THE ANALOGS OF THE RIEMANN TENSOR FOR EXCEPTIONAL STRUCTURES ON SUPERMANIFOLDS

PAVEL GROZMAN, DIMITRY LEITES, IRINA SHCHEPOCHKINA^{1}

Abstract. H. Hertz called any manifold M with a given nonintegrable distribution nonholonomic. Vershik and Gershkovich stated and R. Montgomery proved that the space of germs of any nonholonomic distribution on M with an open and dense orbit of the diffeomorphism group is either (1) of codimension one or (2) an Engel distribution.

No analog of this statement for supermanifolds is formulated yet, we only have some examples: our list (an analog of É. Cartan’s classification) of simple Lie superalgebras of vector fields with polynomial coefficients and a particular (Weisfeiler) grading contains 16 series similar to contact ones and 11 exceptional algebras preserving nonholonomic structures.

Here we compute the cohomology corresponding to the analog of the Riemann tensor for the supermanifolds corresponding to the 15 exceptional simple vectorial Lie superalgebras, 11 of which are nonholonomic. The cohomology for analogs of the Riemann tensor for the manifolds with an exceptional Engel manifolds are computed in [L0]; for the 16 series of contact-like distributions on supermanifolds see [GLS3].

Introduction

The main result. In this paper the ground field is ℂ. Here, for each of the 15 exceptional simple infinite dimensional vectorial Lie superalgebras 𝔤 = ⊕_{i≥−d} 𝔤_{i} in their Weisfeiler grading

we have computed H^{i}(𝔤_{−}; 𝔤) for i = 0, 1, 2, where 𝔤_{−} = ⊕_{i<0} 𝔤_{i}. These cohomology are

especially interesting for i = 2. If d = 1, then H^{2}(𝔤_{−}; 𝔤) can be interpreted as the space of values of the curvature tensor for the G-structure, where G is a Lie supergroup whose Lie superalgebra is 𝔤_{0}; if d > 1 we interprete H^{2}(𝔤_{−}; 𝔤) as the space of values of the recently introduced nonholonomic curvature for the nonintegrable distribution.

To make the text of interest to a wider audience, we would like to review the list of simple vectorial Lie superalgebras and some not so known background, cf. [D], but had to delete this from this text for the lack of space; for the same reason we skipped the results of computations of H^{i}(𝔤_{−}; 𝔤) for i = 0, 1 (these results will be preprinted; H^{1}(𝔤_{−}; 𝔤), though not so interesting, perhaps, as H^{2}(𝔤_{−}; 𝔤), also has an interpretation of interest: its elements represent derivations of 𝔤_{−} into 𝔤, cf. [CK1].)

The results demonstrate one more range of applicability of SuperLie package. It is designed for various computations with Lie superalgebras, not only for computation of (co)homology; for other results and comparison with hand-made calculations, see, e.g., [GLS2]. SuperLie is Mathematica-based which facilitates its usage but imposes in-build Mathematica restrictions. We hope to draw attention to possibilities SuperLie (now installed at MPIM, Bonn; LPT, ENS; Department of Mathematics, University of Stockholm) reveals to its user.

1991 Mathematics Subject Classification. 17A70 (Primary) 17B35 (Secondary).

Key words and phrases. Lie superalgebra, Cartan prolongation, Lie algebra cohomology, nonholonomic structures, Riemannian tensor, Spencer cohomology.

ACCEPTED IN: O. SHEINMAN ET AL (EDS.) PROC. INTERNATIONAL CONFERENCE “FUNDAMENTAL MATHEMATICS TODAY” (DECEMBER 26–29, 2001) DEDICATED TO 10TH ANNIVERSARY OF THE INDEPENDENT UNIV. OF MOSCOW.

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

In particular, our results (as well as similar results of Poletaeva [P1] — [P3] performed by bare hands) vividly demonstrate that in the absence of complete reducibility computer-aided study is indispensable.

0.1. A result of Vershik and Gershkovich. Nonholonomic curvature. R. Montgomery [Mo] proved the following statement whose particular case was stated in [VG1]. Let W_{n}^{k} be the space of germs of k-dimensional distributions at 0 ∈ ℂ^{n}. (Both [Mo] and [VG1] consider the real case but the result is the same.) The group Diff_{n} of germs of diffeomorphisms of ℂ^{n} acts on W_{n}^{k} and it is interesting to find out the conditions for existence of the frame (i.e., pointwise values of the vector fields from a given distribution) that generates a finite dimensional (hence, nilpotent) Lie algebra. The answer:

for 1 < k < n − 1 and (k, n) ≠ (2, 4) any parametrization of any open subset of the space of generic orbits of Diff_{n}-action on W_{n}^{k} requires ≥ k(n−k)−n functions in n indeterminates. The exceptional case k = 1 is trivial. So W_{n}^{k} has an open and dense Diff_{n}-orbit if and only if either (1) k = n − 1 (for n odd this is the contact structure) or (2) (k, n) = (2, 4) (and then the distribution is an Engel one).

In [L0], the notion of nonholonomic curvature is introduced and the above cases (1) and (2) are considered. It turns out that the nonholonomic curvature vanishes if k = n − 1 (this is a reformulation of Darboux theorem on a canonical form of the contact form for n odd) whereas for (k, n) = (2, 4) — the Engel distribution — dim H^{2}(𝔤_{−}; 𝔤) = 2.

Observe that the (infinite dimensional) algebra of symmetries of the exceptional (case (1) or (2)) nonholonomic distribution is simple only if the distribution is of codimension 1 and n is odd. Contemporary mathematicians are often more than necessary fixed on simple Lie algebras and this is, perhaps, an explanation why the Lie algebra preserving an Engel structure (cf. [L0]) was neglected for a long time. Differential geometers, though mildly interested in the cases the total algebra of symmetries is simple, are more interested in cases when it is of finite dimension (simple or not), and therefore their interests are orthogonal to ours as is seen from motivations and results reviewed, e.g., in [Y].

We do not know any super version of the above result of Vershik and Gershkovich [VG1] but we classified simple Lie superalgebras of vector fields ([LSh0], [LSh1], [LSh3], cf. [Ka7]) and, we see that, unlike non-super case, there are 16 series and 11 exceptional simple vectorial Lie superalgebras that preserve nonholonomic distributions. The series will be considered elsewhere.

0.2. A nonholonomic analog of the Riemann tensor. H. Hertz called any manifold with a nonintegrable distribution a nonholonomic one. Until 1989, there was no general definition of the analog of the Riemann tensor for nonholonomic manifolds, cf. lamentations in [VG1] and [WB], though all the ingredients had been discovered ([T], [Y]). Vershik even conjectured [V] that such a general definition does not exist, though in particular cases of small dimension the nonholonomic curvature tensor was computed. In particular, in supergravity.

Recall that SUGRA(N) is a supergravity theory (or equations thereof) on an N-extended Minkowski superspace. Whatever SUGRA(N) and Minkowski superspace are (there are several versions of the definition and, unless N = 1, there is no consensus among physicists which of the definitions is “it”, Manin [M] suggests still other — “exotic” — versions of Minkowski superspaces, and this list of ad hoc superizations of Minkowski space will, clearly, be continued, see e.g., [GL3]), they are superizations of the gravity theory (or Einstein-Hilbert’s equations) on the Minkowski space. So the problem whose existence Wess honestly acknowledged in his lectures [WB] “We do not know how to write the super Riemann tensor”

(on N > 2 extended Minkowski superspaces) sounds strange: take any textbook on differential geometry (say, [St]) and superize definition of the Riemann tensor or, more generally, structure functions — analogs of the Riemannian tensor — for any G-structure according to Sign Rule. This was precisely what A. Schwarz with his colleagues and students suggested to do [S], see also [CAF].

The results of such an approach, however, seem to coincide with the equations physicists write from their mysterious physical considerations only for N = 1 (but actually do not even in this case, cf. [S], [CAF] with [GL1]).

In [GL1], [GL2] it was indicated that the roots of the problem Wess addressed lie not in the prefix “super” which only causes some signs in the classical definitions of the structure functions. The point is that every of numerous versions of Minkowski superspace is nonholonomic, unlike Minkowski space, and since there was no general recipe for computing nonholonomic analog of the curvature tensor, to write SUGRA(N) equations or even determine what is N-extended Minkowski superspace (which of the versions satisfies some natural requirements) was a problem.

Here we will reproduce the definition ([L0], [LP2], [GL1]) of the Riemann tensor R in terms of Lie algebra cohomology rather than in terms of Spencer cohomology (cf. [G], [St]) and give its generalization to nonholonomic case.

