Ker d/Im d is called the homology of d.

3) With every pair (𝔤, V) one can associate an operator d : W ⟶ W for a certain space W = W (𝔤, V) canonically recoverable from (𝔤, V). The space H(d, W) is called the cohomology of the Lie algebra 𝔤 with coefficients in V (a slightly different construction yields homology of the Lie algebra 𝔤 with coefficients in V; furter on we will remind both constructions).

One might think that the problem of calculating H(d, W) is very simple: just find any collection of linearly independent vectors in Ker d complementary to Im d. Any student knows how to do this for matrices of small dimension. But our task amounts to solving a HUGE number of systems of linear equations, especially tough to solve if both Ker d and Im d are infinite dimensional even if H(d, W) is, nevertheless, finite dimensional.

Though d corresponds to a rather sparse matrix (which is good for us), d is introduced recursively which requires repeated loops in quantities able to considerably reduce the life expectancy of the researcher.

Selected subprojectss:

1) Purely mathematical goal: calculate H(d, W) for several types of 𝔤’s and V ’s given. This can be (but seldom is) easy for finite dimensional 𝔤 and V and very tough for infinite dimensional ones. (However, the tougher the problem, the greater recognition :-) .)

E.g., there is a 30 years old problem raised by Gelfand, Kalinin and Fuchs which we belive is ripe to be solved by now. Formally speaking, this problem does not require any preliminary knowledge. In reality, a guided tour along the following textbooks will not hurt:

[Fu]. Regrettably it is NOT a text-book for the first reading. [GM] most lucidly describes spectral sequences and cohomology.

But even those who do not have these books, can consider a tough old problem, to get the feel of computations of Lie algebra cohomology: compute H^{∗}(𝔤) for the Lie algebra 𝔤 of Hamiltoniam vector fields with polynomial coefficients on 2n-diminsional space. Kalinin considered 2n = 2 but did not get eonugh results to make conjectures. Recently these or similar cohomology appeared in various problems, e.g., in a result by I. Gelfand and M. Kontsevich who considered the Lie algebra of Hamiltoniam vector fields with Laurent coefficients; a noncommutative analogs are also of conciderable interest lately.

Though we will deal with Lie algebras, it is adwisable to have some background in groups as well, e.g., from [NS].

At the moment the best, perhaps, text book on Lie algebras for beginners is Fulton, William; Harris, Joe Representation theory. A first course. Graduate Texts in Mathematics,

129. Readings in Mathematics. Springer-Verlag, New York, 1991. xvi+551 pp.; an old review by Dynkin [Dy2] and books by Humphreys and J. F. Adams are worth studying in parallel.

2) Computational problems: test feasibility of computing cohomology as criteria for formal integrability of differential equations. For motivations see [Br] (there exists an abridged version of [Br] in Russian with useful Appendix by Vershik A. and Gershkovich V. and an abridged version in Chinese).

The paper by Vershik [V] is the best (known to us) modern mathematical exposition of nonholonomic problems.

3) Applications to theoretical physics: certain homology H(d, W) for selected pairs (𝔤, V) can be interpreted in terms of supergravity, see Grozman P., Leites D. From supergravity to ball bearings in J. Wess and E. Ivanov (eds.). Lecture notes in Physics, 524, 58–67.

Direction 2. Applications of the classical polynomials of several discrete variables. The equation of the form a(x)y′′+b(x)y′+cy = 0, where a(x) is a polynomial of degree ≤ 2, b(x) is a polynomial of degree ≤ 1 and c is a constant, is called a hypergeometric type equation and its solutions are called functions of hypergeometric type. Examples: Chebychev and Hahn polynomials, in particular, sines and cosines of nx considered as polynomials in x. Polynomial solutions of the hypergeometric equation and of various of its difference approximations are extremely important in various branches of mathematical physics: from thrifty ways to encode signals to various approximations. (Chebyshev discovered his polynomials by studying methods for evaluation and improvement of shell bombarding.)

In tomography, roentgenography (for example) polynomials of SEVERAL variables are desirable. Gauss introduced spherical polynomials to geophysics assuming that Earth can be approximated by a sphere. In the era of satellites this assumption proved to be too rough sometimes; besides, spherical harmonics are out of place in approximation of an arbitrary function without any symmetry.

Modern descriptions of orthogonal polynomials in several variables are usually just sums of products of polynomials in one variable, or a somewhat more sophisticated combinations thereof.

Recently there was found a totally new approach to classical polynomials of (several) discrete variables — solutions of the difference approximation of the above equation and its multidimensional analogs (Leites D., Sergeev A., Orthogonal polynomials of discrete variable and Lie algebras of complex size supermatrices. MPI-1999-36 (http://www.mpimbonn.mpg.de) 49–70; Theor. and Math. Physics, v. 123, no. 2, 582–609).

First of all, we have to define these polynomials, and study their various properties. In order to do it, we also have to compute certain Lie algebra and Lie superalgebra cohomology that describe relations. We also have to consider realizations of various Lie algebras and superalgebras by means of creation and annihilation operators. This is an “easy” part of this project which, however, seems to be too voluminous to be run on a Mathematica-based package.

The “difficult” part is implementation of these polynomials, approximation procedures associated with them, etc.

References: Nikiforov, A. F.; Suslov, S. K.; Uvarov, V. B., Classical orthogonal polynomials of a discrete variable. Translated from the Russian. Springer Series in Computational Physics. Springer-Verlag, Berlin, 1991. xvi+374 pp.; Nikiforov A.; Uvarov V., Special functions of mathematical physics. A unified introduction with applications. Translated from the Russian and with a preface by Ralph P. Boas. With a foreword by A. A. Samarskii. BirkhSuser Verlag, Basel-Boston, MA, 1988. xviii+427 pp.

Advantages and difficulties of the project. There is a sea of problems leading to calculation of H(d, W) = Ker d/Im d for given (𝔤, V) at various levels of difficulty depending on 𝔤 and V. A successful student can continue at Ph.D. level or even go further (apply results to real life problems).

Even if one is interested in actual understanding what one is doing (possible interpretations and applications of the results do require some background in one or several of the following fields: pure mathematics, theoretical physics, analytical mechanics) and to learn a bit will take lots of time, one can start calculating in certain cases immediately. This is, of course, a plus.

However, one can not see connections to real life problems or even compute more efficiently without a course or two in representation theory of Lie algebras. For integrability of differential equations one needs a course in symmetries of differential equations.

A description of the pilot version of SuperLie is given in [GL5]. Here we will reproduce the main points of this paper.

1. Cohomology of Lie superalgebras. Their applications

§0. History (a very biased view)

The notion of Lie algebra cohomology, though known since late 20’s, did not draw much attentions until about 30 years ago. At that time I. M. Gelfand advertised Lie algebra cohomology everywhere, culminating at the Nice Mathematical congress; together with his pupils and colleagues he

(1) gave a purely algebraic proof of Weyl character formula which showed that finite dimensional representations are better understood if considered in the class of certain infinite dimensional modules and interpreted in terms of Lie algebra cohomology,

(2) showed that cohomology of simple infinite dimensional Lie algebras of vector fields are calculable and have application in topology (e.g., characteristic classes of foliations);

(3) realized a deep relation between various cohomology and vacuum weight representations (Borel-Weil-Bott-Kostant-Bernstein-Gelfand-Gelfand’s theorem, in what folows breifly BWZ theorem).

Still, Gelfand and Fuchs were, perhaps, the only ones (except their own and their student’s Ph.D. students) who devoted a significant part of their time to calculation of Lie algebra cohomology. When Gelfand’s interests shifted elsewhere the researchers gradually left the field: almost everything calculable analytically seemed to have been calculated (mainly, by A. Borel, Bott, Kostant and Serre) even before Gelfand and Fuchs entered the field, anyway; Gelfand and Fuchs only considered infinite dimensional Lie algebras (of vector fields) Lepowsky and Wilson (and their followers) considered Kac-Moody algebras. Mainly, only trivial coefficients were discussed in these cases.

Fortunately, there are some calculable problems left for us, as we will see. Besides, having encountered with difficulties calulating cohomology analytically (i.e., with bare hands) we encourage to follow I. M. Gelfand’s advice to use computer-aided study for this problem.

So far, the experience with computer-aided calculations of Lie algebra cohomology is not rich: in early 1970’s D. Kalinin ([GKF]) calcualted some partial results on H^{∗}(𝔤) of the Lie algebra 𝔤 of Hamiltonian vector fields on the plane; recently Roos and Löfwall had obtained a beautiful interpretation of certain Hochschield cohomology ([LR]). Both results are just beginnings of possible fields for study: e.g., even Kalinin’s calculations ([GKF]) are not surpassed yet, which is a challenge and a shame on modern researchers in the field.

It seems that unexpected negative results [GKF] discouraged or even scared researchers; since then nobody did seriously applied computers to the problem.

We think it is a shame that researchers had waited for so long: there are several problems which promise to be applicable in industry and economics, to say nothing about applications in pure mathematics and such romantic branches of physics as supergravity [GL].

At about the same time as Gelfand-inspired mathematicians studied Lie algebra cohomology (S. Shnider), Goldschmidt gave criteria of formal integrability of differential equations in terms of Spencer cohomology, see [Br].

Though such criteria are vital in practice, engineers still use “old-fashioned” numerical methods; nobody uses cohomology. Observe that

(1) Spencer cohomology can be interpreted in terms of Lie algebra ones and the other way round,

(2) with the generalization of the notion of Cartan prolongation, cf. [Sh5], one can extend applicability of Goldschmidt’s criteria to all differential equations whereas the original criteria embraced only one of the two possible cases.

We prefer to use Lie algebra cohomology insted of Spencer ones: the answer, as we will see, can be documented in a more succint form; besides, there are (two or even three) theorems that simplify calulations of Lie algebra (co)homology and none for Spencer ones.

One of the reasons the calculation of (co)homology was abandoned and the limelight shifted away from it: the amount of calculations and the necessity to learn new vocabulary and techniques of representation theory did not encourage to indulge into such a project without extra motivations. We hope to change this habit of engineers and other possible customers; for this we need a number of examples computed and a user-friendly package.

During the past decade or two we were constantly aware of ever increasing role of Lie algebra cohomology. In our research we encountered these (co)homologies virtually everywhere: from supergravity to problems of stability of cars and ball bearings or even markets. We are not alone ton appreciate the role of Lie algebra cohomology: recently international conferences entitled “Cohomological Physics” started to mushroom, compare [HKV], [W] with our manifesto [LPo]. Observe that the study of Lie algebra cohomology from “super” point of view clarified various notions, such as semi-infinite cohomology.

These applications, especially, to supergravity, see [GL] are sufficient motivations to reconsider the list of cohomology to compute.

More explicitly, the known facts about Lie algebra cohomology and interesting open problems can be reviewed as follows. We follow, mainly, [FZ], [BL], [LSc]. For the proof of the statements see [FZ] or more accessible [OV], [K1].

The main types of Lie algebras 𝔤. Here we only consider the types SuperLie can deal with. Resently we started to teach SuperLie how to deal with characteristic p to get new results in this difficult situation, cf. [Ja], see also [BMR].

Each type has an important special case: n = ∞. (For supespaces, the dimension is represented by a pair of numbers each of which can be equal to ∞.)

1. Matrix Lie algebras, i.e., subalgebras of 𝔤𝔩(n).

2. Lie algebras 𝔤(A) with Cartan n × n matrix, defined as 𝔤̃(A)/𝔪 for certain 𝔤̃(A) and the maximal ideal 𝔪 whose intersection with the maximal torus is 0.

3. The vectorial Lie algebras, i.e., subalgebras of 𝔳𝔢𝔠𝔱(n) = 𝔡𝔢𝔯𝕂[x], where x = (x_{1}, …, x_{n}).

4. Lie algebras with Poisson and contact brackets in the space of “functions” in n indeterminates.

5. Stringy Lie superalgebras on n-extended supercircles.

6. 𝔡𝔦𝔣𝔣(n)_{L} = 𝕂[x, ∂], where x = (x_{1}, …, x_{n}), with a Lie or a superLie bracket, and its Lie subalgebras.

7. Quotient algebras, in particular, quotients modulo the center.

8.Subalgebras, in particular, nilpotent and solvable subalgebras.

9. Superizations of the above if not already “super”.

The main types of 𝔤-modules.

0. Induced and coinduced modules.

1. Modules over matrix Lie algebras considered as subalgebras of 𝔤𝔩(V): 1a) the standard (identity) module V; 1b) the relatives of V: superspaces V^{∗}, Π(V), etc.

§1. BACKGROUND 11

1c) the spaces of tensors, e.g., T_{q}^{p}(V) = (⊗^{p}(V)) ⊗ (⊗^{q}(V^{∗})) and its subspaces and quotient spaces. 1d) highest weight modules.

2. Modules over the vectorial Lie algebras: 2a) modules of tensor fields and their duals; 2b) more general modules with vacuum vector (of highest or lowest weight).

3. Submodules.

4. Quotient modules.

§1. Background

All vector spaces considered in what follows are defined over a ground field 𝕂. Unless otherwise stated 𝕂 = ℂ, but we would like to be able to consider sometimes other possibilities: 𝕂 = ℝ, ℚ, 𝔽_{q}, and \overline{𝔽_{q}}, the algebraic closure of 𝔽_{q}; even the case of a ring instead of the field: 𝕂 = ℤ will be encountered.

1.1. A Lie algebra is a vector space 𝔤 with multiplication 𝔤 ⊗ 𝔤 → 𝔤 called bracket. The bracket of two elements is usually denoted by [x, y], not xy, and satisfies the following identitis:

[x, x] = 0 for any x ∈ 𝔤;  (S-S)
[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 for any x, y, z ∈ 𝔤  (J.I.)

The identity (S-S) signifies anti-symmetry, (J.I.) is the Jacobi identity.

By anti-symmetry and bilinearity of [·, ·] we have 0 = [x + y, x + y] = [x, y] + [y, x], i.e., [x, y] is indeed skew-symmetric, but (S-S) is easier to verify and holds even if characteristics of 𝕂 is 2.

1.2. Examples of Lie algebras. Let A be an associative algebra. By means of the subscript L we denote the Lie algebra 𝔤 = A_{L} whose vector subspace is a copy of A and the bracket is given by the formula [x, y] = xy − yx. Clearly, A_{L} is a Lie algebra: (S-S) and (J.I.) are subject to a direct verification.

Particular cases of the above example are important when A is the algebra of matrices, or differential operators or pseudodifferential symbols.

In particular, let EndV be the associative algebra of endomorphisms of a vector space V. The general linear algebra 𝔤𝔩(V) is (EndV)_{L}. We abbreviate Mat(n)_{L} to 𝔤𝔩(n), and, having selected a basis in V, we identify 𝔤𝔩(V) with 𝔤𝔩(n).

Let 𝔤_{+}, 𝔤_{−} and 𝔥 be the subsets of 𝔤 = 𝔤𝔩(n) consisting of all strictly upper triangular, strictly lower triangular or diagonal matrices, respectively. Clearly, 𝔤_{+}, 𝔤_{−} and 𝔥 are Lie algebras.

The classical Lie algebas. The main property of the trace of a matrix is tr[x, y] = 0. Hence, the space of n × n matrices with trace 0 is a Lie algebra; it is called the special linear algebra and denoted by 𝔰𝔩(n).

Let B be a bilinear form on a vector space V. Clearly, the space 𝔞𝔲𝔱(B) of all operators that preserve B, i.e., {x ∈ 𝔤𝔩(V) | B(xu, v) + B(u, xv) = 0 for any u, v ∈ V } is closed under the bracket and so 𝔞𝔲𝔱(B) is a Lie algebra. If B is non-degenerate we distinguish two important subcases:

B is symmetric, then 𝔞𝔲𝔱(B) is called the orthogonal Lie algebra and denoted by 𝔬(V, B). B is skew-symmetric, then 𝔞𝔲𝔱(B) is called the symplectic Lie algebra and denoted by

𝔰𝔭(V, B).

It is well-known that over ℂ a non-degenerate symmetric forms on V are equivalent to each other and the same applies to skew-symmetric forms. So Lie algebras 𝔬(V) and 𝔰𝔭(V) actually depend only on dimV, and we will sometimes denote them by 𝔬(n) and 𝔰𝔭(2m).

The Lie algebras 𝔤𝔩(n), 𝔬(n) and 𝔰𝔭(2m), as well as some (not all!) of their real forms are called classical.

Vectorial Lie algebras. Denote by 𝔳𝔢𝔠𝔱(n) = 𝔡𝔢𝔯𝕂[x], where x = (x_{1}, …, x_{n}), the Lie algebra of derivations of 𝕂[x]. The elements of 𝔳𝔢𝔠𝔱(n) are expressed with the help of partial derivatives as ∑ f_{i}(x)∂_{i}, where ∂_{i} = \frac{∂}{∂x_{i}} and f_{i} ∈ 𝕂[x] for all i.

In what follows any Lie algebra 𝔤 realized as a subalgebra of 𝔤𝔩(n) or 𝔤𝔩(∞) will be called a matrix realization of 𝔤; realized as a subalgebra of 𝔳𝔢𝔠𝔱(n) it will be called a vectorial Lie algebra. Usually, but not always, the adjective matrix or vectorial is applied to a maximal subalgebra of 𝔤𝔩(n) (n can be equal to ∞) or 𝔳𝔢𝔠𝔱(n), respectively.

For more examples see the list of classical Lie superalgebras [LSc] and references therein.

1.3. In view of Example (EndV)_{L}, the bracket in all Lie algebras is often called commutator. We say that elements x and y commute if [x, y] = 0. A Lie algebra 𝔤 is commutative (or abelian) if any two elements of 𝔤 commute. The center of 𝔤 is the set of elements of 𝔤 that commute with all elements of 𝔤.

The bracket in the Lie algebra is just another name for the multiplication in it. Therefore, the following concepts for Lie algebras are defined just as for associative algebras: a subalgebra; an ideal (since the bracket is antisymmetric, any left or right module is two-sided: indeed, the formula am = −ma for a from the algebra and m from the module determines the passage from the left module structure to the right one and vv.); the quotient algebra 𝔤/𝔥 of the Lie algebra 𝔤 modulo the ideal 𝔥; homomorphisms, monomorphisms and epimorphisms of Lie algebras; the (direct) sum of Lie algebras. These notions satisfy all the standard properties.

1.4. Simple and solvable Lie algebras. A nice class of Lie algebras with incredibly many applications is that of simple ones and various their subalgebras. Recall that a Lie algebra 𝔤 is simple, if dim 𝔤 > 1 and 𝔤 has no nontrivial ideals (distnct from 0 and 𝔤 itself).

Actually, the “simple” algebras are usually replaced with even “better” (i.e., more useful in applications) Lie algebras for which the simple Lie algebras serve as a certain “core”: nontrivial central extensions of simple algebras, their algebras of differentiations, etc.

Among simple algebras, the most interesting to study are the ones with sufficiently rich structure for humans to be able to describe their representations, etc. Examples of such subclasses are:

FD) finite dimensional algebras;

PG) ℤ-graded algebras of polynomial growth;

F) their deformations, such as 𝔤𝔩(λ), see [LS], i.e., filtered algebras of polynomial growth. An opposite, in some sence class of Lie algebras is that of solvable ones. Let us describe

them. The Lie algebra with the zero bracket is called an abelian one. The center of 𝔤, denoted by 𝔷(𝔤) is the maximal abelian subalgebra. Slightly more complicated than abelian are nilpotent algebras. To describe them, consider series:

C^{0}𝔤 ⊂ C^{1}𝔤 ⊂ C^{2}𝔤 …,

where

C^{0}𝔤 = {0}; C^{1}𝔤 = 𝔷(𝔤); C^{k+1}𝔤/C^{k}𝔤 = 𝔷(𝔤/C^{k}𝔤) for k ≥ 1.

The Lie algebra 𝔤 is called nilpotent if C^{k}𝔤 = 0 for some k.

Usually, nilpotent Lie algebra is defined as such that L^{k}𝔤 = 0 for some k, where L^{k}𝔤 = [L^{k−1}𝔤, 𝔤] for k ≥ 1. This, equivalent, definition is not so transparent as the above one, but resembles the following one: the Lie algebra 𝔤 is solvable if L^{(k)}𝔤 = 0 for some k, where L^{(k)}𝔤 = [L^{(k−1)}𝔤, L^{(k−1)}𝔤] for k ≥ 1.

The maximal solvable subalgebra is called the radical. The quotient modulo radical is called semisimple Lie algebra. The structure of semisimple Lie algebras is especially simple for finite dimensional Llie algebras over any field of characteristic 0: any semisimple Lie algebra is the direct sum of simple ones.

