The analogs of Riemann and Penrose tensors on supermanifolds

Elena Poletaeva

Department of Mathematics, Lund University, Sweden

, E-mail: elena@maths.lth.se 1991 Mathematics Subject Classification. 58A50, 17B56 (Primary) 53C65, 58H10, 17A70,

17B66, 58H10 (Secondary)

Key words and phrases. Supermanifolds, Riemann tensor, penrose tensor, structure functions.

I warmly thank D. Leites (IAS, 1989), SFB-170 (1990), and MPIM, Bonn during my several visits in the past decade for hospitality. I thank D. Leites for raising the problem; to him, A. Onishchik, J.-L. Brylinsky, V. Serganova and A. Goncharov I am thankful for help.

Abstract. The Spencer cohomology of certain ℤgraded Lie superalgebras are completely computed. This cohomology is interpreted as analogs of Riemann and Penrose tensors on supermanifolds. The results make it manifest that there is no simple generalization of Borel-Weil-Bott’s theorem for Lie superalgebras.

Contents

Introduction 7 Structure functions 7 Spencer cohomology groups 7 Generalized conformal structures 8 The case of a simple Lie algebra 𝔤_{∗} over ℂ 9 Reduced structures 9 Einstein equations 10 F-structures and their structure functions 10 Structure functions on supermanifolds 11 Description of results 11 Algebraic methods 12 §1. Plans for the future; structure functions of Lie superalgebras of Cartan type 13 Terminological conventions 13

Chapter 1. The analogues of the Riemannian tensors for the odd metric on

supermanifolds 15

Periplectic superalgebras and their Cartan prolongations 15 The main theorem 20 Proof of the main theorem 21

Chapter 2. The analogues of Penrose’s tensors 35 Standard ℤ-grading of 𝔰𝔩(m|n) and the corresponding Cartan prolongations 35 Structure functions of Lie superalgebras 𝔳𝔢𝔠𝔱(0|n) and 𝔰𝔳𝔢𝔠𝔱(0|n) 38 Penrose’s tensors 44

Chapter 3. The analogues of the Riemann–Weyl tensors for classical superspaces 53 Spencer cohomology of 𝔰𝔩(m|n) and 𝔭𝔰𝔩(n|n) 53 Spencer cohomology of 𝔭𝔰𝔮(n) 57 Spencer cohomology of 𝔬𝔰𝔭(m|2n) 57 Spencer cohomology of D(α) 58 Spencer cohomology of AB_{3} 58

Appendix. The formula of dimensions of irreducible 𝔰𝔩(n)-modules 61 References 61

Tables 65 Table 1. Irreducible 𝔤𝔩(n)-submodules of 𝔠𝔭𝔢(n) ⊗ V^{∗} 65 Table 2. Irreducible 𝔤𝔩(n)-submodules of H^{q}(V_{0}, 𝔤_{∗}) 65 Table 3. Irreducible 𝔤𝔩(n)-submodules of E_{1}^{p,0} 66 Table 4. Irreducible 𝔤𝔩(n)-submodules of E_{1}^{p,1} 66 Table 5. Irreducible 𝔤𝔩(n)-submodules of E_{1}^{p,2} 66 Table 6. Irreducible 𝔤𝔩(n)-submodules of E_{0}^{p,0} 67 Table 7. Irreducible 𝔤𝔩(n)-submodules of E^{3}V 67 Table 8. Irreducible 𝔤𝔩(m) ⊕ 𝔤𝔩(n)-submodules of 𝔤_{0} ⊗ 𝔤_{−1}^{∗} 67 Table 9. Irreducible 𝔤𝔩(m) ⊕ 𝔤𝔩(n)-submodules of (U^{∗} ⊗ V) ⊗ (U^{∗} ⊗ V) 68 Table 10. Irreducible 𝔤𝔩(n)-submodules of C_{𝔤𝔩(n)}^{k,2} 68 Table 11. Irreducible 𝔤𝔩(n)-submodules of C_{𝔤𝔩(n)}^{k+1,1} 68 Table 12. Irreducible 𝔤𝔩(n)-submodules of 𝔤_{k} 68 Table 13. Structure functions of 𝔰𝔩(m|n) endowed with the standard ℤ-grading 69 Table 14. Irreducible 𝔤𝔩(m) ⊕ 𝔤𝔩(n)submodules of 𝔤_{0} ⊗ Λ^{2}𝔤_{−1}^{∗} 69 Table 15. Irreducible 𝔤𝔩(m) ⊕ 𝔤𝔩(n)-submodules of 𝔤_{1} ⊗ Λ^{2}𝔤_{−1}^{∗} 70 Table 16. Spencer cohomology of 𝔰𝔩(m|n) endowed with a ℤ-grading, where

𝔤_{0} = 𝔠(𝔰𝔩(m|q) ⊕ 𝔰𝔩(n − q)) 70 Table 17. Spencer cohomology of 𝔰𝔩(m|n) endowed with a ℤ-grading, where

𝔤_{0} = 𝔠(𝔰𝔩(m − p|q) ⊕ 𝔰𝔩(p|n − q)) 71 Table 18. Spencer cohomology of 𝔬𝔰𝔭(m|2n) 71

This paper expands the results that appeared mainly in not very accessible [P1-P8] listed below and complements [LPS] which should have been published in 1991. Meanwhile there appeared Grozman’s package SuperLie that confirmed our — not very easy — calculations; for some of Grozman’s independently obtained results (the cases of exceptional algebras) see [LPS].

[P1] Poletaeva, E. I. Spencer cohomology connected with some Lie superalgebras. In: Onishchik A. (ed.) Problems in group theory and homological algebra (Russian), 162–167, Matematika, Yaroslav. Gos. Univ., Yaroslavl, 1988. MR 1 175 010

[P2] Poletaeva, E. Structure functions on the usual and exotic symplectic and periplectic supermanifolds. In: Bartocci C. et al. (eds.) Differential geometric methods in theoretical physics (Rapallo, 1990), 390–395, Lecture Notes in Phys., 375, Springer, Berlin, 1991. MR 92h:58207

[P3] Poletaeva, E. I. Spencer cohomology of Lie superalgebras of vector fields.In: Onishchik A. (ed.) Problems in group theory and homological algebra (Russian), 168–169, Matematika, Yaroslav. Gos. Univ., Yaroslavl, 1990. MR 93c:17036

[P4] Leites, D.; Poletaeva, E. Analogues of the Riemannian structure for classical superspaces. Proceedings of the International Conference on Algebra, Part 1 (Novosibirsk, 1989), 603–612, Contemp. Math., 131, Part 1, Amer. Math. Soc., Providence, RI, 1992. MR 93k:58008

[P5] Poletaeva, E. Analogues of Riemann tensors for the odd metric on supermanifolds. Acta Appl. Math. 31 (1993), no. 2, 137–169. MR 94d:58166

[P6]Poletaeva, E. Penrose’s tensors on super-Grassmannians. Math. Scand. 72 (1993), no. 2, 161–190. MR 94m:58008a

[P7] Poletaeva, E. Penrose’s tensors. II. Math. Scand. 72 (1993), no. 2, 191–211.MR 94m:58008b

[P8] MR96d:58005 Leites, D.; Poletaeva, E. Supergravities and contact type structures on supermanifolds. Second International Conference on Algebra (Barnaul, 1991), 267–274, Contemp. Math., 184, Amer. Math. Soc., Providence, RI, 1995.

Our results also make it clear why one can not just “superize” metric in order to get Einstein-Hilbert’s equations. Their true superizations — various SUGRAs — correspond to ℤ-graded Lie superalgebras of depth d > 1. Such Lie superalgebras were discussed by Yu. Manin [M] but the corresponding structure functions were not calculated yet except in the cases considered by Grozman and Leites in

[GL1] Grozman, P.; Leites, D. From supergravity to ballbearings. Supersymmetries and quantum symmetries (Dubna, 1997), 58–67, Lecture Notes in Phys., 524, Springer, Berlin, 1999 MR 2000j:83090

There is, however, a paper where approach similar to the one described in what follows is applied to ℤ-graded Lie superalgebras of depth d = 1 and the results are interpreted as supergravity since the tensor obtained after deleting all that depends on odd parameters are exactly the standard Riemannian tensor:

[GL2] Grozman, P.; Leites, D., An unconventional supergravity. Noncommutative structures in mathematics and physics (Kiev, 2000), 41–47, NATO Sci. Ser. II Math. Phys. Chem., 22, Kluwer Acad. Publ., Dordrecht, 2001 MR 1 893 452

Introduction

Structure functions

The main object of the study of Riemannian geometry is the properties of the Riemann tensor, which in turn splits into the Weyl tensor, the traceless Ricci tensor, and the scalar curvature. All these tensors are obstructions to the possibility of “flattening” the manifold on which they are considered. The word “splits” above means that at every point of the Riemannian manifold M^{n} for n ≠ 4 the space of values of the Riemann tensor constitutes an O(n)-module which splits into the sum of three irreducible components (for n = 4 there are four of them, because the Weyl tensor splits additionally in this case) [ALV, Kob].

More generally, let G ⊂ GL(n) be any Lie group, not necessarily O(n). A reduction of the principal GL(n)-bundle on M to the principal G-bundle is called a G-structure on M.

Recall that on a manifold with a G-structure there is a canonical connection. For a Riemannian manifold this is the Levi-Civita connection. The so-called structure functions (SFs) constitute the complete set of obstructions to integrability of the canonical connection or, in other words, to the possibility of local flattening of a manifold with G-structure. The Riemann tensor is an example of a SF. Among the most known other examples of SFs are the following ones:

• a conformal structure, G = O(n) × ℝ^{∗}, SFs are called the Weyl tensor;

• Penrose’s twistor theory, G = S(U(2) × U(2)) × ℂ^{∗}, SFs-Penrose’s tensors — split into two components called the “α-forms” and “β-forms”;

• an almost complex structure, G = GL(n; ℂ) ⊂ GL(2n; ℝ), SFs are called the Nijenhuis tensor;

• an almost symplectic structure, G = Sp(2n), no accepted name for SFs.

Spencer cohomology groups

Recall necessary definitions [St, Gu].

The simplest G-structure is the flat G-structure defined as follows. Let V be 𝕂^{n} with a fixed frame. Consider the bundle over V whose fiber over v ∈ V consists of all frames obtained from the fixed one under the G-action, V being identified with T_{v}V.

Obstructions to identification of the (k + 1)-st infinitesimal neighborhood of a point m ∈ M on a manifold M with G-structure and that of a point of the flat manifold V with the above G-structure are called structure functions of order k. The identification is performed inductively and is possible provided the obstructions of lesser orders vanish. At each point of a manifold M SFs take values in certain cohomology groups, called Spencer cohomology groups. The corresponding complex is defined as follows. Let S^{i}V denote the i-th symmetric power of a vector space V and Lie(G) denote the Lie algebra of the Lie group G. Set 𝔤_{−1} = T_{m}M, 𝔤_{0} = 𝔤 = Lie(G) and for i > 0 put:

𝔤_{i} = {X ∈ Hom(𝔤_{−1}, 𝔤_{i−1}) : X(v)(w, …) = X(w)(v, …) for any v, w ∈ 𝔤_{−1}}

= (𝔤_{0} ⊗ S^{i}(𝔤_{−1})^{∗}) ∩ (𝔤_{−1} ⊗ S^{i+1}(𝔤_{−1})^{∗}).

Now set 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) = ⊕_{i≥−1}𝔤_{i}. Suppose that the 𝔤_{0}-module 𝔤_{−1} is faithful .

Then 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) ⊂ 𝔳𝔢𝔠𝔱(n) = der𝕂[[x_{1}, …, x_{n}]], where n = dim 𝔤_{−1}. It can be verified that the Lie algebra structure on 𝔳𝔢𝔠𝔱(n) induces such a structure on 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}). The Lie algebra 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}), usually abbreviated 𝔤_{∗}, will be called the Cartan prolongation of the pair (𝔤_{−1}, 𝔤_{0}).

Let E^{i}V be the i-th exterior power of a vector space V. Set

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

Define the differentials ∂_{𝔤_{0}}^{k,s} : C_{𝔤_{0}}^{k,s} ⟶ C_{𝔤_{0}}^{k,s+1} as follows: for any g_{1}, …, g_{s+1} ∈ 𝔤_{−1}

(∂_{𝔤_{0}}^{k,s}f)(g_{1}, …, g_{s+1}) = ∑_{i}(−1)^{i}[f(g_{1}, …, ĝ_{s+1−i}, …, g_{s+1}), g_{s+1−i}]  (1)

As expected, ∂_{𝔤_{0}}^{k,s}∂_{𝔤_{0}}^{k,s+1} = 0. The cohomology of bidegree (k, s) of this complex is called the (k, s)-th Spencer cohomology group H_{𝔤_{0}}^{k,s}. It turns out that structure functions of order k on a manifold M with G-structure are sections of certain vector bundles over M with fiber over a point m ∈ M isomorphic to H_{𝔤}^{k,2}(T_{m}M), where 𝔤 = Lie(G).

Generalized conformal structures

A generalization of the notion of conformal structure is a G-structure of type X, where X is a classical space, i.e., an irreducible compact Hermitian symmetric space (CHSS). These G-structures were introduced and intensively studied by A. Goncharov, who calculated the corresponding structure functions [G1, G2]. In his examples G is the reductive part of the stabilizer of a point of X. The usual conformal structure is the one that corresponds to X = Q_{n}, a quadric in the projective space. The complex grassmannian X = Gr_{2}^{4} corresponds to Penrose’s twistors.

Recall that Penrose’s idea is to embed the Minkowski space M^{4} into the complex Grassmann manifold Gr_{2}^{4} of planes in ℂ^{4} (or straight lines in ℂℙ^{3}) and to express the conformal structure on M^{4} in terms of the incidence relation of the straight lines in ℂℙ^{3} [Pe].

The conformal structure on M^{4} is given by a field of quadratic cones in the tangent spaces to the points of M^{4}. In Penrose’s case these cones possess two families of two-dimensional flat generators, the so-called “α-planes” and “β-planes.” The geometry of these families is vital for Penrose’s considerations. In particular, the Weyl tensor gets a lucid description in terms of these families.

It is interesting to include 4-dimensional Penrose theory into a more general theory of geometric structures. A. Goncharov has shown that there is an analogous field of quadratic cones for any irreducible compact Hermitian symmetric space X of rank greater then one [G2].

Let S be a simple complex Lie group, P its parabolic subgroup with the Levi decomposition P = GN, i.e., G is reductive and N is the radical of P. As one knows [He], N is Abelian if and only if X = S/P is a CHSS, and in this case G = G_{0} × ℂ^{∗}, where G_{0} is semisimple.

Let P_{x} = G_{x}N_{x} be the Levi decomposition of the stabilizer of x ∈ X in S. Denote by C_{x} the cone of highest weight vectors in the G_{x}-module T_{x}X, i.e., each element in C_{x} is highest with respect to some Borel subgroup in G_{x}. Since s ∈ S transforms C_{x} to C_{sx}, then with X there is associated the cone C(X) ⊂ TēX, where ē is the image of the unit e ∈ S in X.

Let rk(X) > 1, i.e., X ≠ ℂℙ^{n}. Then on a manifold M a generalized conformal structure of type X is given if M is endowed with a family of cones C_{m} and ℂ-linear isomorphisms A_{m} : TēX ⟶ T_{m}M such that A_{m}(C(x)) = C_{m}.

Goncharov has shown that a manifold M with generalized conformal structure of type X is a manifold with a G̃-structure, where G̃ is a group of linear automorphisms of the cone C(X) and the connected component of the identity of this group is precisely G [G2].

The case of a simple Lie algebra 𝔤_{∗} over ℂ

The following remarkable fact, though known to experts, is seldom formulated explicitly [LRC, KN].

Proposition. Let 𝕂 = ℂ, 𝔤_{∗} = 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) be simple. Then only the following cases are possible:

1) 𝔤_{2} ≠ 0, then 𝔤_{∗} is either 𝔳𝔢𝔠𝔱(n) or its special subalgebra 𝔰𝔳𝔢𝔠𝔱(n) of divergence-free vector fields, or its subalgebra 𝔥(2n) of Hamiltonian vector fields.

2) 𝔤_{2} = 0, 𝔤_{1} ≠ 0, then 𝔤_{∗} is the Lie algebra of the complex Lie group of automorphisms of a CHSS (see §3).

Let R(∑_{i} a_{i}π_{i}) be the irreducible 𝔤_{0}-module with the highest weight ∑_{i} a_{i}π_{i}, where π_{i} is the i-th fundamental weight.

Theorem (Serre [St]). In case 1) of Proposition SFs can only be of order 1. More precisely: for 𝔤_{∗} = 𝔳𝔢𝔠𝔱(n) and 𝔰𝔳𝔢𝔠𝔱(n) SFs vanish, for 𝔤_{∗} = 𝔥(2n) nonzero SFs are R(π_{1}) for n = 2, and R(π_{1}) ⊕ R(π_{3}) for n > 2.