Namely, fix a point on the nonholonomic (super)manifold with a nonintegrable distribution D. Let D be given by a system of Pfaff equations and let 𝔊 be the filtered Lie superalgebra preserving this system of equations, 𝔪 the associated graded one. Set 𝔪_{−} = ⊕_{i<0} 𝔪_{i}; by

default we let 𝔤_{0} = 𝔪_{0}, the Lie superalgebra of grading preserving derivations of 𝔤_{−} = 𝔪_{−}. Clearly, 𝔪_{−} is nilpotent, see [VG2], [Y]. Very often an additional structure on D is given; if this is the case, we take for 𝔤_{0} a subalgebra of 𝔪_{0} that preserves this additional structure. Let 𝔤 be the defined below generalized Cartan prolong of the pair (𝔤_{−}, 𝔤_{0}). Then, by the same arguments as in [St], the possible values of the nonholonomic Riemann tensor R at the point span the superspace H^{2}(𝔤_{−}; 𝔤).

0.3. The projective connections and their nonholonomic analogs. The projective connection on the n-dimensional manifold is the one whose group of automorphisms is locally isomorphic to 𝔰𝔩(n + 1) = 𝔤_{−} ⊕ 𝔤𝔩(n) ⊕ (𝔤_{−})^{∗}, cf. [G]. The corresponding structure functions are from H^{2}(𝔤_{−}; 𝔰𝔩(n + 1)).

Similarly, for any ℤ-graded Lie superalgebra 𝔤 of finite depth, let 𝔥 ⊂ 𝔤 be a subalgebra with the same nonpositive part. Then the elements of H^{2}(𝔤_{−}; 𝔥) are analogs of the projective structure functions, especially resembling them if dim 𝔥 < ∞. Such cohomology is considered in [GLS3].

§1. Description of simple vectorial Lie superalgebras

For the lack of space we deleted all the preliminaries. The reader willing to see them is referred to [Sh14], [Sh5] and [LSh3]. For a detailed background see a preprinted version at www.mpim-bonn.mpg.de. Observe only that Π is the shift of parity functor on superspaces, Vol is the space of densities with the generator vol in a fixed coordinate system.

1.1. The exceptional vectorial Lie subsuperalgebras. Here are the terms 𝔤_{i} for i ≤ 0 of 14 of the 15 exceptional algebras, the last column gives dim 𝔤_{−}. Here Λ(n) is the Grassmann superalgebra with n generators; id is the identity representation, Λ(id) its exterior algebra; Vol_{0} is the space of densities with integral 0; and T_{0}^{0}(0) = Vol_{0}/const is well-defined only as module over 𝔰𝔳𝔢𝔠𝔱:

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

Observe that none of the simple W-graded vectorial Lie superalgebras is of depth > 3 and only two algebras are of depth 3: one of the above, 𝔪𝔟(4|5; K), for which we have 𝔪𝔟(4|5; K)_{−3} ≅ Π(id_{𝔰𝔩(2)}), and another one, 𝔨𝔰𝔩𝔢(9|6; CK) = 𝔠𝔨(9|11).

This 𝔠𝔨((9|11) is the 15-th exceptional simple vectorial Lie superalgebra; its non-positive terms are as follows (we assume that the 𝔰𝔩(2)- and 𝔰𝔩(3)-modules are purely even):

𝔠𝔨((9|11)_{0} ≃ (𝔰𝔩(2) ⊕ 𝔰𝔩(3) ⊗ Λ(1)) ⊃ + 𝔳𝔢𝔠𝔱(0|1);
𝔠𝔨((9|11)_{−1} ≃ id_{𝔰𝔩(2)} ⊗ (id_{𝔰𝔩(3)} ⊗ Λ(1));
𝔠𝔨((9|11)_{−2} ≃ id_{𝔰𝔩(3)}^{∗} ⊗ Λ(1);
𝔠𝔨((9|11)_{−3} ≃ Π(id_{𝔰𝔩(2)} ⊗ ℂ).

1.2. A description of 𝔤 as 𝔤¯0 and 𝔤¯1. In [CK2] the exceptional algebras are described as 𝔤 = 𝔤¯0 ⊕ 𝔤¯1. For several series such description is of little value because each homogeneous component 𝔤¯0 and 𝔤¯1 has a complicated structure. For the exceptions (and for twisted polyvector fields) the situation is totally different! Apart from being beautiful, such description (from [CK2]) is useful for the construction of simple Volichenko algebras, cf. [LS].

Recall in this relation a theorem [Gr] that completely describes bilinear differential operators acting in tensor fields and invariant under all changes of coordinates. It turned out that almost all of the first order operators determine a Lie superalgebra on its domain. Some of these superalgebras are simple or close to simple. In the constructions below we use some of these invariant operators. \underline{𝔤 = 𝔨𝔰𝔩𝔢(5|10)}: 𝔤¯0 = 𝔰𝔳𝔢𝔠𝔱(5|0) ≃ dΩ^{4}, 𝔤¯1 = Π(dΩ^{1}) with the natural 𝔤¯0-action on 𝔤¯1 (the Lie derivative) and the bracketing of odd elements being twice their product. (We identify:

∂_{i} = sign(ijklm)dx_{j}dx_{k}dx_{l}dx_{l} for any permutation (ijklm) of (12345).

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

[ω_{1} ⊗ vol^{−1/2}, ω_{2} ⊗ vol^{−1/2}] = (dω_{1} ∧ ω_{2} + ω_{1} ∧ dω_{2}) ⊗ vol^{−1}],

where we identify

dx_{i} ∧ dx_{j} ∧ dx_{k} ⊗ vol^{−1} = sign(ijkl)∂_{l} for any permutation (ijkl) of (1234).

\underline{𝔤 = 𝔳𝔩𝔢(3|6)}: 𝔤¯0 = 𝔳𝔢𝔠𝔱(3|0) ⊕ 𝔰𝔩(2)_{≥0}^{(1)}, where 𝔤_{≥0}^{(1)} = 𝔤 ⊗ ℂ[t] , and 𝔤¯1 = (Ω^{1} ⊗ Vol^{−1/2}) ⊗ id_{𝔰𝔩(2)_{≥0}^{(1)}} with the natural 𝔤¯0-action on 𝔤¯1.

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

\begin{matrix}[(ω_{1} ⊗ v_{1})vol^{−1/2}, (ω_{2} ⊗ v_{2})vol^{−1/2}] = \\ (ω_{1} ∧ ω_{2}) ⊗ (v_{1} ∧ v_{2}) + dω_{1} ∧ ω_{2} + ω_{1} ∧ dω_{2}) ⊗ (v_{1} • v_{2})) vol^{−1}, \end{matrix}

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

dx_{i} ∧ dx_{j} ⊗ vol^{−1} = sign(ijk)\frac{∂}{∂x_{k}} for any permutation (ijk) of (123).

\underline{𝔤 = 𝔪𝔟(3|8)}: 𝔤¯0 = 𝔳𝔢𝔠𝔱(3|0) ⊕ 𝔰𝔩(2)_{≥0}^{(1)}, and 𝔤¯1 = 𝔤_{−1} ⊕ 𝔤_{1}, where

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

clearly, one can interchange 𝔤_{±1}.

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

[(ω_{1} ⊗ v_{1})vol^{−1/2}, (ω_{2} ⊗ v_{2})vol^{−1/2}] = 0,

[(f_{1} ⊗ v_{1})vol^{−1/2}, (f_{2} ⊗ v_{2})vol^{−1/2}] = (df_{1} ∧ df_{2}) ⊗ (v_{1} ∧ v_{2})vol^{−1},

[(f_{1} ⊗ v_{1})vol^{−1/2}, (ω_{1} ⊗ v_{2})vol^{−1/2}] = (f_{1}ω_{1} ⊗ (v_{1} ∧ v_{2}) + (df_{1}ω_{1} + f_{1}dω_{1}) ⊗ (v_{1} • v_{2})) vol^{−1}.

\underline{𝔤 = 𝔨𝔞𝔰}: 𝔤¯0 = 𝔳𝔢𝔠𝔱(1|0) ⊕ 𝔰𝔩(4)_{≥0}^{(1)}, and 𝔤¯1 = 𝔤_{−1} ⊕ 𝔤_{1}, where 𝔤_{−1} = Π (Λ^{2}(id_{𝔰𝔩(2)_{≥0}^{(1)}})) and

𝔤_{1} = Π (S^{2}(id_{𝔰𝔩(2)_{≥0}^{(1)}})); clearly, one can interchange 𝔤_{±1}.