1.5. Modules over Lie algebras. A representation of a Lie algebra 𝔤 in a vector space V is a Lie algebra homomorphism ρ : 𝔤 → 𝔤𝔩(V). The space V or, more exactly, the pair (V, ρ), is called then a 𝔤-module. We often write xv instead of ρ(x) ⊗ v, but dealing with a computer we should be more accurate. Observe that the fact that ρ : 𝔤 → 𝔤𝔩(V) is a Lie algebra homomorphism can be expressed as ρ([x, y]) = [ρ(x), ρ(y)], where [x, y] in the left hand side is the bracket in 𝔤 while [ρ(x), ρ(y)] is the bracket (commutator) in 𝔤𝔩(V).

Examples of 𝔤-modules. If ρ(x) = 0 for any x ∈ 𝔤, then V is the trivial 𝔤-module. Unless otherwise stated, 𝕂 is considered as the one-dimensional trivial 𝔤-module, and every trivial module is one-dimensional, unless specified differently.

The adjoint representation ad : 𝔤 → 𝔤𝔩(𝔤). In this case instead of ρ(x) we write adx; by definition adx(y) = [x, y]. The Jacobi identity implies that ad is a derivation (satisfies the Leibniz rule with the bracket as the product).

The general linear algebra 𝔤𝔩(V) and its subalgebras have a natural representation in the space V; this representation is called the standard or identity representation.

The tensor product V_{1} ⊗ V_{2} (over 𝕂) of two 𝔤-modules has a natural 𝔤-module structure:

x(v_{1} ⊗ v_{2}) = xv_{1} ⊗ v_{2} + v_{1} ⊗ xv_{2} for x ∈ 𝔤, v_{1} ∈ V_{1}, v_{2} ∈ V_{2}.

In this way we define for any 𝔤-module V its tensor, symmetric and exterior powers denoted by ⊗^{n}V, S^{n}(V) and Λ^{n}(V). (Sometimes we write E^{n} instead of Λ^{n} and V^{⊗n} instead of ⊗^{n}V.)

The formula

(xA)(v) = x(Av) − A(xv) for x ∈ 𝔤, A ∈ Hom(V_{1}, V_{2}), v ∈ V_{1}

endows Hom(V_{1}, V_{2}) with a 𝔤-module structure. (The origin of this and above formula: consider the action of the Lie group with Lie algebra 𝔤 and differentiate at the unit.)

The dual module V′ is defined as V′ = Hom(V, 𝕂), where 𝕂 is the one-dimensional trivial 𝔤-module.

The powers of the volume. The 𝔤𝔩(V)-module 𝕂_{λ} given by X(k) = λtr(X) · k for any X ∈ 𝔤𝔩(V) and k ∈ 𝕂_{λ} is well-defined for any finite dimensional V. If λ ∈ ℤ the formula can be integrated to a representation of the Lie group GL(V) in the λ-th tensor power of the space of volume forms.

The standard concepts of the theory of module over algebras such as submodules, quotient modules, direct sums of modules, homomorphisms, irreducible modules and so on, are defined for Lie algebras in just the same way as for associative algebras. Moreover, they can be reduced to the case of associative algebras (see the next subsection). With the help of these notions we construct important induced and coinduced modules over the larger algebra given a module over a subalgebra.

1.6. From Lie algebras to associative ones. Let us construct a functor from the category of Lie algebras to the category of associative algebras with unity. The idea behind this construction is that for the general matrix algebra 𝔤𝔩(V) the product of its elements is defined along with their bracket. For an arbitrary 𝔤, even a matrix one, like 𝔬(V), the product of it elements does not lie in 𝔤, though for any 𝔤-module V corresponding to a representation ρ the element ρ(x)ρ(y) ∈ V is well-defined for any x, y ∈ 𝔤. In order to construct the algebra in which such products lie, we construct an associative algebra U(𝔤) generated by 𝔤.

Let 𝔤 be a Lie algebra. Let T (𝔤) = ⊕_{n≥0} T^{n}(𝔤), where T^{0}(𝔤) = 𝕂 and T^{n}(𝔤) = ⊗^{n}𝔤, be the

tensor algebra of 𝔤. Consider also the two-sided ideal I ⊂ T (𝔤) generated by the elements x ⊗ y − y ⊗ x − [x, y] for x, y ∈ 𝔤. Set U(𝔤) = T (𝔤)/I. The correspondence 𝔤 ↦ U(𝔤) is the desired functor from the Lie algebras to the associative algebras; U(𝔤) is called the universal enveloping algebra of 𝔤.

We will identify the elements of 𝔤 with their images in U(𝔤). This allows us to establish a 1-1 correspondence between the 𝔤-modules and U(𝔤)-modules (we consider unitary U(𝔤)- modules, i.e., such that 1 · m = m for any 𝔤 module M and m ∈ M).

If 𝔤 is abelian, then U(𝔤) is by definition the symmetric algebra S(𝔤). The following theorem shows that, to an extent, this coincidence is true for arbitrary Lie algebras, namely, U(𝔤) and S(𝔤) are still isomorphic as spaces, though not as algebras.

Theorem . (Poincaré–Birkhoff–Witt) Let {x_{i}}_{i∈I} be a basis in 𝔤 indexed by a linearly ordered set I. Then the monomials x_{i_{1}}^{n_{1}}x_{i_{2}}^{n_{2}} … x_{i_{k}}^{n_{k}}, where i_{1} < i_{2} < ⋯ < i_{k} and n_{1}, … n_{k} are non-negative integers, form a basis of U(𝔤).

Corollary . Let 𝔤_{1} and 𝔤_{2} be subalgebras of a Lie algebra 𝔤 such that 𝔤 = 𝔤_{1} ⊕ 𝔤_{2} as a vector space. Then U(𝔤_{1}) and U(𝔤_{2}) are naturally imbedded in U(𝔤), and the linear mapping U(𝔤_{1}) ⊗ U(𝔤_{2}) → U(𝔤) sending x_{1} ⊗ x_{2} to x_{1}x_{2} is an isomorphism of vector spaces.

1.7. Induced and coinduced modules. Let 𝔥 ⊂ 𝔤 be a Lie algebra and its subalgebra, V a module over 𝔥. From V, we construct the induced and coinduced 𝔤-modules by setting

I_{𝔥}^{𝔤}(V) = U(𝔤) ⊗_{U(𝔥)} V,  C_{𝔥}^{𝔤}(V) = Hom_{U(𝔥)}(U(𝔤), V).  (1.7.1)

Clearly, U(𝔤) is both left and right 𝔤-module; hence, I_{𝔥}^{𝔤}(V) and C_{𝔥}^{𝔤}(V) are 𝔤-modules. Thanks to the PBW theorem, it is not difficult to see that, as spaces,

I_{𝔥}^{𝔤}(V) ≃ 𝕂[𝔤/𝔥] ⊗ V,  C_{𝔥}^{𝔤}(V) = 𝕂[(𝔤/𝔥)′] ⊗ V.  (1.7.2)

These formulas make lucid either the basis of I_{𝔥}^{𝔤}(V) and C_{𝔥}^{𝔤}(V) (from (1.7.2)) or the 𝔤-module structure (from (1.7.1)).

1.8. Presentations: generators and relations. A usual way to study algebraic objects is to present them by generators and defining relations. For 𝔤𝔩(n) a standard basis consists of matrix units e_{ij}, i.e., matrices with 1 occupying the (i, j)-th slot and 0 other places. It is subject to a direct verification that

[e_{ij}, e_{kl}] = δ_{jk}e_{il} − δ_{il}e_{jk}.  (1.8.1)

For any simple finite dimensional Lie algebras there is a more convenient shorter presentation via Chevalley generators and Serre relations, see [K1] where a wider class of Kac–Moody algebras is considered. The latter can be succinctly encoded in a Cartan matrix or, even more graphically, a Dynkin diagram. So far, researchers mainly considered Kac–Moody algebras corresponding to extended Dynkin diagrams, until recently Borcherds [Bo], and Gritsenko and Nikulin [GN] demonstrated importance of a still wider class of algebras of the form 𝔤(A). The defining relations for Lie algebras and superalgebras with an arbitrary Cartan matrix is still an open problem even for 3 × 3 matrices, cf. [GL2].

Even for simple classical Lie algebras of vector fields a reasonable and explicit presentation was obtained only recently, [LP3] (compare with implicit description in [F1], [FF1], [FF2]).

Though superization of this result for the Poisson algebras and description of defining relations for OTHER systems of generators are vital, among other problems, for q-quantization of spinor and oscillator representations, this is still an open problem; so far see [GL1],

One can easily obtain a similar explicit description ([GLP]) for nilpotent subalgebras of (twisted) loop algebras and superalgebras which recently appeared in a very interesting paper by Löfval and Roos [LR].

For simple Lie algebras there exists a more convenient than (1.8.1) shorter presentation. As a first (and typical) exibit, consider 𝔤 = 𝔰𝔩(n). Then represent 𝔤 as the sum of the Cartan (diagonal) subalgebra 𝔥 and two maximal nilpotent subalgebras (positive and negative): 𝔤 = 𝔤^{−} ⊕ 𝔥 ⊕ 𝔤^{+}. Let us describe the structure of 𝔤^{+} (observe that 𝔤^{−} ≃ 𝔤^{+}). For i = 1, …, n − 1 set X_{i}^{+} = e_{i,i+1}, X_{i}^{−} = e_{i+1,i}. Then formula (1.8.1) implies that the elements X_{i}^{+} generate the algebra 𝔤^{+} and the X_{i}^{−} generate 𝔤^{−}. It is easy to verify that

[X_{i}^{+}, X_{j}^{+}] = 0 if i − j ≠ ±1 and [X_{j}^{+}, [X_{j}^{+}, X_{i}^{+}]] = 0 if i − j = ±1; (SR^{±}) = (1.8.2)

(similarly for the X_{i}^{−}).

Actually, the Lie algebras 𝔤^{+} and 𝔤^{−} are determined by these relations, i.e., any other relation between the generators is a corollary of (1.8.2) (of course, with the axioms of Lie algebras).

Now consider the interaction of 𝔤^{+} with 𝔤^{−}. Again by (1.8.1), we have [X_{i}^{+}, X_{j}^{−}] = δ_{i,j}H_{i}, where H_{i} = E_{ii} − E_{i+1,i+1}. The algebra 𝔥 is abelian, and the elements H_{1}, …, H_{n−1} form its basis. It is easy to verify that

[H_{i}, X_{i}^{+}] = 2X_{i}^{+}, [H_{i}, X_{i±1}^{+}] = −X_{i±1}^{+} and [H_{i}, X_{j}^{+}] = 0 if |j − i| ≥ 2;

similarly,

[H_{i}, X_{i}^{−}] = −2X_{i}^{−}, [H_{i}, X_{i±1}^{−}] = X_{i±1}^{−} and [H_{i}, X_{j}^{−}] = 0 if |j − i| ≥ 2.

Summing up: we have distinguished in 𝔤 a system of 3(n − 1) generators X_{1}^{−}, …, X_{n−1}^{−}, H_{1}, …, H_{n−1} and X_{1}^{+}, …, X_{n−1}^{+}. All relations given above can be succintly written as follows. Let A = (a_{ij}) be the following (n − 1) × (n − 1)-matrix:

A = (\begin{matrix}2 & −1 & 0 & … & 0 \\ −1 & 2 & −1 & ... & ⋮ \\ 0 & −1 & ... & & 0 \\ ⋮ & ... & ... & ... & −1 \\ 0 & … & 0 & −1 & 2 \end{matrix})

Then

[H_{i}, H_{i}] = 0, [X_{i}^{+}, X_{j}^{−}] = δ_{ij}H_{i}, [H_{i}, X_{j}^{±}] = ±a_{ij}X_{j}^{±};  (SR^{0}) = (1.8.3^{0})
(adX_{i}^{+})^{1−a_{ij}}X_{j}^{+} = 0 and (adX_{i}^{−})^{1−a_{ij}}X_{j}^{−} = 0.  (SR^{±}) = (1.8.3^{±})

1.8.1. It turns out that a good deal of representation theory of classical Lie algebras and their various generalizations can be deduced from the Dynkin diagram and relations (1.8.3), see [Dy1], [Dy2]. These relations actually form the set of defining relations for 𝔤. Any simple finite dimensional Lie algebra can also be defined in a similar way as 𝔤(A) for some matrix A. Since it is quite difficult to verify for concrete algebras that given relations form a system of defining relations, it is more convenient to give another definition of the algebras 𝔤(A).

Let Γ be an abelian group. A Lie algebra 𝔤 is called Γ-graded if 𝔤 = ⊕_{α∈Γ} 𝔤_{α}, where each

𝔤_{α} is a vector subspace of 𝔤 and [𝔤_{α}, 𝔤_{β}] ⊂ 𝔤_{α+β}.

Let A = (a_{ij}) be an n × n-matrix with entries from 𝕂 and Q a free abelian group with a fixed basis α_{1}, …, α_{n}. The Lie algebra 𝔤̃(A) of rank n corresponding to A is a Q-graded Lie algebra with generators X_{1}^{−}, …, X_{n}^{−}, H_{1}, …, H_{n} and X_{1}^{+}, …, X_{n}^{+}, satisfying for any i, j = 1, …, n the following relations:

(a) [H_{i}, H_{j}] = 0, [H_{i}, X_{j}^{±}] = ±a_{ij}X_{j}^{+}, [X_{i}^{+}, X_{j}^{−}] = δ_{ij}H_{i};

(b) H_{1}, …, H_{n} form a basis of the subspace 𝔥 = 𝔤_{0} and 𝔤_{α_{i}} = 𝕂 · X_{i}^{+}, 𝔤_{−α_{i}} = 𝕂 · X_{i}^{−};

(c) Every non-zero Q-graded ideal of 𝔤 has a non-zero intersection with 𝔥.

It is easy to verify that 𝔤̃(A) exists for any matrix A and is determined by A up to an isomorphism; n is called the rank of 𝔤̃(A).

It is known [K1] that for certain matrices A there is a maximal Q-graded ideal 𝔪 of 𝔤̃(A) whose intersection with 𝔥 is zero and such that 𝔤(A) = 𝔤̃(A)/𝔪 is simple. It is a very interesting problem to describe the generators of 𝔪 explicitely. If 𝔤(A) is finite dimensional (or a twisted loop algebra), then these generators are the left hand sides of (1.8.3); for some other cases see [GL2].

The definition readily implies that there is a (unique) antiautomorphism^{1} ı : 𝔤 → 𝔤 defined on the generators by the formula ı(X_{i}^{+}) = X_{i}^{−}, ı(X_{i}^{−}) = X_{i}^{+} and ı(H_{i}) = −H_{i} for i = 1, …, n.

Roots and weights. Let 𝔤 = 𝔤(A). Put R = {α ∈ Q \ {0} | 𝔤_{α} ≠ 0}; the elements of R are called the roots of 𝔤, and the decomposition 𝔤 = 𝔥 ⊕ ( ⊕_{α∈R} 𝔤_{α}) is called the root

decomposition of 𝔤. By (b), each α_{i} is a root for i = 1, …, n; the elements α_{1}, …, α_{n} are called simple roots.

Let 𝔤^{+} be the subalgebra of 𝔤 generated by X_{1}^{+}, …, X_{n}^{+}, let 𝔤^{−} be generated by X_{1}^{−}, …, X_{n}^{−}. It is easy to verify that 𝔤 = 𝔤^{−} ⊕ 𝔥 ⊕ 𝔤^{+} (in all our examples this will be clear). A root α is called positive if 𝔤_{α} ⊂ 𝔤^{+} and negative if 𝔤_{α} ⊂ 𝔤^{−}; the set of positive (negative) roots of 𝔤 is denoted by R^{+} (resp., R^{−}).

Put Q^{+} = {∑ k_{i}α_{i} | k_{i} ∈ ℤ^{+}} and Q^{−} = −Q^{+}; then R^{+} = R ∩ Q^{+}, R^{−} = R ∩ Q^{−}. Though it is clear that R = R^{+} ∪ R^{−}, it is a wonder that all roots split into positive and negfative ones.

Clearly, if α = ∑ k_{i}α_{i} ∈ R^{+}, then the subspace 𝔤_{α} ⊂ 𝔤 is spanned by the elements [X_{i_{1}}^{+}, [X_{i_{2}}^{+}, …, [X_{i_{r−1}}^{+}, X_{i_{r}}^{+}] …]], where each X_{i}^{+} occurs k_{i} times. (A similar statement holds for negative roots.) Therefore, dim 𝔤_{α} < ∞ for any α ∈ R. Clearly, ı(𝔤_{α}) = 𝔤_{−α} for any α ∈ Q. It follows that −α ∈ R for all α ∈ R and dim 𝔤_{α} = dim 𝔤_{−α}; by (b), dim 𝔤_{α} = 1 if ±α is a simple root.

Let F = Q ⊗_{ℤ} ℂ be the vector space with basis α_{1}, …, α_{n}. Define the linear operator F → 𝔥′ (α ↦ \overline{α}) by the formula \overline{α}_{j}(H_{i}) = a_{ij} (i, j = 1, …, n). The description of the root subspaces 𝔤_{α} given above easily implies that [h, x] = \overline{α}(h)x for any h ∈ 𝔥, α ∈ R and x ∈ 𝔤_{α}.

Clearly, the map α ↦ \overline{α} is an isomorphism F ⟶ 𝔥′ if and only if the matrix A is invertible.

The assertion that 𝔰𝔩(n) is a Lie algebra of the form 𝔤(A) of rank n − 1 is almost obvious. In this case R^{+} = {α_{i} + α_{i+1} + ⋯ + α_{j} | 1 ≤ i ≤ j < n}; for α = α_{i} + ⋯ + α_{j} we have 𝔤_{α} = ℂX_{i,j+1}^{+} and 𝔤_{−α} = ℂX_{j+1,i}^{+}. Properties (a) and (b) are obvious, and (c) follows at once from the equality [X_{ij}^{+}, X_{ji}^{+}] = X_{ii}^{+} − X_{jj}^{+} ∈ 𝔥.

1.9. Our definition of 𝔤̃(A) does not require the relations (SR^{±}); these relations make sense only if a_{ij} is a nonpositive integer for i ≠ j. An important class of Lie algebras satisfying these relations is the following one.

1Recall that a linear map a : 𝔤 → 𝔤 is an antiautomorphism if a([x, y]) = [a(y), a(x)].

The matrix A = (a_{ij}) is called a Cartan matrix if a_{ii} = 2 or 0, a_{ij} is a nonpositive integer for i ≠ j and a_{ij} = 0 implies a_{ji} = 0. If A is symmetrizable, i.e., if there is an invertible diagonal matrix D such that AD is symmetric, then by (a), D can be chosen so that all its diagonal entries are positive rational numbers.

The Lie algebras 𝔤(A) or slightly larger algebras, see [K1], are called the (generalized) Kac-Moody algebras.

The matrix A is called decomposable if under some simultaneous permutation of rows

and columns it takes the form (\begin{matrix}A_{1} & 0 \\ 0 & A_{2} \end{matrix}). Clearly, in this case 𝔤(A) = 𝔤(A_{1}) ⊕ 𝔤(A_{2}).

Proposition . Let 𝔤 = 𝔤(A).

(a) The center 𝔷 of 𝔤 consists of the elements of 𝔥 that commute with all X_{i}^{+} and X_{i}^{−} (i = 1, …, n); the center is 0 if and only if A is invertible.

(b) If A is a Cartan matrix then [𝔤, 𝔤] = 𝔤.

(c) If A is an indecomposable Cartan matrix, then any proper graded ideal of 𝔤 is contained in 𝔷. In particular, if A is invertible, then 𝔤 has no proper graded ideals.

1.10. Let us show that 𝔬(n) and 𝔰𝔭(2n) are of the form 𝔤(A).

For finite dimensional vector spaces V and W we have a natural isomorphism Hom(V, W) ≃ V′ ⊗ W (the element ξ ⊗ w ∈ V′ ⊗ W corresponds to the operator v ↦ ξ(v)w for any ξ ∈ V′ and w ∈ W). Clearly, if V and W are 𝔤-modules then this isomorphism is compatible with 𝔤-module structures defined in 1.5. In particular, we identify ⊗^{2}V′ with Hom(V, V′); since V = (V′)′, we also have ⊗^{2}V = Hom(V′, V). Therefore, Λ^{2}V and S^{2}V may be considered not only as quotients of but also as submodules of Hom(V′, V), while Λ^{2}V′ and S^{2}V′ are submodules of Hom(V, V′).

Let V be an n-dimensional vector space and 𝔤 one of the algebras 𝔰𝔭(2n), 𝔬(2n) or 𝔬(2n + 1).