When 𝔤_{∗} is a simple finite dimensional Lie algebra over ℂ computation of SFs becomes an easy corollary of the Borel-Weil-Bott (BWB) theorem in a form due to W. Shmid [Sh], cf. work of A. Goncharov [G2]. Indeed, by definition,

⊕_{k}H_{𝔤_{0}}^{k,2} = H^{2}(𝔤_{−1}, 𝔤_{∗}).

The BWB theorem implies that as a 𝔤_{0}-module, H^{2}(𝔤_{−1}, 𝔤_{∗}) has as many components as H^{2}(𝔤_{−1}). Thanks to commutativity of 𝔤_{−1} one has H^{2}(𝔤_{−1}) = E^{2}𝔤_{−1}^{∗}, which facilitates the count of components. The BWB theorem also gives the formula for the highest weights of these components.

Reduced structures

Let X = S/P, where Lie(S) = 𝔤_{∗} = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} and Lie(P) = 𝔤_{0} ⊕ 𝔤_{1}, be a CHSS. Let 𝔤̂_{0} be the semisimple part of 𝔤_{0} = Lie(G). A Ĝ-structure, where Lie(Ĝ) = 𝔤̂_{0}, will be referred to as a Riemannian structure of type X. To reduce the structure group G to its semisimple part Ĝ is an action similar to distinguishing a metric from a conformal class on a conformal manifold.

The structure functions of the Ĝ-structures form an analogue of the Riemann tensor for the metric. They include the structure functions of the G-structure and several other irreducible components, some of which are analogues of the traceless Ricci tensor or the scalar curvature.

More precisely, the structure functions of the G-structure are defined as the part of the structure functions of the Ĝ-structure obtained by a reduction of the G-structure that does not depend on the choice of reduction. In other words, this is a generalized conformally invariant part of the structure functions of the Ĝ-structure.

Since in the case of the Riemannian structure 𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}) = 𝔤_{−1} ⊕ 𝔤̂_{0},then there only exist SFs of orders 1 and 2. Though the BWB theorem doesn’t work in this case, SFs are describable thanks to the following proposition:

Proposition. [G2]. 1) H1,2 ; 𝔤̂_{0} = H_{𝔤_{0}}^{1,2};

2) H2,2 ; 𝔤̂_{0} = H_{𝔤_{0}}^{2,2} ⊕ S^{2}(𝔤_{−1}^{∗}).

The Riemannian structure in the classical case of Riemannian geometry will be considered next.

Einstein equations

Let G = O(n). In this case 𝔤_{1} = 𝔤_{−1} and a 1-dimensional subspace is distinguished in S^{2}(𝔤_{−1})^{∗}. The sections through this subspace constitute a Riemannian metric g on M. The usual way to determine a metric on M is to define a matrix-valued function, but actually this function with values in symmetric matrices depends only on one functional parameter. The values of the Riemann tensor at a point of M constitute an O(n)-module H^{2}(𝔤_{−1}, 𝔤_{∗}), which contains a trivial component. Let a section through it be denoted by R. This trivial component is naturally realized as a submodule in a module isomorphic to S^{2}(𝔤_{−1})^{∗}.

Thus, there exist two matrix-valued functions: g and R, both preserved by O(n). Now let R correspond to the Levi-Civita connection. The process of restoring R from g involves differentiations and in this way one gets a nonlinear pde, which constitutes one of the two conditions called Einstein equations [L4, LSV, LPS]:

R = λg, where λ ∈ R.  (EE_{0})

The other condition is that the other component belonging to S^{2}(𝔤_{−1})^{∗}, the traceless Ricci tensor Ric, vanishes:

Ric = 0.  (EE_{ric})

There is a close relation between G-structures and so-called F -structures, which are also of interest, in particular, because of their application to Penrose’s geometry. This relation will now be explained.

F-structures and their structure functions

Recall that the notion of F -structure is a generalization of the notion of distribution, i.e., a subbundle in T M and the SFs of an F -structure generalize the notion of the Frobenius form [G2].

Let V = T_{m}M, F ⊂ Gr_{k}(V) be a manifold with a transitive action of a subgroup G_{F} ⊂ GL(V), 𝔽(M) be a subbundle of Gr_{k}(T M), where the fiber of Gr_{k}(T M) is Gr_{k}(T_{m}M). The bundle 𝔽(M) ⟶ M is called an F-structure on M, if for any point m of M there is a linear isomorphism I_{m} : V ⟶ T_{m}M, which induces a diffeomorphism I_{m}(F) = 𝔽(m). A submanifold Z ⊂ M of dimension k such that T_{z}Z ⊂ 𝔽(z) for any z ∈ Z is called an integral submanifold. An F -structure is integrable if for any z ∈ Z and for any subspace V (z) ⊂ 𝔽(z) there is an integral manifold Z with T_{z}Z = V (z).

SFs of an F -structure are defined as follows. For f ∈ F let V_{f} ⊂ V be the subspace corresponding to f. Set

(T_{f}F)_{−1} = V/V_{f}, (T_{f}F)_{0} = T_{f}F.

Define

(T_{f}F)_{s} = ((T_{f}F)_{s−1} ⊗ V_{f}^{∗}) ∩ ((T_{f}F)_{s−2} ⊗ S^{2}V_{f}^{∗})

for s > 0, and

C_{T_{f} F}^{k,s} = (T_{f}F)_{k−s} ⊗ E^{s}V_{f}^{∗}

Define the differentials as in (1). Then the cohomology groups H_{T_{f} F}^{k,s} are naturally defined. It turns out that the obstruction to integrability of order k of an F -structure on a manifold M is a section of a certain vector bundle over 𝔽(M) with fiber over a point ψ ∈ 𝔽(m) isomorphic to H_{T_{I{−1 \atop m}(ψ)}F}^{k,2} . Moreover, there exists a map H_{𝔤_{F}}^{k,s} ⟶ H_{T_{f} F}^{k,s}, where 𝔤_{F} = Lie(G_{F})

[G2].

The relation between SFs of a G_{F} -structure and the obstructions to integrability of an F -structure generalizes a theorem of Penrose, which states that the anti-selfdual part of the Weyl tensor on a 4-dimensional manifold with a conformal structure vanishes if and only if α-surfaces exist, in other words, the metric is α-integrable [AHS, Gi].

More precisely, for a generalized conformal structure of type X, where X = Gr_{m}^{m+n}(ℂ), there exist two families of m and n-dimensional flat generators–analogues of Penrose’s αplanes and β-planes. When neither m nor n is equal to 1, i.e., the grassmannian is not a projective space, SFs decompose into the direct sum of two components, which are analogues of the self-dual and anti-self-dual parts of the Weyl tensor on a 4-dimensional manifold with a conformal structure. The integrability of each of two families of generators is equivalent to the vanishing of the corresponding component of the SFs.

Structure functions on supermanifolds

The necessary background on Lie superalgebras and supermanifolds is gathered in [L1, L2, L3, K1, M].

The classical superspaces (homogeneous compact Hermitian symmetric superspaces), which are the super analogues of CHSS, considered by Goncharov, are listed in [S1].

The above definitions of SFs are generalized to Lie superalgebras via the sign rule. However, in the super case new phenomena appear, which have no analogues in the classical case:

• Cartan prolongations of (𝔤_{−1}, 𝔤_{0}) and of (Π𝔤_{−1}, 𝔤_{0}) are essentially different;

• faithfulness of the 𝔤_{0}-action on 𝔤_{−1} is violated in natural examples of supergrassmannians of subsuperspaces in an (n, n)-dimensional superspace when the center z of 𝔤_{0} acts trivially. This will be explained in §9.

• the formulation of Serre’s theorem and of the Proposition of §4 fails to be literally true for Lie superalgebras.

Description of results

In Chapter 1 I compute the SFs for the odd analogue of the metric on the supermanifolds and for several related G-structures (see §2 of Chapter 1). In this case 𝔤_{0} = Lie(G) is the periplectic Lie superalgebra, the special periplectic Lie superalgebra, or their central extensions. It turns out that unlike the classical case of Riemannian geometry, the 𝔤_{0}-module H_{𝔤_{0}}^{k,2} is not completely reducible, and I describe the Jordan-Hölder series for this module. Thus, my computations show that there is no analogue of (EE_{0}) for the odd metric.

In Chapter 2 and Chapter 3 I obtain an explicit description of the Spencer cohomology groups H_{𝔤_{0}}^{k,2} for simple finite-dimensional complex classical Lie superalgebras endowed with ℤ-grading of depth 1: 𝔤 = ⊕_{i≥−1}𝔤_{i}, where 𝔤_{0} is the zero-th part of the grading.

It is known [K2, S2] that all such ℤ-gradings are of the form 𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1}, except for the case when 𝔤 is the special periplectic superalgebra considered in Chapter 1. Thus, the cohomology groups H_{𝔤_{0}}^{k,2} constitute the space of values of SFs of G-structures corresponding to homogeneous compact Hermitian symmetric superspaces, where G is a reductive complex Lie supergroup of classical type and 𝔤_{0} = Lie(G). The groups Hk,2 ; 𝔤̂_{0} correspond to structures of Riemannian type.

An important particular case is 𝔤 = 𝔰𝔩(m|n), where m ≠ n, corresponding to general supergrassmannians.

In Chapter 2 I consider a ℤ-grading of 𝔤 for which 𝔤_{0} is a reductive Lie algebra. Thus, the 𝔤_{0}-module H_{𝔤_{0}}^{k,2} is completely reducible, and for m, n > 2 decomposes into the direct sum of two irreducible components — super analogues of Penrose’s tensors for the usual complex grassmannians (see Theorem 3.1 of Chapter 2).

The case 𝔤 = 𝔰𝔩(n|n) is also interesting, because I discovered a phenomenon which has no an analogue in the classical case. Indeed, the center z of 𝔤_{0} acts trivially on 𝔤_{−1}. If one retains the same definition of the Cartan prolongation, then it has the form of the semidirect sum S^{∗}(𝔤_{−1}^{∗}) ⊂ + 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}/z) (the ideal is S^{∗}(𝔤_{−1}^{∗})) with the natural ℤ-grading and Lie superalgebra structure, but this Lie superalgebra is not a subsuperalgebra of 𝔳𝔢𝔠𝔱(dim 𝔤_{−1}) anymore (see Theorem 1.1 of Chapter 2).

In Chapter 3 I describe the Spencer cohomology groups for the other ℤ-gradings of depth 1 of 𝔰𝔩(m|n) and 𝔭𝔰𝔩(n|n) (see Theorem 1.4 and Theorem 1.5 of Chapter 3). These theorems show that the superspace of SFs can be not completely reducible, and I get the answer in terms of nonsplit exact sequences of 𝔤_{0}-modules.

Finally, in the cases when m or n are equal to 1, I get SFs of the Lie superalgebra of vector fields 𝔳𝔢𝔠𝔱(m|n) or of divergence-free vector fields 𝔰𝔳𝔢𝔠𝔱(m|n) (see Theorem 2.1 of Chapter 2 and Theorem 1.3 of Chapter 3).

Theorem 2.3 of Chapter 3 shows that the SFs for queer grassmannians constitute a module looking exactly the same as that for grassmannians of generic dimensions.

The case 𝔤 = 𝔬𝔰𝔭(m|2n) is similar to Riemannian geometry. The SFs constitute an irreducible 𝔤_{0}-module, which is an analogue of the Weyl tensor, and the superspace of the SFs for the reduced structure decomposes into the direct sum of three components–the super analogues of the Weyl tensor, the traceless Ricci tensor, and the scalar curvature. I find the highest weights of these components (see Theorem 3.3 of Chapter 3).

Finally, I describe the Spencer cohomology groups for exceptional Lie superalgebras D(α) and AB_{3} (see Theorem 4.3 and Theorem 5.3 of Chapter 3, respectively).

Algebraic methods

As in the classical case (Lie theory), computation of Spencer cohomology groups reduces to certain problems of representation theory. However, in the super case computations become much more complicated, because of the absence of complete reducibility. I could not directly apply the usual tools for computing (co)homology (spectral sequences and the Borel-Weil-Bott theorem) to superalgebras and had to retreat a step and apply these tools to the even parts of the considered Lie superalgebras. Then, using certain necessary conditions, I verified whether two modules over a Lie superalgebra that could be glued into an indecomposable module were glued or not.

My method of computing the structure functions is based on the Hochcshild-Serre spectral sequence [Fu]. Let 𝔤 = 𝔤_{0} ⊕ 𝔤_{1} be a Lie superalgebra and M be a 𝔤-module. On the superspace of k-dimensional cochains C^{k} = C^{k}(𝔤, M) define a filtration:

F^{0}C^{k} = C^{k} ⊃ F^{1}C^{k} ⊃ F^{2}C^{k} ⊃ … F^{j}C^{k} ⊃ … ⊃ F^{k+1}C^{k} = 0,

where F^{j}C^{k} = {c ∈ C^{k}|c(g_{1}, …, g_{i}, …, g_{k}) = 0 if k − j + 1 arguments belong to 𝔤_{0}}, 0 ≤ j ≤ k + 1. Using this filtration define the usual corresponding spectral sequence E_{r}^{p,q} [GM]. Thus, H^{2}(𝔤, M) = ⊕_{p+q=2}E_{∞}^{p,q} = E_{3}^{2,0} ⊕ E_{3}^{1,1} ⊕ E_{4}^{0,2}.

In particular, E_{1}^{p,q} = H^{q}(𝔤_{0}, M ⊗ S^{p}𝔤_{1}^{∗}) [Fu]. Since, in the case of Spencer cohomology, 𝔤 = 𝔤_{−1} = (𝔤_{−1})_{0} ⊕ (𝔤_{−1})_{1} is a commutative Lie superalgebra, then

E_{1}^{p,q} = H^{q}((𝔤_{−1})_{0}, 𝔤_{∗} ⊗ S^{p}(𝔤_{−1})_{1}^{∗}) = H^{q}((𝔤_{−1})_{0}, 𝔤_{∗}) ⊗ S^{p}(𝔤_{−1})_{1}^{∗}.

Then in special cases I use the BWB theorem to compute H^{q}((𝔤_{−1})_{0}, 𝔤_{∗}) as a module over (𝔤_{0})_{0}.

§1. Plans for the future; structure functions of Lie superalgebras of Cartan type

It is interesting to compute the Spencer cohomology groups H_{𝔤_{0}}^{k,s} for simple finite-dimensional Lie superalgebras of Cartan type with nonstandard ℤ-gradings of depth 1. For example, let 𝔤 = 𝔳𝔢𝔠𝔱(0|n). Define the ℤ–grading as follows: 𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1}, where 𝔤_{0} = Λ(n − 1) ⊂ + 𝔳𝔢𝔠𝔱(0|n − 1), Λ(n − 1) is a commutative ideal in 𝔤_{0} isomorphic to the Grassmann algebra as the 𝔳𝔢𝔠𝔱(0|n−1)-module, 𝔤_{1} ≅ Π(𝔳𝔢𝔠𝔱(0|n−1)) as the 𝔤_{0}-module, 𝔤_{−1} ≅ Π(Λ(n−1)) as the 𝔤_{0}-module, and the action of Λ(n − 1) on 𝔤_{−1} is the usual multiplication in Λ(n − 1).

It is much more difficult to compute the Spencer cohomology groups for Cartan Lie superalgebras than for classical ones, because in this case the number of the irreducible quotient modules in the Jordan-Hölder series does depend on n. Moreover, even for small n computations seem to be very complicated, because of the absence of complete reducibility [LPS].

Terminological conventions

𝔠𝔤 is the trivial central extension of the Lie superalgebra 𝔤; ⊂ + stands for the semidirect sum of the algebras with the ideal to the left; Π is the functor of the change of parity; p(X) is the parity of homogeneous element X of a superspace; S^{i}V is the i-th symmetric power of a vector (super)space V; S^{∗}V = ⊕_{k≥0}S^{k}V; E^{i}V is the i-th exterior power of a vector (super)space V; ⟨1⟩ is a trivial module over a Lie superalgebra; I will consider the following Lie superalgebras [L2, L3]: 𝔤𝔩(m|n) = Mat(m|n; ℂ), 𝔰𝔩(m|n) = {X ∈ 𝔤𝔩(m|n)| strX = 0}, 𝔭𝔰𝔩(n|n) = 𝔰𝔩(n|n)/⟨1_{2n}⟩, where 1_{2n} is the unit matrix; 𝔳𝔢𝔠𝔱(m|n) = 𝔡𝔢𝔯ℂ[[x]] is the Lie superalgebra of vector fields; 𝔰𝔳𝔢𝔠𝔱(m|n) = {D ∈ 𝔳𝔢𝔠𝔱(m|n)| ÷ (D) = 0} is the Lie superalgebra of divergence-free

vector fields;

𝔥(0|n) = {D ∈ 𝔳𝔢𝔠𝔱(0|n)| Dw = 0 for the Hamiltonian form w = ∑_{i=1}^{n}(dξ_{i})^{2}} is the Lie superalgebra of Hamiltonian vector fields;

Other Lie superalgebras will be defined in Chapter 1 and Chapter 3.