§2. Main result

The above description of the exceptional algebras is nice to visualize them, but in calculations we have used sometimes the description of the elements from [Sh14], which we do not reproduce to save space. Theorem . The 𝔤_{0}-modules H^{2}(𝔤_{−}; 𝔤) are given by the lists of modules over the semisimple part of (𝔤_{0})¯0 given in §3.

Comments. 1) important: observe that if the 𝔤_{0}-module from H^{2}(𝔤_{−}; 𝔤) is indecomposable, then, in equations like Einstein equations or Wess-Zumino constraints, there is no need to vanish all its irreducible components: it suffices to vanish only the modules that generate all.

2) The highest weights are given with respect to the standard basis of Cartan subalgebra of the maximal semisimple part of (𝔤_{0})¯0, or, if this part is 𝔰𝔩(n), with respect to the matrix units E_{ii} of the 𝔤𝔩(n). The degree of the cocycle is given relative to the grading of 𝔤. The weight of vector A is denoted by w(A). “mult” denotes the multiplicity of the corresponding module in the space of closed/exact forms. If it is not equal to r/0, the respective cohomologies are not pure (are defined up to a coboundary) and it may well happen that our choice of the representative may be beautified. The expression d[v]^{2} means (dv)^{2}. Few more notations appear in respective places.

3) For 𝔤 of depth d the degrees of cohomology (interpreted as orders k of the structure functions responsible for obstructions to flattening the structure under investigation up to kth infinitesimal neighborhood, cf. [St]) may range from 2 − d, which we will indicate for d > 1. Recall that the structure functions of order k are defined provided structure functions of lesser orders vanish. It happens sometimes (the case of Riemannian metric) that in these lesser orders there are no cohomology (torsion-free property of Levi-Civita connection). Otherwise that vanishing conditions on low-order structure functions are analogs of Wess-Zumino constraints in supergravity, cf. [WZ].

4) Due to a theorem formulated only for d = 1 (by Serre for Lie algebras and by Serganova for superalgebras, cf. [LPS]), the order of nonzero structure functions from H^{2}(𝔤_{−1}; 𝔤) is always equal to 1 under certain conditions (involutivity). As we will see, the considered Lie superalgebras of depth 1 are all involutive. The yet nonexisting analog of Serre’s theorem for d > 1 is more complicated: cohomology may be non-vanishing in several degrees. If the lowest degrees are absent, we indicate this by writing “torsion-free” by analogy with the Riemannian manifolds.

§3. The 𝔤_{0}-modules H^{2}(𝔤_{−}; 𝔤)

𝔤 = 𝔳𝔩𝔢(4|3). Cohomology: a single irreducible 𝔤_{0}-module in deg = 1, dim = (24|24).

Set: w(u_{i}) = (1, 1, 1) − ε_{i}, w(y) = (0, 0, 0), w(ξ_{i}) = ε_{i} and ∂_{0} = \frac{∂}{∂y}, ∂_{i} = \frac{∂}{∂u_{i}}, δ_{i} = \frac{∂}{∂ξ_{i}}. The 𝔤𝔩(3)-highest vectors are:

𝔤𝔩(3)-highest vectors 𝔤𝔩(3)-weight dim mult
∂_{1} d[∂_{2}] ∧ d[∂_{3}]  (2, 0, 0)  (6|0) 3/2
∂_{0} d[∂_{0}] ∧ d[δ_{1}]  (1, 0, 0)  (0|3) 5/4
∂_{1} d[∂_{3}] ∧ d[δ_{1}]  (2, 0, −1) (0|15) 2/1
∂_{1} d[∂_{3}] ∧ d[∂_{0}]  (1, 0, −1) (8|0) 3/2
∂_{1} d[δ_{1}] ∧ d[δ_{1}]  (2, −1, −1) (10|0) 1/0
∂_{1} d[∂_{0}] ∧ d[δ_{1}]  (1, −1, −1) (0|6) 1/0

𝔤 = 𝔳𝔩𝔢(4|3; 1). 2 − d = 0.

Cohomology in deg = 1, torsion-free, dim = (20|20):

The (𝔤_{0})¯0-highest weight vectors (with respect to 𝔰𝔩(2) ⊕ 𝔤𝔩(2)) are as follows: (the first coordinate of the weight is given with respect to a copy of 𝔰𝔩(2) realized as 𝔬(3), with half-integer weights; the last two coordinates are with respect to a copy of 𝔤𝔩(2)). In this realization w(u_{1}) = (1, 1, 1), w(u_{2}) = (1, 0, 1), w(u_{3}) = (1, 1, 0), w(y) = (−1, 0, 0), w(ξ_{1}) = (0, 0, 0), w(ξ_{2}) = (0, 1, 0), w(ξ_{3}) = (0, 0, 1). Denote the elements of 𝔤_{−} as follows: g_{1} = ∂_{2};

g_{2} = ∂_{3}; g_{3} = δ_{2}; g_{4} = δ_{3}; g_{5} = −u_{2}∂_{1} + ξ_{1}δ_{2}; g_{6} = −u_{3}∂_{1} + ξ_{1}δ_{3}; g_{7} = −yδ_{3} − ξ_{1}∂_{2} + ξ_{2}∂_{1};

g_{8} = yδ_{2} − ξ_{1}∂_{3} + ξ_{3}∂_{1}; g_{9} = ∂_{1}. We have

N  𝔰𝔩(2) ⊕ 𝔤𝔩(2)-highest vectors  𝔰𝔩(2) ⊕ 𝔤𝔩(2)-weight dim mult
[1] 2 ∂_{2}dg_{1} ∧ dg_{5} + ∂_{2}dg_{2} ∧ dg_{6} + ∂_{3}dg_{2} ∧ dg_{5}  (0, 1, 0)  (2|0) 5/4
[2]  ∂_{2}dg_{2} ∧ dg_{5}−  (0, 2, −1)  (4|0) 2/1
[3] −δ_{3}dg_{3} ∧ dg_{8} + (yδ_{3} − ξ_{1}∂_{2} + ξ_{2}∂_{1})dg_{8} ∧ dg_{8}  (1, 2, −1)  (0|8) 3/2
[4]  ∂_{2}dg_{5} ∧ dg_{6}  (0, 1, 0)  (2|0) 1/0
[5] 2∂_{2}dg_{5} ∧ dg_{7} + ∂_{2}dg_{6} ∧ dg_{8} + ∂_{3}dg_{5} ∧ dg_{8}  (1, 1, 0)  (0|4) 2/1
[6]  ∂_{2}dg_{5} ∧ dg_{8}  (1, 2, −1)  (0|8) 1/0
[7]  −δ_{3}dg_{5} ∧ dg_{8} − ∂_{2}dg_{8} ∧ dg_{8}  (2, 2, −1)  (12|0) 1/0

𝔤_{0}-modules:

[A] = [2] + [3] + [6] + [7] of dim = (16|16) and [B] = [A] + [1] + [4] + [5] of dim = (20|20); the modules [A] and [B]/[A] are irreducible.

𝔤 = 𝔳𝔩𝔢(4|3; K). 2 − d = 0.

The 𝔤_{0}-highest weight vectors are as follows: 𝔤_{0} = 𝔰𝔩(2) ⊕ 𝔤𝔩(3); the first coordinate of the weight is given with respect to 𝔰𝔩(2) realized as x_{−} = ∂_{y}, x_{+} = y^{2}∂_{y} + y ∑_{i} ξ_{i}δ_{i} + ξ_{1}ξ_{2}∂_{3} − ξ_{1}ξ_{3}∂_{2} + ξ_{2}ξ_{3}∂_{1}; 𝔤𝔩(3) is realized as x_{j}^{i} = −u_{j}∂_{i} + ξ_{i}δ_{j} (i ≠ j) and x_{i}^{i} = −u_{i}∂_{i} + ξ_{i}δ_{i} + ∑_{k} u_{i}∂_{i}. In this realization w(u_{1}) = (2, 0, 1, 1), w(u_{2}) = (2, 1, 0, 1), w(u_{3}) = (2, 1, 1, 0), w(y) = (−2, 0, 0, 0), w(ξ_{1}) = (0, −1, 0, 0), w(ξ_{2}) = (0, 0, −1, 0), w(ξ_{3}) = (0, 0, 0, −1)).

Cohomology: a single irreducible 𝔤_{0}-module in deg = 0, dim = (30|0):

𝔰𝔩(2) ⊕ 𝔤𝔩(3)-highest vectors 𝔰𝔩(2) ⊕ 𝔤𝔩(3)-weight dim mult
∂_{1}d[δ_{1}]^{2}  (2, 2, −1, −1)  (30|0) 1/0

𝔤 = 𝔳𝔞𝔰(4|4). The weights are given relative to 𝔤𝔩(4) ⊂ 𝔤_{0}. In this realization, the weight of u_{i} is ε_{i} − \frac{1}{2}(1, 1, 1, 1) and the weight of ξ_{i} is −ε_{i}; we set ∂_{i} = \frac{∂}{∂u_{i}}, δ_{i} = \frac{∂}{∂ξ_{i}} for 1 ≤ i ≤ 4.