Then 𝔤 admits a ℤ-grading displayed in the following table:

\begin{matrix}𝔤 & k = −2 & k = −1 & k = 0 & k = 1 & k = 2 \\ 𝔰𝔭(2n) & – & S^{2}V′ & 𝔤𝔩(V) & S^{2}V & – \\ 𝔬(2n) & – & Λ^{2}V′ & 𝔤𝔩(V) & Λ^{2}V & – \\ 𝔬(2n + 1) & Λ^{2}V′ & V′ & 𝔤𝔩(V) & V & Λ^{2}V \end{matrix}

Proposition . Let 𝔤 be a simple finite dimensional Lie algebra and A the corresponding Cartan matrix (the matrices A are given in [OV]). Then A is invertible and indecomposable.

1.11. The Killing form B on 𝔤 is defined by

B(x, y) = tr ( ad x ∘ ad y), x, y ∈ 𝔤.

Proposition . The form B is symmetric, non-degenerate and invariant (the latter means that B([x, y], z) = B(x, [y, z]) for any x, y, z ∈ 𝔤).

1.12. The affine Kac–Moody algebra 𝔤^{(1)} (over ℂ) corresponding to 𝔤 is defined as follows. As a vector space 𝔤^{(1)} is (𝔤⊗ℂ[t, t^{−1}]) ⊃ + ℂc, where ℂ[t, t^{−1}] is the ring of Laurent polynomials in t. By definition c lies in the center of 𝔤^{(1)}, and the bracket of two monomials is defined by

[x ⊗ t^{i}, y ⊗ t^{j}] = \begin{cases}[x, y] ⊗ t^{i+j} & if i + j ≠ 0, \\ [x, y] + iB(x, y)c & if i + j = 0 \end{cases} for any x, y ∈ 𝔤.

The Lie algebra axioms can be verified directly. We shall write xt^{k} instead of x ⊗ t^{k} and simply x if k = 0.

Remark . The quotient algebra 𝔤^{(1)}/ℂc = 𝔤 ⊗ ℂ[t, t^{−1}] is called a loop algebra. (The meaning of the term stems from the fact that the loop algebra can be naturally realized as the algebra of 𝔤-valued functions on the unit circle with finite Fourier series, and the pointwise bracket.)

1.13. We want to prove that every affine algebra 𝔤^{(1)} is a Kac–Moody algebra. For this we need some additional information on 𝔤.

Proposition . Let 𝔤 be a simple finite dimensional Lie algebra and R the set of roots of 𝔤. Then 𝔤 has the maximal root γ, that is, γ − α ∈ Q^{+} for all α ∈ R.

In 𝔥_{γ} = [𝔤_{γ}, 𝔤_{−γ}], there is a unique element h_{γ}^{∨} such that \overline{γ}(h_{γ}^{∨}) = 2. If the elements X_{γ}^{+} ∈ 𝔤_{γ} and X_{γ}^{−} ∈ 𝔤_{−γ} are chosen so that [X_{γ}^{+}, X_{γ}^{−}] = h_{γ}^{∨}, then B(X_{γ}^{+}, X_{γ}^{−}) = \frac{1}{2}B(h_{γ}^{∨}, h_{γ}^{∨}) > 0.

In Table we give the explicit values of the root γ, linear form \overline{γ} and vectors h_{γ}^{∨}, X_{γ}^{+} and X_{γ}^{−} for all the classical 𝔤 (in the notation introduced above).

\begin{matrix}𝔤 & γ & \overline{γ} & h^{∨}X_{γ}^{+} & X_{γ}^{+} & X_{γ}^{−} \\ 𝔰𝔩(n) & α_{1} + ⋯ + α_{n−1} & ξ_{1} − ξ_{n} & X_{11}^{+} − X_{nn}^{+} & X_{1n}^{+} & X_{n1}^{+} \\ 𝔰𝔭(2n) & 2(α_{1} + ⋯ + α_{n−1}) + α_{n} & 2ξ_{1} & X_{11}^{+} & v_{1} ⊗ v_{1} & v′_{1} ⊗ v′_{1} \\ 𝔬(2n) & α_{1} + 2(α_{2} + ⋯ + α_{n−2}) + α_{n−1} + α_{n} & ξ_{1} + ξ_{2} & X_{11}^{+} + X_{22}^{+} & v_{1} ∧ v_{2} & −v′_{1} ∧ v′_{2} \\ 𝔬(2n + 1) & α_{1} + 2(α_{2} + ⋯ + α_{n}) & ξ_{1} + ξ_{2} & X_{11}^{+} + X_{22}^{+} & v_{1} ∧ v_{2} & −v′_{1} ∧ v′_{2} \end{matrix}

The maximal root γ always exists.

1.14. Proposition . If 𝔤 is a simple finite dimensional Lie algebra of rank n then the affine Lie algebra 𝔤^{(1)} is a Kac–Moody algebra of rank n + 1. The corresponding Cartan matrices are given by the extended Dynkin diagrams.

§2. Homology and cohomology of Lie algebras

We are thankful to A. Dzhumadildaev who drew our attention to the fact that all mathematical textbooks give a “bad” formula for the differential, the one with an extra loop from the computers’ point of view.

2.0.1. Cohomology. Let M be a left 𝔤-module. The space of q-cochains on 𝔤 with values (coefficients) in M is the space

C^{q}(𝔤; M) = M ⊗ Λ^{q}(𝔤′) for q ≥ 0 and C^{q}(𝔤; M) = 0 for q < 0

of skew-symmetric linear in every variable functions on 𝔤 with values in M. In particular, C^{0}(𝔤; M) = M. The differential d_{q} : C^{q}(𝔤; M) ⟶ C^{q+1}(𝔤; M) is determined as 0 for q < 0 and for q ≥ 0 by the formula (the hatted variable is to be omitted)

\begin{matrix}d_{q}(f)(g_{1}, …, g_{q+1}) = ∑_{1≤s<t≤q+1}(−1)^{s}f(g_{1}, …, g_{s−1}, [g_{s}, g_{t}], g_{s+1}, …, ĝ_{t}, …, g_{q+1}) \\ + ∑_{1≤s≤q+1}(−1)^{s−1}g_{s}f(g_{1}, …, ĝ_{s}, …, g_{q+1}) \end{matrix}

for any m ∈ M, g_{1}, …, g_{q+1} ∈ 𝔤.

Theorem . d_{q+1} ∘ d_{q} = 0.

§2. HOMOLOGY AND COHOMOLOGY OF LIE ALGEBRAS 19

(A shorthand of this Theorem: d^{2} = 0, where d = ∑ d_{q} is considered in V = ⊕ C^{q}.) The space of q-cohomology of 𝔤 with values (coefficients) in M is

H^{q}(𝔤; M) = Ker d_{q+1}/Imd_{q}.

For the trivial module of coefficients we abbreviate H^{q}(𝔤; 𝕂) to H^{q}(𝔤).

2.0.2. Homology. Let M be a right 𝔤-module. The space of q-chains on 𝔤 with values (coefficients) in M is

C_{q}(𝔤; M) = M ⊗ Λ^{q}(𝔤) for q ≥ 0 and C_{q}(𝔤; M) = 0 for q < 0.

In particular, C_{0}(𝔤; M) = M. The differential d_{q} : C_{q}(𝔤; M) ⟶ C_{q−1}(𝔤; M) determines as 0 for q ≤ 0 and for q > 0 by the formula

\begin{matrix}d_{q}(m ⊗ (g_{1} ∧ ⋯ ∧ g_{q})) = ∑_{1≤s<t≤q}(−1)^{s}m ⊗ (g_{1} ∧ ⋯ ∧ g_{s−1} ∧ [g_{s}, g_{t}] ∧ g_{s+1} ∧ ⋯ ∧ ĝ_{t} ∧ ⋯ ∧ g_{q}) \\ + ∑_{1≤s≤q}(−1)^{s−1}mg_{s} ⊗ (g_{1} ∧ ⋯ ∧ ĝ_{s} ∧ ⋯ ∧ g_{q}) \end{matrix}

for any m ∈ M, g_{1}, …, g_{q} ∈ 𝔤.

Theorem . d_{q} ∘ d_{q+1} = 0.

(A shorthand: d^{2} = 0, where d = ∑ d_{q} is considered in ⊕C_{q}.) The space of q-homology of 𝔤 with values (coefficients) in M is

H_{q}(𝔤; M) = Ker d_{q−1}/Imd_{q}.

For the trivial module of coefficients we abbreviate H_{q}(𝔤; 𝕂) to H_{q}(𝔤).

If 𝔤 and M are finite dimensional, then

H_{q}(𝔤; M) ≃ (H^{q}(𝔤; M′))′.

2.0.3. (Co)homology and (co)induced modules. For an 𝔥-module M we have ([Fu])

H^{q}(𝔤; C_{𝔥}^{𝔤}(M)) ≃ H^{q}(𝔥; M); H_{q}(𝔤; I_{𝔥}^{𝔤}(M)) ≃ H_{q}(𝔥; M).

Hereafter 𝔤 is a simple Lie (super)algebra and M an irreducible 𝔤-module unless otherwise stated.

2.1.1. Theorem . dim 𝔤 < ∞. Then

H^{∗}(𝔤) = Λ[ξ_{3}, …, ξ_{2k_{n}+1}],

where the numbers 2k_{i} for 1 ≤ i ≤ rk𝔤 are the so-called exponents listed, e.g., in [OV];

H^{∗}(𝔤; M) = 0 if M = L^{λ} is any nontrivial irreducible and dim M < ∞

This theorem is a classical result, cf. [FF2]. For the general modules even irreducible, but not finite dimensional, little is known.

In certain particular cases cohomology with coefficients in irreducible infinite dimensional modules (not only with vacuum vector) are computed.

2.1.2. Theorem . ([FF2]) Let 𝔤 be vectorial. Then for certain more or less explicitely described ideals I_{v} and I_{k} we have

H^{∗}(𝔳𝔢𝔠𝔱(n)) ≃ H^{∗}(𝔰𝔩(n + 1))/I_{v}, H^{∗}(𝔨(2n + 1)) ≃ H^{∗}(𝔰𝔭(2n + 2))/I_{k}.

These results are rather new. For the remaining two series of simple vectorial Lie superalgebras from Cartan’s list we know nothing. In 1971 I. Gelfand, D. Kalinin and D. Fuchs used computer to estimate cohomology of 𝔥(2), the Lie algebra of Hamiltonian vector fields on the plane, from below. This result is not surpassed even by modern computers, though Fuchs announced a result towards the proof of finite dimension of the cohomology in question. A recent attempt [KK2] to compute this cohomology failed: the complexity of calculations grows too steeply and new ideas are a must.

Denote by 𝔡(𝔤) the Lie algebra of derivations of 𝔤. The cohomology with trivial coefficients are known for 𝔡(𝔰𝔳𝔢𝔠𝔱(n)) [FF2]. For 𝔡(𝔥(2n)), it is known that they are finite dimensional, but, strangely, no explicit formula is available, though to derive it should not be difficult.

Thanks to the first theorems of cohomology theory, H^{∗}(𝔤; coind(M)) ≃ H^{∗}(𝔤_{0}; M) for any 𝔤_{0}-module M. Therefore, among nontrivial coefficients in irreducible modules it suffices to consider the exceptional, i.e., not coinduced modules. One of the most interesting nontrivial coefficients for 𝔰𝔳𝔢𝔠𝔱(n), 𝔭𝔬(2n) and 𝔥(2n) is the adjoint module and other non-induced modules (of closed differential forms, etc.).

Only in a particular case, 𝔤 = 𝔳𝔢𝔠𝔱(1) (with polynomial coefficients), cohomology are computed for various coefficients [FF2], e.g.,

dim H^{q}(𝔤; F′_{λ}) = dim H^{q}(𝔤; F_{λ}) = \begin{cases}1 & if λ = \frac{1}{2}(3q^{2} ± q) or λ = \frac{1}{2}(3(q − 1)^{2} ± (q − 1)) \\ 0 & otherwise. \end{cases}

2.1.3. Cohomology of loop algebras. Feigin’s thesis (Selecta Math. Soviet. 7 (1988), no. 1, 49–62) contains the description of these cohomology with trivial coefficients only.

For nontrivial coefficients no systematic treatment is known to us and virtually nothing is known of (co)homology of Kac–Moody algebras or twisted loop algebras, except interesting, though too general (without any down-to-earth applications) results by P. Zusmanovich aborted ca 10 years ago. (We sincerely hope his revitalized interest in the topic will last a while.) Nothing is known even about deformations, i.e., H^{2}(𝔤; 𝔤).)

2.1.4. Cohomology of 𝔴𝔦𝔱𝔱 and 𝔳𝔦𝔯. These were studied by Feigin and Fuchs and Poletaeva, see [FF2] and [P4]. For cohomology of four series of their superversions ([GLS1]) little is known.

2.1.5. Cohomology of 𝔤𝔩(λ). The only result is [F2].

2.1.6. Cohomology of “stripy” Lie algebras: 𝔤𝔩(∞) and 𝔬(2∞), 𝔬(2∞ + 1), 𝔰𝔭(2∞). These algebras of infinite matrices with support in a stripe along the main diagonal were investigated by Feigin and Tsygan, see a review in [FF2]. Little is known.

2.2. Lie superalgebras 𝔤. Investigated are only trivial coefficients and H^{i}(𝔤; 𝔤) for I ≤ 2 in certain cases.

For the finite dimensional simple and Lie superalgebras and their “almost simple” relatives the known results with trivial coefficients are announced in [FL] (in certain cases proof is given in [Fu] and [Kz]). In some cases results of [FL] should be corrected, cf. [KK1], and several series are still to be considered.

The cohomology of exceptional simple Lie superalgebras with trivial coefficients are computed by a wonderful and ingenious method due to Gruson [Gr1]–[Gr3], a student of M. Duflo. This method has, however, limited applications (at least, so far).

Nothing is known on nontrivial coefficients, except when they constitute a “typical” module and 𝔤 possesses an invariant nondegenerate supersymmetric bilinear form (in which case the arguments from Lie algebra theory are applicable).

Of the simple infinite dimensional vectorial Lie superalgebras results are available only for the general series 𝔳𝔢𝔠𝔱(m|n), see [AF], for trivial coefficients only and with some restrictions on n and m.

2.2.1. Lie superalgebras. dim 𝔤 < ∞.

H^{∗}(𝔤𝔩(m|n)) ≃ \begin{cases}H^{∗}(𝔤𝔩(m)) & if m ≥ n \\ H^{∗}(𝔤𝔩(n)) & otherwise. \end{cases}

H^{∗}(𝔭𝔤𝔩(n|n)) ≃ H^{∗}(𝔤𝔩(n)) ⊗ 𝕂[h_{2}], where h_{2} represents the extension 𝔤𝔩(n|n) ⟶ 𝔭𝔤𝔩(n|n).

H^{∗}(𝔰𝔩(m|n)) ≃ \begin{cases}H^{∗}(𝔰𝔩(m)) & if m ≥ n \\ H^{∗}(𝔰𝔩(n)) & otherwise. \end{cases}

H^{∗}(𝔭𝔰𝔩(n|n)) ≃ H^{∗}(𝔰𝔩(n)) ⊗ 𝕂[h_{2}],

where h_{2} represents the extension 𝔰𝔩(n|n) ⟶ 𝔭𝔰𝔩(n|n). for n > 2

H^{∗}(𝔭𝔰𝔩(2|2)) ≃ ℂ[x_{2}, y_{2}, h_{2}, ξ_{3}]/(x_{2}y_{2} − h_{2}^{2}).

Observe that the cocycle ξ_{3} was missed in the original calculations [FL] and the answer was rectified at our request by A. Shapovalov with the help of SuperLie, see also [KK1]. The 2-cocycles represent the following central extensions:

x_{2} : (\begin{matrix}A_{1} & B_{1} \\ C_{1} & D_{1} \end{matrix}) , (\begin{matrix}A_{2} & B_{2} \\ C_{2} & D_{2} \end{matrix}) ↦ trB_{1}B_{2};

y_{2} : (\begin{matrix}A_{1} & B_{1} \\ C_{1} & D_{1} \end{matrix}) , (\begin{matrix}A_{2} & B_{2} \\ C_{2} & D_{2} \end{matrix}) ↦ trC_{1}C_{2}

h_{2} : (\begin{matrix}A_{1} & B_{1} \\ C_{1} & D_{1} \end{matrix}) , (\begin{matrix}A_{2} & B_{2} \\ C_{2} & D_{2} \end{matrix}) ↦ tr(B_{1}C_{2} + C_{1}B_{2})

Clearly, the degrees of the cocycles are:

deg x_{2} = 2, deg y_{2} = −2, deg h_{2} = 0

and, therefore, resemble 𝔰𝔩(2) very much. Having defined 𝔰𝔩(2) structure on the space of these cocycles, we get the superalgebra 𝔬𝔰𝔭_{α}(4|2) for α = 0.

H^{∗}(𝔰𝔩(1|1)) ≃ ℂ[ξ_{1}, η_{1}]/(ξ_{1}η_{1}).

H^{∗}(𝔭𝔢(n)) ≃ ℂ[ξ_{1}, …, ξ_{4k+1}, … ξ_{2n−1}].

H^{∗}(𝔰𝔭𝔢(2n + 3)) ≃ H^{∗}(𝔭𝔢(2n + 3)).

H^{∗}(𝔰𝔭𝔢(2n + 4)) ≃ H^{∗}(𝔭𝔢(2n + 4)) ⊗ \begin{cases}𝕂[x_{n+2}] & if n ≡ 0 (mod 4) \\ 𝕂[ξ_{n+2}] & if n ≡ 2 (mod 4) \end{cases} .

H^{∗}(𝔰𝔭𝔢(2)) ≃ 𝕂[x_{1}, y_{2}]/(x_{1}^{2} − y_{2}).

To describe the cohomology of 𝔮 series, let s_{1}, …, s_{n} be the elementary symmetric polynomials in n even indeterminates x, i.e.,

s_{1} = x_{1} + ⋯ + x_{n},
. . . . . . . . . . . . . . . . . .
s_{1} = x_{1} … x_{n};

let S_{1}, …, S_{n} be the elementary symmetric polynomials in the squares of n even indeterminates x, i.e.,

S_{1} = x_{1}^{2} + ⋯ + x_{n}^{2},
. . . . . . . . . . . . . . . . . .
S_{1} = x_{1}^{2} … x_{n}^{2}.

Denote by x_{i} the generators of S.(Π(𝔤¯1))^{𝔤¯0}. We have then

H^{∗}(𝔮(n)) ≃ ℂ[s_{1}, …, s_{n}]/(S_{1}, …, S_{n});
H^{∗}(𝔰𝔮(n)) ≃??;
H^{∗}(𝔭𝔮(n)) ≃??;
H^{∗}(𝔭𝔰𝔮(n)) ≃??

The cohomology of exceptional Lie superalgebras are computed by an ingeneous method due to Caroline Gruson [Gr1]-[Gr3] and the answer is:

H^{∗}(𝔞𝔤_{2}) ≃ H^{∗}(𝔤_{2});
H^{∗}(𝔞𝔟_{3}) ≃ H^{∗}(𝔬(7));
H^{∗}(𝔬𝔰𝔭_{λ}(4|2)) ≃ H^{∗}(𝔬(4)) ≃ H^{∗}(𝔰𝔩(2) ⊕ 𝔰𝔩(2)).

It seems, however, that her method is almost exhausted as far as simple algebras are concerned, and, anyway, we have to consider various nontrivial coefficients; to do this by hands is hardly possible.

2.2.2. Vectorial Lie superalgebras. Of the vectorial Lie superalgebras the general results are available for the general series only:

H^{∗}(𝔳𝔢𝔠𝔱(m|n)) ≃ H^{∗}(𝔳𝔢𝔠𝔱(0|n)) ≃ H^{∗}(S^{2n−1})

the first isomorphism being induced by an embedding 𝔳𝔢𝔠𝔱(0|n) ⟶ 𝔳𝔢𝔠𝔱(m|n).

H^{∗}(𝔳𝔢𝔠𝔱(n|n)) ≃ H^{∗}(Σ^{2n}U(n)),

where Σ denotes the suspension, U(n) is the unitary group.

By the same reasons as for Lie algebra case [Fu], the cohomology of 𝔡(𝔤) for the vectorial Lie superalgebras 𝔤 are finite dimensional; observe that sometimes 𝔡(𝔤) = 𝔤 (e.g., for 𝔳𝔢𝔠𝔱(m|n), 𝔨(2m + 1|n), 𝔪(n)). In these cases, especially for the exceptional algebras one can compute the final answer.