CHAPTER 1 The analogues of the Riemannian tensors for the odd metric on supermanifolds

Periplectic superalgebras and their Cartan prolongations

Let z = 1_{2n} be the unit matrix and τ = diag(1_{n}, − 1_{n}).

Let P be a nondegenerate supersymmetric odd bilinear form on a superspace V. Clearly, dim V = (n, n). Define the odd analogue of the symplectic Lie algebra, the periplectic Lie superalgebra 𝔭𝔢(n), and its special subsuperalgebra 𝔰𝔭𝔢(n), setting

\begin{matrix}𝔭𝔢(n) = \{X ∈ 𝔤𝔩(n|n)| X^{st}P + (−1)^{p(X)}P X = 0\}, \\ 𝔰𝔭𝔢(n) = 𝔭𝔢(n) ∩ 𝔰𝔩(n|n). \end{matrix}

Thus,

𝔭𝔢(n) = 𝔰𝔭𝔢(n) ⊂ +⟨τ⟩.

Denote by ε_{1}, …, ε_{n} the standard basis of the space dual to the space of diagonal matrices in 𝔤𝔩(n) ⊂ 𝔭𝔢(n). Denote by V_{λ} the irreducible 𝔤𝔩(n)-module with highest weight λ and highest vector v_{λ} and by X_{λ} the irreducible 𝔭𝔢(n)-module with highest weight λ and an even highest vector.

Let V = V_{0} ⊕ V_{1} be the standard (identity) 𝔭𝔢(n)-module, e_{1}, …, e_{n} be a basis of V_{0}, and f_{1}, …, f_{n} be a basis of V_{1} with respect to which the form P on V takes the form P = antidiag(1_{n}, 1_{n}). With respect to this basis the elements X ∈ 𝔭𝔢(n) are represented by matrices of the standard format (n, n) [L2]:

X = diag(A, −A^{t}) + antidiag(B, C), where A ∈ 𝔤𝔩(n), B^{t} = B, C^{t} = −C.

In what follows we will often use a natural abbreviation: e.g., B_{1,n} stands for the matrix X whose components A and C are zero and all the entries of B are also zero except for (1, n)-th and (n, 1)-st.

Denote by ẽ_{1}, …, ẽ_{n} and f̃_{1}, …, f̃_{n} the basis of V^{∗} dual to the above basis of V, i.e., f̃_{i}(e_{j}) = ẽ_{i}(f_{j}) = δ_{ij}. Since the form P preserved by 𝔭𝔢(n) is odd, then V^{∗} and Π(V) are isomorphic as 𝔭𝔢(n)-modules. Notice that as 𝔭𝔢(n)-modules,

𝔭𝔢(n) ≅ Π(E^{2}V^{∗}).

1.1. Lemma. a) There exists a ℤ-grading of the Lie superalgebra 𝔭𝔢(n + 1) of the form

𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} ⊕ 𝔤_{2},

where

𝔤_{−1} = V,
𝔤_{0} = 𝔠𝔭𝔢(n),
𝔤_{1} = V^{∗} = Π(V),
𝔤_{2} = Π(⟨1⟩).

b) There exists a ℤ-grading of the Lie superalgebra 𝔰𝔭𝔢(n + 1) of the form

𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} ⊕ 𝔤_{2},

where

𝔤_{−1} = V,
𝔤_{0} = 𝔰𝔭𝔢(n) ⊂ +⟨τ + nz⟩,
𝔤_{1} = V^{∗} = Π(V),
𝔤_{2} = Π(⟨1⟩).

Proof. Let W = W_{0} ⊕ W_{1} be the standard (identity) 𝔭𝔢(n + 1)-module, e_{1}, …, e_{n+1} be a basis of W_{0}, and f_{1}, …, f_{n+1} be a basis of W_{1} with respect to which the form P on W takes the form P = antidiag(1_{n+1}, 1_{n+1}). Denote by ẽ_{1}, …, ẽ_{n+1} and f̃_{1} …, f̃_{n+1} the basis of W^{∗} dual to the above basis of W, e.g., f̃_{i}(e_{j}) = ẽ_{i}(f_{j}) = δ_{ij}.

Note that

𝔭𝔢(n + 1) = Π(E^{2}W^{∗}) = Π(E^{2}W_{1} ⊕ W_{0} ∧ W_{1} ⊕ S^{2}W_{0}).

Thus,

𝔭𝔢(n + 1) = ⟨e_{i}ẽ_{j}, e_{i} ∧ f̃_{j}, f_{i} ∧ f̃_{j}⟩(1 ≤ i, j ≤ n + 1), where

\begin{matrix}e_{i}ẽ_{j} = (1/2)(e_{i} ⊗ ẽ_{j} + e_{j} ⊗ ẽ_{i}), \\ e_{i} ∧ f̃_{j} = (1/2)(e_{i} ⊗ f̃_{j} − f_{j} ⊗ ẽ_{i}), \\ f_{i} ∧ f̃_{j} = (1/2)(f_{i} ⊗ f̃_{j} − f_{j} ⊗ f̃_{i}). \end{matrix}  (1.1.1)

Note that the commutator in 𝔭𝔢(n + 1) is defined as follows:

[w_{i} ∧ w̃_{j}, w_{s} ∧ w̃_{t}] = (1/2)(w̃_{j}(w_{s})(w_{i} ∧ w̃_{t}) − (−1)^{p(w_{s})}w̃_{t}(w_{i})(w_{s} ∧ w̃_{j})),  (1.1.2)

where w_{l} ∈ {e_{1}, …, e_{n+1}; f_{1}, …, f_{n+1}} for 1 ≤ l ≤ n + 1.

Let V = V_{0} ⊕ V_{1} = ⟨e_{1}, …, e_{n}; f_{1}, …, f_{n}⟩ be (n, n)-dimensional subsuperspace in W. Then

Π(E^{2}W_{1}) = Π(E^{2}V_{1}) ⊕ V_{1} ∧ ⟨f̃_{n+1}⟩,
Π(W_{0} ∧ W_{1}) = Π(V_{0} ∧ V_{1}) ⊕ V_{0} ∧ ⟨f̃_{n+1}⟩ ⊕ V_{1} ∧ ⟨ẽ_{n+1}⟩ ⊕ ⟨e_{n+1} ∧ f̃_{n+1}⟩,
Π(S^{2}W_{0}) = Π(S^{2}V_{0}) ⊕ V_{0} ∧ ⟨ẽ_{n+1}⟩ ⊕ ⟨e_{n+1}ẽ_{n+1}⟩.

Set

\begin{matrix}𝔤_{−1} = V_{0} ∧ ⟨f̃_{n+1}⟩ ⊕ V_{1} ∧ ⟨f̃_{n+1}⟩, \\ 𝔤_{0} = Π(E^{2}V_{1} ⊕ V_{0} ∧ V_{1} ⊕ S^{2}V_{0}) ⊕ ⟨e_{n+1} ∧ f̃_{n+1}⟩, \\ 𝔤_{1} = V_{0} ∧ ⟨ẽ_{n+1}⟩ ⊕ V_{1} ∧ ⟨ẽ_{n+1}⟩, \\ 𝔤_{2} = ⟨e_{n+1}ẽ_{n+1}⟩. \end{matrix}  (1.1.3)

According to (1.1.2), formulas (1.1.3) indeed define a ℤ-grading of 𝔭𝔢(n + 1), described in Lemma 1.1.

In order to define a ℤ-grading of 𝔰𝔭𝔢(n + 1) we set

\begin{matrix}𝔤_{0} = Π(E^{2}V_{1} ⊕ S^{2}V_{0}) ⊕ ⟨∑_{i,j=1,n} a_{ij}e_{i} ∧ f̃_{j}| ∑_{i=1,n} a_{ii} = 0⟩⊕ \\ ⟨(∑_{i=1,n} e_{i} ∧ f̃_{i}) − ne_{n+1} ∧ f̃_{n+1}⟩. \end{matrix}

Note that by (1.1.2) we have 2[e_{n+1}∧f̃_{n+1}, 𝔤_{i}] = i𝔤_{i} for (−1 ≤ i ≤ 2). Hence, −2e_{n+1}∧f̃_{n+1} = z. Since by (1.1.1) 2 ∑_{i=1,n} e_{i} ∧ f̃_{i} = τ, then

2( ∑_{i=1,n} e_{i} ∧ f̃_{i} − ne_{n+1} ∧ f̃_{n+1}) = τ + nz.

Thus, 𝔤_{0} = 𝔰𝔭𝔢(n) ⊂ +⟨τ + nz⟩. This proves Lemma 1.1. □

1.2. Theorem. Let 𝔤_{−1} = V. Then

a) If 𝔤_{0} = 𝔰𝔭𝔢(n), 𝔭𝔢(n), 𝔠𝔰𝔭𝔢(n) or 𝔰𝔭𝔢(n) ⊂ +⟨aτ + bz⟩, where a, b ∈ ℂ are such that a, b ≠ 0 and b/a ≠ n, then 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) = 𝔤_{−1} ⊕ 𝔤_{0}.

b) If 𝔤_{0} = 𝔠𝔭𝔢(n) or 𝔰𝔭𝔢(n) ⊂ +⟨τ + nz⟩, then 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) is either 𝔭𝔢(n + 1) or 𝔰𝔭𝔢(n + 1), respectively, in the ℤ-grading described in Lemma 1.1.

Proof. Let us consider the case where 𝔤_{0} = 𝔠𝔭𝔢(n). By Lemma 1.1 we have

𝔭𝔢(n + 1) = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} ⊕ 𝔤_{2},

where

𝔤_{−1} = V,
𝔤_{0} = 𝔠𝔭𝔢(n),
𝔤_{1} = V^{∗} = Π(V),
𝔤_{2} = Π(⟨1⟩).

Therefore,

𝔭𝔢(n + 1) ⊂ 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}).  (1.2.1)

In fact, since 𝔰𝔭𝔢(n + 1) is a simple Lie superalgebra, then it is transitive, (i.e., if there exists g ∈ 𝔤_{i}(i ≥ 0) such that [𝔤_{−1}, g] = 0, then g = 0). It follows that 𝔤_{i} ⊂ 𝔤_{i−1} ⊗ 𝔤_{−1}^{∗}. The Jacobi identity implies 𝔤_{i} ⊂ 𝔤_{i−2} ⊗ S^{2}𝔤_{−1}^{∗}. □

Let us find 𝔤_{1}.

1.3. Lemma. As a 𝔤𝔩(n)-module, 𝔤_{0}⊗𝔤_{−1}^{∗} is the direct sum of irreducible 𝔤𝔩(n)-submodules whose highest weights and highest vectors are listed in Table 1.

Convention. Let v, w be elements of a vector space. Set

vw = (v ⊗ w + (−1)^{p(v)p(w)}w ⊗ v)/2, v ∧ w = (v ⊗ w − (−1)^{p(v)p(w)}w ⊗ v)/2.

Proof of Lemma 1.3. consists of:

a) a verification of the fact that vectors v from Table 1 are indeed highest with respect to 𝔤𝔩(n), i.e., A_{i,j}v = 0 for i < j,

b) a calculation of dimensions of the corresponding irreducible submodules by the formula from the Appendix. □

Let us show with the help of Table 1 that if λ ≠ ε_{1}, −ε_{n}, then v_{λ} ∉ 𝔤_{−1} ⊗S^{2}𝔤_{−1}^{∗}. Indeed, if λ = −ε_{n−1} − 2ε_{n}, then

v_{λ}(e_{n})(e_{n−1}) = −f_{n}/2, v_{λ}(e_{n−1})(e_{n}) = 0;

if λ = −ε_{n−2} − ε_{n−1} − ε_{n}, then

v_{λ}(e_{n})(e_{n−1}) = f_{n−2}/2, v_{λ}(e_{n−1})(e_{n}) = −f_{n−2}/2;

if λ = ε_{1} − 2ε_{n}, then

v_{λ}(e_{n})(f_{1}) = f_{n}/2, v_{λ}(f_{1})(e_{n}) = 0;

if λ = ε_{1} + ε_{2} − ε_{n}, then

v_{λ}(f_{2})(e_{n}) = −e_{1}/2, v_{λ}(e_{n})(f_{2}) = 0;

if λ = 3ε_{1}, then

v_{λ}(f_{1})(f_{1}) = e_{1} ≠ 0;

if λ = 2ε_{1} + ε_{2}, then

v_{λ}(f_{1})(f_{1}) = e_{2}/2 ≠ 0.

Let λ = ε_{1} − ε_{n−1} − ε_{n}. According to Table 1, 𝔤_{0} ⊗ 𝔤_{−1}^{∗} contains two highest vectors of weight λ. Let

v_{λ} = k_{1}f_{n−1} ∧ f̃_{n} ⊗ ẽ_{1} + k_{2}(f_{n−1} ∧ ẽ_{1} ⊗ f̃_{n} − f_{n} ∧ ẽ_{1} ⊗ f̃_{n−1}), where k_{1}, k_{2} ∈ ℂ,

be a linear combination of these vectors. The condition

v_{λ}(e_{n})(e_{n−1}) = v_{λ}(e_{n−1})(e_{n})

implies k_{2} = 0. Then the condition

v_{λ}(e_{n})(f_{1}) = v_{λ}(f_{1})(e_{n})

implies k_{1} = 0.

Let λ = 2ε_{1} − ε_{n}. Let

v_{λ} = k_{1}f_{n} ∧ ẽ_{1} ⊗ ẽ_{1} + k_{2}e_{1}ẽ_{1} ⊗ f̃_{n}, where k_{1}, k_{2} ∈ ℂ,

be a linear combination of the highest vectors of weight λ which belong to 𝔤_{0} ⊗ 𝔤_{−1}^{∗}. The condition v_{λ}(f_{1})(f_{1}) = 0 implies k_{1} = 0. Then the condition v_{λ}(e_{n})(f_{1}) = v_{λ}(f_{1})(e_{n}) implies k_{2} = 0. Therefore, if λ ≠ ε_{1}, − ε_{n}, then v_{λ} ∉ 𝔤_{−1} ⊗ S^{2}𝔤_{−1}^{∗}, hence v_{λ} ∉ 𝔤_{1}.

Let λ = ε_{1}. According to Table 1, 𝔤_{0} ⊗ 𝔤_{−1}^{∗} has four highest vectors of weight λ. Let

v_{λ} = k_{1} ∑_{i=1}^{n} f_{i} ∧ ẽ_{i} ⊗ ẽ_{1} + k_{2} ∑_{i=1}^{n} f_{i} ∧ ẽ_{1} ⊗ ẽ_{i} + k_{3} ∑_{i=1}^{n} e_{1}ẽ_{i} ⊗ f̃_{i} + k_{4} ∑_{i=1}^{n} e_{i}f̃_{i} ⊗ ẽ_{1},

where k_{1}, k_{2}, k_{3}, k_{4} ∈ ℂ, be their linear combination. Note that v_{λ} ∈ 𝔤_{−1} ⊗ S^{2}𝔤_{−1}^{∗} if and only if the following conditions are satisfied:

v_{λ}(f_{1})(f_{1}) = 0,

v_{λ}(f_{1})(f_{i}) = −v_{λ}(f_{i})(f_{1}) for i ≠ 1,
v_{λ}(f_{1})(e_{1}) = v_{λ}(e_{1})(f_{1}),
v_{λ}(f_{1})(e_{i}) = v_{λ}(e_{i})(f_{1}) for i ≠ 1,
v_{λ}(f_{i})(e_{i}) = v_{λ}(e_{i})(f_{i}) for i ≠ 1,

which determine, respectively, the following system of linear equations:

k_{1} + k_{2} + k_{4} = 0,

k_{1} + k_{4} = −k_{2},
(−k_{1} − k_{2} + k_{4})/2 = k_{3},
− k_{1} + k_{4} = k_{3},
− k_{2} = k_{3}.

The solution of this system is

k_{1} = 0, k_{3} = −k_{2} = k_{4}.

Therefore,

v_{ε_{1}} = − ∑_{i=1}^{n} f_{i} ∧ ẽ_{1} ⊗ ẽ_{i} + ∑_{i=1}^{n} e_{1}ẽ_{i} ⊗ f̃_{i} + ∑_{i=1}^{n} e_{i}f̃_{i} ⊗ ẽ_{1} ∈ 𝔤_{1}.  (1.3.1)

Since 𝔤_{1} is a 𝔭𝔢(n)-module and v_{ε_{1}} is an odd vector, 𝔤_{1} = V^{∗}.

Let us find 𝔤_{2}.