Cohomology: in deg = 1, dim = (40|40):

N  𝔤𝔩(4)-highest vectors  weight  dim mult
[1] δ_{1}d[∂_{2}] ∧ d[∂_{3}] − δ_{2}d[∂_{1}] ∧ d[∂_{3}] + δ_{3}d[∂_{1}] ∧ d[∂_{2}] (0, 0, 0, −1) (0|4) 4/3
[2]  ∂_{4}d[∂_{1}] ∧ d[∂_{2}]  (\frac{1}{2}, \frac{1}{2}, −\frac{1}{2}, −\frac{3}{2}) (20|0) 3/2
[3]  ∂_{4}d[δ_{4}] ∧ d[δ_{4}]  (\frac{1}{2}, \frac{1}{2}, \frac{1}{1}, −\frac{5}{2}) (20|0) 1/0
[4]  δ_{1}d[δ_{4}] ∧ d[δ_{4}]  (1, 0, 0, −2) (0|36) 2/1

The module is reducible but indecomposable. Vector [1] is the highest weight vector in the quotient module.

𝔤 = 𝔨𝔞𝔰. 2 − d = 0. For brevity, instead of K_{f} we write simply f and df^{2} means (df)^{2}.

Cohomology: a single irreducible module in deg = 1, hence torsion free. The 𝔬(6)-weights are:

𝔬(4)-highest vectors  𝔬(6)-weight dim mult
ξ_{1}ξ_{2}d[ξ_{3}] ∧ d[η_{1}] − ξ_{1}η_{3}d[η_{1}] ∧ d[η_{2}] + ξ_{2}η_{3}d[η_{1}] ∧ d[η_{1}] (2, 1, −1) (45|0) 1/0

𝔤 = 𝔨𝔞𝔰(; 1). 2 − d = 0. Nontrivial cohomology are in degrees 0 and 1.

The weights are given relative to 𝔬(6), but the corresponding vectors are heighest only with respect to 𝔬(4) = 𝔬(6) ∩ 𝔤_{0}.

Cohomology in deg = 0, dim = (6|6):

N 𝔬(4)-highest vectors 𝔬(6)-weight dim mult
[1] d[ξ_{1}ξ_{3}] ∧ d[ξ_{1}η_{2}] (−2, 1, −1) (3|0) 1/0
[2] d[ξ_{1}η_{2}] ∧ d[ξ_{1}η_{3}]  (−2, 1, 1) (3|0) 1/0
[3] ξ_{1}d[ξ_{1}ξ_{3}] ∧ d[ξ_{1}η_{2}] (−1, 1, −1) (0|3) 2/1
[4] ξ_{1}d[ξ_{1}η_{2}] ∧ d[ξ_{1}η_{3}] (−1, 1, −1) (0|3) 2/1

[2] ⊕ [4] is 𝔤_{0}-irreducible; [1] and [3] are glued as 𝔤_{0}-modules.

Cohomology in deg = 1, dim = (8|8):

N  𝔬(4)-highest vectors  𝔬(6)-weight dim mult
[1] ξ_{1}d[ξ_{1}η_{2}] ∧ (ξ_{2}d[ξ_{3}] − η_{3}d[η_{2}]) (0, 2, −1) (0|8) 2/1
[2] ξ_{1}d[η_{2}] ∧ (ξ_{2}d[ξ_{3}] − η_{3}d[η_{2}]) (1, 2, −1) (8|0) 1/0

A single irreducible 𝔤_{0}-module.

𝔤 = 𝔨𝔞𝔰(; 3η). Cohomology in degrees 1 and 2.

deg = 1: a single irreducible 𝔤_{0}-module of dim = (12|12) (the vectors are highest with respect to 𝔤𝔩(3) = 𝔬(6) ∩ 𝔤_{0}, the weights are the same as in 𝔬(6)):

N  𝔤𝔩(3)-highest vectors  weight dim mult
[1]  d[η_{1}η_{2}] ∧ d[η_{1}η_{3}]  (2, 1, 1) (3|0) 1/0
[2] d[η_{1}] ∧ d[η_{2}η_{3}] − d[η_{2}] ∧ d[η_{1}η_{3}] + d[η_{3}] ∧ d[η_{1}η_{2}] (1, 1, 1) (0|1) 2/1
[3]  d[η_{1}] ∧ d[η_{1}η_{2}]  (2, 1, 0) (0|8) 2/1
[4]  d[1] ∧ d[η_{1}η_{2}]  (1, 1, 0) (3|0) 3/2
[5]  d[η_{1}]^{2}  (2, 0, 0) (6|0) 3/2
[6]  d[1] ∧ d[η_{1}]  (1, 0, 0) (0|3) 5/4

deg = 2: dim = (15|16), a single irreducible 𝔤_{0}-module:

N  𝔤𝔩(3)-highest vectors  weight dim mult
[1]  ξ_{1}η_{3}d[η_{2}] ∧ d[η_{1}η_{2}] − ξ_{2}η_{3}d[η_{1}] ∧ d[η_{1}η_{2}]  (2, 2, −1) (0|10) 1/0
[2] \begin{matrix}−ξ_{1}η_{3}d[η_{1}] ∧ d[η_{2}] + ξ_{2}η_{3}d[η_{1}]^{2} + ξ_{1}η_{1}η_{3}d[η_{1}] ∧ d[η_{1}η_{2}] \\ +2ξ_{1}η_{2}η_{3}d[η_{2}] ∧ d[η_{1}η_{2}] − ξ_{2}η_{2}η_{3}d[η_{1}] ∧ d[η_{1}η_{2}] \end{matrix} (2, 1, −1) (15|0) 2/1
[3]  η_{3}(η_{1}d[η_{1}] + η_{2}d[η_{2}]) ∧ (ξ_{1}d[η_{2}] − ξ_{2}d[η_{1}])  (1, 1, −1) (0|6) 3/2

𝔤 = 𝔨𝔞𝔰(; 3ξ). Cohomology in deg = 1, dim = (52|52) (the vectors are highest with respect to 𝔤𝔩(3) = 𝔬(6) ∩ 𝔤_{0}, the weights are given in the following table with respect to 𝔬(6)), where

𝔤_{0}-modules:

[A] = [2′] + [3′] + [4] + [5′] + [6′] + [7′] of dim = (12|12), where [2′], [3′], [5′], [6′], [7′] are generated by [2_{1}] − [2_{2}], [3_{1}] − 2[3_{2}] + [3_{3}], 2[5_{1}] − [5_{2}] + [5_{3}], [6_{1}] + [6_{2}], and [7_{1}] + [7_{2}], respectively;

[B] = [8′] + [9′] + [10] + [11] + [12] + [13] of dim = (24|24), where [8′] and [9′] are generated by [8_{1}] + 2[8_{2}], and [9_{1}] + 2[9_{2}], respectively;

[C] = [B] + [3′′] + [5′′] + [6_{1}] + [7_{1}] + [8_{1}] + [9_{1}] of dim = (36|36), where [3′′] and [5′′] are generated by [3_{1}] − [3_{2}] and [5_{2}] − 3[5_{1}], respectively;

[D] = [A] + [C] of dim = (48|48);

[E] = [D] + [1] + [2′′] + [3′′′] + [5′′′] of dim = (52|52).

The modules [A], [B], [C]/[B] and [E]/[D] are irreducible, dim([C]/[B]) = (12|12) and dim([E]/[D]) = (4|4).