2.3. Theorem . Borel-Weil-Bott-Kostant-Bernstein-Gelfand-Gelfand’s theorem (BWZ) Let M be an irreducible finite dimensional 𝔤-module; W the Weyl group of 𝔤, let l(w), the length of w, be the minimal number of generators of W needed to express w. Then

dim H^{i}(𝔫; M) = {w ∈ W (𝔤) | l(w) = (i)}.

A similar statement holds for smaller nilpotent subalgebras 𝔫_{i_{1}...i_{k}} generated by all the (say positive) Chevalley generators, except X_{i_{1}}, …, X_{i_{k}}. About cohomology of algebras 𝔫_{i_{1}...i_{k}} with coefficients in modules which are not 𝔤-modules, e.g., in the adjoint module of in 𝔤/𝔫_{i_{1}...i_{k}}^{±}, little is known; mainly these are statements for i = 0, 1, 2, cf. Tolpygo’s papers [T], [TL] and Premet’s results below for a review, new results and open problems.

2.4. Singular vectors and invariant differential operators. As shown in [Fu], description of vacuum (singular) vectors in Verma modules is equivalent to calculation of certain cohomology; in turn these vectors are a first step in the classification of differential operators acting in tensor fields and invariant with respect to various groups of diffeomorphisms ([BL]). All physical laws are expressed in terms of invariant differential operators, their classification is important (problem Veblen raised on 1928 Mathematical congress).

2. Analytical results

In this chapter we reproduce some of analytical results (from Premet’s letters to Leites in 1989) as a motivation for computer-aded study . Premet’s results cover earlier results by Leger and Luks and complement Tolpygo’s results [TL]. Their complete proof deserves to be published and we hope this note will prompt even further studies.

§1. Background

Hereafter the ground field 𝕂 is ℂ, unless otherwise stated. We consider cohomology H^{i}(𝔫; M), where 𝔫 is a nilpotent subalgebra of a simple Lie algebra or superalgebra 𝔤, and M an 𝔫-module which is not necessarily a 𝔤-module.

Our motivations:

— the Riemann tensor can be formulated in terms of H^{2}(𝔫; 𝔤) for a particular pairs 𝔫 and 𝔤, the ones corresponding to the ℤ-grading of 𝔤 of the form 𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} in which case 𝔫 = 𝔤_{−1}. Moreover, the result, i.e., H^{2}(𝔫; 𝔤) is convenient to represent succinctly as a 𝔤_{0}-module, see [Go].

Such a ℤ-grading (𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1}) corresponds to some minimal nilpotent subalgebras 𝔫 = 𝔤_{−1} of 𝔤 and pure curiosity requires to consider other minimal (and not only minimal) nilpotent subalgebras. This insatible curiosity became grounded on solid applications when the cohomology H^{2}(𝔫; 𝔤) for the remaining minimal and, generally, any other 𝔫 had been inerepreted as analogs of Riemann tensor for non-holonomic structures [L3].

When such an interpretation is at hand, one naturally passes to all nilpotent subalgebras 𝔫 of any simple 𝔤; some of them are direct analogs of constructions Manin studied in supersetting [Ma].

In a very inspiring paper [V] Vershik launched a program: “study non-holonomic structures”, in particular, analogs of Riemann tensor for them.

Vershik aborted these studies, presumably because he himself did not believe that the general definition of such a tensor exists (see [V]) and nobody followed his research.

M. Zelikin informed us that recently Agrachev also introduced analogs of of Riemann tensor for non-holonomic structures associated with optimal control (so far we did not see anything published). For various non-holonomic structures associated with optimal control see, e.g., [Ag].

Apart from appearing in the left hand side of Einstein Equations, the Riemann tensor on a manifold, being a curvature, is responsible for stability of the dynamical system on this manifold ([Ar]); the cohomology similar to H^{2}(𝔫; 𝔤) but formualted in equivalent (but less convenient for calcualtions) terms of Spencer cohomology are criteria for formal integrability of differential equations [Br].

These areas of possible applications seem to be already sufficient to proceed with calculations for various examples. Still, there are more areas; one of them is economics, see [SV].

— desire to represent known results on cohomology of nilpotent subalgebras of simple Lie algebras in more graphic form than classical results. Indeed, H^{∗} possesses several superalgebra structures. What are they? To formulate the problem, we need some preliminaries.

A recent result of Löfvall and Roos [LR] is an illustration of what we had in mind as a model of description: in [LR] they reduce description of infinite dimensional nilpotent Lie superalgebras 𝔤 to simple finite dimensional ones, 𝔥, having realized that 𝔤 = 𝔥 × tℂ[t]. (There answer can be represented still more conveniently if 𝔤 is represented as an 𝔥-module, cf. [GLP].)

1.1. Terminological conventions. Let 𝔤 be a simple Lie algebra. The 𝔤-module V with highest (lowest) weight ξ will be denoted by V^{ξ} (resp. V_{ξ}) or R(ξ). Let R be the root system of 𝔤, B its base (system of simple roots).

Recall (see 1.8 above) that every 𝔤 can be recovered from an integer r × r matrix (A_{ij}) (called the Cartan matrix) as the quotient of the free Lie algebra generated by X_{1}^{±}, …, X_{r}^{±} called Chevalley generators of 𝔤 to which one usually adds H_{i} = [X_{i}^{+}, X_{i}^{−}] modulo the relations

[X_{i}^{+}, X_{j}^{−}] = δ_{ij}H_{i}; [H_{i}, H_{j}] = 0;
[H_{i}, X_{j}^{±}] = ±A_{ij}X_{j}^{±};
(adX_{i}^{±})^{1−A_{ij}}(X_{j}^{±}) = 0.

Clearly, by rescaling (H_{i} ⟶ λH_{i}) the Chevalley generators we may assume that A_{ii} = 2 or 0 for all the i. For finite dimensional Lie algebras the Cartan matrices are very special, in particular, A_{ii} = 2 for all i.

It is easy to see that 𝔤 = 𝔫^{−} ⊕ 𝔥 ⊕ 𝔫^{+}, where 𝔫^{±} are generated by the X_{i}^{±} and 𝔥 = Span(H_{1}, …, H_{r}).

Let I = {i_{1}, …, i_{k}} be a subset of I = {1, … r}, and 𝔭_{I}^{+} the subalgebra of 𝔤 generated by the X_{i}^{±} for i ∈ I and X_{i}^{+} for i ∈ {1, … r} \ I. It is called a (positive) parabolic subalgebra.

Let 𝔫_{I}^{−} be the opposite nilpotent algebra, i.e., 𝔫_{I}^{−} is the ideal of 𝔫^{−} generated by X_{i}^{−} for i ∈ {1, … r} \ I.

In particular, 𝔫^{±} = 𝔫_{I}^{±} are the maximal nilpotent subalgebras. Clearly, 𝔭_{I}^{±} = 𝔩_{I} ⊕ 𝔫_{I}^{±}, where 𝔩_{I} (called the Levi subalgebra) is the direct sum of a semisimple Lie algebra and a subalgebra 𝔥_{𝔩_{I}}^{comp} of 𝔥 complementing the Cartan subalgebra 𝔥_{𝔩_{I}} of 𝔩_{I}, so that 𝔥_{𝔩_{I}}^{comp}⊕ 𝔥_{𝔩_{I}} = 𝔥.

Having selected the Chevalley generators X_{i_{1}}^{±}, …, X_{i_{k}}^{±} corresponding to roots α_{i_{1}}^{±}, …, α_{i_{k}}^{±}, let R_{I} ⊂ R be a subsystem with base B_{I} corresponding to the remaining simple roots.

Let W = W (R) be the Weyl group, i.e., the group generated by the reflections s_{i} in simple roots. For any I, define the subgroup W_{I} ⊂ W corresponding to R_{I} (with base B_{I}) as

W_{I} = {w ∈ W | w^{−1}(B \ B_{0}) > 0}.

Define the length l(w) of any element w ∈ W as the minimal number of the simple reflections s_{i} required to represent w. Denote the subset of elements of length i in W by W (B_{I})^{i} and its cardinality by k_{I}(i). In particular, if B_{I} = B, let k(i) be the number of elements from W of length i.

For the definition of the exponents and several other standard notations that might arise see [OV] or [Bo].

1.2. General statements on cohomology.

1.2.1. Theorem . 1) Let M be an irreducible finite dimensional 𝔤-module. Then H^{∗}(𝔤; M) = 0 if M ≠ 𝕂.

2) H^{∗}(𝔤) ≃ E[ξ_{1}, …, ξ_{r}], where r = rk𝔤 and deg ξ_{i} = e_{i} + 1. (E^{i} is the operator of the ith exterior power; E = ∑ E^{i}.)

1.2.2. Cohomology of nilpotent subalgebras. There are two types of modules: with values in 𝔤-modules and other modules.

Theorem . 1) (Borel-Weil-Bott-Kostant-Bernstein-Gelfand-Gelfand’s theorem (BWZ) Let M be an irreducible finite dimensional 𝔤-module. Then

dim H^{i}(𝔫; M) = k(i).

2) A similar statement holds for lesser, non-maximal, nilpotent subalgebras 𝔫_{I}^{±}, where I = {i_{1}, …, i_{k}}: Let M be an irreducible finite dimensional 𝔤-module. Then

dim H^{j}(𝔫_{I}^{±}; M) = k_{I}(j).

Observe, that there is a more economic way to describe the spaces H^{j}(𝔫_{I}^{±}; M). Namely, both 𝔫_{I}^{±} and M are 𝔩_{I}-modules; hence, so is H^{j}(𝔫_{I}^{±}; M). Since 𝔩_{I} is semisimple, every its finite dimensional module is completely reducible and every irreducible component is completely described by its highest (or lowest) weight and the corresponding vector.

For example, let 𝔫_{I} = 𝔫_{i} be a minimal algebra corresponding to #(I) = 1 and M = 𝔤. Suppose that 𝔫_{i} is commutative (this is the case when G/P, where G corresponds to 𝔤 and P to 𝔭_{i}, is a classical domain, e.g., the Grassmannian Gr_{k}^{n+k}). Then H^{j}(𝔫_{i}; M) is a 𝔰𝔩(k) ⊕ 𝔰𝔩(n + k)-module. Moreover, since dim H^{j}(𝔫_{i}; M) does not depend on M, it suffices to describe it in the simplest case of the trivial module, M = 𝕂. In this case the cohomology are easy to compute thanks to commutativity of 𝔫_{i}:

H^{j}(𝔫_{i}) = E^{j}(𝔫_{i}).

For j = 1, 2 these cocomology have interesting applications and they are our prime concern, though the complete information, i.e., H^{∗} = ⊕H^{j} is also important as a base for computating cohomology of infinite dimensional algebras and for purely mathematical interest.

About cohomology of algebras 𝔫_{i_{1}...i_{k}}^{±} with coefficients in modules which are not 𝔤modules, e.g., in the adjoint module, or in 𝔤/𝔫_{i_{1}...i_{k}}^{±}, little is known, see [T] and Premet’s theorem below.

Observe that though the BWB theorem is very neat for generic irreducible 𝔤-modules M, for H^{∗}(𝔫) the description is rather unnatural. Indeed, H^{∗}(𝔫) possesses an algebra structure. Who describes a polynomial or exterior algebra, or their quotients, via dimensions of homogeneous components?!

But here the situation is the same: H^{∗}(𝔫) and H^{∗}(𝔫; 𝔫) possess natural algebraic structures and one of the reasonable descriptions of any algebra is via its generators and defining relations.

The algebraic structures we have in mind are as follows.

If M possesses an associative algebra structure the multiplication in M and the 𝔤-action being related by the Leibniz rule (say, M is the 1-dimensional trivial module), then H^{∗}(𝔤 : M) possesses an associative superalgebra structure with respect to the product of cochains.

For the adjoint coefficients, for any a ∈ C^{p}(𝔤; 𝔤), b ∈ C^{q}(𝔤; 𝔤) define ab ∈ C^{p+q−1}(𝔤; 𝔤) by setting

ab(x_{1}, …, x_{p+q−1}) = ∑ A(a(b(x_{1}, …, x_{q}), x_{q+1}, …, x_{p+q−1})),

where A is the alternation, i.e.,

\begin{matrix}A(f(x_{1}, …, x_{q}), x_{q+1}, …, x_{p+q−1})) = \\ \frac{1}{(p+q−1)!} ∑_{σ∈S_{p+q−1}} sign(σ; x)(f(x_{σ(1)}, …, x_{σ(q)}), x_{σ(q+1)}, …, x_{σ(p+q−1)}), \end{matrix}

where sign(σ; x) is a cocycle on S_{p+q−1} described in [L2], no. 23.

Set

[a, b] = ab − (−1)^{(p(a)−1)(p(b)−1)}ba for any a, b ∈ C^{∗}(𝔤; 𝔤).  (∗)

Theorem . Π(H^{∗}(𝔤; 𝔤)) with the multiplication (∗) is a Lie superalgebra.

Observe that the multiplication is defined on cocycles and the statement is made about their cohomology classes.

§2. Premet’s Theorems

2.1. Let Dynkin diagram of B be as follows:

[figure: •; •; ∘; •; •; •; ∘; ∘; ∘; •; •; ∘; •; •; •; ∘; •; •; ∘; ∘]

and let B_{I} consist of black nodes. Let us represent B_{I} as the union of connected subdiagrams:

B_{I} = B_{I}^{(1)} ∐ … ∐ B_{I}^{(s)},

where s (in our example s = 5) is the number of connected components of the Dynkin diagram D_{I} of B_{I} and where B_{I}^{(i)} corresponds to the i-th connected components of D_{I} (counted from left to right). Set

c_{0} = card B, c_{i} = card ({α ∈ B \ B_{I} | (α, B_{I}^{(i)}) ≠ 0}) − 1.

Clearly, if B_{I} ≠ B, then c_{i} ∈ {0, 1, 2}. For example, for the diagram of 𝔬(20) depicted above we have:

c_{0} = 20, c_{1} = 0, c_{2} = 1, c_{3} = 1, c_{4} = 1, c_{5} = 2.

The following statement is obvious.

Statement . 1) c_{i} = 2 if and only if R is of type D_{n}, E_{6}, E_{7}, E_{8}, one of the endpoints of D_{I}^{(i)} is a branching point for D, and the remaining endpoint of D_{I}^{(i)} is not an endpoint of D.

2) c_{i} = 0 if and only if all but one of the end vertices of the diagram of B_{I}^{(i)} are the end vertices for the diagram of R.

2.2. Let 𝔤 be a simple Lie algebra of rank r > 1; let 𝔭 = 𝔫_{−}⊕𝔩 be its parabolic subalgebra, where 𝔩 is the Levi subalgebra, 𝔩^{(1)} the derived algebra of the Levi subalgebra, 𝔷 = 𝔷(𝔩), the center of 𝔩. (Recall, that 𝔭 is generated by the Chevalley generators X_{i}^{±}, i = 1, …, r, of 𝔤 except several X_{i}^{+}; 𝔩 is generated by all the remaining X_{i}^{±}; finally, 𝔩 = 𝔩^{(1)} ⊕ 𝔷.)

Theorem . H^{2}(𝔫_{−}; 𝔫_{−} ⊕ 𝔩^{(1)}) = H^{2}(𝔫_{−}; 𝔤) ⊕ H^{1}(𝔫_{−}; (𝔫_{−} ⊕ 𝔷)^{∗}).

Let B_{1} = B \ B_{I} and W (B_{1})^{(2)} = {w ∈ W (B_{1}) | l(w) = 2}.

Set 𝔫_{−}^{ab} = (𝔫_{−}/𝔫_{−}^{(1)})^{∗} = H^{1}(𝔫_{−}).

Corollary . 1) The following sequence is exact:

\begin{matrix}0 ⟶ 𝔫_{−}^{ab} ⟶ 𝔷^{∗} ⊗ 𝔫_{−}^{ab} ⟶ H^{1}(𝔫_{−}; 𝔤/(𝔫_{−} ⊕ 𝔩^{(1)})) ⟶ H^{1}(𝔫_{−}; 𝔫_{−}^{∗}) ⟶ \\ H^{2}(𝔫_{−}) ⊕ ⊕_{w∈W (B_{1})^{(2)}} R(ρ − w(ρ)) ⟶ 0. \end{matrix}

2) If dim 𝔷 = 1, then the sequence

0 ⟶ H^{1}(𝔫_{−}; (𝔫_{−} ⊕ 𝔷)^{∗}) ⟶ H^{1}(𝔫_{−}; 𝔫_{−}^{∗}) ⟶ H^{2}(𝔫_{−}) ⟶ 0

§2. PREMET’S THEOREMS 29

is exact. In particular, a) if 𝔫_{−} is abelian, then

H^{2}(𝔫_{−}; 𝔫_{−} ⊕ 𝔩^{(1)}) = H^{2}(𝔫_{−}; 𝔤) ⊕ S^{2}(𝔫_{−}^{∗});

b) if 𝔫_{−} is a Heisenberg algebra, then

H^{2}(𝔫_{−}; 𝔫_{−} ⊕ 𝔩^{(1)}) = H^{2}(𝔫_{−}; 𝔤) ⊕ S^{2}(𝔫_{−}/𝔷(𝔫_{−}))^{∗}.

2.3. Lemma . Let E = L^{λ} be an irreducible (finite dimensional) 𝔤-module with highest weight λ such that E ≃ E^{∗}. Let V be a 𝔭_{−}-invariant subspace in E which contains E_{−} = ⊕ ; μ=∑ k_{i}α_{i}, k_{i}<0 E_{μ}. Then for any i < rk𝔤 we have the 𝔩-module isomorphism:

H^{i}(𝔫_{−}; V) = H^{i}(𝔫_{−}; E) ⊕ H^{i−1}(𝔫_{−}; E/V).

Proof of Lemma 2.3. Our main tool is the following version of the BWB theorem.

Theorem . (BWZ). H^{i}(𝔫_{−}; E) is the direct sum of 𝔩-modules with lowest weghts −w_{i}(λ+ ρ) + ρ for i = 1, …, k(i); each such module enters with multiplicity 1.

As is well known [FF], with every short exact sequence of 𝔫_{−}-modules

0 ⟶ V ⟶ E ⟶ E/V ⟶ 0

there is associated the long exact cohomology sequence

0 ⟶ H^{0}(𝔫_{−}; V) ⟶ H^{0}(𝔫_{−}; E) ⟶^{ϕ_{0}} H^{0}(𝔫_{−}; E/V) ⟶

H^{1}(𝔫_{−}; V) ⟶ H^{1}(𝔫_{−}; E) ⟶^{ϕ_{1}} H^{1}(𝔫_{−}; E/V) ⟶ …

⟶ H^{i}(𝔫_{−}; V) ⟶ H^{i}(𝔫_{−}; E) ⟶^{ϕ_{i}} H^{i}(𝔫_{−}; E/V) ⟶ …

Let us prove that the weight −w(λ+ρ)+ρ can not be a weight of the 𝔩-module H^{i}(𝔫_{−}; E/V) for i < rk 𝔤. Indeed, each weight of H^{i}(𝔫_{−}; E/V), which is a quotient of a submodule of ∧^{i}(𝔫_{−})^{∗} ⊗ E/V, is of the form

γ_{1} + … + γ_{k} + μ,

where γ_{1}, …, γ_{k} are distinct positive roots which do not belong to the root lattice Q(B_{I}), and μ is a weight of E of the form μ = ∑_{a_{i}≥0} a_{i}α_{i}.

Suppose that

−w(λ + ρ) + ρ = γ_{1} + … + γ_{k} + μ.

Then −μ − w(λ) − w(ρ) + ρ = γ_{1} + … + γ_{k}. Set

R_{W^{−}} = {a ∈ R_{+} | w^{−1}(α) > 0} and R_{W^{+}} = {a ∈ R_{+} | w^{−1}(α) > 0}.

Suppose that γ_{1}, …, γ_{s} ∈ R_{W^{−}}, γ_{s+1}, …, γ_{k} ∈ R_{W^{+}}.

Recall that

ρ − w(ρ) = ∑_{β∈R_{W^{−}}} β.

Since E = E^{∗}, the weight −μ is a weight of E; but then

\begin{matrix}w^{−1}(γ_{1} + … + γ_{k}) = [w^{−1}(γ_{1}) + … + w^{−1}(γ_{s})] + [w^{−1}(γ_{s+1}) + … + w^{−1}(γ_{k})] = \\ (−λ − w^{−1}(μ)) + w^{−1}( ∑_{β∈R_{W^{−}}} β). \end{matrix}

Since −w^{−1}(μ) is a weight, λ + w^{−1}(μ) ∈ Q_{+}(R). Hence,

−(λ + w^{−1}(μ)) + w^{−1}( ∑_{β∈R_{W^{−}}} β) ∈ −Q_{+}(R).