1.4. Lemma. There exist the following nonsplit sequences of 𝔭𝔢(n)-modules:

0 ⟶ X_{2ε_{1}} ⟶ E^{2}V^{∗} ⟶ Π(⟨1⟩) ⟶ 0,  (1.4.1)
0 ⟶ Π(⟨1⟩) ⟶ S^{2}V^{∗} ⟶ X_{ε_{1}+ε_{2}} ⟶ 0.  (1.4.2)

Proof. First of all recall that Π(E^{2}V^{∗}) and 𝔭𝔢(n) itself are isomorphic 𝔭𝔢(n)-modules and there exists the following nonsplit sequence of 𝔭𝔢(n)-modules

0 ⟶ 𝔰𝔭𝔢(n) ⟶ 𝔭𝔢(n) ⟶ ⟨τ⟩ ⟶ 0.  (1.4.3)

Note that as a 𝔤𝔩(n)-module, E^{2}V^{∗} is isomorphic to

E^{2}V_{0}^{∗} ⊕ V_{0}^{∗} ∧ V_{0} ⊕ S^{2}V_{0}, where S^{2}V_{0} = V_{2ε_{1}}, v_{2ε_{1}} = ẽ_{1}^{2}.  (1.4.4)

Since B_{i,j}v_{2ε_{1}} = 0, then v_{2ε_{1}} is a 𝔭𝔢(n)-highest vector. From the simplicity of 𝔰𝔭𝔢(n) we get (1.4.1) after the change of parity.

Let us prove (1.4.2). Notice that as 𝔤𝔩(n)-modules,

S^{2}V^{∗} = S^{2}V_{0}^{∗} ⊕ V_{0}^{∗} · V_{0} ⊕ E^{2}V_{0}, where

\begin{matrix}S^{2}V_{0}^{∗} = V_{−2ε_{n}}, v_{−2ε_{n}} = f̃_{n}^{2}, \\ V_{0}^{∗} · V_{0} = V_{ε_{1}−ε_{n}} ⊕ ⟨v_{0}⟩, v_{ε_{1}−ε_{n}} = f̃_{n}ẽ_{1}, v_{0} = ∑_{i=1}^{n} f̃_{i}ẽ_{i}, \\ E^{2}V_{0} = V_{ε_{1}+ε_{2}}, v_{ε_{1}+ε_{2}} = ẽ_{1} ∧ ẽ_{2}. \end{matrix}  (1.4.5)

Note that ⟨v_{0}⟩ is the trivial 1-dimensional 𝔭𝔢(n)-module. Indeed,

B_{i,j}(v_{0}) = −ẽ_{j} ∧ ẽ_{i} + −ẽ_{i} ∧ ẽ_{j} = 0, C_{i,j}(v_{0}) = f̃_{i}f̃_{j} + f̃_{j}(−f̃_{i}) = 0.

Since B_{i,j}v_{ε_{1}+ε_{2}} = 0, then v_{ε_{1}+ε_{2}} is a 𝔭𝔢(n)-highest vector.

Let us prove that as 𝔤𝔩(n)-modules,

X_{ε_{1}+ε_{2}} ≅ V_{ε_{1}+ε_{2}} ⊕ V_{ε_{1}−ε_{n}} ⊕ V_{−2ε_{n}}.

Indeed,

\begin{matrix}C_{2,n}(v_{ε_{1}+ε_{2}}) = v_{ε_{1}−ε_{n}}, − B_{2,n}(v_{ε_{1}−ε_{n}}) = v_{ε_{1}+ε_{2}}, \\ C_{1,n}(v_{ε_{1}−ε_{n}}) = v_{−2ε_{n}}, B_{1,n}(v_{−2ε_{n}}) = v_{ε_{1}−ε_{n}}. \end{matrix}

Finally, we get

v_{0} = \frac{1}{2} ∑_{i=1}^{n−1} B_{i,n}A_{n,i} + nB_{n,n}v_{−2ε_{n}}.

This proves (1.4.2) and Lemma 1.4. □

Let us prove that if v_{λ} from 𝔤_{1} ⊗ 𝔤_{−1}^{∗} is a 𝔭𝔢(n)-highest vector of weight either λ = 2ε_{1} or ε_{1} + ε_{2}, then v_{λ} ∉ 𝔤_{0} ⊗ S^{2}𝔤_{−1}^{∗}. In fact, if λ = 2ε_{1}, then by (1.3.1) and (1.4.4) we have

v_{λ} = (− ∑_{i=1}^{n} f_{i} ∧ ẽ_{1} ⊗ ẽ_{i} + ∑_{i=1}^{n} e_{1}ẽ_{i} ⊗ f̃_{i} + ∑_{i=1}^{n} e_{i}f̃_{i} ⊗ ẽ_{1}) ⊗ ẽ_{1}.

Then

v_{λ}(f_{1})(f_{1}) = −f_{1} ∧ ẽ_{1} + ∑_{i=1}^{n} e_{i}f̃_{i} ≠ 0.

Therefore, v_{λ} ∉ 𝔤_{0} ⊗ S^{2}𝔤_{−1}^{∗}.

If λ = ε_{1} + ε_{2}, then by (1.3.1) and (1.4.5)

\begin{matrix}v_{λ} = (− ∑_{i=1}^{n} f_{i} ∧ ẽ_{1} ⊗ ẽ_{i} + ∑_{i=1}^{n} e_{1}ẽ_{i} ⊗ f̃_{i} + ∑_{i=1}^{n} e_{i}f̃_{i} ⊗ ẽ_{1}) ⊗ ẽ_{2} \\ − (− ∑_{i=1}^{n} f_{i} ∧ ẽ_{2} ⊗ ẽ_{i} + ∑_{i=1}^{n} e_{2}ẽ_{i} ⊗ f̃_{i} + ∑_{i=1}^{n} e_{i}f̃_{i} ⊗ ẽ_{2}) ⊗ ẽ_{1}. \end{matrix}

Thus,

v_{λ}(f_{2})(e_{i}) = e_{1}ẽ_{i} ≠ 0 and v_{λ}(e_{i})(f_{2}) = 0.

Hence, v_{λ} ∉ 𝔤_{0} ⊗ S^{2}𝔤_{−1}^{∗}.

Let λ = 0. According to Lemma 1.4, the Jordan-Hölder series of the 𝔭𝔢(n)-module 𝔤_{1} ⊗ 𝔤_{−1}^{∗} contains two 𝔭𝔢(n)-modules with highest weight 0. By Lemma 1.4 the sequence (1.4.1) is nonsplit and we have already proved that 𝔤_{2} has no irreducible 𝔭𝔢(n)-module with highest weight 2ε_{1}. Therefore, either 𝔤_{2} consists of one trivial 𝔭𝔢(n)-module or 𝔤_{2} = 0. But by (1.2.1)

𝔭𝔢(n + 1) ⊂ 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}).

Hence, 𝔤_{2} = Π(⟨1⟩).

Finally, let us show that 𝔤_{3} = 0. By definition

𝔤_{3} = (𝔤_{2} ⊗ 𝔤_{−1}^{∗}) ∩ (𝔤_{1} ⊗ S^{2}𝔤_{−1}^{∗}).

Note that

𝔤_{2} ⊗ 𝔤_{−1}^{∗} = Π(⟨1⟩) ⊗ V^{∗} ≅ V,

as 𝔭𝔢(n)-modules. By (1.1.3) the 𝔭𝔢(n)-highest vector in 𝔤_{2} ⊗ 𝔤_{−1}^{∗} is v = e_{n+1}ẽ_{n+1} ⊗ ẽ_{1}. By the explicit formula (1.1.2) of multiplication in 𝔭𝔢(n + 1) we have

v(f_{1})(e_{1}) = [e_{n+1}ẽ_{n+1}, e_{1} ∧ f̃_{n+1}] = e_{1}ẽ_{n+1} ∈ 𝔤_{1}.

On the other hand, v(e_{1})(f_{1}) = 0. Therefore, v ∉ 𝔤_{1} ⊗ S^{2}𝔤_{−1}^{∗}. Hence, 𝔤_{3} = 0.

Thus, Theorem 1.2 is proved for 𝔤_{0} = 𝔠𝔭𝔢(n). This result and part b) of Lemma 1.1 imply the statement of Theorem 1.2 for 𝔤_{0} = 𝔰𝔭𝔢(n) ⊂ +⟨τ + nz⟩.

Let us prove that 𝔤_{1} = 0 for 𝔤_{0} = 𝔰𝔭𝔢(n), 𝔭𝔢(n), 𝔠𝔰𝔭𝔢(n), or 𝔰𝔭𝔢(n) ⊂ +⟨aτ + bz⟩, where a, b ∈ ℂ, a, b ≠ 0, and b/a ≠ n.

Indeed, as has been shown, 𝔤_{1} = Π(V) for 𝔤_{0} = 𝔠𝔭𝔢(n) or 𝔰𝔭𝔢(n) ⊂ +⟨τ +nz⟩, and by (1.3.1) the corresponding 𝔰𝔭𝔢(n)-highest vector is

v_{ε_{1}} = − ∑_{i=1}^{n} f_{i} ∧ ẽ_{1} ⊗ ẽ_{i} + ∑_{i=1}^{n} e_{1}ẽ_{i} ⊗ f̃_{i} + ∑_{i=1}^{n} e_{i}f̃_{i} ⊗ ẽ_{1}.

Then

v_{ε_{1}}(f_{1}) = −f_{1} ∧ ẽ_{1} + ∑_{i=1}^{n} e_{i}f̃_{i} ∈ 𝔰𝔭𝔢(n) ⊂ +⟨τ + nz⟩.

Note that

−f_{1} ∧ ẽ_{1} + ∑_{i=1}^{n} e_{i}f̃_{i} ∉ 𝔰𝔭𝔢(n).

Hence v_{ε_{1}}(f_{1}) ∉ 𝔤_{0} for 𝔤_{0} = 𝔰𝔭𝔢(n) ⊂ +⟨aτ + bz⟩, where a, b ∈ ℂ, a ≠ 0 and b/a ≠ n, or a = 0. Therefore,

𝔤_{1} = 0 for 𝔤_{0} = 𝔰𝔭𝔢(n), 𝔭𝔢(n), 𝔠𝔰𝔭𝔢(n), or 𝔰𝔭𝔢(n) ⊂ +⟨aτ + bz⟩,

where a, b ∈ ℂ, a, b ≠ 0 and b/a ≠ n. Thus, in these cases

𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) = 𝔤_{−1} ⊕ 𝔤_{0}.

The main theorem

The following theorem describes SFs for the odd analogues of Riemannian metric and various conformal versions.

Theorem. For the G-structures with the following Lie (G) = 𝔤_{0} the nonzero SFs are of orders not exceeding 2 and as follows:

order 1: if 𝔤_{0} = 𝔰𝔭𝔢(n) or 𝔰𝔭𝔢(n) ⊂ +⟨τ + nz⟩ we have V^{∗}; order 2:if 𝔤_{0} = 𝔰𝔭𝔢(n), where n > 3, we have the following nonsplit exact sequence of 𝔰𝔭𝔢(n)-modules:

0 ⟶ X_{ε_{1}+ε_{2}} ⟶ H_{𝔤_{0}}^{2,2} ⟶ Π(X_{2ε_{1}+2ε_{2}}) ⟶ 0;

if 𝔤_{0} = 𝔰𝔭𝔢(3) another space is added to the SFs: we have the following nonsplit exact sequence of 𝔰𝔭𝔢(3)-modules:

0 ⟶ X ⟶ H_{𝔤_{0}}^{2,2} ⟶ Π(X_{3ε_{1}}) ⟶ 0,

where X is determined from the following nonsplit exact sequence of 𝔰𝔭𝔢(3)-modules:

0 ⟶ X_{ε_{1}+ε_{2}} ⟶ X ⟶ Π(X_{2ε_{1}+2ε_{2}}) ⟶ 0;

if 𝔤_{0} = 𝔰𝔭𝔢(n) ⊂ +⟨aτ + bz⟩, where a, b ∈ ℂ are such that a = 0, b ≠ 0 or a ≠ 0, b/a ≠ n, then for n > 2 we have the following nonsplit exact sequence of 𝔰𝔭𝔢(n)-modules:

0 ⟶ H_{𝔰𝔭𝔢(n)}^{2,2} ⟶ H_{𝔤_{0}}^{2,2} ⟶ X_{2ε_{1}} ⟶ 0;

if 𝔤_{0} = 𝔰𝔭𝔢(n) ⊂ +⟨τ + nz⟩, then for n > 3 H_{𝔤_{0}}^{2,2} = Π(X_{2ε_{1}+2ε_{2}}) is an irreducible 𝔰𝔭𝔢(n)-module, for n = 3 we have the following nonsplit exact sequence of 𝔰𝔭𝔢(3)-modules:

0 ⟶ Π(X_{2ε_{1}+2ε_{2}}) ⟶ H_{𝔤_{0}}^{2,2} ⟶ Π(X_{3ε_{1}}) ⟶ 0;

if 𝔤_{0} = 𝔠𝔭𝔢(n), n > 2, then

H_{𝔠𝔭𝔢(n)}^{2,2} = Π(S^{2}(E^{2}V/Π(⟨1⟩))/E^{4}V),

more precisely, we have the following nonsplit exact sequence of 𝔰𝔭𝔢(n)-modules:

0 ⟶ H_{𝔰𝔭𝔢(n) ⊂ +⟨τ+nz⟩}^{2,2} ⟶ H_{𝔤_{0}}^{2,2} ⟶ X_{2ε_{1}} ⟶ 0.

Proof of the main theorem

3.1. Calculation of SFs of order 1. Recall that the bidegree of the differentials in the Spencer complex is (−1, 1). We will often refer to the following

Lemma. Let (𝔤_{−1}, 𝔤_{0}) be an arbitrary pair, where 𝔤_{−1} is a faithful module over a Lie superalgebra 𝔤_{0}, and let

𝔤_{k−1} ⊗ 𝔤_{−1}^{∗} ⟶^{∂_{𝔤_{0}}^{k+1,1}} 𝔤_{k−2} ⊗ E^{2}𝔤_{−1}^{∗} ⟶^{∂_{𝔤_{0}}^{k,2}} 𝔤_{k−3} ⊗ E^{3}𝔤_{−1}^{∗}  (k ≥ 1)

be the corresponding Spencer cochain sequence. Then

Im∂_{𝔤_{0}}^{k+1,1} ≅ (𝔤_{k−1} ⊗ 𝔤_{−1}^{∗})/𝔤_{k}.  (3.1.1)

Proof. By the definition of the Cartan prolongation

𝔤_{k} = (𝔤_{k−1} ⊗ 𝔤_{−1}^{∗}) ∩ (𝔤_{k−2} ⊗ S^{2}𝔤_{−1}^{∗}).

Let c ∈ 𝔤_{k−1} ⊗ 𝔤_{−1}^{∗}. Then c ∈ 𝔤_{k} if and only if c(g_{1})g_{2} = (−1)^{p(g_{1})p(g_{2})}c(g_{2})g_{1} for any (homogeneous) g_{1}, g_{2} ∈ 𝔤_{−1}. On the other hand

∂_{𝔤_{0}}^{k+1,1}c(g_{1}, g_{2}) = −(−1)^{p(g_{1})p(g_{2})}c(g_{2})g_{1} + c(g_{1})g_{2}.

Hence, Ker∂_{𝔤_{0}}^{k+1,1} = 𝔤_{k}. This proves the Lemma. □

In particular, to define H_{𝔤_{0}}^{1,2} we have the following Spencer cochain sequence:

𝔤_{0} ⊗ 𝔤_{−1}^{∗} ⟶^{∂_{𝔤_{0}}^{2,1}} 𝔤_{−1} ⊗ E^{2}𝔤_{−1}^{∗} ⟶^{∂_{𝔤_{0}}^{1,2}} 0, where Ker∂_{𝔤_{0}}^{2,1} = 𝔤_{1}.

Let us prove that

H_{𝔤_{0}}^{1,2} = 0 if either 𝔤_{0} = 𝔠𝔭𝔢(n) or 𝔤_{0} = 𝔰𝔭𝔢(n) ⊂ +⟨aτ + bz⟩,

where a = 0, b ≠ 0 or a ≠ 0, b/a ≠ n.

Let 𝔤_{0} = 𝔰𝔭𝔢(n) ⊂ +⟨τ⟩ = 𝔭𝔢(n). Since 𝔭𝔢(n) and Π(E^{2}V^{∗}) are isomorphic 𝔤_{0}-modules, the 𝔭𝔢(n)-module 𝔤_{0} ⊗ 𝔤_{−1}^{∗} is isomorphic to

Π(E^{2}V^{∗}) ⊗ V^{∗} ≅ E^{2}V^{∗} ⊗ V ≅ 𝔤_{−1} ⊗ E^{2}𝔤_{−1}^{∗} = Ker∂_{𝔤_{0}}^{1,2}.