N  𝔤𝔩(3)-highest vectors  weight  dim mult
[1]  d[ξ_{1}ξ_{2}ξ_{3}]^{2}  (−2, −2, −2) (1|0) 1/0
[2_{1}]  d[ξ_{2}ξ_{3}] ∧ d[ξ_{1}ξ_{2}ξ_{3}]  (−1, −2, −2) 2 · (0|3) 2/0
[2_{2}]  ξ_{1}d[ξ_{1}ξ_{2}ξ_{3}]^{2}
[3_{1}]  d[ξ_{3}] ∧ d[ξ_{1}ξ_{2}ξ_{3}]
[3_{2}] d[ξ_{1}ξ_{3}] ∧ d[ξ_{2}ξ_{3}] (−1, −1, −2) 3 · (3|0) 4/1
[3_{3}]  ξ_{1}ξ_{2}d[ξ_{1}ξ_{2}ξ_{3}]^{2}
[4]  ξ_{1}d[ξ_{2}ξ_{3}] ∧ d[ξ_{1}ξ_{2}ξ_{3}]  (0, −2, −2) (6|0) 1/0
[5_{1}]  d[1] ∧ d[ξ_{1}ξ_{2}ξ_{3}]
[5_{2}] d[ξ_{1}] ∧ d[ξ_{2}ξ_{3}] − d[ξ_{2}] ∧ d[ξ_{1}ξ_{3}] + d[ξ_{3}] ∧ d[ξ_{1}ξ_{2}] (−1, −1, −1) 3 · (0|1) 6/3
[5_{3}]  ξ_{1}ξ_{2}ξ_{3}d[ξ_{1}ξ_{2}ξ_{3}]^{2}
[6_{1}]  d[ξ_{3}] ∧ d[ξ_{2}ξ_{3}]  (0, −1, −2) 2 · (0|8) 4/2
[6_{2}]  ξ_{1}ξ_{2}d[ξ_{2}ξ_{3}] ∧ d[ξ_{1}ξ_{2}ξ_{3}]
[7_{1}]  d[1] ∧ d[ξ_{2}ξ_{3}]  (0, −1, −1) 2 · (3|0) 7/5
[7_{2}]  ξ_{1}d[1] ∧ d[ξ_{1}ξ_{2}ξ_{3}]
[8_{1}]  d[ξ_{3}]^{2}  (0, 0, −2) 2 · (6|0) 4/2
[8_{2}]  ξ_{1}ξ_{2}d[ξ_{1}ξ_{3}] ∧ d[ξ_{2}ξ_{3}]
[9_{1}]  d[1] ∧ d[ξ_{3}]  (0, 0, −1) 2 · (0|3) 8/6
[9_{2}]  ξ_{1}ξ_{2}ξ_{3}d[ξ_{1}ξ_{3}] ∧ d[ξ_{2}ξ_{3}]
[10]  ξ_{1}d[ξ_{3}]^{2}  (1, 0, −2)  (0|15) 2/1
[11]  ξ_{1}d[1] ∧ d[ξ_{3}]  (1, 0, −1)  (8|0) 4/3
[12]  ξ_{1}ξ_{2}d[ξ_{3}]^{2}  (1, 1, −2)  (10|0) 1/0
[13]  ξ_{1}ξ_{2}d[1] ∧ d[ξ_{3}]  (1, 1, −1)  (0|6) 1/0

𝔤 = 𝔪𝔟(4|5). 2 − d = 0. Here w(u_{0}) = 0, w(u_{1}) = (1, 0), w(u_{2}) = (−1, 1), w(u_{3}) = (0, −1), w(ξ_{i}) = −w(u_{i}).

Cohomology: in deg = 1, hence, torsion free. A single irreducible 𝔤_{0}-module of dim = (12|12) glued of the following 𝔰𝔩(3)-modules:

N  𝔤𝔩(3)-highest vectors  weight dim mult
[1]  u_{0}d[u_{0}] ∧ d[ξ_{0}]  (0, 0) (0|1) 4/3
[2] u_{0}d[u_{0}] ∧ d[u_{3}] − ξ_{0}d[u_{3}] ∧ d[ξ_{0}] (0, 1) (3|0) 6/5
[3]  u_{0}d[u_{3}] ∧ d[ξ_{0}]  (0, 1) (0|3) 3/2
[4]  u_{0}d[ξ_{0}] ∧ d[ξ_{1}]  (1, 0) (3|0) 3/2
[5] u_{0}d[u_{3}] ∧ d[ξ_{1}] + u_{1}d[u_{3}] ∧ d[ξ_{0}] (1, 1) (0|8) 3/2
[6]  u_{0}d[u_{3}]^{2} − ξ_{3}d[u_{3}] ∧ d[ξ_{0}]  (0, 2) (6|0) 1/0

𝔤 = 𝔪𝔟(4|5; 1). 2 − d = 0.

Cohomology in deg = 0, dim = (22|22):

N  𝔰𝔩(2) ⊕ 𝔰𝔩(2)-highest vectors  weight dim mult
[1]  ξ_{1}d[u_{2}ξ_{1}] ∧ d[u_{3}ξ_{1}]  (0, 0)  (0|1)  6/5
[2_{1}] d[u_{2}] ∧ d[u_{0}u_{3} − ξ_{1}ξ_{2}] − d[u_{3}] ∧ d[u_{0}u_{2} + ξ_{1}ξ_{3}]  (0, 1) 2 × (2|0) 6/4
[2_{2}]  d[u_{2}ξ_{1}] ∧ d[u_{3}ξ_{1}]
[3] d[u_{0}u_{2} + ξ_{1}ξ_{3}] ∧ d[u_{3}ξ_{1}] − d[u_{0}u_{3} − ξ_{1}ξ_{2}] ∧ d[u_{2}ξ_{1}] (0, 2)  (0|3)  2/1
[4]  d[ξ_{2}] ∧ d[u_{3}ξ_{1}]  (2, 1)  (6|0)  4/3
[5_{1}]  d[ξ_{2}] ∧ d[u_{0}u_{3} − ξ_{1}ξ_{2}]  (2, 2) 2 × (0|9) 3/1
[5_{2}]  d[u_{0}u_{3} − ξ_{1}ξ_{2}] ∧ d[u_{3}ξ_{1}]
[6]  d[u_{0}u_{3} − ξ_{1}ξ_{2}]^{2}  (2, 3) (12|0) 1/0

𝔤_{0}-modules:

[A] = [4] + [5_{1}] + [5_{2}] + [6] of dim = 18|18;

[B] = [2_{2}] + [3] + [A] of dim = 20|21;

[C] = [1] + [2_{1}] + [B] of dim = 22|22. Irreducible modules: [A], [B]/[A] (of dim = 2|3) and [C]/[B] (of dim = 2|1).

𝔤 = 𝔪𝔟(4|5; K). 2 − d = −1.

The 𝔤_{0}-highest weight vectors are as follows: 𝔤_{0} = 𝔰𝔩(2) ⊕ 𝔤𝔩(3); the first coordinate of the weight is given with respect to 𝔰𝔩(2) realized as x_{+} = q_{0}(τ + q_{0}ξ_{0} − ∑_{i=1}^{3} q_{i}ξ_{i}) + 2ξ_{1}ξ_{2}ξ_{3}, x_{−} = ξ_{0}; 𝔤𝔩(3) is realized as x_{j}^{i} = q_{i}ξ_{j} (i ≠ j) and x_{i}^{i} = τ + q_{i}ξ_{i} − q_{0}ξ_{0}. In this realization w(q_{0}) = (1, −1, −1, −1), w(u_{1}) = (−1, 1, 0, 0), w(u_{2}) = (−1, 0, 1, 0), w(u_{3}) = (−1, 0, 0, 1), w(ξ_{0}) = (−2, 1, 1, 1), w(ξ_{1}) = (0, −1, 0, 0), w(ξ_{2}) = (0, 0, −1, 0), w(ξ_{3}) = (0, 0, 0, −1)).

Cohomology: a single irreducible module in deg = −1:

𝔰𝔩(2) ⊕ 𝔤𝔩(3)-highest vectors weight  dim mult
u_{0}dq^{3} ∧ dq_{3}  (3, 0, 0, −2) (24|0) 1/0

𝔤 = 𝔨𝔰𝔩𝔢(9|6; K). 2 − d = 0. Cohomology are in degrees 0 and 1 and constitute irreducible modules. (The weights are given in 𝔤𝔩-basis of matrix diagonal units.)

deg  𝔤𝔩(5)-highest vectors  𝔤𝔩(5)-weight  dim mult
0  ∂_{5}d[πdx_{4}dx_{5}]^{2}  (0, 0, 0, −2, −3) (175|0) 1/0
1 \begin{matrix}∑ ∂_{i}d[πdx_{4}dx_{5}] ∧ d[∂_{i}] + ∑ πdx_{i}dx_{j}(d[πdx_{i}dx_{4}] ∧ d[πdx_{j}dx_{5}]− \\ d[πdx_{i}dx_{5}] ∧ d[πdx_{j}dx_{4}] − d[πdx_{i}dx_{j}] ∧ d[πdx_{4}dx_{5}]) \end{matrix} (0, 0, 0, −1, −1) (0|10) 3/2

𝔤 = 𝔨𝔰𝔩𝔢(9|6; CK). 2 − d = −1. Cohomology are in degrees −1, 0, and 1.