On the other hand,

w^{−1}(γ_{s+1}) + … + w^{−1}(γ_{k}) ∈ Q_{+}(R)

and all the w^{−1}(γ_{i}) for i ≤ s enter, as summands, ∑_{β∈R_{W^{−}}} w^{−1}β and, therefore, cancell.

We finally get:

Q_{+}(R) ∋ w^{−1}(γ_{s+1}) + … + w^{−1}(γ_{k}) =

− (λ + w^{−1}(μ)) + w^{−1}( ∑_{β∈R_{W^{−}}\{γ_{1},...,γ_{s}}} β) ∈ −Q_{+}(R).

Thus, s = k, λ = −w^{−1}(μ). In other words,

−μ = w(λ) = − ∑ a_{i}α_{i} for a_{i} ≥ 0.

Since λ is a dominant weight, then λ = ∑ m_{i}α_{i}, where all the m_{i} ∈ ℚ are positive. This is true for any fundamental weight, as follows from the tables from [Bo].

By hypothesis, l(w) < rk 𝔤 and, therefore,

w = s_{α_{1}} … s_{α_{r}}, where r < rk 𝔤, B_{W} = {α_{i_{1}}, …, α_{i_{r}}} ⊂ B.

For any x ∈ Q(B) ⊗_{ℤ} ℝ we have x − w(x) = − ∑_{α∈B_{W}} n_{α}α. But then

(x − w(x), ω_{j}) = 0 for some j ≤ rk 𝔤.

Therefore, (λ, ω_{j}) = (w(λ), ω_{j}). Now notice that

(λ, ω_{j}) = m_{j}(α_{j}, ω_{j}) > 0, (w(λ), ω_{j}) = −a_{j}(α_{j}, ω_{j}) ≤ 0.

This contradiction shows that ϕ_{i} = 0 for i < rk 𝔤. This, in turn, means that there exist short exact sequences of 𝔩-homomorphisms

0 ⟶ H^{i−1}(𝔫_{−}; E/V) ⟶ H^{i}(𝔫_{−}; V) ⟶ H^{i}(𝔫_{−}; E) ⟶ 0 for i = 1, …, rk𝔤 − 1.

Lemma 2.3 is proved. □

Proof of Theorem 2.2. Set E = 𝔤, V = 𝔫_{−} ⊕ 𝔩^{(1)}. By Lemma 2.3,

H^{2}(𝔫_{−}; 𝔫_{−} ⊕ 𝔩^{(1)}) = H^{2}(𝔫_{−}; 𝔤) ⊕ H^{1}(𝔫_{−}; 𝔤/(𝔫_{−} ⊕ 𝔩^{(1)})).

It remains to verify that 𝔤/(𝔫_{−} ⊕ 𝔩^{(1)}) = (𝔫_{−} ⊕ 𝔷)^{∗}. Indeed, the Killing form B establises an isomorphism 𝔤 = 𝔤^{∗} and, therefore,

(𝔤/(𝔫_{−} ⊕ 𝔩^{(1)}))^{∗} = {x ∈ 𝔤 | B(x, 𝔫_{−} ⊕ 𝔩^{(1)}) = 0} = 𝔫_{−} ⊕ 𝔷,

where 𝔷 = {z ∈ 𝔩 | B(z, h_{α}) = 0 for any α ∈ B_{I}}, as required. □ Proof of Corollary 2.2. From the short exact sequence

0 ⟶ (𝔫_{−} ⊕ 𝔩)/(𝔫_{−} ⊕ 𝔩^{(1)}) ⟶ 𝔤/(𝔫_{−} ⊕ 𝔩^{(1)})) ⟶ 𝔤/(𝔫_{−} ⊕ 𝔩) ⟶ 0,

where 𝔤/(𝔫_{−} ⊕ 𝔩) = 𝔫_{−}^{∗} and (𝔫_{−} ⊕ 𝔩)/(𝔫_{−} ⊕ 𝔩^{(1)}) = 𝔷^{∗}, we get the long exact sequence

\begin{matrix}0 ⟶ 𝔫_{−}^{ab} ⟶ 𝔷^{∗} ⊗ H^{1}(𝔫_{−}) ⟶ H^{1}(𝔫_{−}; 𝔤/(𝔫_{−} ⊕ 𝔩^{(1)})) ⟶ H^{1}(𝔫_{−}; 𝔫_{−}^{∗}) ⟶^{ψ} \\ ⟶ H^{2}(𝔫_{−}; 𝔷^{∗}) (= H^{2}(𝔫_{−}) ⊗ 𝔷^{∗})) ⟶ … \end{matrix}

To compute the image of the map ψ, consider the 𝔩-module

M = {f ∈ Hom (𝔫_{−}, 𝔤/(𝔫_{−} ⊕ 𝔩^{(1)})) | Xf(Y) − Y f(X) − f([X, Y]) ∈ 𝔷^{∗} for any X, Y ∈ 𝔫_{−}}.

From the general properties of the long exact sequences ([FF]) we deduce that Im ψ = M/Z^{1}(𝔫_{−}; 𝔤/(𝔫_{−} ⊕ 𝔩^{(1)})).

By the BWZ theorem,

H^{2}(𝔫_{−}) = ⊕_{l(w)=2, w(B_{I})>0} E(w(ρ) − ρ).

Since M is the direct sum of its weight subspaces relative to 𝔥, let us study the subspaces M_{ρ−w(ρ)}. The weight of f ∈ M is equal to w(ρ) − ρ if and only if it sends 𝔫_{−γ} to (𝔤/(𝔫_{−} ⊕ 𝔩^{(1)}))ρ − w(ρ) − γ. Therefore, either ρ − w(ρ) − γ = γ′ ∈ R or ρ − w(ρ) − γ = 0.

The second option is ruled out since ρ − w(ρ) is not a root: indeed, l(w) = 2 ≠ 1.

In the first case ρ − w(ρ) = γ + γ′, where γ, γ′ ∈ R_{+}. If γ = γ′, then ρ − w(ρ) = 2γ, implying w^{−1}(γ) < 0. But card (R_{W}^{−}) = 2 and, ρ − w(ρ) = 2γ = γ_{1} + γ_{2}, where γ_{1}, γ_{2} ∈ R_{W}^{−}. Since one of the γ_{i} is equal to γ, so is the other one. This contradicts to the hypothesis: γ_{1} ≠ γ_{2}.

Thus, ρ − w(ρ) = γ + γ′, where γ ≠ γ′. It is not difficult to deduce from this that w^{−1}(γ) < 0, w^{−1}(γ′) < 0. Hence, f(𝔫_{−δ}) ≠ 0 only if w^{−1}(δ) < 0 for f ∈ M_{ρ−w(ρ)}. In this case f(𝔫_{−δ}) ∈ (𝔤/(𝔫_{−} ⊕ 𝔩^{(1)}))_{δ′}, where w^{−1}(δ′) < 0. Besides, since δ, δ′ ∉ R, we have [X_{−δ}, X_{−δ′}] = 0 for the root vectors X_{−δ}, X_{−δ′}. (Indeed, otherwise δ, δ′ ∈ R_{W}^{−} and l(w) > 2.)

On the other hand, any map f_{w} : 𝔫_{−δ} ⟶ 𝔤/(𝔫_{−} ⊕ 𝔩^{(1)}) which vanishes outside the subspace 𝔫_{−γ} ⊕ 𝔫_{−γ′}, where {γ, γ′} = R_{W}^{−}, and such that the weight of f_{w}(𝔫_{−γ}) is equal to γ′ whereas that of f_{w}(𝔫_{−γ′}) is γ belongs, clearly, to M_{ρ−w(ρ)}. This means that dim M_{ρ−w(ρ)} ≤ 2.

Let w = s_{α_{1}}s_{α_{2}}, where α_{2} ∈ B_{1} and (α_{1}, α_{2}) ≠ 0. Then dim M_{ρ−w(ρ)} = 1, as the following calculation shows: let f(X_{−α_{2}}) = cX′_{α_{1}+α_{2}}, f(X_{−(α_{1}+α_{2})}) = c′X′_{α_{2}}, where the primed vectors belong to the quotient space. Then for appropriate c, c′ we have

X_{−(α_{1}+α_{2})}f(X_{−α_{2}}) − X_{−α_{2}}f(X′_{−(α_{1}+α_{2})}) =

− cH′_{α_{1}+α_{2}} + c′H′_{α_{2}} ∈ ℂH′_{α_{1}} ⊂ (𝔫_{−} ⊕ 𝔩)/(𝔫_{−} ⊕ 𝔩^{(1)}).

If w = s_{α_{1}}s_{α_{2}}, where α_{1}, α_{2} ∈ B_{I}, then dim M_{ρ−w(ρ)} = 2. Since for any such w we have M_{ρ−w(ρ)} ∩ Z^{2}(𝔫_{−}; 𝔤/(𝔫_{−} ⊕ 𝔩^{(1)})) ≠ 0, we get the desired heading 1) of the corollary.

The general statement of heading 2) is straightforward; its particulart case a) follows from evident remarks: H^{1}(𝔫_{−}; 𝔫_{−}^{∗}) = 𝔫′_{−} ⊗ 𝔫′_{−}, H^{2}(𝔫_{−}) = Λ^{2}(𝔫′_{−}).

Proof of heading 2b). Set 𝔫′_{−} = 𝔫_{−}/𝔷(𝔫_{−}). Clearly, 𝔫′_{−} is a trivial 𝔫′_{−}-module. Therefore, the sequence

0 ⟶ 𝔫′_{−} ⟶ 𝔫′_{−} ⟶ ℂ ⟶ 0

is exact, hence, so is the sequence

0 ⟶ ℂ ⟶ 𝔫′_{−} ⊗ H^{1}(𝔫_{−}) ⟶ H^{1}(𝔫_{−}; 𝔫′_{−}) ⟶^{f} H^{2}(𝔫_{−}) ⟶ ….

It is not difficult to notice that the image of any cocycle 𝔫_{−} ⟶ 𝔫′_{−} under f belongs to Λ^{2}(𝔫′_{−}), hence, f is zero. But then H^{1}(𝔫_{−}; 𝔫′_{−}) is isomorphic to the subspace (𝔫′_{−} ⊗ 𝔫′_{−})_{0} of traceless operators in 𝔫′_{−} ⊗ 𝔫′_{−}. It remains to notice that H^{2}(𝔫_{−}) ⊕ ℂ = Λ^{2}(𝔫′_{−}). □

2.4. Let IR be the number of irreducible components in the 𝔤_{0}-module H^{2}(𝔤_{−}; 𝔤_{∗}).

Theorem . IR = \frac{1}{2}c_{0}(c_{0} + 1) + ∑_{i≠0} c_{i}.

Proof. Since IR = card {w ∈ W_{I} | l(w) = 2}, let us list the length 2 elements of W_{I}. Let W (R_{I}) be the Weyl group of the root system R_{I}. Clearly,

W (R_{I})_{2} = {w ∈ W (R_{I}) | l(w) = 2} ⊂ (W_{0})_{2}.

If s_{i}s_{j} ∈ (W_{0})_{2} \ W (R_{I})_{2}, then s_{j} ∈ B_{I} and s_{i} ∉ B_{I}. (Indeed, if s_{j} ∉ B_{I}, then s_{i}s_{j}(a_{j}) = −s_{i}(a_{j}) < 0 which is false.) Furthermore, (α_{i}, α_{j}) ≠ 0 since otherwise s_{i}s_{j}(α_{i}) = −α_{i} < 0.

Let rk R = r. Let R = R_{1} ∐ … ∐ R_{s}, be the presentation as the union of connected components. Let B = {α_{1}, …, α_{r}} be a base of R. Then

W (R)_{2} = {w ∈ W (R) | l(w) = 2} = {s_{i}s_{j} for i ≠ j} =
{s_{i}s_{j} for i < j} ∐{s_{i}s_{j} for i > j | (α_{i}, α_{j}) ≠ 0}.

It follows that

card W (R)_{2} = \frac{c_{0}(c_{0} − 1)}{2} + card \{vertices of the Dynkin diagram of R\}

= \frac{c_{0}(c_{0} − 1)}{2} + c_{0} − s.

If α ∈ R \ R_{I} and (α, R_{I}^{(i)}) ≠ 0 for some i then there exists a unique β ∈ R_{I}^{(i)} such that (α, β) ≠ 0. Indeed, if there is two such roots, say, β_{1} and β_{2}, then the Dynkin diagram contains a cycle

α —β_{1} —γ_{1} —γ_{2} — … —γ_{s} —β_{2} —α —

Indeed: D_{I}^{(i)} is connected and α ∉ R_{I}^{(i)}, hence, γ_{1}, …, γ_{s} ∈ D_{I}^{(i)}.

All this demonstrates that

card (W_{I})_{2} = card (W (R_{I}))_{2} + ∑_{i≠0} c_{i} + 1

= \frac{1}{2}c_{0}(c_{0} − 1) + c_{0} − s + ∑_{i≠0} c_{i} + s = \frac{1}{2}c_{0}(c_{0} − 1) + ∑_{i≠0} c_{i}. □

§3. Premet’s generalization of a theorem of Leger and Luks

Premet found the following generalization of Lemma 2.3.

Theorem . Let E(λ) = L_{λ} be an irreducible (finite dimensional) 𝔤-module with highest weight λ such that E = E^{∗}. Let V be a subspace in E which is 𝔭_{−}-invariant and contains

E_{−} = ⊕ ; μ=∑ k_{i}α_{i},k_{i}<0 E_{μ}.

Let P (V) be the set of weights of V. Then the following sequence is exact:

0 ⟶ ⊕_{{l(w)=i−1 \atop ww_{0}(λ)∉P (V)}} E(−w(λ + ρ) + ρ) ⟶ H^{i−1}(𝔫_{−}; E(λ)/V)

⟶ H^{i}(𝔫_{−}; V) ⟶ ⊕_{{l(w)=i \atop ww_{0}(λ)∈P (V)}} E(−w(λ + ρ) + ρ) ⟶ 0 for any i < rk𝔤.

§4. Open problems to start with

4.1. For any simple finite dimensional Lie algebra 𝔤 and its nilponent subalgebra 𝔑 = 𝔫_{i_{1}...i_{k}} for k = 0, …, k = rk𝔤, describe the associative algebra H^{∗}(𝔑) and H^{∗}(𝔑; Λ.(𝔑)), and the Lie superalgebras H^{∗}(𝔑; 𝔑) and H^{∗}(𝔑; 𝔤).

Comment. How to describe the algebra (via generators and relations) or as a semidirect product of a radical and a semisimple algebra and how to describe this semisimple Lie superalgebra is not clear at the moment. The result is unpredictable, so one has to test various cases to see what we obtain. So far, particular results are due to Kostant, Leger and Luks, Premet. Tolpygo deduced asymptotic formulas to compute the dimension and generating functions of the above cohomology. Consideration of algebraic structure of these cohomology — a totally new approach — might clarify the picture. Unfortunately, some

§4. OPEN PROBLEMS TO START WITH 33

preliminary results due to I. Shereshevskii demonstrate that the general picture will not be easy to describe even in this approach.

4.2. For every simple finite dimensional Lie superalgebra 𝔤 describe its cohomology with coefficients in simplest nontrivial atypical modules (identity, adjoint, their symmetric powers).

Among most interesting particular cases: H^{∗}(𝔤) and H^{∗}(𝔤; 𝔤). Describe associative algebra H^{∗}(𝔤) and the Lie superalgebras H^{∗}(𝔤; 𝔤) via generators and relations.

3.3. For every simple ℤ-graded Lie superalgebra 𝔤 = ⊕_{i≥−d} 𝔤_{i} and its subalgebra 𝔤_{−} =

⊕_{i<0} 𝔤_{i} compute H^{i}(𝔤_{−}; 𝔤) (say, as a 𝔤_{0}-module) for i ≤ 2.

Certain interpretations: These cohomology (especially for i = 2) can be considered as obstruction to stability of the dynamical system with 𝔤 as its algebra of symmetries; as obstructions of formal integrability of differential equations with 𝔤 as its algebra of symmetries; the 𝔤_{0}-invariants of H^{2}(𝔤_{−}; 𝔤) describe filtered deformations of 𝔤.

For vectorial Lie algebras these cohomology are implicitly described by Serre, cf. [St]. Explicit description is of interest even for Lie algebras, cf. [LP1], [LP2], [P1]. For Lie superalgebras see [CK], [GLS2].

3. From Supergravity to ball bearings

Disclaimer. The authors are not experts in either analytical mechanics or physics. Therefore our examples are very simple and scanty (Internet lists thousands of entries for nonholonomic and hundreds more on a synonym, sub-riemannian). We were just lucky to observe that various problems listed below (we tried to provide with ample references to start with, we particularly like and suggest [VG], [A2], [Br], [Gr], [H], [M], [NF], [3], and, especially, a user-friendly paper [N]) can be solved (at least theoretically) by means of computing certain Lie algebra cohomology. The Sign Rule inserts superflavour.

The talk will require no preliminary knowledge and will be addressed mainly to students who do not know too much to be prejudiced against unusual methods with only few romantic results to back up our claims of their usefulness.

Here is our story.

§1. Nonholonomic dynamics: History

Supergravity. In 1972 Leites introduced [L0] the what is now called superscheme or algebraic supervariety. (Unfortunately, short communications to Russian Mathematical Surveys were not translated into English at that time, but, even if they were, the text was too short. Even its expansion rewritten from algebraic language to C^{∞} one [BL] did not get much acclaim.) This (and [L1]) was an answer to a question of Berezin together with whom he was slightly ahead of time, see Minlos’ recollections in [D].

Now supermanifold theory is fully recognized thanks to its part, supersymmetry, considered in theoretical physics as the language of the future unified theory of all fundamental forces. Wess and Zumino were the first to realize and explain to others that we live on a supermanifold and thus made one of the most fundamental steps in science, not only in physics. Physicists realized that there is not one, but a whole hierarchy of standard Minkowski superspaces and a number of exsotic models (see [3], [SS], [Ma]), labeled for the standard (for the moment) model by an inner parameter N running (conventionally) 1 to 8. The most interesting was supposed to be the largest model, for N = 8.

Ensuing enthusiasm was overwhelming (at least, among physicists). Classification of simple finite dimensional Lie superalgebras performed by Kaplansky, Nahm–Rittenberg– Scheunert, and skillfully rounded up by Kac — strikingly resembling that of simple Lie algebras — convinced many mathematicians that this is “it” in a sense.

Mass media (from Pravda to New York Times, except, perhaps, Ren Min Ribao) heralded that soon Einstein’s dream — Grand Unified Theory (now called SUSY GUTs for its supersymmetry) — will come true. But eventually fanfare subsided. The reason: SUGRA equations for N = 1 were established during 1974 – 76, N = 2 (and to an extent N = 3) were supposed to have been written in 1984. After much work in various research centers, it was Ogievetsky’s group who realized that to fulfill natural requirements (see §4) one has to enlarge the Minkowski space underlying Minkowski superspace by adding the what was called “harmonic” coordinate. How to do it for N > 3 was unclear, see [GI], [GIOS].

To apply for grants subsidizing studies of nonexisting (as far as supersymmetry was concerned) equations became more difficult and most of the researchers left the field. What was the problem? (Not with grants, all is clear here; what is the mathematics difficulty?)

One of the best experts, Wess, honestly lectured [3]: “ we do not know how to express the Riemann tensor (the left hand side of the equations to be written) in the general case”.

In [Aea], Witten wrote: “Direct experimental confirmation of supersymmetry is one of the prime missions of the proposed Superconducting supercollider.” And then added: “More fundamentally, I believe that the main obstacle is that the core geometrical ideas — which must underline string theory the way Riemannian geometry underlines general relativity — have not yet been unearthed.”

We believe that we were lucky to have discovered some of these ideas [L3]. They are related to (a) nonholonomic mechanics, (b) the presentation of the curvature tensor via cohomology and (c) the classification of simple Lie superalgebras of vector fields — the analog of É. Cartan’s classification, [LS].

Relation with nonholonomic mechanics is occasioned by the fact that Minkowski superspaces (whatever they are, see [3], [SS], [Ma]; in recent dissident studies of M. Vasiliev [2] the inner parameter N conventionally running 1 to 8, can tend to ∞; Manin [Ma] suggested to consider series of other, nonstandard and and exceptional exotic models) are nonholonomic, unlike the Minkowski space which is holonomic. (For a holonomic analog of Minkowski superspace see [GL7].)

The importance of Lie algebra cohomology for physics in general was in the air lately; conferences entitled “Cohomological Physics” started to mushroom. In particular, in supergravity the importance of cohomology is clearly stated by several authors (e.g., [CDF]) but the description of the curvature tensor in nonholonomic case was never given until recently [LP2], [GL3] and this is the whole point totally missed in attempts to describe Riemannian tensor on Minkowski superspaces as if they were holonomic (compare [LP2], where these computations are made for a deliberately wrong according to today’s views, since it is (holonomic, version of Minkowski superspace, with real SUGRA [GL3]) and such attempts stopped at N = 1, see [RS].