By part a) of Theorem 1.2 we have 𝔤_{1} = 0. Therefore, by (3.1.1)

Im∂_{𝔤_{0}}^{2,1} ≅ 𝔤_{0} ⊗ 𝔤_{−1}^{∗} ≅ Ker∂_{𝔤_{0}}^{1,2},

i.e., H_{𝔭𝔢(n)}^{1,2} = 0.

Let 𝔤_{0} = 𝔰𝔭𝔢(n) ⊂ +⟨aτ + bz⟩, where a = 0, b ≠ 0 or a ≠ 0, b/a ≠ n. By Theorem 1.2, 𝔤_{1} = 0 for such 𝔤_{0}. Note that dim 𝔤_{0} = dim 𝔭𝔢(n). Therefore,

dim Im∂_{𝔤_{0}}^{2,1} = dim 𝔤_{0} ⊗ 𝔤_{−1}^{∗} = dim 𝔭𝔢(n) ⊗ V^{∗} = dim E^{2}V^{∗} ⊗ V = dim Ker∂_{𝔤_{0}}^{1,2}.

Hence, H_{𝔤_{0}}^{1,2} = 0.

Let 𝔤_{0} = 𝔠𝔭𝔢(n). Then by part b) of Theorem 1.2 𝔤_{1} = V^{∗}. Note that

𝔤_{0} ⊗ 𝔤_{−1}^{∗} = (𝔭𝔢(n) ⊕ ⟨z⟩) ⊗ V^{∗} = 𝔭𝔢(n) ⊗ V^{∗} ⊕ ⟨z⟩ ⊗ V^{∗} ≅ V ⊗ E^{2}V^{∗} ⊕ V^{∗}.

Therefore,

Im∂_{𝔤_{0}}^{2,1} = V ⊗ E^{2}V^{∗} = Ker∂_{𝔤_{0}}^{1,2}.

Hence, H_{𝔠𝔭𝔢(n)}^{1,2} = 0.

Let us prove that

H_{𝔤_{0}}^{1,2} = V^{∗} if 𝔤_{0} = 𝔰𝔭𝔢(n) or 𝔰𝔭𝔢(n) ⊂ +⟨τ + nz⟩.

Let 𝔤_{0} = 𝔰𝔭𝔢(n). By part a) of Theorem 1.2 𝔤_{1} = 0. Therefore,

Im∂_{𝔤_{0}}^{2,1} ≅ 𝔤_{0} ⊗ 𝔤_{−1}^{∗} = 𝔰𝔭𝔢(n) ⊗ V^{∗}.

As has been shown for the case 𝔤_{0} = 𝔠𝔰𝔭𝔢(n) = 𝔰𝔭𝔢(n) ⊕ ⟨z⟩, we have

𝔤_{0} ⊗ 𝔤_{−1}^{∗} = (𝔰𝔭𝔢(n) ⊕ ⟨z⟩) ⊗ V^{∗} ≅ V ⊗ E^{2}V^{∗}.

Therefore,

Ker∂_{𝔰𝔭𝔢(n)}^{1,2}/Im∂_{𝔰𝔭𝔢(n)}^{2,1} ≅ V^{∗}.

Hence H_{𝔰𝔭𝔢(n)}^{1,2} = V^{∗}.

Finally, let 𝔤_{0} = 𝔰𝔭𝔢(n) ⊂ +⟨τ + nz⟩. By part b) of Theorem 1.2 𝔤_{1} = V^{∗}. Since

𝔤_{0} ⊗ 𝔤_{−1}^{∗} = (𝔰𝔭𝔢(n) ⊂ +⟨τ + nz⟩) ⊗ V^{∗},

the Jordan-Hölder series for the 𝔰𝔭𝔢(n)-module Im∂_{𝔤_{0}}^{2,1} contains the same irreducible quotient modules as that for the 𝔰𝔭𝔢(n)-module 𝔰𝔭𝔢(n) ⊗ V^{∗}. Since

Ker∂_{𝔤_{0}}^{1,2} = V ⊗ E^{2}V^{∗} ≅ (𝔰𝔭𝔢(n) ⊕ ⟨z⟩) ⊗ V^{∗},

then H_{𝔤_{0}}^{1,2} = V^{∗}. This proves Theorem 2.1 in the case of the SFs of the first order.

3.2. Hochschild-Serre spectral sequence. The continuation of the proof of Theorem 2.1 is based on the Hochschild-Serre spectral sequence. Let us recall the corresponding formulations in a form convenient for us, since the case of Lie superalgebras is hardly reflected in the literature (one might think that the union of [Fu] and [GM] should suffice, but the sign rule applied to the Lie algebra case does not completely solve the problem).

Let 𝔤 = 𝔤_{0} ⊕ 𝔤_{1} be a Lie superalgebra and M be a 𝔤-module. On the superspace of k-dimensional cochains C^{k} = C^{k}(𝔤, M) define a filtration:

F^{0}C^{k} = C^{k} ⊃ F^{1}C^{k} ⊃ F^{2}C^{k} ⊃ … ⊃ F^{j}C^{k} ⊃ … ⊃ F^{k+1}C^{k} = 0,

where

F^{j}C^{k} = {c ∈ C^{k}| c(g_{1}, …, g_{i}, …, g_{k}) = 0 if k − j + 1 arguments belong to 𝔤_{0}},

0 ≤ j ≤ k + 1. Set

Z_{r}^{p,q} = {c ∈ F^{p}C^{p+q}| dc ∈ F^{p+r}C^{p+q+1}}.  (3.2.1)

Finally, set

E_{r}^{p,q} = Z_{r}^{p,q}/(Z_{r−1}^{p+1,q−1} + dZ_{r−1}^{p−r+1,q+r−2}).  (3.2.2)

Notice that the differential d induces the differentials

d_{r}^{p,q} : E_{r}^{p,q} ⟶ E_{r}^{p+r,q−r+1}  (3.2.3)

and E_{r+1}^{p,q} = H^{p,q}(E_{r}) [GM].

Since d(F^{j}C^{k}) ⊂ F^{j}C^{k+1}, on H^{k} = H^{k}(𝔤, M) we get the induced filtration such that F^{p}H^{k}/F^{p+1}H^{k} = E_{∞}^{p,q}, where p + q = k.

We want to compute the group H^{2}(V, 𝔤_{∗}), where V = V_{0} ⊕ V_{1} is the standard 𝔭𝔢(n)- module, 𝔤_{∗} = 𝔤_{∗}(V, 𝔠𝔭𝔢(n)). The Hochschild-Serre spectral sequence corresponding to the subalgebra V_{0} converges to H^{2}(V, 𝔤_{∗}). Thus, H^{2}(V, 𝔤_{∗}) = ⊕_{p+q=2}E_{∞}^{p,q} and in order to compute the limit terms of the spectral sequence E_{∞}^{p,q} we have to consider three cases:

1) p = 2, q = 0. Then by formula (3.2.3) we have

E_{1}^{1,0} ⟶^{d_{1}^{1,0}} E_{1}^{2,0} ⟶^{d_{1}^{2,0}} E_{1}^{3,0},

E_{2}^{0,1} ⟶^{d_{2}^{0,1}} E_{2}^{2,0} ⟶^{d_{2}^{2,0}} 0,  (3.2.4)

0 ⟶^{d_{3}^{−1,2}} E_{3}^{2,0} ⟶^{d_{3}^{2,0}} 0.

Therefore, E_{∞}^{2,0} = E_{3}^{2,0}.

2) p = 1, q = 1. Then by formula (3.2.3) we have

E_{1}^{0,1} ⟶^{d_{1}^{0,1}} E_{1}^{1,1} ⟶^{d_{1}^{1,1}} E_{1}^{2,1},

0 ⟶^{d_{2}^{−1,2}} E_{2}^{1,1} ⟶^{d_{2}^{1,1}} E_{2}^{3,0},  (3.2.5)

0 ⟶^{d_{3}^{−2,3}} E_{3}^{1,1} ⟶^{d_{3}^{1,1}} 0.

Therefore, E_{∞}^{1,1} = E_{3}^{1,1}.

3) p = 0, q = 2. Then by formula (3.2.3) we have

0 ⟶^{d_{1}^{−1,2}} E_{1}^{0,2} ⟶^{d_{1}^{0,2}} E_{1}^{1,2},

0 ⟶^{d_{2}^{−2,3}} E_{2}^{0,2} ⟶^{d_{2}^{0,2}} E_{2}^{2,1},  (3.2.6)

0 ⟶^{d_{3}^{−3,4}} E_{3}^{0,2} ⟶^{d_{3}^{0,2}} E_{3}^{3,0},

0 ⟶^{d_{4}^{−4,5}} E_{4}^{0,2} ⟶^{d_{4}^{0,2}} 0.

Therefore, E_{∞}^{0,2} = E_{4}^{0,2}.

3.3. Continuation of the proof. Notice that by [Fu] E_{1}^{p,q} = H^{q}(𝔤_{0}, M ⊗ S^{p}𝔤_{1}^{∗}). Since in our case V = V_{0} ⊕ V_{1} is a commutative Lie superalgebra, then

E_{1}^{p,q} = H^{q}(V_{0}, 𝔤_{∗} ⊗ S^{p}V_{1}^{∗}) = H^{q}(V_{0}, 𝔤_{∗}) ⊗ S^{p}V_{1}^{∗}.  (3.3.1)

Let us calculate H^{q}(V_{0}, 𝔤_{∗}) for q = 0, 1, 2. By Lemma 1.1

𝔤_{∗} = 𝔭𝔢(n + 1) = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} ⊕ 𝔤_{2}, where
𝔤_{−1} = V_{0} ⊕ V_{1}, 𝔤_{0} = 𝔠𝔭𝔢(n) = 𝔰𝔩(n) ⊕ Π(S^{2}V_{0}) ⊕ Π(E^{2}V_{1}) ⊕ ⟨τ⟩ ⊕ ⟨z⟩,
𝔤_{1} = Π(V_{1}) ⊕ Π(V_{0}),
𝔤_{2} = Π(⟨1⟩).

(3.3.2

Recall that as an 𝔰𝔩(n + 1)-module,

𝔭𝔢(n + 1) ≅ 𝔰𝔩(n + 1) ⊕ E^{2}W_{0}^{∗} ⊕ S^{2}W_{0} ⊕ ⟨ d ⟩,

where d is diag(1_{n+1}, − 1_{n+1}) and W_{0} is the standard 𝔰𝔩(n + 1)-module. Clearly,

E^{2}W_{0}^{∗} = Π(E^{2}V_{1}) ⊕ V_{1},
S^{2}W_{0} = Π(S^{2}V_{0}) ⊕ Π(V_{0}) ⊕ 𝔤_{2},
𝔰𝔩(n + 1) = 𝔰𝔩(n) ⊕ ⟨τ + nz⟩ ⊕ V_{0} ⊕ Π(V_{1}),
d = τ − z.

Let ε_{1}, …, ε_{n+1} be the standard basis of the dual space to the space of the diagonal matrices in 𝔤𝔩(n + 1) and the ordering is performed so that

Δ_{+} = {ε_{i} − ε_{j}, i < j}, Δ_{−} = {ε_{i} − ε_{j}, i > j}.

Let E_{ε_{i}−ε_{j}}(i ≠ j) be the corresponding root vectors. Then V_{0} is the subspace of 𝔰𝔩(n + 1) generated by

E_{ε_{1}−ε_{n+1}}, E_{ε_{2}−ε_{n+1}}, …, E_{ε_{n}−ε_{n+1}}.

Let V_{λ} be the irreducible 𝔰𝔩(n + 1)-module with highest weight λ. The BWB theorem says [Kos] that there exists a 1-1 correspondence between the irreducible components of H^{q}(V_{0}, V_{λ}), considered as 𝔤𝔩(n)-module, and elements w ∈ W (𝔰𝔩(n + 1)) of length q from the Weyl group of the Lie algebra 𝔰𝔩(n + 1) such that

w(Δ_{−}) ∩ Δ_{+} ⊂ {ε_{1} − ε_{n+1}, ε_{2} − ε_{n+1}, …, ε_{n} − ε_{n+1}}.  (3.3.3)

Moreover, the highest weight of the 𝔤𝔩(n)-module corresponding to w is equal to w(λ+ρ)−ρ, where ρ = (∑_{α∈Δ_{+}} α)/2.

Notice that E^{2}W_{0}^{∗}, S^{2}W_{0}, 𝔰𝔩(n + 1), ⟨τ − z⟩ are all irreducible 𝔰𝔩(n + 1)-modules with highest weights, respectively,

ε_{1} + … + ε_{n−1}, 2ε_{1}, 2ε_{1} + ε_{2} + … + ε_{n}, 0.

Let us find the highest weights of irreducible 𝔤𝔩(n)-submodules of H^{q}(V_{0}, V_{λ}) for each of the indicated λ.

1) q = 0. The only element of the Weyl group of length 0 is the unit. Hence w(λ+ρ)−ρ = λ.

2) q = 1. Let ⟨α_{1}, …, α_{n}⟩, where

α_{1} = ε_{1} − ε_{2}, …, α_{i} = ε_{i} − ε_{i+1}, …, α_{n} = ε_{n} − ε_{n+1}

be the system of simple roots. The elements of the Weyl group of length 1 are reflections corresponding to the simple roots:

r_{α_{i}} : α ⟶ α − \frac{2(α_{i}, α)}{(α_{i}, α_{i})}α_{i}.

Since the only element r_{α_{i}} satisfying (3.3.3) is r_{α_{n}}, then w(λ + ρ) − ρ = r_{α_{n}}(λ) − α_{n}. For

λ = ε_{1} + … + ε_{n−1}, 2ε_{1}, 2ε_{1} + ε_{2} + … + ε_{n}, 0

this expression is equal to, respectively:

−2ε_{n}, ε_{1} − ε_{2} − … − ε_{n−1} − 2ε_{n}, − ε_{2} − ε_{3} − … − ε_{n−1} − 3ε_{n}, − ε_{1} − ε_{2} − … − ε_{n−1} − 2ε_{n}.

3) q = 2. The elements of length 2 are of the form r_{α_{i}}r_{α_{j}}. The only such element satisfying (3.3.3) is r_{α_{n}}r_{α_{n−1}}. Then

w(λ + ρ) − ρ = r_{α_{n}}r_{α_{n−1}}(λ) − α_{n−1} − 2α_{n}.

For λ = ε_{1} + … + ε_{n−1}, 2ε_{1}, 2ε_{1} + ε_{2} + … + ε_{n}, 0 this expression is equal to, respectively:

− 2ε_{1} − 2ε_{2} − … − 2ε_{n−2} − 4ε_{n−1} − 4ε_{n},

− 3ε_{2} − 3ε_{3} (if n = 3) or − 2ε_{2} − … − 2ε_{n−2} − 3ε_{n−1} − 3ε_{n} (if n > 3), − ε_{1} − 3ε_{2} − 4ε_{3} (if n = 3) or − ε_{1} − 2ε_{2} − … − 2ε_{n−2} − 3ε_{n−1} − 4ε_{n} (if n > 3), − 2ε_{1} − … − 2ε_{n−2} − 3ε_{n−1} − 3ε_{n}.

Remark. We have obtained the weights with respect to 𝔤𝔩(n) = 𝔰𝔩(n) ⊕ ⟨τ + nz⟩ embedded into 𝔰𝔩(n + 1). Now it is not difficult to rewrite these weights as the highest ones with respect to 𝔭𝔢(n)_{0} = 𝔰𝔩(n) ⊕ ⟨τ⟩. We collect all our results in the following

3.4.

Lemma. 𝔤𝔩(n) = 𝔭𝔢(n)_{0} module H^{q}(V_{0}, 𝔤_{∗}) (q = 0, 1, 2) is the direct sum of irreducible submodules with the highest weights and highest vectors listed in Table 2.

3.5. Lemma. Let V_{λ} be an irreducible 𝔤𝔩(n)-module with highest weight λ. Then E_{1}^{p,0}(p = 1, 2, 3), E_{1}^{p,1}(p = 0, 1, 2), E_{1}^{p,2} (p = 0, 1) are the direct sums of irreducible 𝔤𝔩(n)-submodules with highest weights given in the corresponding columns of Tables 3, 4, and 5, respectively.

Proof. By formula (3.3.1) the following 𝔤𝔩(n)-modules are isomorphic: E_{1}^{p,q} ≅ H^{q}(V_{0}, 𝔤_{∗}) ⊗ S^{p}V_{0}. Making use of the description of H^{q}(V_{0}, 𝔤_{∗}) as a 𝔤𝔩(n)-module given in Lemma 3.4 we find the decomposition of the indicated tensor product into irreducible components described in Tables 3, 4, and 5. □

3.6. Lemma. E_{0}^{p,0}(p = 1, 2, 3) are the direct sums of the irreducible 𝔤𝔩(n)-modules with highest weights described in the corresponding columns of Table 6.