Cohomology in deg = −1, dim = (36|36):

N  𝔰𝔩(2) ⊕ 𝔰𝔩(3)-highest vectors  weight dim mult
[1]  ∂_{2}d[πdx_{2}dx_{5}]^{2}  (3, 0, 2) (24|0) 1/0
[2]  ∂_{2}d[πdx_{2}dx_{5}] ∧ d[x_{5}∂_{1}]  (3, 0, 2) (0|24) 1/0
[3]  ∂_{2}d[x_{4}∂_{1}] ∧ d[x_{5}∂_{1}]  (3, 1, 0) (12|0) 1/0
[4] ∂_{2}d[πdx_{2}dx_{4}] ∧ d[x_{5}∂_{1}] − ∂_{2}d[πdx_{2}dx_{5}] ∧ d[x_{4}∂_{1}] (3, 1, 0) (0|12) 1/0

𝔤_{0}-modules:

[A] = [1] + [2] of dim = 24|24;

RIEMANN TENSOR FOR EXCEPTIONAL SUPERMANIFOLDS 11

[B] = [A] + [3] + [4] of dim = 36|36. Irreducible modules: [A] and [B]/[A] (of dim = 12|12).

Cohomology in deg = 0, single 𝔤_{0}-module, dim = 10|10:

N  𝔰𝔩(2) ⊕ 𝔰𝔩(3)-highest vectors  weight dim mult
[1]  ∂_{5}d[x_{5}∂_{1}] ∧ d[x_{5}∂_{2}]  (0, 0, 3) (10|0) 1/0
[2] \begin{matrix}∂_{5}d[πdx_{1}dx_{5}] ∧ d[x_{5}∂_{1}] + ∂_{5}d[πdx_{2}dx_{5}] ∧ d[x_{5}∂_{2}]− \\ πdx_{3}dx_{4}d[x_{5}∂_{1}] ∧ d[x_{5}∂_{2}] \end{matrix} (0, 0, 3) (0|10) 1/0

Cohomology: in deg = 1, a single 𝔤_{0}-module, dim = (6|6):

N  𝔰𝔩(2) ⊕ 𝔰𝔩(3)-highest vectors  weight dim mult
[1] \begin{matrix}−∂_{2}d[x_{5}∂_{2}] ∧ d[∂_{1}] + ∑ ∂_{i}d[x_{5}∂_{1}] ∧ d[∂_{i}]+ \\ ∑ (πdx_{1}dx_{j}d[πdx_{i}dx_{j}] ∧ d[x_{5}∂_{i}] + x_{j}∂_{i}d[x_{j}∂_{1}] ∧ d[x_{5}∂_{i}]) \end{matrix} (1, 0, 1) (6|0) 8/7

[2]  \begin{matrix}− ∑ ∂_{i}d[πdx_{2}dx_{5}] ∧ d[∂_{i}]+ \\ ∑ πdx_{i}dx_{j}d[πdx_{2}dx_{5}] ∧ d[πdx_{i}dx_{j}]− \\ _{∑1≤i≤2}_{∑3≤j≤5}x_{j}∂_{i}d[πdx_{2}dx_{5}] ∧ d[x_{j}∂_{i}] \end{matrix}  (1, 0, 1) (0|6) 8/7

𝔤 = 𝔨𝔰𝔩𝔢(9|6). 2 − d = 0. Cohomology in deg = 1, hence, torsion-free. dim = (168|167)

N  𝔤𝔩(3)-highest vectors  weight  dim  mult
[1]  \begin{matrix}∑ (−1)^{p(σ)}∂_{σ(1)}d[πdx_{σ(2)}dx_{σ(3)}] ∧ d[x_{σ(4)}∂_{5}]− \\ 2 ∑ πdx_{i}dx_{j}d[x_{i}∂_{5}] ∧ d[x_{j}∂_{5}] \end{matrix}  (0, 0, 0)  (0|1)  2/1
[2_{1}]  ∑ ∂_{i}d[∂_{i}] ∧ d[x_{4}∂_{5}] + ∂_{4}d[∂_{i}] ∧ d[x_{i}∂_{5}]  (0, 0, 1) 2 × (4|0) 5/3
[2_{2}] \begin{matrix}2∂_{4}(−d[πdx_{1}dx_{2}] ∧ d[πdx_{3}dx_{4}] + d[πdx_{1}dx_{3}] ∧ d[πdx_{2}dx_{4}]− \\ d[πdx_{1}dx_{4}] ∧ d[πdx_{2}dx_{3}]) + ∑ πdx_{i}dx_{j}d[πdx_{i}dx_{j}] ∧ d[x_{4}∂_{5}] \end{matrix}
[3]  ∑ ∂_{4}d[∂_{i}] ∧ d[πdx_{i}dx_{4}] − ∑ πdx_{σ(1)}dx_{σ(2)}d[∂_{σ(3)}] ∧ d[x_{4}∂_{5}] (0, 0, 2)  (0|10)  2/1
[4_{1}]  _{∑1≤i≤4}(∂_{i}d[∂_{i}] ∧ d[πdx_{3}dx_{4}] + πdx_{1}dx_{2}d[∂_{i}] ∧ d[x_{i}∂_{5}])  (0, 1, 0) 2 × (0|6) 6/4
[4_{2}]  \begin{matrix}∑ πdx_{i}dx_{i}d[πdx_{i}dx_{j}] ∧ d[πdx_{3}dx_{4}]− \\ 2 ∑ (−1)^{sign(σ)}πdx_{1}dx_{2} + d[πdx_{1}dx_{σ(2)}] ∧ d[πdx_{σ(3)σ(4)}] \end{matrix}
[5_{1}]  ∂_{4}d[x_{3}∂_{5}] ∧ d[x_{4}∂_{5}]  (0, 1, 1) 2 × (20|0) 4/2
[5_{2}] −2∂_{4}d[∂_{1}] ∧ d[∂_{2}] − x_{1}∂_{5}d[∂_{2}] ∧ d[x_{4}∂_{5}] + x_{2}∂_{5}d[∂_{1}] ∧ d[x_{4}∂_{5}]
[6]  ∂_{4}d[πdx_{3}dx_{4}] ∧ d[x_{4}∂_{5}]  (0, 1, 2)  (0|45)  1/0
[7]  \begin{matrix}πdx_{1}dx_{2}d[x_{3}∂_{5}] ∧ d[x_{4}∂_{5}] − ∂_{3}d[πdx_{3}dx_{4}] ∧ d[x_{4}∂_{5}]+ \\ ∂_{4}d[πdx_{3}dx_{4}] ∧ d[x_{3}∂_{5}] \end{matrix}  (0, 2, 0)  (0|20)  2/1
[8]  πdx_{1}dx_{2}d[πdx_{3}dx_{4}] ∧ d[x_{4}∂_{5}] − ∂_{4}d[πdx_{3}dx_{4}]^{2}  (0, 2, 1)  (60|0)  1/0
[9]  ∑(∂_{i}d[∂_{1}] ∧ d[∂_{i}] + x_{1}∂_{5}d[∂_{i}] ∧ d[x_{i}∂_{5}])  (1, 0, 0)  (4|0)  5/4
[10]  \begin{matrix}∑ (−1)^{p(σ)}(∂_{σ(2)}d[πdx_{σ(3)}dx_{σ(4)}] ∧ d[x_{4}∂_{5}]+ \\ ∂_{4}d[πdx_{σ(2)}dx_{σ(3)}] ∧ d[x_{σ(3)}∂_{5}]) \end{matrix}  (1, 0, 1)  (0|15)  4/3
[11]  ∂_{4}d[∂_{1}] ∧ d[x_{4}∂_{5}]  (1, 0, 2)  (36|0)  4/3
[12] 2x_{1}∂_{5}d[x_{3}∂_{5}] ∧ d[x_{4}∂_{5}] − ∂_{3}d[∂_{1}] ∧ d[x_{4}∂_{5}] + ∂_{4}d[∂_{1}] ∧ d[x_{3}∂_{5}] (1, 1, 0)  (20|0)  4/3
[13]  πdx_{1}dx_{2}d[∂_{1}] ∧ d[x_{4}∂_{5}] + ∂_{4}d[∂_{1}] ∧ d[πdx_{3}dx_{4}]  (1, 1, 1)  (0|64)  3/2

𝔤_{0}-modules:

[A] = [2′] + [4′] + [9], dim = (8|6), where [2′] and [4′] are generated by 2[2_{1}] − [2_{2}] and 2[4_{1}] − [4_{2}], respectively;

[B]=[A]+[1], dim = (8|7);

[C] = [1] + ⋯ + [13], dim = (168|167).

The modules [A], [B]/[A] (of dim = (0|1)), and [C]/[B] (of dim = (160|160)) are irreducible.