Mechanics. In XIX century Hertz [H] in his attempts to get rid of the notion “force” (we recommend to read Feynman’s discussions of the problem [FLS]) divided the dynamical systems into holonomic (no constraints on velocities) and nonholonomic (with nonintegrable constraints on velocities). Analytical study of holonomic systems has progressed much further than that of nonholonomic. Moreover, only linear constraints are usually considered.

Though nonholonomic systems are important in various branches of applied physics and engineering, they have not been sufficiently studied in mathematics, even the best text books on analytical mechanics just mention several examples and pass to holonomic systems, cf. [A1], [A2]. One of the reasons: the qualitative study, such as stability problems, require the notion of curvature tensor which is not defined for the nonholonomic systems in the literature, cf. [St], the standard textbook for this purpose.

Examples of nonholonomic systems. 1) In mechanics. The ball on a rough plane; ballbearings and gyroscopes; any vehicle with wheels or any body rolling without gliding over another body (the point at which the moving body is tangent to the surface of the other body has zero velocity). Any car with cruise control switched ON is an example of nonholonomic system with a nonlinear constraint. For further (numerous) examples we refer the reader to Internet, of thousands entries there we would like to point out [FK], [K], [E],

[Gl] and references therein.

§1. NONHOLONOMIC DYNAMICS: HISTORY 37

Related problems: general description of the movement and stability of a rapidly rotating hole body with a liquid or gyroscope inside it; orbital stability of a missile with liquid inside it. (The linear approximation does not work here.)

For several interesting practical applications (e.g., how the cat, when dropped, uses nonholonomicity to fall on its paws or how nonholonomicity influences the parking car) see [M], [E], [Fe].

In electromechanics (see, e.g., the book by Nejmark and Fufaev [NF] or [BF]) an early example was provided by Gaponov who, in 1952, found that a conductor connected to a gliding contact (as in a rotor collector) is equivalent to imposing nonholonomic constraints on the distribution of the electric current.

For comprehensive reviews of nonholonomic problems, see the books [NF] and [VG]. They list numerous problems which go back to Euler, Gauss, Carnot, Hertz, Carathéodory, Appell, Schouten, Synge, and many others.

2) The magneto-hydrodynamics: waves in plasma. Study of stability of such waves is extremely important for the development of thermonuclear power plants.

3) Recent studies of Komech, Spohn and Kunze (e.g., [KS]) show that electron represents a nonholonomic system with a (Coulomb, Maxwell, etc.) gauge as a nonholonomic constraint.

4) Numerous works on hypoelliptic differential equations (by Hörmander, Melin, Malliavin, Bismut, Bell; for a review see, e.g., [Ar]) also lead to nonholonomic distributions.

Statistical thermodynamics and economics. The well-known from school relation between pressure, temperature and volume of the ideal gas is an example of a nonholonomic constraint. This seems to be one of the first non-trivial but easy-to-compute examples (the very first ones are contact and Engel distributions, cf. [L3]). Recently V. Sergeev expressed certain market models in thermodynamical terms. Having developped ideas of Hyek, Sergeev suggested mathematical models which gives, among other things, a transparent explanation of such well known phenomena as economic growth in China and Vietnam and economic decline in various post Soviet republics [SV]. This is already somthing, but what fascinates us in Sergeev’s studies is a clear formulation of economics as a non-holonomic system.

Related problems.

• Vershik was probably the first who noticed similarity of various distributions of solid bodies that appeared in optimal control (corresponding to the nonlinear nonholonomic constraints and fields of cones, spheres, etc.) with the “well-known” nonholonomic problems, cf. [CF], [VG] and Agrachev’s plenary talk at the 1994 International Congress of Mathematicians [Ag].

• Supersymmetry appears even in seemingly non-super questions, like the study of the spectrum of the Schrödinger operator [L2] or relation between the Schrödinger operator and the KdV (Korteveg de Vries) operator [L4]. It does not always bring in nonholonomicity, but rather often, and usually unexpectedly.

• Recent studies of the Gelfand–Dickey bracket lead us to the discovery of a new class of simple Lie algebras of polynomial growth — generalization of Lie algebras of matrices of “complex size” [GL], [GL2]. Simultaneously, together with Shchepochkina, we announced two classifications of simple Lie superalgebras: (A) of vector fields [LS] and (B) stringy or superconformal superalgebras [GLS1].

Solution of these problems [GLS1], [LS] was considerably speedified by our computeraided study.

• Related with Lie superalgebras of complex size generalizations of orthogonal polynomials (especially, hopefully, in several variables), see our talk with A. Sergeev at SF-2000 [LSe].

Amazingly, all the above topics are interrelated by common mathematics: Lie algebra cohomology and nonholonomic structures. M. Vasiliev augmented our amazement: in one of his papers on super versions of Dunkl operators and Calogero–Sutherland model he (with coauthors) showed [1] that generalized Lie superalgebras of supermatrices of complex supersize play the role of Poincaré algebras in N-extended SUGRAs for N > 8.

Accordingly, our research splits into several topics united by the usage of a common Mathematica-based package SuperLie written by Grozman.

1. Study of nonholonomic mechanics: from supergravity to ballbearings. Attempts to understand applications to stochastic analysis, media with dislocations, and optimal control. Study of stability via cohomology.

2. Study of representation theory; in particular, classification of simple Lie superalgebras of vector fields, stringy or superconformal, and generalizations of Lie algebras of matrices of complex size. Description of defining relations for them via cohomology. Applications to Lie algebras and superalgebras and their representations over fields of prime characteristic and over integers.

3. Application of representation theory to the study of integrability of differential equations (KdV, KP, Schrödinger, Liouville, Toda lattice, etc. and their superizations). Criteria of integrability expressed in terms of Lie algebra cohomology.

4. Classification of invariant differential operators.

In this talk we will briefly report two types of results: on SUGRA and on invariant operators.

§2. Formulation of problems

According to Arnold–Kozlov–Neishtadt [A2], the behavior of nonholonomic systems is often “surprising”, and its quantitative study is handicapped by the lack of adequate tools. Let us recall some examples from [A2].

Consider a skate on the inclined plane. Where do you think it will move if pushed not directly downwards, but sideways? If we ignore loss of energy due to friction, it will never reach the floor but will oscillate between certain horizontal lines on the plane. Similarly, consider a ball rolling along the wall inside a vertical tube. It seems natural to expect the ball to descend on a spiral trajectory with increasing steepness. In “reality”, however, (if to assume conservation of energy regardless of friction can be considered realistic) the ball will perform oscillations between two fixed vertical planes.

Though individual solutions of nonholonomic systems are usually known, the stability questions are solved ad hoc.

Stability problems. The stability of a holonomic system can be studied in terms of the Riemann curvature tensor. The sign of the curvature indicates whether the geodesics converge or diverge: compare meridians on a sphere (positive curvature) with those on a trumpet = Lobachevsky plane (negative curvature).

For nonholonomic manifolds there was no such tool: in the literature the definition of the analogue of the Riemann tensor is only given in a few particular cases of little practical value, cf. [VG], [T]. For the traditional analytical approach see [RK].

Difficulties. In the last decades, dozens of publications on dynamical problems in nonholonomic mechanics appeared each year in technical journals, but very few in Mathematical journals. The main reason: lack of adequate language needed to even formulate the problem of calculating the local curvature tensor (the analogue of the Riemann tensor) in intrinsic terms. The bulky coordinate expressions used by classics in 1920’s, as well as now, hinder

§3. WHAT IS DONE: A NOTION, A FORMULA, AND A PACKAGE 39

further progress. Vershik and Gerschkovich gloomily stated [VG] that there is no language to even formulate the integrability problems for nonholonomic systems with linear constraints (to say nothing about nonlinear constraints).

This is one of the reasons why this important field was abandoned by mathematicians, and even the interest and occasional works of such leaders as Faddeev, Griffiths, Chern and Arnold (see [VG], [Br], [A2]) did not attract any followers. Perhaps, only the latest works of Marsden [Ms] do not satisfy this pessimistic description.

§3. What is done: a notion, a formula, and a package

We know the importance of the notion of curvature in drawing very exact maps; application to stability were already mentioned. How to compute the curvature? In text books on geometry this is done in terms of Spencer cohomology. Same cohomology are used in Goldschmidt’s criteria for (formal) integrability of differential equations.

Observe that these criteria [Br] are only given for “a half” of the cases. The point is that by a theorem of Cartan the symmetries of each differential equation are induced by either point transformations or contact transformations [KLV]. The latter case belongs to the realm of the simplest nonholonomic manifold — a contact one — and, apart from partial results of [T], nothing was known in this case. Our notions allow one to extend Goldschmidt’s criteria to all differential equations.

The notion and the formula: Lie Algebra Cohomology. In lectures at a school at ICTP, Trieste, in 1990, Leites reformulated the definition of the Riemann tensor in terms of Lie algebra cohomology for any ℤ-graded Lie algebra of finite depth (see §4). The notion of generalized Cartan prolongation [LS] is crutial here. In these terms, the problems discussed above can be posed precisely, and, in principle, solved. In particular, we describe the space of possible values of Riemannian tensor as H^{2}(𝔤_{−}; (𝔤_{−}, 𝔤_{0})_{∗}); this is “the formula”. It even becomes possible to generalize the description of the local curvature tensor for nonholonomic manifolds with any constraints, like the fields of surfaces.

The Package. During 1992–94, Grozman developed a Mathematica-based pilot package SuperLie for computing Lie algebras and their cohomology. Since then the package was constantly developped and now SuperLie makes it possible to determine (1) Lie algebras matrix algebras, algebras of vector fields, algebras defined via generators and relations, via Cartan matrices, via Poisson or contact (Legendre) bracket, etc., as subalgebras of any of the above algebras; (2) various modules over these Lie algebras (tensors, with vacuum vector; submodules, quotients, duals, etc.), and calculate bracket and action on the Lie algebra on its module, (3) solve systems of equations with scalar and (linear) vector unknowns; find ker and Im of a linear operator; (4) calculate exterior differential in various complexes, and their homology and cohomology.

The mentioned above reformulation of Spencer cohomology in terms of Lie algebra cohomology means that much of the time-consuming calculations become considerably simplified and new ways to diminish the amount of calculations appear. However, the computation effort is still overwhelming: to compute the curvature tensor even for the simplest practical examples of, e.g., a ballbearing, or a bike, we need to develop the package further and complete its documentation to make available to engineers. On top of that symbolic computations in Mathematica is irritatingly slow (though hardly any of the wide spread computer languages is faster). Still, though it seems that an ad hoc written code for each particular task would be faster than our package, none managed so far.

The advantage of SuperLie is its universality and versatility. (For example, in [Gz],

[LKW] bilinear differential operators acting in tensor fields invariant with respect to the general and volume preserving diffeomorphism groups are classified; further advancement “by hands” seems hardly possible. Recently with the help of SuperLie Grozman obtained new results in hamiltonian and contact cases, cf. §5.)

In spite of some drawbacks of the existing pilot package, we used it to correct several results found analytically by mathematicians (certain cohomology from [FL], [Fu]) and by physicists (certain Wess–Zumino constraint in supergravity [3] turned out to be redundant). The last example is of particular interest: it convinces that computer aded study is a must here since this redundunce was missed by hundreds of physicists working with SUGRA around the globe for two decades.

The results, obtained completely or partly with the help of SuperLie are described in [GL], [GL2], [GLP], [GLS2]-[GLS4] and used also in [LS], [L4]. Several more are in preparation.

Rival and not so rival teams. An alternative package (not so overwhelming but allegedly much faster) was being developed for several years in JINR, Dubna, by V. Kornyak, our coworker, in C. (So far his package can not do anything SuperLie can not.) For results of REDUCE-fans from Twente University see [LP] and refs. in [GLP].

§4.0. The left hand sides of SUGRAs ([GL3])

Here we briefly explain how to solve the problem in deriving supergravity equations, SUGRA(N) — the analogues of Einstein’s equations (EE) on an N-extended Minkowski superspace M(N) and what are our the criteria for distinguishing suitable Minkowski superspaces among other supermanifolds. Our considerations are in vacuum. As in twistor theory, we consider complex case, the physical reality to be recovered on a suitable real form of M(N). Our requirements: SUGRA(N) should be

(A) a differential equation of order ≤ 2 on the components;

(B) the component expansion of SUGRA(N) should contain the ordinary Einstein’s equations. For simplicity we assume that the supergroup of motions of M(N) is G = SL(N|4) (though other possibilities can not be eliminated, cf. [GL2], [Ma]), and we wonder: what is the stationary subgroup P for which M(N) = SL(N|4)/P?

As N grows, it becomes clear (by comparing the spectrum of cohomology for these supergroups of motion with the underlying Riemann tensor, as modules over O(4)) that to justify the above requirements (A) and (B) is only possible if we diminish P, as GIKOS did, cf. [GI]. Then for N > 3 we see that the underlying manifold of M(N) is the direct product of several copies of the Minkowski space M (times, perhaps, an auxiliary space of a yet unclear merit); so SUGRA splits into the usual Einstein equations on each copy of M glued together by odd superfields. In particular, for N = 4 there are two copies of M.

For N = 8, in a most symmetric model, there are three copies of M, one of them distinguished (say, “our world”), the other two – perfectly interchangeable (in the model considered here; there are other possibilities) – mirroring, say, “heaven” and “hell”). These extra copies of the Universe (they MUST appear in our approach) embody an idea first, perhaps, voiced by A. Sakharov [Sa] on the necessity to consider several copies of our Universe to describe physical reality. Another feature of nonholonomic nature of Minkowski superspace is the preferred direction of time, an observation we derive by directly looking at the rattleback (Celtic stone); the universality of this observation for nonholonomic systems follows from recent elementary, and therefore wonderful, studies by A. Nordmark [N].

§4.1. STRUCTURE FUNCTIONS FOR NONHOLONOMIC STRUCTURES 41

We start first with a presentation of Einstein’s equations in a form convenient in what follows — as equations on conformally noninvariant components of the Riemann tensor, and represent the Riemann tensor as a section of the bundle whose fiber is certain Lie algebra cohomology. This is equivalent to the standard modern treatment of G-structures in differential geometry that uses Spencer homology [St] but can be generalized to treat nonholonomic structures, such as SUGRA. Then we generalize the notion of the G-structure to embrace nonholonomic manifolds, i.e., manifolds with nonintegrable distributions, see [H]. The structure of a nonholonomic manifold is encountered quite often. The simplest example is the contact structure on a manifold, cf. [A1]. The applications range from the Cat’s Problem to electro-mechanical devices [NF], [BF], [E], [CF], [K], [KS], [M]. For a moving account of nonholonomic problems and their history see [VG].

One can calculate Spencer cohomology for any supermanifold with G-structure (such attempts are numerous in the literature) but the tensors obtained do not match the one physicists consider, cf. [3]. for nonholonomic manifolds and supermanifolds we have to consider generalized Spencer cohomology which we formulate in terms of Lie algebra cohomology.

We also offer a general method to derive constraints — analogues of Wess–Zumino constraints — for any N. Funnily, one of the conventional constraints for N = 1 is redundant: its cohomology class is zero. This demonstrates that a computer-aided study [GL5] is a must here: even when we know what to do and how, the amount of computations is too vast for a human. Even as skilled one as Poletaeva or physicists [GI].

Though the notion of supermanifolds is past its 25-th birthday [L1], certain basics are, regrettably, insufficiently well-known yet. So we will recall them.

§4.1. Structure functions for nonholonomic structures

1.1. Nonholonomic (super)manifolds ([VG]). Let M be a manifold with a distribution D. Let

D = D_{1} ⊂ D_{2} ⊂ D_{3} ⊂ … ⊂ D_{d}  (1.0)

be the sequence of strict inclusions, where D_{i}(x) = D_{i−1}(x) ∪ [D_{1}(x), D_{i−1}(x)] for every x ∈ M and d is the least number for which the sequence (1.0) stabilizes, i.e., such that D_{d}(x) ∪ [D_{1}(x), D_{d}(x)] = D_{d}(x). The number d = d(M, D) is called the nonholonomicity degree. In case D_{d} = T M the manifold M is called completely nonholonomic. Each pair: (M, D) with a nonintegrable D will be referred to as a nonholonomic manifold if d ≠ 1. Here we will only consider completely nonholonomic (super)manifolds (the incomplete case corresponds to nontransitive generalized Cartan prolongs, cf. below, and the modifications needed to adjust general machinery to it are obvious).

Let n_{i}(x) = dim D_{i}(x). The distribution D is called regular if all the dimensions n_{i} are constant functions on M. In what follows we will only consider regular distributions.

With the tangent bundle over a nonholonomic manifold (M, D) we can naturally associate a sheaf of nilpotent Lie algebras as follows. At a point x ∈ M set

𝔫(x) = {X ∈ 𝔳𝔢𝔠𝔱(M) | X preserves the Pfaff system that distinguishes D}.  (1.1)

Clearly, 𝔫(x) is a well-defined Lie algebra and one can show that, as spaces,

𝔫(x) = ⊕_{−d≤i≤−1} 𝔫_{i}(x), where 𝔫_{−i}(x) ≃ D_{i}(x)/D_{i−1}(x), D_{0} = 0.  (1.2)

Here d is the nonholonomicity degree of (M, D).

1.1.1. A flat (G, 𝔫(x))-structure. This is a straightforward generalization of the notion of the flat G-structure (the latter implies, in particular, that every 𝔫(x) is commutative, but not only that: the curvature tensor must also vanish). Namely, let M be just ℝ^{n} endowed with a nilpotent ℤ-graded Lie algebra structure, call it 𝔫. Let G be a subgroup of homogeneous (preserving the grading (1.2)) automorphisms of 𝔫. Let us identify the tangent space at a point m ∈ M with M by means of a transformation from G. The preimages 𝔫_{−1}(m) of 𝔫_{−1} under this identification determine a distribution on M. This distribution together with the G-action on the accompanying flag 𝔫(m) at each m ∈ M will be called a flat (G, 𝔫)-structure. Observe that this “nonholonomic flatness” and the obstructions to flatness introduced below differ drastically from their conventional counterparts: any contact manifold is nonholonomically flat (Darboux theorem) whereas it is never flat in terms of Gstructures: the torsion is proportional to the symplectic form on the leaves of the distribution.

In order to gauge the measure of this nonholonomic nonflatness, we need an analogue of Spencer cohomological complex, which, in turn, requires the following notion.

1.2. Generalized Cartan’s prolongs. Given a ℤ-graded nilpotent Lie algebra 𝔤_{−} = ⊕_{0>i≥−d} 𝔤_{i} and a Lie subalgebra 𝔤_{0} ⊂ 𝔡𝔢𝔯 𝔤_{−} which preserves the ℤ-grading of 𝔤_{−}, define the

i-th prolong of (𝔤_{−}, 𝔤_{0}) for i > 0 to be (here S. = ⊕_{k≥0} S^{k} and V^{∗} is the dual of V):

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

where the subscript in the right hand side singles out the component of degree i and the intersection is well-defined thanks to the fact that 𝔤_{0} ⊂ 𝔡𝔢𝔯 𝔤_{−} ⊂ 𝔤_{−}^{∗} ⊗ 𝔤_{−}.

Define the generalized Cartan’s prolong (𝔤_{−}, 𝔤_{0})_{∗} as ⊕_{i≥−d} 𝔤_{i}. By the routine arguments,

(𝔤_{−}, 𝔤_{0})_{∗} is a Lie algebra. By the same arguments as for the G-structures, cf. [St], [Gz], [T], the space H^{2}(𝔤_{−}; (𝔤_{−}, 𝔤_{0})_{∗}) is the space of obstructions to flatness of the nonholonomic supermanifold (M, D) and the elements of H^{2}(𝔤_{−}; (𝔤_{−}, 𝔤_{0})_{∗}) will be called (in analogy with the case d = 1) structure functions, though to avoid misunderstanding we will often add adjective nonholonomic.

The space of nonholonomic structure functions naturally splits into homogeneous components whose degree is induced by the ℤ-grading of (𝔤_{−}, 𝔤_{0})_{∗}; generally, the minimal order of nonholonomic structure functions is 2 − d. Let C^{s}(𝔤_{−}; (𝔤_{−}, 𝔤_{0})_{∗}) = ⊕_{k} C^{k,s}(𝔤_{−}; (𝔤_{−}, 𝔤_{0})_{∗})

be this splitting on the cochain level; the corresponding cohomology H^{k,s}(𝔤_{−}; (𝔤_{−}, 𝔤_{0})_{∗}) are precisely the analogues of the Spencer cohomology and coincide with them for 𝔤_{−} = 𝔤_{−1}.