Proof. By formula (3.2.2) we have

E_{0}^{p,0} = Z_{0}^{p,0} = 𝔤_{∗} ⊗ S^{p}V_{1}^{∗} = 𝔤_{∗} ⊗ S^{p}V_{0}.

Making use of the description of 𝔤_{∗} as 𝔤𝔩(n)-module given in (3.3.2) we find the decomposition of the indicated tensor products into direct sum of the irreducible components described in Table 6. □

3.7. Lemma. E_{∞}^{2,0} is an irreducible 𝔤𝔩(n)-module with highest weight 2ε_{1} + 2ε_{2}.

Proof. First, recall that H_{𝔠𝔭𝔢(n)}^{1,2} = 0 by the already proved part of Theorem 2.1 for the case of SFs of order 1. Making use of Lemma 3.4, we note that if E_{∞}^{2,0} had contained a 𝔤𝔩(n)-submodule belonging to either V_{−ε_{n}} ⊗ S^{2}V_{0} or V_{ε_{1}} ⊗ S^{2}V_{0}, then this submodule would have belonged to H_{𝔠𝔭𝔢(n)}^{1,2}. □

Therefore, with the help of Table 3 we deduce that E_{∞}^{2,0} has no 𝔤𝔩(n)-submodules with highest weights 2ε_{1} − ε_{n}, ε_{1}, 3ε_{1}, 2ε_{1} + ε_{2}.

Let us show that E_{∞}^{2,0} has no irreducible 𝔤𝔩(n)-submodules with highest weights 4ε_{1}, 3ε_{1} + ε_{2}, and 2ε_{1} either. More precisely, let us show that even E_{2}^{2,0} does not have them.

Recall that the corresponding differentials act as follows:

E_{1}^{1,0} ⟶^{d_{1}^{1,0}} E_{1}^{2,0} ⟶^{d_{1}^{2,0}} E_{1}^{3,0}.

Note that according to (3.2.2), E_{1}^{p,0} = Z_{1}^{p,0} for p = 1, 2, 3. Let us show that Kerd_{1}^{2,0} has no components with weights 4ε_{1}, 3ε_{1} + ε_{2}, and 2ε_{1}. It follows from Tables 2 and 3 that the corresponding highest vectors in E_{1}^{2,0} are

v_{4ε_{1}} = e_{1}ẽ_{1} ⊗ ẽ_{1}^{2}, v_{3ε_{1}+ε_{2}} = e_{1}ẽ_{2} ⊗ ẽ_{1}^{2} − e_{1}ẽ_{1} ⊗ ẽ_{1}ẽ_{2}, v_{2ε_{1}} = (τ − z) ⊗ ẽ_{1}^{2}.

We remind the reader that the differentials d in our case are the same as the differentials ∂_{𝔤_{0}}^{k,s}. Notice that if c ∈ E_{1}^{2,0} then

d_{1}^{2,0}c(f_{1}, f_{1}, f_{1}) = dc(f_{1}, f_{1}, f_{1}) = −3c(f_{1}, f_{1})(f_{1}).

Therefore,

d_{1}^{2,0}v_{4ε_{1}}(f_{1}, f_{1}) = 3e_{1}, d_{1}^{2,0}v_{3ε_{1}+ε_{2}}(f_{1}, f_{1}) = (3/2)e_{2}, d_{1}^{2,0}v_{2ε_{1}}(f_{1}, f_{1}) = −6f_{1}.  (3.7.1)

Hence,

v_{4ε_{1}}, v_{3ε_{1}+ε_{2}}, v_{2ε_{1}} ∉ Kerd_{1}^{2,0}.

Finally, let us prove that in E_{∞}^{2,0} there is an irreducible 𝔤𝔩(n)-submodule of highest weight 2ε_{1} + 2ε_{2}. Notice that E_{2}^{2,0} has submodule of highest weight 2ε_{1} + 2ε_{2}, since according to Table 3, this module is contained in E_{1}^{2,0} and is not contained in either E_{1}^{1,0} or E_{1}^{3,0}. Recall that the corresponding differentials act as follows:

E_{2}^{0,1} ⟶^{d_{2}^{0,1}} E_{2}^{2,0} ⟶^{d_{2}^{2,0}} 0.

Therefore, Kerd_{2}^{2,0} = E_{2}^{2,0} has a component of weight 2ε_{1} + 2ε_{2}. By Table 4 E_{1}^{0,1} has no components of weight 2ε_{1} + 2ε_{2}. Hence, neither E_{2}^{0,1} nor Imd_{2}^{0,1} have such a component . Therefore, it must be in E_{3}^{2,0} = E_{∞}^{2,0}.

3.8. Lemma. a) As a 𝔤𝔩(n)-module, E_{∞}^{1,1} can only have the irreducible submodules with the following highest weights, each of multiplicity not greater then 1:

2ε_{1}, 2ε_{1} + ε_{2} − ε_{n}, ε_{1} + ε_{2} − 2ε_{n}, and ε_{1} − ε_{n};

b) E_{∞}^{1,1} has an irreducible 𝔤𝔩(n)-submodule with highest weight 2ε_{1}.

Proof. By Theorem 2.1 for the case of SFs of order 1 and Tables 2 and 4 we see that E_{∞}^{1,1} has no irreducible 𝔤𝔩(n)-submodules of highest weight ε_{1} − 2ε_{n} and −ε_{n}, since they would have corresponded to SFs of order 1.

Let us show that there are no components of weight 2ε_{1} −2ε_{n} or 0 in E_{∞}^{1,1}, more precisely, that even E_{2}^{1,1} does not have them. Recall that the corresponding differentials act as follows:

E_{1}^{0,1} ⟶^{d_{1}^{0,1}} E_{1}^{1,1} ⟶^{d_{1}^{1,1}} E_{1}^{2,1}.

By (3.2.2) we have

\begin{matrix}E_{1}^{0,1} = Z_{1}^{0,1}/(Z_{0}^{1,0} + dZ_{0}^{0,0}), \\ E_{1}^{1,1} = Z_{1}^{1,1}/(Z_{0}^{2,0} + dZ_{0}^{1,0}), \\ E_{1}^{2,1} = Z_{1}^{2,1}/(Z_{0}^{3,0} + dZ_{0}^{2,0}). \end{matrix}  3.8.1

By Tables 2 and 4 the highest vectors of weights 2ε_{1} − 2ε_{n} and 0 in E_{1}^{1,1} are, respectively,

v_{2ε_{1}−2ε_{n}} = (e_{1} ∧ f̃_{n}) ⊗ ẽ_{1} ∧ f̃_{n} and v_{0} = ∑_{i=1}^{n}(τ − z) ⊗ f̃_{i} ∧ ẽ_{i}.

We see that

dv_{2ε_{1}−2ε_{n}}(e_{n}, f_{1}, f_{1}) = −v_{2ε_{1}−2ε_{n}}(f_{1}, f_{1})(e_{n}) − 2v_{2ε_{1}−2ε_{n}}(e_{n}, f_{1})(f_{1}) = −f_{n}/2 ≠ 0,
dv_{0}(e_{1}, f_{1}, f_{1}) = −v_{0}(f_{1}, f_{1})(e_{1}) − 2v_{0}(e_{1}, f_{1})(f_{1}) = 2f_{1} ≠ 0.

Suppose that

dv_{2ε_{1}−2ε_{n}} ∈ Z_{0}^{3,0} + dZ_{0}^{2,0}.

Then there exist highest 𝔤𝔩(n)-vectors v′_{2ε_{1}−2ε_{n}} ∈ Z_{0}^{3,0} and v′′_{2ε_{1}−2ε_{n}} ∈ dZ_{0}^{2,0} of weight 2ε_{1} −2ε_{n} such that

dv_{2ε_{1}−2ε_{n}} = v′_{2ε_{1}−2ε_{n}} + v′′_{2ε_{1}−2ε_{n}}.

Since e_{n} ∈ V_{0}, then v′_{2ε_{1}−2ε_{n}}(e_{n}, f_{1}, f_{1}) = 0. Hence v′′_{2ε_{1}−2ε_{n}} ≠ 0 and therefore, dZ_{0}^{2,0} has an irreducible 𝔤𝔩(n)-submodule of highest weight 2ε_{1} − 2ε_{n}.

Similarly, having assumed that dv_{0} ∈ Z_{0}^{3,0}+dZ_{0}^{2,0}, we deduce that dZ_{0}^{2,0} has an irreducible 𝔤𝔩(n)-submodule of weight 0. Note that according to Table 6, E_{0}^{2,0} has no submodules of highest weight 2ε_{1}−2ε_{n} or 0. Since E_{0}^{2,0} = Z_{0}^{2,0}, then Z_{0}^{2,0} and dZ_{0}^{2,0} have no such components either. Therefore, dv_{2ε_{1}−2ε_{n}} and dv_{0} do not belong to Z_{0}^{3,0} + dZ_{0}^{2,0}. Thanks to (3.8.1) this implies that

d_{1}^{1,1}v_{2ε_{1}−2ε_{n}} ≠ 0 and d_{1}^{1,1}v_{0} ≠ 0.

Hence, Kerd_{1}^{1,1}, and therefore, E_{2}^{1,1}, have no irreducible 𝔤𝔩(n)-submodules of highest weight 2ε_{1} − 2ε_{n} and 0.

Let us prove now that E_{∞}^{1,1} = E_{3}^{1,1} has no irreducible 𝔤𝔩(n)-submodule of highest weight 3ε_{1} − ε_{n}. Notice that E_{2}^{1,1} has such a submodule, since by Table 4 it is contained in E_{1}^{1,1} and is not contained in either E_{1}^{0,1} or E_{1}^{2,1}. Tables 2 and 4 imply that the 𝔤𝔩(n)-highest vector in E_{1}^{1,1} of weight 3ε_{1} − ε_{n} is v_{3ε_{1}−ε_{n}} = e_{1}ẽ_{1} ⊗ f̃_{n} ∧ ẽ_{1}.

Recall that the corresponding differentials act as follows:

0 ⟶^{d_{2}^{−1,2}} E_{2}^{1,1} ⟶^{d_{2}^{1,1}} E_{2}^{3,0}.

By formula (3.2.2) we have

\begin{matrix}E_{2}^{1,1} = Z_{2}^{1,1}/(Z_{1}^{2,0} + dZ_{1}^{0,1}), \\ E_{2}^{3,0} = Z_{2}^{3,0}/dZ_{1}^{2,0}. \end{matrix}  (3.8.2)

Thanks to formulas (3.8.1) we see that the 𝔤𝔩(n)-highest vector in E_{2}^{1,1} of weight 3ε_{1} − ε_{n} is

w_{3ε_{1}−ε_{n}} = v_{3ε_{1}−ε_{n}} + v′_{3ε_{1}−ε_{n}} + v′′_{3ε_{1}−ε_{n}},

where v′_{3ε_{1}−ε_{n}} and v′′_{3ε_{1}−ε_{n}} are 𝔤𝔩(n)-highest vectors in Z_{0}^{2,0} and dZ_{0}^{1,0}, respectively.

Since by (3.2.1) dw_{3ε_{1}−ε_{n}} ∈ Z_{2}^{3,0}, then dw_{3ε_{1}−ε_{n}}(e_{n}, f_{1}, f_{1}) = 0. Since v′′_{3ε_{1}−ε_{n}} ∈ dZ_{0}^{1,0}, then dv′′_{3ε_{1}−ε_{n}} = 0. We have

dv_{3ε_{1}−ε_{n}}(e_{n}, f_{1}, f_{1}) = −2v_{3ε_{1}−ε_{n}}(e_{n}, f_{1})(f_{1}) = −e_{1} ≠ 0.

Therefore, v′_{3ε_{1}−ε_{n}} ≠ 0. Looking at Table 6 we see that the unique highest vector of weight 3ε_{1} − ε_{n} in Z_{0}^{2,0} is e_{1} ∧ f̃_{n} ⊗ ẽ_{1}^{2}. Hence, v′_{3ε_{1}−ε_{n}} = ke_{1} ∧ f̃_{n} ⊗ ẽ_{1}^{2}, where k ∈ ℂ^{∗}. Note that since v_{3ε_{1}−ε_{n}}(f_{1}, f_{1}) = 0, then dv_{3ε_{1}−ε_{n}}(f_{1}, f_{1}, f_{1}) = 0. Therefore,

\begin{matrix}dw_{3ε_{1}−ε_{n}}(f_{1}, f_{1}, f_{1}) = dv′_{3ε_{1}−ε_{n}}(f_{1}, f_{1}, f_{1}) = \\ − 3v′_{3ε_{1}−ε_{n}}(f_{1}, f_{1})(f_{1}) = (−3/2)kf_{n} ≠ 0. \end{matrix}

Note that dw_{3ε_{1}−ε_{n}} ∉ dZ_{1}^{2,0}. In fact, by Table 3 E_{1}^{2,0} has no irreducible 𝔤𝔩(n)-component with highest weight 3ε_{1}−ε_{n}. Since E_{1}^{2,0} = Z_{1}^{2,0}, then Z_{1}^{2,0} and dZ_{1}^{2,0} have no such component either. Therefore, by (3.8.2) we have d_{2}^{1,1}w_{3ε_{1}−ε_{n}} ≠ 0. Hence, Kerd_{2}^{1,1} = E_{3}^{1,1} has no components with highest weight 3ε_{1} − ε_{n}.

Let us prove that the irreducible component with highest weight ε_{1} − ε_{n} cannot be contained in E_{∞}^{1,1} with multiplicity greater than 1. Note that E_{1}^{1,1} has two components of weight ε_{1} −ε_{n}. According to Tables 2 and 4, one of the 𝔤𝔩(n)-highest vectors of weight ε_{1} −ε_{n} in E_{1}^{1,1} is v_{ε_{1}−ε_{n}} = (τ − z) ⊗ f̃_{n} ∧ ẽ_{1}. We see that

dv_{ε_{1}−ε_{n}}(e_{n}, f_{1}, f_{1}) = −2v_{ε_{1}−ε_{n}}(e_{n}, f_{1})(f_{1}) = −(τ − z)(f_{1}) = 2f_{1} ≠ 0.

Suppose that dv_{ε_{1}−ε_{n}} ∈ Z_{0}^{3,0} + dZ_{0}^{2,0}. Then there exist 𝔤𝔩(n)-highest vectors v′_{ε_{1}−ε_{n}} ∈ Z_{0}^{3,0} and v′′_{ε_{1}−ε_{n}} ∈ dZ_{0}^{2,0} of weight ε_{1} − ε_{n} such that

dv_{ε_{1}−ε_{n}} = v′_{ε_{1}−ε_{n}} + v′′_{ε_{1}−ε_{n}}.

Since e_{n} ∈ V_{0}, then v′_{ε_{1}−ε_{n}}(e_{n}, f_{1}, f_{1}) = 0. Therefore, v′′_{ε_{1}−ε_{n}} ≠ 0. Now note that

dv_{ε_{1}−ε_{n}}(e_{1}, f_{1}, f_{1}) = −v_{ε_{1}−ε_{n}}(f_{1}, f_{1})(e_{1}) − 2v_{ε_{1}−ε_{n}}(e_{1}, f_{1})(f_{1}) = 0.

Since e_{1} ∈ V_{0}, then v′_{ε_{1}−ε_{n}}(e_{1}, f_{1}, f_{1}) = 0. Therefore,

v′′_{ε_{1}−ε_{n}}(e_{1}, f_{1}, f_{1}) = 0.  (3.8.3)

By Table 6 E_{0}^{2,0} = Z_{0}^{2,0} contains a unique highest vector of weight ε_{1} − ε_{n}, namely,

∑_{i=1}^{n} f_{n} ∧ f̃_{i} ⊗ ẽ_{i}ẽ_{1}.

Then

v′′_{ε_{1}−ε_{n}} = kd(∑_{i=1}^{n} f_{n} ∧ f̃_{i} ⊗ ẽ_{i}ẽ_{1}), where k ∈ ℂ^{∗}.

Note that in this case

\begin{matrix}v′′_{ε_{1}−ε_{n}}(e_{1}, f_{1}, f_{1}) = kd(∑_{i=1}^{n} f_{n} ∧ f̃_{i} ⊗ ẽ_{i}ẽ_{1})(e_{1}, f_{1}, f_{1}) = \\ − k(∑_{i=1}^{n} f_{n} ∧ f̃_{i} ⊗ ẽ_{i}ẽ_{1})(f_{1}, f_{1})(e_{1}) = (k/2)f_{n} ≠ 0, \end{matrix}

which contradicts (3.8.3). Thus, dv_{3ε_{1}−ε_{n}} ∉ Z_{0}^{3,0} + dZ_{0}^{2,0}. Then (3.8.1) yields d_{1}^{1,1}v_{ε_{1}−ε_{n}} ≠ 0. Therefore, the component of highest weight ε_{1} − ε_{n} can not be contained in Kerd_{1}^{1,1} and hence, in E_{∞}^{1,1}, with multiplicity exceeding 1. So part a) of Lemma 3.8 is proved.