𝔤 = 𝔨𝔰𝔩𝔢(9|6; 2). 2 − d = 0. Cohomology are in degrees 0 and 1.

Cohomology in deg = 0, dim = (140|140):

N  𝔰𝔩(3) ⊕ 𝔰𝔩(2)-highest vectors  weight  dim  mult
[1]  πdx_{4}dx_{5}d[πx_{5}dx_{4}dx_{5}]^{2}  (0, 0, 2)  (0|3)  2/1
[2]  ∂_{3}d[∂_{4}] ∧ d[πx_{4}dx_{4}dx_{5}] + ∂_{3}d[∂_{5}] ∧ d[πx_{5}dx_{4}dx_{5}]  (0, 1, 0)  (0|3)  4/3
[3_{1}]  ∂_{3}d[∂_{4}] ∧ d[πx_{5}dx_{4}dx_{5}]  (0, 1, 2) 2 · (0|9) 5/3
[3_{2}]  πdx_{4}dx_{5}d[x_{5}∂_{1}] ∧ d[x_{5}∂_{2}]
[3_{3}]  ∂_{3}d[πx_{5}dx_{4}dx_{5}]^{2}  (0, 1, 2)  (9|0)  2/1
[4]  \begin{matrix}∂_{3}d[πdx_{3}dx_{4}] ∧ d[πx_{5}dx_{4}dx_{5}]− \\ ∂_{3}d[πdx_{3}dx_{5}] ∧ d[πx_{4}dx_{4}dx_{5}] \end{matrix}  (0, 2, 0)  (6|0)  1/0
[5_{1}]  ∂_{3}d[∂_{4}] ∧ d[πdx_{3}dx_{5}]  (0, 2, 2) (0|18)  2/0
[5_{2}]  ∂_{3}d[πdx_{3}dx_{5}] ∧ d[πx_{5}dx_{4}dx_{5}]  (0, 2, 2) 2 · (18|0) 2/1
[5_{3}]  ∂_{3}d[x_{5}∂_{1}] ∧ d[x_{5}∂_{2}]
[6]  ∂_{3}d[πdx_{3}dx_{5}]^{2}  (0, 3, 2) (30|0)  1/0
[7]  \begin{matrix}πdx_{4}dx_{5}d[x_{4}∂_{1}] ∧ d[πx_{5}dx_{4}dx_{5}]− \\ πdx_{4}dx_{5}d[x_{5}∂_{1}] ∧ d[πx_{4}dx_{4}dx_{5}] \end{matrix}  (1, 0, 0)  (3|0)  3/2
[8]  πdx_{4}dx_{5}d[x_{5}∂_{1}] ∧ d[πx_{5}dx_{4}dx_{5}]  (1, 0, 2)  (9|0)  3/2
[9_{1}]  ∂_{3}d[∂_{4}] ∧ d[x_{4}∂_{1}] + ∂_{3}d[∂_{5}] ∧ d[x_{5}∂_{1}]  (1, 1, 0)  (8|0)  3/2
[9_{2}] ∂_{3}d[x_{4}∂_{1}] ∧ d[πx_{5}dx_{4}dx_{5}] − ∂_{3}d[x_{5}∂_{1}] ∧ d[πx_{4}dx_{4}dx_{5}] (1, 1, 0)  (0|8)  1/0
[10_{1}]  ∂_{3}d[∂_{4}] ∧ d[x_{5}∂_{1}]  (1, 1, 2) (24|0)  3/2
[10_{2}]  ∂_{3}d[x_{5}∂_{1}] ∧ d[πx_{5}dx_{4}dx_{5}]  (1, 1, 2) (0|24)  1/0
[11]  ∂_{3}d[πdx_{3}dx_{4}] ∧ d[x_{5}∂_{1}] − ∂_{3}d[πdx_{3}dx_{5}] ∧ d[x_{4}∂_{1}]  (1, 2, 0) (0|15)  1/0
[12]  ∂_{3}d[πdx_{3}dx_{5}] ∧ d[x_{5}∂_{1}]  (1, 2, 2) (0|45)  1/0
[13]  πdx_{4}dx_{5}d[x_{4}∂_{1}] ∧ d[x_{5}∂_{1}]  (2, 0, 0)  (0|6)  2/1
[14]  ∂_{3}d[x_{4}∂_{1}] ∧ d[x_{5}∂_{1}]  (2, 1, 0) (15|0)  1/0

The column “mult” shows the multiplicity of the homogeneous (even or odd) component of given weight.

𝔤_{0}-modules:

[A] = [3_{1} − 3_{2}] + [5_{1}] + [5_{2} + 5_{3}] + [6] + [10_{1}] + [12] of dim = 72|72;

[B] = [A] + [2] + [4] + [7] + [9_{1}] + [9_{2}] + [11] + [13] + [14] of dim = 104|104;

[C] = all. The modules [A], [B]/[A] (of dim = (32|32)), and [C]/[B] (of dim = (36|36)) are irreducible.

Cohomology in deg = 1, dim = (8|8): the 𝔰𝔩(3) ⊕ 𝔰𝔩(2)-highest weights are as follows: even ones are (0, 0, 1), (1, 0, 1); odd ones are (0, 0, 1), (0, 1, 1). The corresponding highest weight vectors are too complicated to be included here.

Acknowledgment. P.G., D. L. and I. Shch. acknowledge financial support of TBSS, Stockholm; Université Marseille-Aix and MPIM, Bonn, where the final molding had been performed; and RFBR grant 01-01-00490a, respectively.

References

[CAF] Castellani L., D’Auria R., Fré P.,Supergravity and superstrings. A geometric perspective. Vol. 1–3. Superstrings. World Scientific Publishing Co., Inc., Teaneck, NJ, 1991. pp. i–xvi and 1–2162

[CK1] Cheng Shun-Jen; Kac V., Generalized Spencer Cohomology and filtered Deformations of ℤ-graded Lie Superalgebras, math-RT/9805039; Adv. Theor. Math. Phys. 2 (1998), no. 5, 1141–1182

[CK2] Cheng S., Kac V., Structure of some ℤ-graded Lie superalgebras of vector fields, Transformation groups, v. 4, 1999, 219–272

[D] Deligne P. et al (eds.) Quantum fields and strings: a course for mathematicians. Vol. 1, 2. Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997. AMS, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 1999. Vol. 1: xxii+723 pp.; Vol. 2: pp. i–xxiv and 727–1501

[G] Goncharov A., Infinitesimal structures related to hermitian symmetric spaces, Funct. Anal. Appl., 15, n.3 (1981), 23–24 (in Russian); a detailed version: id., Generalized conformal structures on manifolds. In: [L3], #11 and Selecta Math. Soviet. 6, 1987, no. 4, 307–340

[Gr] Grozman P. , Classification of bilinear invariant operators on tensor fields. Funktsional. Anal. i Prilozhen. 14 (1980), no. 2, 58–59; English translation: Functional Anal. Appl. 14 (1980), no. 2, 127–128

[GL1] Grozman P., Leites D., Supergravities and N-extended Minkowski superspaces for any N. In: Wess J., Ivanov E. (eds.) Supersymmetries and quantum symmetries. Proc. International Conference in memory of V. Ogievetsky, June 1997, Lecture Notes in Physics 524, Springer, 1999, 58–67

[GL2] Grozman P., Leites D., Mathematica-aided study of Lie algebras and their cohomology. From supergravity to ball bearings and magnetic hydrodynamics In: Keränen V. (eds.) The second International Mathematica symposium, Rovaniemi, 1997, 185–192

[GL3] Grozman P., Leites D., An unconventional supergravity, In: S. Duplij and J. Wess (eds.) Noncommutative structures in mathematics and physics Proc. NATO Advanced Research Workshop, Kiev, 2000. Kluwer, 41–48; hep-th/0202115

[GLS2] Grozman P., Leites D., Shchepochkina I., Invariant differential operators on supermanifolds and The Standard Model. math.RT/0202193; In: M. Olshanetsky, A. Vainstein (eds.) Multiple facets of quantization and supersymmetry. Michael Marinov Memorial Volume, World Sci., 2002

[GLS3] Grozman P., Leites D., Shchepochkina I., The analogs of the Riemann tensor for the 16 series of nonholonomic structures on supermanifolds, in preparation

[Ka7] Kac V., Classification of infinite-dimensional simple linearly compact Lie superalgebras. Adv. Math. 139 (1998), no. 1, 1–55; Kac V., Classification of infinite-dimensional simple groups of supersymmetries and quantum field theory. GAFA 2000 (Tel Aviv, 1999). Geom. Funct. Anal. 2000, Special Volume, Part I, 162–183