1.3. Example. Let 𝔤 = 𝔠𝔰𝔭(2n), 𝔤_{−1} = R(π_{1}; 1), 𝔤_{−2} = R(0; 0); then (𝔤_{−}, 𝔤_{0})_{∗} = 𝔨(2n + 1) and (here E is the exterior power operator)

C_{(𝔤_{−},𝔤_{0})_{∗}}^{k,2} = 𝔤_{k−2} ⊗ E^{2}(𝔤_{−1}^{∗}) ⨁ 𝔤_{k−2−1} ⊗ E^{2−1}(𝔤_{−1}^{∗}) ⊗ 𝔤_{−2}^{∗}.

The number k here is the order of the structure functions. Unlike the holonomic case when order k structure functions are obstructions to identification of the k-th infinitesimal neighborhood of the point of M with that on the flat manifold, obstructions to flatness in nonholonomic case are not as lucidly interpreted.

Riemann-like and Weyl-like structures. Let 𝔤̂_{0} be the semisimple part of 𝔤_{0} and 𝔤̂_{∗} = (𝔤_{−}; 𝔤̂_{0})_{∗}. The elements of H^{2}(𝔤_{−}; 𝔤̂_{∗}) are nonholonomic analogues of the Riemann tensor whereas the elements of H^{2}(𝔤_{−}; (𝔤_{−}, 𝔤_{0})_{∗}) are nonholonomic analogues of the Weyl tensor, the structure functions of the conformal structure.

Theorem . (Weyl-like structures) For (𝔤_{−}, 𝔤_{0})_{∗} = 𝔨(2n + 1) all nonholonomic structure functions vanish.

§4.2. STRUCTURE FUNCTIONS OF THE N-EXTENDED MINKOWSKI SUPERMANIFOLD 43

Remark . This is a conceptual reformulation of Darboux’s theorem on the absence of parameters for the contact form.

1.4. Structure functions for projective structures and their analogs. Let 𝔤 = 𝔤𝔩(n) and 𝔤_{−1} its standard (identity) representation. Then (𝔤_{−}, 𝔤_{0})_{∗} = 𝔳𝔢𝔠𝔱(n) and all structure function vanish ([LPS]); but if for coefficients of the above cohomology we take the subalgebra 𝔥 = 𝔰𝔩(n + 1) ⊂ 𝔳𝔢𝔠𝔱(n) instead of (𝔤_{−}, 𝔤_{0})_{∗}, then the corresponding structure functions are nonzero and provide us with obstructions to integrability of what is called a projective connection ([Go]).

More generally, for a ℤ-graded subalgebra 𝔥 ⊂ (𝔤_{−}, 𝔤_{0})_{∗}, such that 𝔥_{i} = 𝔤_{i} for i ≤ 0 we will call ⊕_{k} H^{k,2}(𝔤_{−}; 𝔥) nonholonomic structure functions of the generalized projective structure of

type 𝔥. Here are examples:

Theorem . (in notations of [OV]; in R(φ; k) the value on the center of 𝔤_{0} stands after semicolon) 1) Let (𝔤_{−}, 𝔤_{0})_{∗} = 𝔳𝔢𝔠𝔱(n), 𝔥 = 𝔰𝔩(n + 1). Then structure functions of order 1 and 2 vanish, structure functions of order 3 constitute the 𝔤_{0}-module R(π_{1} + π_{2} + π_{n−1}; 2).

2) Let (𝔤_{−}, 𝔤_{0})_{∗} = 𝔨(2n + 1), 𝔥 = 𝔰𝔭(2n + 2). Then the space of structure functions is R(π_{1} + π_{2}; 3) of order 3.

1.5. Structure function for contact structures on supermanifolds. Let ω_{0} = ∑ dp_{i}dq_{i} + ∑ ε_{j}(dξ_{j})^{2} and ω_{1} = ∑ dπ_{i}dq_{i} be the canonical even and odd, respectively, 2forms (see [LPS]). Denote by 𝔨(2n + 1|m), resp. 𝔪(n), the Lie superalgebra which preserve the contact structures given by the odd form dt + α and the even dτ + β, where dα = ω_{0}, dβ = ω_{1}, respectively.

Theorem . For 𝔨(2n + 1|m), 𝔪(n) all nonholonomic structure function vanish.

For the deformation 𝔰𝔪_{λ}(n) (see [GLS2]) of the Lie superalgebra of divergence-free contact fields the analog of the above theorem fails: there are nonzero nonholonomic structure functions.

§4.2. Structure functions of the N-extended Minkowski supermanifold

Recall that the ground field is ℂ, the complex numbers.

The G-structure of the Minkowski space can be viewed as either (pseudo) Riemannian or, equivalently, twistor structure. “Straightforward” superizations of these structures are distinct. They are considered in [Ma] (see also refs. therein), or [LPS], or [GL3]. With some exceptions for N = 1, neither of these superizations gives rise to what physicists accept as supergravity. The reason is that a Minkowski superspace is still another superization of the Minkowski space and is naturally endowed with a nonholonomic structure.

2.1. What is an N-extended complexified Minkowski supermanifold. First of all, recall what was the “standard” model of complexified and a compactified Minkowski superspace M(N) before the GIKOS group (Galperin, Ivanov, Kalitsyn, Ogievetsky, Sokachev) showed that one can not write SUGRA on it. Recall also that the “physical reasons” for the restrictions N ≤ 4 for the Yang-Mills and N ≤ 8 for the supergravity theories ([GI], [3]) were put to doubt in [2].

Recapitulations ([L2]). A supermatrix is a rectangular table with elements from a supercommutative superalgebra C with given sets of parities P_{row} and P_{col} of its rows and columns. The size of a matrix is P_{row} × P_{col}. Usually, the parities are chosen so that the even rows and columns come first followed by the odd ones; such matrices are said to be of the standard format. For the square matrices we will only consider the cases when P_{row} = P_{col} and will denote this set of parities by P ar. The parity of the matrix unit — the matrix with an element c ∈ C in the (i, j)-th position and 0’s elsewhere — is defined to be p(c) + P_{row}(i) + P_{col}(j). Hereafter C = ℂ.

Let 𝔤𝔩 (P ar) be the set of square matrices of size P ar×P ar. It is immediately clear, that for distinct P ar’s with the same p and q the Lie superalgebras 𝔤𝔩(P ar) have nonisomorphic maximal nilpotent (say, upper triangular) subalgebras, though the algebras 𝔤𝔩(P ar) themselves are isomorphic. It often suffices to consider the standard format only; then 𝔤𝔩(P ar) is abbreviated to 𝔤𝔩(p|q), where p is the number of 0’s and q is the number of 1’s in P ar. In supergravity we MUST consider all formats.

The Lie superalgebra 𝔤𝔩(P ar) is generated by the generators of its maximal nilpotent subalgebras, the upper and the lower triangular ones. Since P_{row} = P_{col}, the Lie subsuperalgebra of upper triangular matrices in 𝔤𝔩(P ar) is isomorphic to the Lie subsuperalgebra of lower triangular matrices and we will confine ourselves to one of them, denoted by 𝔫. The generators of 𝔫 are the elements just above (below) the main diagonal; we will denote the even generators of 𝔫 by white nodes and the odd generators by “grey” nodes, the nodes corresponding to commuting generators are disconnected, otherwise they are joined by a segment, cf. [GL2].

For instance, for the standard format, i.e., 𝔤𝔩(p|q), we have:

(\begin{matrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix})  \underbrace{0 − ⋯ − 0}_{p−1 nodes} − ⊗ − \underbrace{0 − ⋯ − 0}_{q−1 nodes}  (2.0)

Consider the Lie supergroup SL(N|4) and its parabolic subsupergroup P corresponding to the two marked odd simple roots in the following system of simple roots (this means that P is generated by all the simple roots except for the marked negative ones):

0 − −− ⊗^{+} − − − \underbrace{0 − − ⋯ − −0}_{N−1 nodes} − − − ⊗^{+} − − −0  (2.1)

Let G = SL(N) × SL_{L}(2) × SL_{R}(2) × ℂ^{∗} be the Lie group corresponding to the 0-th term of the ℤ-grading described by diagram (2.1), i.e., the degree of a marked root is equal to 1 the other simple roots being of degree 0; here indices L and R distinguish the “left” copy of SL(2) from its “right” twin. In this case 𝔤_{0} = 𝔤 = 𝔬(4) ⊕ 𝔤𝔩(N) = 𝔰𝔩_{L}(2) ⊕ 𝔰𝔩_{R}(2) ⊕ 𝔤𝔩(N),

𝔤_{−} = ⊕_{−1≥i≥−2} 𝔤_{i} with 𝔤_{−1} = (id_{L} ⊗ Id) ⊕ (id_{R} ⊗ Id^{∗}), 𝔤_{−2} = id_{L} ⊗ id_{R}^{∗}, where id_{j} is the

space of the standard (identity) representation of 𝔰𝔩_{j}(2); j = L, R and Id is the space of the identity representation of 𝔰𝔩(N). The corresponding matrix representaion of 𝔭 = Lie(P) is of format 2|N|2:

(\begin{matrix}A_{11} & 0 & 0 \\ Q_{21} & A_{22} & 0 \\ A_{31} & Q_{32} & A_{33} \end{matrix}) with A_{11}, A_{33} ∈ 𝔰𝔩(2); A_{22} ∈ 𝔰𝔩(N)  (N_{st})

(the odd elements are denoted by Q); it was first found for a particular real form of 𝔤̂_{∗} = 𝔤_{−} ⊕ 𝔤̂_{0}.

The N-extended Minkowski superspace M(N) is SL(N|4)/P endowed with the natural (G, 𝔤_{−})-structure. The conventional versions of the Minkowski superspace correspond to a certain real form of the (complex) superspace M̂(N) with the (Ĝ, 𝔤_{−})-structure, i.e., the reduced (G, 𝔤_{−})-structure. Clearly,

M̂(N) = P/Ĝ, where Ĝ = SL(N) × SL_{L}(2) × SL_{R}(2).

GIKOS [GI] guessed (and we can prove) that these M(N) never satisfy our requirements (A) and (B) on SUGRA for N > 1. GIKOS considered an enlargement R̂(N) of M̂(N) defined as

R̂(N) = P/Ĝ′, where Ĝ′ = Q × SL_{L}(2) × SL_{R}(2) and Q is a parabolic subgroup of SL(N).

In other words, from P we pass to a smaller parabolic subsupergroup, P′, whose diagram has several middle roots marked as well. To satisfy (A) and (B), we have to test various P′s. This is impossible without a computer.

We can prove that the diagram (2.0) regardless of the number of middle roots marked can not satisfy requirements (A) and (B). In particular, it is well known to physicists that even for N = 1 the standard model does not satisfy (B). So in order to satisfy requirements (A) and (B) we consider nonstandard formats, some of them will be dubbed as “standard”.

For N = 8 the following two possibilities seem to be distinguished by their symmetry:

0− ⊗^{+} −0− 0^{+} −0− 0^{+} −0− 0^{+} −0− ⊗^{+} −0 P ar = (001111111100)  (8_{“standard”})

and

0− 0^{+} −0− ⊗^{+} −0− 0^{+} −0− ⊗^{+} −0− 0^{+} −0 P ar = (111100001111)  (8_{triality})

The “standard” model for N > 0 was originally of the form (N_{st}).

Even though supergravitationists did use cumbersome notations and worked with bare hands, GIKOS guessed that in (N_{st}) for N > 1 there is no chance to get structure functions satisfies (A) and (B) and in order to get EE-like equations, on has to enlarge the Minkowski superspace by taking for N = 2 a smaller A_{22}:

(\begin{matrix}A_{11} & 0 & 0 \\ Q_{21} & A_{22} & 0 \\ A_{31} & Q_{32} & A_{33} \end{matrix}) , where A_{22} = (\begin{matrix}A_{22}^{11} & 0 \\ A_{22}^{21} & A_{22}^{22} \end{matrix}) .

Elsewhere we will discuss the assumptions of Haag-Łopuszanski-Sohnius’ theorem which lead to the Poincaré supergroup and its enlargement, SL(N|4). It could be that by, yet unmentioned, physical reasons some of the quotients of SL(N|4) have to be ruled out; so far these reasons are not formulated explicitly. At the moment we accept the correctness of the choice of the “twistor supergroup” SL(N|4) but investigate several possible quotients corresponding to various matrix realizations or, which is the same, to connected components of the flag supermanifolds. The stationary subgroups we will consider in this paper are the simplest ones: the Lie groups, subgroups of O(4) × SL(N). The stationary Lie supergroups are to be considered.

Notice that even for the same stationary subgroup O(4)×SL(N) we can consider several realizations:

(\begin{matrix}A_{11} & 0 & 0 \\ Q_{21} & A_{22} & 0 \\ A_{31} & Q_{32} & A_{33} \end{matrix}) or (\begin{matrix}A_{11} & 0 & 0 \\ A_{21} & A_{22} & 0 \\ Q_{31} & Q_{32} & A_{33} \end{matrix}) .

The experience with the analogues of Einstein’s equations on symmetric spaces [LPS] teaches us to consider the models of Minkowski superspace whose stationary subgroup is a product of several copies of SL(2): otherwise the equations will not be of order 2.

With all these conventions we first have to study the following examples of Minkowski superspaces which we will denote more instructively by M̂(P ar) rather than M(N):

\underline{For N = 1}:

(\begin{matrix}A_{11} & 0 & 0 \\ Q_{21} & A_{22} & 0 \\ A_{31} & Q_{32} & A_{33} \end{matrix}) or (\begin{matrix}A_{11} & 0 & 0 \\ A_{21} & A_{22} & 0 \\ Q_{31} & Q_{32} & A_{33} \end{matrix}) , where A_{22} (resp. A_{33}) = 0

and with P ar = (00100) and (00001), respectively.

\underline{For N = 2}:

(\begin{matrix}A_{11} & 0 & 0 \\ Q_{21} & A_{22} & 0 \\ A_{31} & Q_{32} & A_{33} \end{matrix}) , or (\begin{matrix}A_{11} & 0 & 0 \\ A_{21} & A_{22} & 0 \\ Q_{31} & Q_{32} & A_{33} \end{matrix}) , or (\begin{matrix}A_{11} & 0 & 0 & 0 \\ Q_{21} & A_{22} & 0 & 0 \\ A_{31} & Q_{32} & A_{33} & 0 \\ Q_{41} & A_{42} & Q_{43} & A_{44} \end{matrix}) ,

where P ar = (001100), (000011), and (100001), respectively. In the last case A_{11} = A_{44} = 0, otherwise A_{ii} ∈ 𝔰𝔩(2).

\underline{For N = 4}:

(\begin{matrix}A_{11} & 0 & 0 \\ Q_{21} & A_{22} & 0 \\ A_{31} & Q_{32} & A_{33} \end{matrix}) , where A_{22} = (\begin{matrix}A_{22}^{11} & 0 \\ A_{22}^{21} & A_{22}^{22} \end{matrix}) and P ar = (00111100), (N = 4_{1})

(this corresponds to two copies of the Minkowski space united by a common superstructure) and

(\begin{matrix}A_{11} & 0 & 0 & 0 \\ Q_{21} & A_{22} & 0 & 0 \\ A_{31} & Q_{32} & A_{33} & 0 \\ Q_{41} & A_{42} & Q_{43} & A_{44} \end{matrix})  (N = 4_{2})

There is, naturally, the possibility with P ar = (00001111), as always.

\underline{For N = 8} the following cases seem to be the most interesting (symmetric) ones:

(\begin{matrix}A_{11} & 0 & 0 \\ Q_{21} & A_{22} & 0 \\ A_{31} & Q_{32} & A_{33} \end{matrix}) , where A_{ii} = (\begin{matrix}A_{ii}^{11} & 0 \\ A_{ii}^{21} & A_{ii}^{22} \end{matrix}) ,  (N = 8)

with P ar = (111100001111). This model corresponds to three copies of the Minkowski space appended by a 16-dimensional Gr_{4}^{8}, analogue of GIKOS’ “harmonic” space and united by a common superstructure. Another possibility is

(\begin{matrix}A_{11} & 0 & 0 \\ Q_{21} & A_{22} & 0 \\ A_{31} & Q_{32} & A_{33} \end{matrix}) , where A_{22} = (\begin{matrix}A_{22}^{11} & 0 & 0 & 0 \\ A_{22}^{21} & A_{22}^{22} & 0 & 0 \\ A_{22}^{31} & A_{22}^{32} & A_{22}^{33} & 0 \\ A_{22}^{41} & A_{22}^{42} & A_{22}^{43} & A_{22}^{44} \end{matrix})  (N = 8_{1})

with P ar = (001111111100). Ever less symmetric possibilities are:

(\begin{matrix}A_{11} & 0 & 0 \\ A_{21} & A_{22} & 0 \\ Q_{31} & Q_{32} & A_{33} \end{matrix}) , where A_{33} = (\begin{matrix}A_{33}^{11} & 0 & 0 & 0 \\ A_{33}^{21} & A_{33}^{22} & 0 & 0 \\ A_{33}^{31} & A_{33}^{32} & A_{33}^{33} & 0 \\ A_{33}^{41} & A_{33}^{42} & A_{33}^{43} & A_{33}^{44} \end{matrix})  (N = 8_{2})

with P ar = (000011111111) and

(\begin{matrix}A_{11} & 0 & 0 & 0 \\ Q_{21} & A_{22} & 0 & 0 \\ A_{31} & Q_{32} & A_{33} & 0 \\ Q_{41} & A_{42} & Q_{43} & A_{44} \end{matrix}) , where A_{44} = (\begin{matrix}A_{33}^{11} & 0 & 0 \\ A_{33}^{21} & A_{33}^{22} & 0 \\ A_{33}^{31} & A_{33}^{32} & A_{33}^{33} \end{matrix})  (N = 8_{3})

with P ar = (001100111111).

2.2. Theorem . In Table (below) there are listed all the orders and weights of all the structure functions for the indicated M̂(P ar).

The corresponding cocycles are listed in a detailed version. We numbered the cocycles so that the i-th cocycle of weight w = (w_{1}, …, w_{k}) and degree d is denoted by R_{i}^{d}(w_{1}, …, w_{k}). We drop some of the indices if they are clear from the contents.

Clearly, there are more candidates on the role of M̂(P ar) for the N we have considered and for other N as well; the package allows the reader to perform corresponding calculations.

Remark . We could have stuck to the yet “standard” model with Ĝ = SL_{L}(2) × SL(N) × SL_{R}(2) which preserves ε_{L} ⊗ vol ⊗ ε_{R}, where ε_{i} for i = L, R and vol are the volume elements preserved by SL_{i}(2) and SL(N), respectively. Then we could consider an equation similar to the equations (EE_{0}) from [LPS]; namely, consider a connection compatible with ε_{L} ⊗ vol ⊗ ε_{R} and for a component R of weight 0 in the space of nonholonomic structure functions with respect to Ĝ take

R = λε_{L} ⊗ vol ⊗ ε_{R}.  (EE_{0})

There might be several obstacles in writing (EE_{0}): such a tensor R might not exist, or, if it exists, the corresponding equation might be not of order 2, or the valency of R could be not the same as that of the right hand side, the tensor preserved by G. The equations that appear in this way are considered in [LPS]; they correspond to higher spins, favorites of M. Vasiliev [2].

To interpret the supergravity in the same way as we have treated the Einstein Equations [LPS], define the supergravity equations as follows. On M̂(N), the stationary subgroup (which coincides with Ĝ) of the point preserves ε_{1} ⊗ ε_{2} ⊗ ⋯ ⊗ ε_{k}, where ε_{i} is the volume preserved by SL_{i}(2), the i-th copy of SL(2), in the 2-dimensional identity representation.

If there are several, say k, tensors of weight 0 we can take for R their linear combination and the coefficients of this combination determine a parameter which runs over ℙ^{k−1}.

Example: N = 1. The tensor R depends on a parameter, the ratio a : b which runs over the projective line ℙ^{1}. Physicists call this parameter the Gates–Sigel parameter. On 𝔤_{−1}, there is the inner product given by the bracket. Notice that this product is even for P ar = (00100) and odd for P ar = (00001). The tacit choice was P ar = (00100). For P ar = (00100) the metric g on the Minkowski space M is the product of spinorial metrics ε_{L} and ε_{R} on the the maximal isotropic (with respect to the pairing) subspaces of 𝔤_{−1}, cf. [Ma].

The equation (EE_{0}) takes the form

aR_{1}(00) + bR_{2}(00) = λg.  (EE_{0}(a : b))

From the explicit form of the cocycles it is clear that the component expansion of the above equation does not contain (EE_{0}) for P ar = (00100), and only R_{1}(00) for P ar = (00001) has the right expansion.