Let us prove that E_{∞}^{1,1} = E_{3}^{1,1} contains an irreducible 𝔤𝔩(n)-submodule of highest weight 2ε_{1}. Note that E_{2}^{1,1} does contain such a submodule, since by Table 4 it is contained in E_{1}^{1,1} and is not contained in either E_{1}^{0,1} or E_{1}^{2,1}.

Let u_{2ε_{1}} be the 𝔤𝔩(n)-highest vector of weight 2ε_{1} in E_{1}^{1,1}. By (3.8.1) u_{2ε_{1}} can be chosen so that

u_{2ε_{1}}(v_{1}, v_{2}) = 0 for any v_{1}, v_{2} ∈ V_{1}.

According to (3.8.1), the 𝔤𝔩(n)-highest vector of weight 2ε_{1} in E_{2}^{1,1} is

w_{2ε_{1}} = u_{2ε_{1}} + t_{2ε_{1}} + s_{2ε_{1}},  (3.8.4)

where t_{2ε_{1}} ∈ Z_{0}^{2,0} and s_{2ε_{1}} ∈ dZ_{0}^{1,0} are 𝔤𝔩(n)-highest vectors. If dw_{2ε_{1}} = 0, then

w_{2ε_{1}} ∈ Kerd_{2}^{1,1} = E_{3}^{1,1}

and therefore, E_{∞}^{1,1} has an irreducible 𝔤𝔩(n)-submodule with highest weight 2ε_{1}. Suppose that dw_{2ε_{1}} ≠ 0. Let us prove that then

dw_{2ε_{1}} ∈ dZ_{1}^{2,0}.  (3.8.5)

Recall that Z_{1}^{2,0} = E_{1}^{2,0} and by Table 3 E_{1}^{2,0} has one highest vector of weight 2ε_{1}, namely, v_{2ε_{1}}. Let us show that

dw_{2ε_{1}} = kdv_{2ε_{1}}, where k ∈ ℂ^{∗}.  (3.8.6)

Note that by (3.2.1) dw_{2ε_{1}} ∈ Z_{2}^{3,0} and dZ_{1}^{2,0} ⊂ Z_{0}^{3,0}. Therefore, in order to prove (3.8.6) it suffices to show that

dw_{2ε_{1}}(v_{1}, v_{2}, v_{3}) = kdv_{2ε_{1}}(v_{1}, v_{2}, v_{3}), where k ∈ ℂ^{∗}, for any v_{1}, v_{2}, v_{3} ∈ V_{1}.

We have

du_{2ε_{1}}(v_{1}, v_{2}, v_{3}) = 0 for any v_{1}, v_{2}, v_{3} ∈ V_{1}.

Since s_{2ε_{1}} ∈ dZ_{0}^{1,0}, then ds_{2ε_{1}} = 0. Therefore,

dw_{2ε_{1}}(v_{1}, v_{2}, v_{3}) = dt_{2ε_{1}}(v_{1}, v_{2}, v_{3}) for any v_{1}, v_{2}, v_{3} ∈ V_{1}.  (3.8.7)

Since t_{2ε_{1}} ∈ Z_{0}^{2,0}, then dt_{2ε_{1}} ∈ Z_{0}^{2,1}. Hence dt_{2ε_{1}} = t′_{2ε_{1}} + t′′_{2ε_{1}},where t′_{2ε_{1}}, t′′_{2ε_{1}} are 𝔤𝔩(n)-highest vectors from Z_{0}^{2,1} such that

t′_{2ε_{1}}(v_{1}, v_{2}, v_{3}) = 0 for all v_{1}, v_{2}, v_{3} ∈ V_{1} and t′′_{2ε_{1}} ∈ Z_{0}^{3,0}.

Since by hypothesis dw_{2ε_{1}} ≠ 0, then t′′_{2ε_{1}} ≠ 0. Since v_{2ε_{1}} ∈ Z_{1}^{2,0}, then dv_{2ε_{1}} ∈ Z_{0}^{3,0}.

In Lemma 3.7 we have proved that dv_{2ε_{1}} ≠ 0 (see (3.7.1)). By Table 6 Z_{0}^{3,0} = E_{0}^{3,0} has only one irreducible 𝔤𝔩(n)-submodule with highest weight 2ε_{1}. Hence t′′_{2ε_{1}} = kdv_{2ε_{1}}, where k ∈ ℂ^{∗}. Therefore,

dt_{2ε_{1}}(v_{1}, v_{2}, v_{3}) = kdv_{2ε_{1}}(v_{1}, v_{2}, v_{3}) for any v_{1}, v_{2}, v_{3} ∈ V_{1}.

Thus, by (3.8.7)

dw_{2ε_{1}}(v_{1}, v_{2}, v_{3}) = kdv_{2ε_{1}}(v_{1}, v_{2}, v_{3}) for any v_{1}, v_{2}, v_{3} ∈ V_{1}

and formula (3.8.6) is proved. Then by (3.8.2) d_{2}^{1,1}w_{2ε_{1}} = 0. Therefore, w_{2ε_{1}} ∈ Kerd_{2}^{1,1} = E_{3}^{1,1}. Thus, E_{∞}^{1,1} contains an irreducible 𝔤𝔩(n)-submodule with highest weight 2ε_{1}. This proves part b) of Lemma 3.8. □

3.9. Lemma. Only the following highest weights of irreducible 𝔤𝔩(n)-submodules can be encountered among those in E_{∞}^{0,2}:

−2ε_{n−1} − 2ε_{n}, 2ε_{1} − ε_{n−1} − ε_{n}, ε_{1} − ε_{n−1} − 2ε_{n}, − ε_{n−1} − ε_{n}.

Proof. By Table 5 E_{1}^{0,2} is a direct sum of irreducible 𝔤𝔩(n)-components with the indicated highest weights.

Note that due to Table 2 H^{2}(V, 𝔤_{∗}) can only possess first and second order SFs. By the statement of Theorem 2.1 for SFs of order 1, H_{𝔠𝔭𝔢(n)}^{1,2} = 0. Therefore, by Lemmas 7, 8, and 9 H_{𝔠𝔭𝔢(n)}^{2,2} can only contain irreducible 𝔤𝔩(n)-submodules with the following highest weights:

\begin{matrix}2ε_{1} + 2ε_{2}, 2ε_{1}, 2ε_{1} + ε_{2} − ε_{n}, ε_{1} + ε_{2} − 2ε_{n}, ε_{1} − ε_{n}, \\ − 2ε_{n−1} − 2ε_{n}, 2ε_{1} − ε_{n−1} − ε_{n}, ε_{1} − ε_{n−1} − 2ε_{n}, − ε_{n−1} − ε_{n} \end{matrix}

each with multiplicity not greater than one, and the components with highest weights 2ε_{1} + 2ε_{2} and 2ε_{1} are contained in H_{𝔠𝔭𝔢(n)}^{2,2} with multiplicity one each.

Recall that by definition H_{𝔤_{0}}^{2,2} is determined by the sequence

𝔤_{1} ⊗ 𝔤_{−1}^{∗} ⟶^{∂_{𝔤_{0}}^{3,1}} 𝔤_{0} ⊗ E^{2}𝔤_{−1}^{∗} ⟶^{∂_{𝔤_{0}}^{2,2}} 𝔤_{−1} ⊗ E^{3}𝔤_{−1}^{∗}.

Thus, 𝔤_{0} ⊗ E^{2}𝔤_{−1}^{∗} for

𝔤_{0} = 𝔠𝔭𝔢(n) or 𝔰𝔭𝔢(n) ⊂ +⟨τ + nz⟩

is equal to

𝔠𝔭𝔢(n) ⊗ E^{2}𝔤_{−1}^{∗} or (𝔰𝔭𝔢(n) ⊂ +⟨τ + nz⟩) ⊗ E^{2}𝔤_{−1}^{∗},

respectively. Note that Im∂_{𝔠𝔭𝔢(n)}^{3,1} and Im∂_{𝔰𝔭𝔢(n) ⊂ +⟨τ+nz⟩}^{3,1} are isomorphic 𝔰𝔭𝔢(n)-modules by part b) of Theorem 1.2 and (3.1.1).

Note also that by Lemma 1.4 the Jordan-Hölder series for E^{2}V^{∗} contains 𝔰𝔭𝔢(n)-modules with highest weights 2ε_{1} and 0. Therefore, the Jordan-Hölder series of the 𝔰𝔭𝔢(n)-module H_{𝔠𝔭𝔢(n)}^{2,2}, as compared with that of H_{𝔰𝔭𝔢(n) ⊂ +⟨τ+nz⟩}^{2,2}, can additionally contain only the irreducible 𝔰𝔭𝔢(n)-modules with highest weights 2ε_{1} and 0. But we have shown that H_{𝔠𝔭𝔢(n)}^{2,2} has no trivial 𝔤𝔩(n)-submodule and therefore, H_{𝔠𝔭𝔢(n)}^{2,2} and H_{𝔰𝔭𝔢(n) ⊂ +⟨τ+nz⟩}^{2,2} can only differ by an irreducible 𝔰𝔭𝔢(n)-submodule with highest weight 2ε_{1}, which, being considered as 𝔰𝔩(n)-module, is the sum of irreducible 𝔰𝔩(n)-submodules with highest weights

2ε_{1}, ε_{1} − ε_{n}, and − ε_{n−1} − ε_{n} (see (1.4.4)).
□

3.10. Lemma. H_{𝔰𝔭𝔢(n) ⊂ +⟨τ+nz⟩}^{2,2} has no irreducible 𝔰𝔩(n)-submodule with highest weight 2ε_{1}.

Proof. Making use of Lemma 3.8, let us prove that E_{∞}^{1,1} considered as an 𝔰𝔩(n)-module has no irreducible component with highest weight 2ε_{1} in the case where 𝔤_{0} = 𝔰𝔭𝔢(n) ⊂ +⟨τ+nz⟩.

Indeed, in Lemma 3.8 we have shown that if w_{2ε_{1}} is the 𝔰𝔩(n)-highest vector of weight 2ε_{1} in E_{2}^{1,1} such that dw_{2ε_{1}} ≠ 0 then dw_{2ε_{1}} ∈ dZ_{1}^{2,0} (see (3.8.5)).

According to Table 2, H^{0}(V_{0}, 𝔤_{∗}(𝔤_{−1}, 𝔤_{0})) has no irreducible 𝔰𝔩(n)-submodule with highest weight 0 when 𝔤_{0} = 𝔰𝔭𝔢(n) ⊂ +⟨τ + nz⟩ Therefore, by Table 3 E_{1}^{2,0} has no irreducible 𝔰𝔩(n)- submodule with highest weight 2ε_{1}. Since Z_{1}^{2,0} = E_{1}^{2,0}, then Z_{1}^{2,0} and dZ_{1}^{2,0} have no such component either. Thus, by (3.8.2) d_{2}^{1,1}w_{2ε_{1}} ≠ 0 and therefore, w_{2ε_{1}} ∉ E_{3}^{1,1} = E_{∞}^{1,1}. It remains to show that dw_{2ε_{1}} ≠ 0. Recall that w_{2ε_{1}} = u_{2ε_{1}} + t_{2ε_{1}} + s_{2ε_{1}}, where u_{2ε_{1}} ∈ E_{1}^{1,1}, t_{2ε_{1}} ∈ Z_{0}^{2,0} and s_{2ε_{1}} ∈ dZ_{0}^{1,0} are 𝔰𝔩(n)-highest vectors (see (3.8.4)). By Tables 2 and 4

u_{2ε_{1}} = 2 ∑_{i=1}^{n} e_{1}ẽ_{i} ⊗ f̃_{i} ∧ ẽ_{1} − (n + 1) ∑_{i=1}^{n} e_{1}ẽ_{1} ⊗ f̃_{i} ∧ ẽ_{i}.  (3.10.1)

Therefore,

\begin{matrix}du_{2ε_{1}}(e_{1}, f_{1}, f_{1}) = −u_{2ε_{1}}(f_{1}, f_{1})(e_{1}) − 2u_{2ε_{1}}(e_{1}, f_{1})(f_{1}) = \\ − 2(e_{1}ẽ_{1} − (n + 1)e_{1}ẽ_{1}/2)(f_{1}) = (n − 1)e_{1} ≠ 0. \end{matrix}

Since s_{2ε_{1}} ∈ dZ_{0}^{1,0}, then ds_{2ε_{1}} = 0. Hence, if dw_{2ε_{1}} = 0 then t_{2ε_{1}} ≠ 0 and since u_{2ε_{1}} is an even vector, then vector t_{2ε_{1}} must be even.

By Table 6 Z_{0}^{2,0} = E_{0}^{2,0} has 4 irreducible 𝔰𝔩(n)-submodules with highest weight 2ε_{1}. The corresponding highest vectors are

\begin{matrix}(∑_{i=2}^{n} e_{i} ∧ f̃_{i} − (n − 1)e_{1} ∧ f̃_{1}) ⊗ ẽ_{1}^{2} − n ∑_{i=2}^{n} e_{1} ∧ f̃_{i} ⊗ ẽ_{1}ẽ_{i}, \\ τ ⊗ ẽ_{1}^{2}, z ⊗ ẽ_{1}^{2}, and 𝔤_{2} ⊗ ẽ_{1}^{2}. \end{matrix} .

Only the first three of these vectors are even. Therefore, if we confine ourselves to the case 𝔤_{0} = 𝔰𝔭𝔢(n) ⊂ +⟨τ + nz⟩, we see that there should be two 𝔰𝔩(n)-highest vectors of weight 2ε_{1} in E_{0}^{2,0}. Let

\begin{matrix}t_{2ε_{1}} = k_{1}((∑_{i=2}^{n} e_{i} ∧ f̃_{i} − (n − 1)e_{1} ∧ f̃_{1}) ⊗ ẽ_{1}^{2} − n ∑_{i=2}^{n} e_{1} ∧ f̃_{i} ⊗ ẽ_{1}ẽ_{i}) \\ + k_{2}((τ + nz) ⊗ ẽ_{1}^{2}), where k_{1}, k_{2} ∈ ℂ, \end{matrix}

be a linear combination of these vectors. Note that

\begin{matrix}du_{2ε_{1}}(e_{2}, f_{2}, f_{1}) = −u_{2ε_{1}}(f_{2}, f_{1})(e_{2}) − u_{2ε_{1}}(e_{2}, f_{1})(f_{2}) − u_{2ε_{1}}(e_{2}, f_{2})(f_{1}) = \\ − 2(e_{1}ẽ_{2}/2)(f_{2}) + (n + 1)(e_{1}ẽ_{1}/2)(f_{1}) = −e_{1}/2 + (n + 1)e_{1}/2 = ne_{1}/2, \end{matrix}

\begin{matrix}dt_{2ε_{1}}(e_{2}, f_{2}, f_{1}) = −t_{2ε_{1}}(f_{2}, f_{1})(e_{2}) − t_{2ε_{1}}(e_{2}, f_{1})(f_{2}) − t_{2ε_{1}}(e_{2}, f_{2})(f_{1}) = \\ (−1/2)k_{1}ne_{1} ∧ f̃_{2}(e_{2}) = (−1/4)k_{1}ne_{1}. \end{matrix}

Therefore, if dw_{2ε_{1}} = 0, then k_{1} = 2. Observe that

\begin{matrix}dt_{2ε_{1}}(e_{1}, f_{1}, f_{1}) = −t_{2ε_{1}}(f_{1}, f_{1})(e_{1}) − 2t_{2ε_{1}}(e_{1}, f_{1})(f_{1}) = \\ − k_{1}(n − 1)(e_{1} ∧ f̃_{1})(e_{1}) + k_{2}(τ + nz)(e_{1}) = (−k_{1}(n − 1)/2 + k_{2}(n + 1))e_{1}. \end{matrix}

Since du_{2ε_{1}}(e_{1}, f_{1}, f_{1}) = (n − 1)e_{1}, then

−(n − 1) + k_{2}(n + 1) + (n − 1) = 0.

Hence, k_{2} = 0. But then

\begin{matrix}dw_{2ε_{1}}(f_{1}, f_{1}, f_{1}) = dt_{2ε_{1}}(f_{1}, f_{1}, f_{1}) = −3t_{2ε_{1}}(f_{1}, f_{1})(f_{1}) = \\ − 3k_{1}(n − 1)e_{1} ∧ f̃_{1}(f_{1}) = (3/2)k_{1}(n − 1)f_{1} = 3(n − 1)f_{1} ≠ 0. \end{matrix}

This proves Lemma 3.10. □

Lemma 3.10 implies that H_{𝔠𝔭𝔢(n)}^{2,2} and H_{𝔰𝔭𝔢(n) ⊂ +⟨τ+nz⟩}^{2,2} differ by an irreducible 𝔰𝔭𝔢(n)-module with highest weight 2ε_{1}. Thus, H_{𝔰𝔭𝔢(n) ⊂ +⟨τ+nz⟩}^{2,2} can only contain irreducible 𝔰𝔩(n)-submodules with highest weights

\begin{matrix}2ε_{1} + 2ε_{2}, 2ε_{1} + ε_{2} − ε_{n}, ε_{1} + ε_{2} − 2ε_{n}, \\ − 2ε_{n−1} − 2ε_{n}, 2ε_{1} − ε_{n−1} − ε_{n}, ε_{1} − ε_{n−1} − 2ε_{n} \end{matrix}

each with multiplicity not greater than 1, and the multiplicity of the submodule with highest weight 2ε_{1} + 2ε_{2} is precisely 1.