[L0] Leites D., The Riemann tensor for nonholonomic manifolds, Homology, Homotopy and Applications, vol. 4 (2), 2002, 397–407 (math.RT/0202213)

[LP2] Leites, D., Poletaeva E., Supergravities and contact type structures on supermanifolds. Second International Conference on Algebra (Barnaul, 1991), Contemp. Math., 184, Amer. Math. Soc., Providence, RI, 1995, 267–274

[LPS] Leites, D., Poletaeva E., Serganova V., On Einstein equations on manifolds and supermanifolds, J. Nonlinear Math. Phys., v. 9, no. 4, 2002, 394–425

[LSh0] Leites D., Shchepochkina I., Towards classification of simple vectorial Lie superalgebras. In: [L5], 31/1988-14; Leites D., Toward classification of classical Lie superalgebras. In: Nahm W., Chau L. (eds.) Differential geometric methods in theoretical physics (Davis, CA, 1988), NATO Adv. Sci. Inst. Ser. B Phys., 245, Plenum, New York, 1990, 633–651; Leites D., Shchepochkina I., Quivers and Lie superalgebras, Czech. J. Phys. v. 47, no. 12, 1997, 1221–1229

[LSh1] Leites D., Shchepochkina I., Classification of simple Lie superalgebras of vector fields, preprint mpimbonn, no.

[LSh3] Leites D., Shchepochkina I., How to quantize antibracket, preprint ESI-875 (www.esi.ac.at); Theor. and Math. Physics, v. 126, no. 3, 339–369; math.RT/0202181

[LS] Leites D., Serganova V., Symmetries wider than supersymmetries (with V. Serganova) In: S. Duplij and J. Wess (eds.) Noncommutative structures in mathematics and physics, Proc. NATO Advanced Research Workshop, Kiev, 2000. Kluwer, 13–30; Leites D., Kochetkov Yu., Simple finite dimensional Lie algebras in characteristic 2 related to superalgebras and on a notion of finite simple group. In: L. A. Bokut’ , Yu. L. Ershov and A. I. Kostrikin (eds.) Proceedings of the International Conference on Algebra. Part 1., Novosibirsk, August 1989, Contemporary Math. 131, Part 1, AMS, 1992, 59–67 MR 93g:17035

[M] Manin Y. I., Gauge field theory and complex geometry. Translated from the 1984 Russian original by N. Koblitz and J. R. King. Second edition, 289. Springer-Verlag, Berlin, 1997. xii+346 pp

[Mo] Montgomery R., Generic distributions and Lie algebras of vector fields. J. Differential Equations 103 (1993), no. 2, 387–393

[P1] Poletaeva E., Spencer cohomology of some Lie superalgebras. In: A. L. Onishchik et. al. (eds.) Problems in group theory and homological algebra, Yaroslav. Gos. Univ., Yaroslavl, 1988, 162–167; id., Spencer cohomology of Lie superalgebras of vector fields. In: A. L. Onishchik et. al. (eds.) Problems in group theory and homological algebra, Yaroslav. Gos. Univ., Yaroslavl, 1990, 168–169 [P2] Poletaeva E., Structure functions on the usual and exotic symplectic and periplectic supermanifolds .

U. Bruzzo et. al.(eds.) Differential geometric methods in theoretical physics, Lecture Notes in Physics 375, Springer, Berlin, 1991, 390–395

[P3] Poletaeva E., Analogues of the Riemann tensors for the odd metric on supermanifolds. Acta Appl. Math. 31, no 2, 1993, pp 137-169; id., Penrose’s tensors on supergrassmannians. Math. Scand. 72, 1993, no 2, pp 161-190; id., Penrose’s tensors. II. Math. Scand. 72, 1993, no 2, 191–211

[S] Schwarz, A. S. Supergravity, complex geometry and G-structures. Comm. Math. Phys. 87 (1982/83), no. 1, 37–6: Rosly, A. A.; Schwarz, A. S. Geometry of N = 1 supergravity. Comm. Math. Phys. 95 (1984), no. 2, 161–184. II. Comm. Math. Phys. 96 (1984), no. 3, 285–309. Baranov, M. A.; Roslyi, A. A.; Shvarts, A. S. The relation between gravity and supergravity. Teoret. Mat. Fiz. 64 (1985), no. 1, 7–16. English translation: Theoret. and Math. Phys. 64 (1985), no. 1, 649–655

[S1] Serganova V., Automorphisms of simple Lie superalgebras. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 3, 585–598; Feigin B. L., Leites D. A.,Serganova,V. V., Kac-Moody superalgebras. Group theoretical methods in physics, Vol. 1–3 (Zvenigorod, 1982), 631–637, Harwood Academic Publ., Chur, 1985

[Sh5] Shchepochkina I., The five exceptional simple Lie superalgebras of vector fields, hep-th 9702120; id.,Five simple exceptional Lie superalgebras of vector fields. Funktsional. Anal. i Prilozhen. 33 (1999), no. 3, 59–72, 96; translation in Funct. Anal. Appl. 33 (1999), no. 3, 208–219

[Sh14] Shchepochkina I., Five exceptional simple Lie superalgebras of vector fields and their fourteen regradings. Representation Theory (electronic journal of AMS), v. 3, 1999, 3 (1999), 373–415

[St] Sternberg S., Lectures on differential geometry, Chelsey, 2nd edition, 1985

[T] Tanaka N., On differential systems, graded Lie algebras and pseudogroups. J. Math. Kyoto Univ. 10, 1970, 1–82; id, On infinitesimal automorphisms of Siegel domains, J. Math. Soc. Japan 22, 1970, 180– 212; id., On the equivalence problems associated with simple graded Lie algebras. Hokkaido Math. J., 8 (1979), no. 1, 23–84

[V] Vershik A. M., Classical and nonclassical dynamics with constraints. (Russian) In: Yu. Borisovich, Yu. Gliklikh (eds.) Geometry and topology in global nonlinear problems, 23–48, Novoe Global. Anal., Voronezh. Gos. Univ., Voronezh, 1984 English translation in: Lect. Notes in Math., 1108, 1984, 278–301

[VG1] Vershik A. M., Gershkovich V. Ya., Estimation of the functional dimension of the orbit space of germs of distributions in general position. (Russian) Mat. Zametki 44, no. 5, 1988, 596–603, 700; English translation in: Math. Notes 44, no. 5-6, 1988, 806–810

[VG2] Vershik A. M., Gershkovich V. Ya., A bundle of nilpotent Lie algebras over a nonholonomic manifold (nilpotentization). (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 172 (1989), Differentsialnaya Geom. Gruppy Li i Mekh. Vol. 10,21–40, 169 translation in J. Soviet Math. 59 (1992), no. 5, 1040–1053; id., Nonholonomic manifolds and nilpotent analysis. J. Geom. Phys. 5 (1988), no. 3, 407–452

[WB] Wess J., Bagger J., Supersymmetry and supergravity. Princeton Series in Physics. Princeton University Press, Princeton, N.J., 1983. i+180 pp

[WZ] Wess, J., Zumino, B., Supergauge transformations in four dimensions. Nuclear Phys. B70 (1974), 39– 50; Wess J., Supersymmetry-supergravity. Topics in quantum field theory and gauge theories (Proc. VIII Internat. GIFT Sem. Theoret. Phys., Salamanca, 1977), pp. 81–125, Lecture Notes in Phys., 77, Springer, Berlin-New York, 1978; Wess J., Zumino B., Superspace formulation of supergravity. Phys. Lett. B 66 (1977), no. 4, 361–364; Wess J., Supersymmetry/supergravity. Concepts and trends in particle physics (Schladming, 1986), 29–58, Springer, Berlin, 1987; Wess J., Introduction to supersymmetric theories. Frontiers in particle physics ’83 (Dubrovnik, 1983), 104–131, World Sci. Publishing, Singapore, 1984; Wess J., Bagger J., Supersymmetry and supergravity. Second edition. Princeton Series in Physics. Princeton University Press, Princeton, NJ, 1992. x+259 pp.

[Y] Yamaguchi K., Differential systems associated with simple graded Lie algebras. Progress in differential geometry, Adv. Stud. Pure Math., 22, Math. Soc. Japan, Tokyo, 1993, 413–494

Department of Mathematics, University of Stockholm, Roslagsv. 101, Kräftriket hus 6, SE-106 91, Stockholm, Sweden; mleites@matematik.su.se; ^{1}On leave of absence from the Independent University of Moscow, Bolshoj Vlasievsky per, dom 11, RU-121 002 Moscow, Russia; Ira@Paramonova.mccme.ru