Notice immediately, that for (EE_{0}(a : b)) to be well-defined we must demand that all structure functions of orders > 2 vanish. These conditions are the Wess-Zumino constraints. We have fewer of them than, say, in [3]: one of the constraints is “harmless”, its cohomology class is zero (like torsion of the Levi–Civita connection).

What shall we take for nonholonomic analogs of Ricci flatness? As for N = 0, these should be the vanishing conditions on the part of the Riemann tensor which does not belong to the conformal or Weyl tensor. For N > 0 there are several such components and we can equate to zero either or all of them. The equations are well-defined as integrability condition provided the constraints vanish.

If all structure functions of order 2 (and of lesser orders) vanish, the higher obstructions are well-defined and we can write an equation on them. For example, for P ar = (00100) we can equate to zero one or both of the equations:

R(3, 0) = 0, R(0, 3) = 0.

Different choices correspond to different supergravities (minimal, flexible, etc.).

On how many parameters do these equation actually depend? The analysis will be also performed in the detailed version, let us only say here that the the above equations should be considered for a symmetric connection compatible with the structure preserved by the stationary group of point of M̂(N). There is only one such connection for all the M̂(N). For N = 0 this suffices to single out the symmetric 10-dimensional part of the vierbein. For N > 0 we still have too many components after having singled out the symmetric part of the connection, but the requirement of vanishing of the structure function of orders lesser than that of EE (in our cases such are orders < 2) kills more components.

Table

deg(SF)\P ar  (0000)  (00100)  (00001)
0  not defined (3, 1), (1, 3)  —
1  —  (1, 0), (0, 1) (1, 1) ^{1 +1}, (0, 1)_{1}, (0, 3)_{1}
2  (2, 2), (0, 0) (1, 1), (0, 0)^{2}  (0, 0)^{2}, (2, 2),
(4, 0), (0,4)  (1, 0)_{1}, (3, 2)_{1}
3  not defined (3, 0), (0, 3)  —
deg(SF)\P ar  (001100)  (100001)  (000011)
-1  not defined  (0, 2), (2, 0)  not defined
0  (222), (123), (321)  (0, 1)_{1}, (1, 0)_{1}  (222), (123)_{1}, (301)_{1}
1  (110)_{1}, (011)_{1}  (0, 0) , (2, 2)^{2}  (110) , (031)_{1}
(1, 1)^{2}, (1, 0)_{1}^{2}, (0, 1)_{1}^{2}  (011)_{1}
2  (020)^{2}, (200)  (2, 2), (0, 0)^{3}  (000), (220), (002)
(101), (002)  (1, 0)_{1}, (0, 1)_{1}  (200) , (020), (101)_{1}
≥ 3  —  not defined  —
deg(SF)\P ar  (00111100)  (00001111)
-1  (1021), (1023), (1201), (2112),  (1122), (2211), (1001)_{1}
(3201), (1122)_{1}, (2211)_{1}  (1221)_{1}, (2112)_{1}, (3001)_{1}, (1003)_{1}
0  —  —
1  (0110), (0011)_{1}^{2 +1}, (1100)_{1}^{2 +1} (0011) ^{2 +1}, (1100)^{2}, (0110)_{1}^{2 +1}
(0000), (0220), (2000)  (0000)^{2}, (0022), (2200),
2 (1111), (0101)_{1}, (10010)_{1}, (0200), (0020)
(0002), (0020), (0200)  (0101)_{1}, (1010)_{1}
≥ 3  —  —

deg(SF)\P ar  (111100001111)
-3  (100001)^{2}, (100003), (300001), (100201), (102001)
(100221), (122001), (101112), (211101), (210012), (111111)
-2  (100122), (221001), (101211), (112101), (111012)
(210111), (030001), (100030)
−1  —
0  —
1  (000011)^{4}, (001100)^{4}, (110000)^{4}, (000110)_{1}^{4}, (011000)_{1}^{4}
(000000)^{3}, (000020), (000022)
2  (000200), (002200), (002000), (020000)
(220000), (000101)_{1}, (001010)_{1}, (010100)_{1}, (101000)_{1}
≥ 3  —

Notations. The cocycles which also correspond to the “conformal” case – on shell – are boxed ; the cocycles of small orders are all conformally invariant, such orders are boxed;

the cocycles which only exist in the conformal case are doubleboxed ; the suscript 1 singles out odd cocycles; the exponent denotes the multiplicity of the cocycle; the multiplicity of conformally invariant vectors is boxed.

For structure functions of degree > 2 the entry “not defined” in the Table refers to the “Riemannian” case, i.e., to the Ĝ-structure. It so happened that in these cases there are no conformal cohomologies which facilitates out plotting of the table. The dash — indicates that either there are no structure functions in this order or (if the degree is > 2 and the case is the Riemannian one) they are not defined.

Clearly, the Table is invariant under the change of parities 1 ↔ 0 in P ar.

4. Invariant differential operators [Gz], [GLS3]

§5. Bilinear differential operators invariant with repect to symplectomorphisms

5.1. Let ρ be a representation of the group Sp(2m; ℝ) in a V_{ρ}. A tensor field of type ρ on a 2m-dimensional symplectic manifold M is an object t defined in each local coordinate system x, in which the symplectic form is of the canonical form ω = ∑_{i≤m} dx_{i} ∧ dx_{2m+1−i}, by

the vector t(x) ∈ V_{ρ} so that the passage to other coordinates, y (with the same property), is defined by the formula

t(y(x)) = ρ (\frac{∂y(x)}{∂x}) t(x).

Traditionally (see reviews [K1], [K2]) the fibers of the tensor bundles were considered finite dimensional, but Leites showed recently [LKW] that on supermanifolds it is natural and fruitful to consider infinite dimensional fibers: this leads to semi-infinite cohomology of supermanifolds. Similar problem for symplectic manifolds was not studied yet.

The space of smooth tensor fields of type ρ will be denoted by T (ρ) or by T (λ), where λ = (λ_{1}, …, λ_{m}) is the lowest (or, for finite dimensional representations, highest, for convenience) weight of the irreducible representation ρ.

In what follows the letters ρ, σ, τ will denote irreducible representations of Sp(2m; ℝ) and letters λ, μ, ν their highest weights. (We should have concidered lowest weight only, but in this preliminary report we stick to finite dimensional representations.)

Examples of spaces of tensor fields: a) T (0) = C^{∞}(M);

b) T (1, 0, …, 0) ≅ 𝔳𝔢𝔠𝔱 ≅ Ω^{1} is the space of vector fields or (which is the same on any symplectic manifold thanks to the nondegeneracy of ω) the space of 1-forms on M;

c) ∏^{r} = T (\underbrace{1, …, 1}_{r−many}, 0, …, 0) the space of primitive r-forms.

On the space T_{c}(ρ) of tensor fields of type ρ with compact support, as indicated by the subscript, there is an invariant inner product

< χ, θ >= ∫_{M} < χ(x), θ(x) > ω_{0}^{m},  (IP)

where < ·, · > in the integrand is the Sp(2m; ℝ)-invariant inner product on V_{ρ}.

A differential operator B : T (ρ_{1}) ⊗ T (ρ_{2}) ⊗ ⋯ ⊗ T (ρ_{n}) ⟶ T (τ) is called n-ary (unary, binary, etc. for n = 1, 2, respectively). Such an operator B is called Diff(M)-invariant if it is uniquely expressed in all coordinate systems; it is Diff_{ω}(M)-invariant if it is uniquely expressed in all coordinate systems in which the symplectic form is of the standard form.

5.2. The unary Diff_{ω}(M)-invariant differential operators. All such operators are classifieded by A. N. Rudakov ([R1], [R2]):

0-th order: the multiplication by a number;

1-st order : the derivations of the primitive forms d_{+} : ∏^{r} ⟶ ∏^{r+1} and d_{−} : ∏^{r+1} ⟶ ∏^{r} (0 ≤ r ≤ m − 1). These operators are compositions of the exterior differential d : Ω^{p} ⟶ Ω^{p+1} and the projection onto the space of primitive forms; recall that Ω^{p} = ∏^{p} ⊕ ∏^{p−2} ⊕ … for p ≤ m and Ω^{p} ≅ Ω^{2m−p};

2-nd order: d_{2} = d_{+} ∘ d_{−} : ∏^{r} ⟶ ∏^{r}(1 ≤ r ≤ m).

Remark . Rudakov’s theorem implies that other invariant operators that might spring to mind (d_{−} ∘ ωd_{−}, etc.) are multiples of the described ones.

5.3. The binary Diff_{ω}(M)-invariant differential operators. Clearly, the operators B^{∗1} : T (τ) × T (ρ_{2}) ⟶ T (ρ_{1}) and B^{∗2} : T (ρ_{1}) × T (τ) ⟶ T (ρ_{2}) conjugate to (or 1- and 2-duals of) the invariant differential operator B : T (ρ_{1}) ⊗ T (ρ_{2}) ⟶ T (τ) with respect to the inner product (IP) are also differential and invariant ones.

0-th order operators are obviously those of the form

Z(χ, θ) = pr(χ(x) ⊗ θ(x)),

where pr : V_{ρ} ⊗ V_{σ} ⟶ V_{τ} is the projection of the tensor product onto an irreducible component.

1-st order operators:

Theorem . Any bilinear 1-st order (with respect to all arguments) Diff_{ω}(M)-invariant differential operator B : T (λ) ⊗ T (μ) ⟶ T (ν) is a linear combination of the following cases P1 – P8 (some of which host several distinct operators) and the operators obtained from them by 1- and 2-dualizaton and transposition of the arguments.

P1) λ = ( \underbrace{1, …, 1}_{p−many 1’s}, 0, …, 0); weights μ and ν differ by a unit in r places, r ≡ p + 1

mod 2. For r ≤ p + 1 there exist operators of the form Z(d_{+}ω, θ) and for r ≤ p − 1 there exist operators of the form Z(d_{−}ω, θ).

P2) The Lie derivative being restricted onto Sp(2m; ℝ)-irreducible subspaces splits into several operators of the form Z(d_{±}ω, θ) and an operator

L : V ect × T (ρ) ⟶ T (ρ)

which cannot be reduced to operators of the form P1).

Remark . Observe that if ξ ∈ 𝔥(M) ⊂ 𝔳𝔢𝔠𝔱(M) is a Hamiltonian vector field, then, by identifying 𝔥 with dΩ^{0}, we see that d_{+}ξ = d_{−}ξ = 0 and in this case L coincides with the Lie derivative. Therefore, L determines a representation of the Lie algebra 𝔥(M) in the space T (ρ). It is not difficult to show that the invariance of B is equivalent to its 𝔥(M)-invariance:

L(ξ, B(χ, θ)) = B(L(ξ, χ), θ) + B(χ, L(ξ, θ))

for any χ ∈ T (ρ), θ ∈ T (σ), ξ ∈ 𝔥(M).

P3) S^{k}𝔳𝔢𝔠𝔱×S^{l}𝔳𝔢𝔠𝔱 ⟶ S^{k+l−1}𝔳𝔢𝔠𝔱 (clearly S^{k}𝔳𝔢𝔠𝔱 ≅ T (k, 0, …, 0)) is the Poisson bracket (a.k.a. the symmetric Schouten’s concomitant) on (polynomial in momenta) functions on T^{∗}M.

P4) λ, μ, ν are of the form (2, 1, …, 1, 0, …, 0) each, with p, q and r non-zero numerical marks, respectively, such that p+q+r ≡ 0 mod 2, |p−q| ≤ r ≤ p+q, and p+q+r ≤ 2m+2.

If all inequalities are strict, then there exist four distinct operators defined on the spaces of such fields, otherwise there exist only two distinct operators. For p + q + r ≤ 2m two of these four or two operators are obtained as restrictions of the Nijenhuis bracket, or its conjugates, onto the subspaces

T (2, 1, …, 1, 0, …, 0) ⊂ Ω^{p} ⊗_{C^{∞}(M)} 𝔳𝔢𝔠𝔱.

§5. BILINEAR DIFFERENTIAL OPERATORS INVARIANT WITH REPECT TO SYMPLECTOMORPHISMS 53

Remark . The remaining two operators (i.e., the ones which are not the restrictions of the Nijenhuis bracket) are new. We do not know anything about them except that they exist and the same applies to the following two cases P5) and P6).

P5) λ, μ are of the same form as for P4), ν = (3, 1, 1, …, 1, 0, …, 0). There exists one operator for |p − q| + 1 ≤ r ≤ p + q − 1, p + q + r ≡ 1 mod 2, p + q + r ≤ 2m + 1.

P6) λ, μ are the same as in 4), ν = (2, 2, 1, …, 1, 0, …, 0) with r non-zero entries. The operator exists under the same conditions on p, q, r as for P5).

P7) ν = (1, …, 1, 0, …, 0); whereas λ, μ and conditions on p, q, r are the same as in 5). In this case there exists a unique operator which is not reducible to operators of the form d_{±}Z. It is a restriction of the Nijenhuis bracket.

P8) λ = (2, 0, …, 0); whereas μ and ν differ from each other by a unit at one place. There exists a unique such operator. Further on I’ll give arguments which enable one to express it, in principle, explicitly.

5.4. 2nd order operators. Grozman could not classify such operators so far. However, he was lucky to find one new invariant operator, denoted in the literature Gz:

Gz : T (2, 0, …, 0) × T (2, 0, …, 0) ⟶ T (2, 0, …, 0).

For m = 1 Grozman got the explicit expression for the operator Gz in 1976. Let me us reproduce it. In coordinates x, y we have ω = dx ∧ dy. Then

Gz : a · dx^{2} + 2b · dxdy + c · dy^{2}, a′ · dx^{2} + 2b′ · dxdy + c′ · dy^{2} ↦
\frac{∂^{2}g}{∂x^{2}} dx^{2} + 2\frac{∂^{2}g}{∂x∂y}dxdy + \frac{∂^{2}g}{∂y^{2}} dy^{2} + (\{c, a′\} − \{a, c′\})dxdy+
(\frac{∂^{2}a}{∂y^{2}} − 2\frac{∂^{2}b}{∂x∂y} + \frac{∂^{2}c}{∂x^{2}} ) (a′dx^{2} + 2bdxdy + c′dy^{2}) + (\{a, b′\} − \{b, a′\})dx^{2}+
(\frac{∂^{2}a}{∂y^{2}} − 2\frac{∂^{2}b}{∂x∂y} + \frac{∂^{2}c}{∂x^{2}} ) (adx^{2} + 2bdxdy + cdy^{2})(\{b, c′\} − \{c, b′\})dy^{2},

where g = ac′ − 2bb′ + ca′ and {·, ·} is the Poisson bracket. An explicit form of Gz for m > 1 is to be found.

5.5. Sketch of the proof of Theorem. Set y_{i} = x_{2m+1−i}(1 ≤ i ≤ m), ∂_{i} = \frac{∂}{∂x_{i}} , δ_{i} = \frac{∂}{∂y_{i}}. Denote the elements of the Lie algebra 𝔰𝔭(2m; ℝ) ⊂ 𝔥(M) by

e^{ii} = y_{i}∂_{i}, e_{ii} = x_{i}δ_{i}, e_{j}^{i} = x_{j}∂_{i} + y_{i}δ_{j}.

Then, clearly,

e^{ij} = e^{ji} = y_{i}∂_{j} + y_{j}∂_{i} and e_{ij} = e_{ji} = x_{i}δ_{j} + x_{j}δ_{i} for i ≠ j.

Let I(ρ) be the space of differential operators from T (ρ) into C^{∞}(M) with constant coefficients, i.e.,

I(ρ) = {∑ P_{i}(∂, δ)u_{i} | u_{i} ∈ V_{ρ}^{∗} ≅ V_{ρ}}.

The grading in I(ρ) is induced by that in the space of polynomials P_{i}’s, i.e., I(ρ)_{0} ≅ V_{ρ}. Define the pairing I(ρ) × T (ρ) ⟶ ℝ by the formula

< P u, x >= P (< u, χ(x) >)|_{x=0}.

On I(ρ), define the 𝔥(M)-action, dual to the action on T (ρ), via L. Now, to describe the invariant operators it suffices to find all the 𝔥(M)-morphisms I(τ) ⟶ I(ρ) ⊗_{ℝ} I(σ). It turns out that such a morphism is completely defined by the image of the highest vector v ∈ V_{τ} = I(τ)_{0}. Here we have fixed a Borel subalgebra {∑_{i≤j} a_{ij}x_{j}∂_{i}} ⋂ 𝔰𝔭(2m; ℝ) so that

w ∈ I(ρ) ⊗_{ℝ} I(σ) can be the image of a highest weight singular vector if and only if

e_{i+1}^{i}w = 0 for 1 ≤ i ≤ m − 1 and e^{m,m}w = 0 (conditions on w to be a highest vector)

and

(x_{1}^{2}δ_{1})w = 0.  (conditions of singularity of the vector)

The degree of w ∈ I(ρ) ⊗_{ℝ} I(σ) is equal to the order of the corresponding differential operator. The general form of a vector of degree 1 is

w = ∑_{i≤m} ∂i^{′}z_{i}^{0} + δi^{′}t_{i}^{0} + ∂i^{′′}z_{i}^{1} + δi^{′′}t_{i}^{1},

where z_{i}^{j}, t_{i}^{j} ∈ V_{ρ} ⊗ V_{σ}, ∂′(u ⊗ v) = ∂u ⊗ v, ∂′′(u ⊗ v) = u ⊗ ∂v. If w is a highest vector, then all vectors z, t are expressed in terms of z_{1}^{0}, z_{1}^{1} which should satisfy

e_{i+1}^{i}z_{1}^{j} = 0 for 2 ≤ i ≤ m − 1, (e_{2}^{1})^{2}z_{1}^{j} = 0, e^{m,m}z_{1}^{j} = 0.

The condition (x_{1}^{2}δ_{1})w = 0 is equivalent to the equation

e1,1^{′}z_{1}^{0} + e1,1^{′′}z_{1}^{1} = 0,

where (double) prime means that the operator acts only on the first (second) multiple of the tensor product.

Grozman succeeded to define all the cases, where the above system possesses a solution in V_{ρ} ⊗ V_{σ}; though in certain cases he was not able to find the solution itself.

Here is an example of a sucessfully solved case (case 8)):

λ = (2, 0, …, 0), ν = (μ_{1}, …, μ_{k−1}, μ_{k} + 1, μ_{k+1}, …);

the case ν_{k} = μ_{k} − 1 is dual to this one. Let u_{0} ∈ V_{ρ} be a highest vector, then

u_{0} ⊗ v − \frac{1}{2} ∑_{2≤i≤k} e_{1}^{i}u_{0} ⊗ e_{i}^{1}v ∈ V_{ρ} ⊗ V_{σ}

is a highest vector of weight (ν_{1} + 1, ν_{2}, …, ν_{m}). We conclude that

z_{1}^{0} = u_{0} ⊕ e_{11}v − ∑_{2≤i≤k} e_{1}^{i}u_{0} ⊗ e_{11}e_{i}^{1}v − \frac{1}{2} ∑_{2≤i<j≤m} e_{1}^{i}e_{1}^{j}u_{0} ⊗ (e_{ij} + e_{1j}e_{1}^{i} + e_{1i}e_{j}^{1})v−

\frac{1}{2} ∑_{2≤i≤k}(e_{1}^{i})^{2}u_{0} ⊗ (e_{ii} + e_{1i}e_{i}^{1})v + \frac{1}{2} ∑_{2≤i≠j≤m} e_{1}^{i}e_{1j}u_{0} ⊗ (e_{i}^{j} + e_{1}^{j}e_{i}^{1})v + (ν_{i} + e_{1}^{i}e_{i}^{1})v+

\frac{1}{4} ∑_{2≤i≤m} e_{ii}(e_{1}^{i})^{2}u_{0} ⊗ +\frac{ν_{1}−1}{2} ∑_{2≤i≤k} e_{11}e_{1}^{i}u_{0} ⊗ e_{i}^{1}v,

and

z_{1}^{1} = −e_{11}u_{0} ⊗ v + ∑_{2≤i≤k} e_{11}e_{1}^{i}u_{0} ⊗ e_{i}^{1}v.

5.6. Conjectures. 1) The operator Gz is a particular case of a more general operator :

Gz_{r,s} : T (2, \underbrace{1, …, 1}_{r−many}, 0, …, 0) × T (2, \underbrace{1, …, 1}_{s−many}, 0, …, 0) ⟶ T (2, \underbrace{1, …, 1}_{r+s−many}, 0, …, 0).

2) Operators of order > 2 are compositions of operators of orders ≤ 2. There are no operators of order > 5.

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