3.11. Lemma. The irreducible 𝔭𝔢(n)-module with highest weight 2ε_{1} + 2ε_{2} is the direct sum of irreducible 𝔤𝔩(n)-modules with the following highest weights:

a) for n > 3:

\begin{matrix}2ε_{1} + 2ε_{2}, 2ε_{1} + ε_{2} − ε_{n}, ε_{1} + ε_{2} − 2ε_{n}, \\ − 2ε_{n−1} − 2ε_{n}, 2ε_{1} − ε_{n−1} − ε_{n}, ε_{1} − ε_{n−1} − 2ε_{n}; \end{matrix}

b) for n = 3:

2ε_{1} + 2ε_{2}, 2ε_{1} + ε_{2} − ε_{3}, ε_{1} + ε_{2} − 2ε_{3}.

Proof. Let us consider the 𝔭𝔢(n)-module S^{2}(S^{2}V). Note that v_{2ε_{1}+2ε_{2}} = (e_{1}e_{2})^{2}−(e_{1}^{2})(e_{2}^{2}) is a 𝔭𝔢(n)-highest vector. Indeed, B_{i,j}v_{2ε_{1}+2ε_{2}} = 0 for any i and j. Set

v_{2ε_{1}+ε_{2}−ε_{n}} = (e_{1}e_{2})(e_{1}f_{n}) − (e_{1}^{2})(e_{2}f_{n}),
v_{ε_{1}+ε_{2}−2ε_{n}} = (e_{1}f_{n}) ∧ (e_{2}f_{n}),
v_{−2ε_{n−1}−2ε_{n}} = (f_{n−1} ∧ f_{n})(f_{n−1} ∧ f_{n}),
v_{2ε_{1}−ε_{n−1}−ε_{n}} = (e_{1}f_{n−1}) ∧ (e_{1}f_{n}) − (e_{1}^{2})(f_{n−1} ∧ f_{n}),
v_{ε_{1}−ε_{n−1}−2ε_{n}} = (f_{n−1} ∧ f_{n})(e_{1}f_{n}).

Notice that these vectors are the 𝔤𝔩(n)-highest ones. Moreover,

\begin{matrix}C_{2,n}v_{2ε_{1}+2ε_{2}} = 2v_{2ε_{1}+ε_{2}−ε_{n}}, & (3.11.1) \\ B_{2,n}v_{2ε_{1}+ε_{2}−ε_{n}} = v_{2ε_{1}+2ε_{2}}, & (3.11.2) \\ C_{1,n}v_{2ε_{1}+ε_{2}−ε_{n}} = −3v_{ε_{1}+ε_{2}−2ε_{n}}, & (3.11.3) \\ B_{1,n}v_{ε_{1}+ε_{2}−2ε_{n}} = −v_{2ε_{1}+ε_{2}−ε_{n}}. & (3.11.4) \end{matrix}

If n > 3, then additionally

C_{2,n−1}v_{2ε_{1}+ε_{2}−ε_{n}} = v_{2ε_{1}−ε_{n−1}−ε_{n}},
B_{2,n−1}v_{2ε_{1}−ε_{n−1}−ε_{n}} = v_{2ε_{1}+ε_{2}−ε_{n}},
C_{1,n}v_{2ε_{1}−ε_{n−1}−ε_{n}} = −3v_{ε_{1}−ε_{n−1}−2ε_{n}},
B_{1,n}v_{ε_{1}−ε_{n−1}−2ε_{n}} = −v_{2ε_{1}−ε_{n−1}−ε_{n}},
C_{1,n−1}v_{ε_{1}−ε_{n−1}−2ε_{n}} = v_{−2ε_{n−1}−2ε_{n}},
B_{1,n−1}v_{−2ε_{n−1}−2ε_{n}} = 2v_{ε_{1}−ε_{n−1}−2ε_{n}}.

Therefore, if n > 3, then the irreducible 𝔭𝔢(n)-module with highest weight 2ε_{1} +2ε_{2} contains irreducible 𝔤𝔩(n)-modules with highest weights

\begin{matrix}2ε_{1} + 2ε_{2}, 2ε_{1} + ε_{2} − ε_{n}, ε_{1} + ε_{2} − 2ε_{n}, \\ − 2ε_{n−1} − 2ε_{n}, 2ε_{1} − ε_{n−1} − ε_{n}, ε_{1} − ε_{n−1} − 2ε_{n}. \end{matrix}

We have already shown that the 𝔰𝔭𝔢(n)-module H_{𝔰𝔭𝔢(n) ⊂ +⟨τ+nz⟩}^{2,2} does contain irreducible 𝔰𝔩(n)- submodules with these highest weights exactly, their multiplicities are not greater than 1, and the multiplicity of 𝔰𝔩(n)-submodule with highest weight 2ε_{1} + 2ε_{2} is precisely one.

From Tables 2 and 3 we see that the corresponding 𝔰𝔩(n)-highest vector is

v_{2ε_{1}+2ε_{2}} = (e_{1}ẽ_{1}) ⊗ (ẽ_{2}ẽ_{2}) + (e_{2}ẽ_{2}) ⊗ (ẽ_{1}ẽ_{1}) − 2(e_{1}ẽ_{2}) ⊗ (ẽ_{1}ẽ_{2}).  (3.11.5)

Hence, v_{2ε_{1}+2ε_{2}} is an odd 𝔰𝔭𝔢(n)-highest vector. So part a) of Lemma 3.11 and Theorem 2.1 for the case where 𝔤_{0} = 𝔰𝔭𝔢(n) ⊂ +⟨τ + nz⟩, n > 3, are proved.

Let n = 3. Consider the 𝔰𝔭𝔢(3)-module E^{3}V. As an 𝔰𝔩(3)-module, this module is isomorphic to

E^{3}(V_{0} ⊕ V_{0}^{∗}) = E^{3}V_{0} ⊕ (E^{2}V_{0})(V_{0}^{∗}) ⊕ V_{0}(S^{2}V_{0}^{∗}) ⊕ S^{3}V_{0}^{∗}.

Therefore, E^{3}V is the direct sum of irreducible 𝔰𝔩(3)-modules with the highest weights and highest vectors listed in Table 7. Note that the vectors of weights 0 and ε_{1} are the 𝔰𝔭𝔢(3)- highest ones. Therefore, the Jordan-Hölder series of the 𝔰𝔭𝔢(3)-module E^{3}V contains as quotient modules the trivial and the standard ones. Notice that the vector of weight −2ε_{3} is the 𝔰𝔭𝔢(3)-highest one in the corresponding quotient module which can only contain 𝔰𝔩(3)- submodules with highest weights −2ε_{3}, ε_{1} − 2ε_{3}, and − 3ε_{3}.

Since v_{2ε_{1}+2ε_{2}} is the 𝔰𝔩(3)-highest vector of weight −2ε_{3}, then the relations (3.11.1)– (3.11.4) imply part b) of Lemma 3.11. □

3.12. Lemma. For n = 3 we have the following nonsplit exact sequence of 𝔰𝔭𝔢(3)-modules

0 ⟶ Π(X_{2ε_{1}+2ε_{2}}) ⟶ H_{𝔰𝔭𝔢(3) ⊂ +⟨τ+3z⟩}^{2,2} ⟶ Π(X_{3ε_{1}}) ⟶ 0.  (3.12.1)

Proof. By part b) of Lemma 3.11 and (3.11.5) we see that H_{𝔰𝔭𝔢(3) ⊂ +⟨τ+3z⟩}^{2,2} contains an irreducible 𝔰𝔭𝔢(3)-module with highest weight −2ε_{3}, which being considered as an 𝔰𝔩(3)- module, is the sum of irreducible 𝔰𝔩(3)-components with highest weights −2ε_{3}, ε_{1}−2ε_{3}, and− 3ε_{3}.

In addition to these 𝔰𝔩(3)-components, H_{𝔰𝔭𝔢(3) ⊂ +⟨τ+3z⟩}^{2,2} can only contain irreducible 𝔰𝔩(3)- components with the following highest weights:

3ε_{1}, 2ε_{1}, and 2ε_{1} − ε_{3}.

Let us show that these components are indeed contained in H_{𝔰𝔭𝔢(3) ⊂ +⟨τ+3z⟩}^{2,2}, and that their sum is an irreducible 𝔰𝔭𝔢(3)-quotient module with highest weight 3ε_{1}.

First, note that E_{∞}^{0,2} = E_{4}^{0,2} has an irreducible 𝔤𝔩(n)-module with highest weight 2ε_{1} − ε_{n−1} − ε_{n}. In fact, by Table 5 such a submodule is contained in E_{1}^{0,2} but is not contained in E_{1}^{1,2}. Therefore, by (3.2.6) it is contained in E_{2}^{0,2}.

According to Table 4, in E_{1}^{2,1} there is no submodule with highest weight 2ε_{1} − ε_{n−1} − ε_{n}, hence such a submodule is not contained in E_{2}^{2,1} either. Therefore, by (3.2.6) the submodule with this highest weight is contained in E_{3}^{0,2}.

According to Table 6, E_{0}^{3,0} has no submodule with highest weight 2ε_{1} − ε_{n−1} − ε_{n}, hence it is not contained in E_{3}^{3,0} either. Therefore, by (3.2.6) it is contained in E_{∞}^{0,2} = E_{4}^{0,2}.

By Tables 2 and 5 the 𝔤𝔩(n)-highest vectors in E_{1}^{0,2} of weights

−2ε_{n−1} − 2ε_{n}, 2ε_{1} − ε_{n−1} − ε_{n}, and ε_{1} − ε_{n−1} − 2ε_{n}

are, respectively,

v_{−2ε_{n−1}−2ε_{n}} = (f_{n−1} ∧ f̃_{n}) ⊗ (f̃_{n−1} ∧ f̃_{n}),
v_{2ε_{1}−ε_{n−1}−ε_{n}} = (e_{1}ẽ_{1}) ⊗ (f̃_{n−1} ∧ f̃_{n}), and
v_{ε_{1}−ε_{n−1}−2ε_{n}} = (e_{1} ∧ f̃_{n}) ⊗ (f̃_{n−1} ∧ f̃_{n}).

Note that

\begin{matrix}C_{1,n}v_{2ε_{1}−ε_{n−1}−ε_{n}} = −2v_{ε_{1}−ε_{n−1}−2ε_{n}}, \\ B_{1,n}v_{ε_{1}−ε_{n−1}−2ε_{n}} = −v_{2ε_{1}−ε_{n−1}−ε_{n}} − (e_{1} ∧ f̃_{n}) ⊗ (f̃_{n−1} ∧ ẽ_{1}), \end{matrix}

\begin{matrix}C_{1,n−1}v_{ε_{1}−ε_{n−1}−2ε_{n}} = v_{−2ε_{n−1}−2ε_{n}}, \\ B_{1,n−1}v_{−2ε_{n−1}−2ε_{n}} = v_{ε_{1}−ε_{n−1}−2ε_{n}} + (f_{n−1} ∧ f̃_{n}) ⊗ (ẽ_{1} ∧ f̃_{n}). \end{matrix}

Therefore, for n = 3 the components with highest weights 3ε_{1}, 2ε_{1}, and 2ε_{1} − ε_{3} constitute an irreducible quotient module with highest weight 3ε_{1}.

From Tables 2 and 4 we see that the 𝔤𝔩(n)-highest vector in E_{1}^{1,1} of weight 2ε_{1} + ε_{2} − ε_{n} is

v_{2ε_{1}+ε_{2}−ε_{n}} = (e_{1}ẽ_{2}) ⊗ (ẽ_{1} ∧ f̃_{n}) − (e_{1}ẽ_{1}) ⊗ (ẽ_{2} ∧ f̃_{n}).

Observe that

(B_{1,2}A_{2,1} − (1/2)B_{2,2})(v_{2ε_{1}−ε_{2}−ε_{3}}) = 2v_{2ε_{1}+ε_{2}−ε_{3}}.

Therefore, the sequence (3.12.1) is nonsplit. This proves Lemma 3.12, and Theorem 2.1 in the case where 𝔤_{0} = 𝔰𝔭𝔢(3) ⊂ +⟨τ + 3z⟩. □

Recall that the Jordan-Hölder series of 𝔰𝔭𝔢(n)-module H_{𝔠𝔭𝔢(n)}^{2,2}, as compared to that of H_{𝔰𝔭𝔢(n) ⊂ +⟨τ+nz⟩}^{2,2}, contains in addition the 𝔰𝔭𝔢(n)-component with highest weight 2ε_{1}. Recall that by (3.10.1) the highest 𝔤𝔩(n)-vector with weight 2ε_{1} in E_{1}^{1,1} is

u_{2ε_{1}} = 2 ∑_{i=1}^{n} e_{1}ẽ_{i} ⊗ f̃_{i} ∧ ẽ_{1} − (n + 1) ∑_{i=1}^{n} e_{1}ẽ_{1} ⊗ f̃_{i} ∧ ẽ_{i}.

Note that

C_{n−1,n}(u_{2ε_{1}}) = −2(n + 1)v_{2ε_{1}−ε_{n−1}−ε_{n}} + 2(e_{1} ∧ f̃_{n−1} ⊗ f̃_{n} ∧ ẽ_{1} − e_{1} ∧ f̃_{n} ⊗ f̃_{n−1} ∧ ẽ_{1}).

This proves Theorem 2.1 in the case where 𝔤_{0} = 𝔠𝔭𝔢(n).

The proof of Theorem 2.1 in the case where 𝔤_{0} = 𝔰𝔭𝔢(n) follows from the fact that the Jordan-Hölder series of 𝔰𝔭𝔢(n)-module H_{𝔰𝔭𝔢(n)}^{2,2}, as compared to that of H_{𝔰𝔭𝔢(n) ⊂ +⟨τ+nz⟩}^{2,2}, contains in addition the 𝔰𝔭𝔢(n)-component with highest weight ε_{1} + ε_{2}.

Finally, the proof of Theorem 2.1 in the case when 𝔤_{0} = 𝔰𝔭𝔢(n) ⊂ +⟨aτ + bz⟩, where a, b ∈ ℂ are such that a = 0, b ≠ 0, or a ≠ 0, b/a ≠ n, follows from the fact that the Jordan-Hölder series of 𝔰𝔭𝔢(n)-module Ker∂_{𝔰𝔭𝔢(n) ⊂ +⟨aτ+bz⟩}^{2,2}, as compared to that of Ker∂_{𝔰𝔭𝔢(n)}^{2,2}, contains in addition the 𝔰𝔭𝔢(n)-component with highest weight 2ε_{1}.

CHAPTER 2 The analogues of Penrose’s tensors

In this chapter Λ^{i}V is the i-th exterior power of a vector (super)space V.

Standard ℤ-grading of 𝔰𝔩(m|n) and the corresponding Cartan prolongations

Let V = V (m|0) and U = U(0|n) be the standard (identity) 𝔤𝔩(m)- and 𝔤𝔩(n)-modules. (Hereafter 𝔤𝔩(m) = 𝔤𝔩(m|0), 𝔤𝔩(n) = 𝔤𝔩(0|n), etc.)

In what follows we will consider the standard (compatible) ℤ-grading of 𝔤 = 𝔰𝔩(m|n) with m ≤ n and let the degrees of all even roots be zero. This yields the ℤ-grading of the form:

𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1}, where 𝔤_{0} = 𝔰𝔩(m) ⊕ 𝔰𝔩(n) ⊕ ℂ, 𝔤_{−1} = 𝔤_{1}^{∗} = U ⊗ V^{∗}.

Let 𝔤̂_{0} be the Levi subalgebra of 𝔤_{0}, i.e., 𝔤̂_{0} = 𝔰𝔩(m) ⊕ 𝔰𝔩(n). The weights are given with respect to the bases ε_{1}, …, ε_{m} and δ_{1}, …, δ_{n} of the dual spaces to the maximal tori of 𝔤𝔩(m|n). Let e_{1}, …, e_{m} be the weight basis of V and f_{1}, …, f_{n} be the weight basis of U. Let ẽ_{1}, …, ẽ_{m} and f̃_{1}, …, f̃_{n} be the bases of the dual spaces to V and U, respectively, normed so that ẽ_{i}(e_{j}) = f̃_{i}(f_{j}) = δ_{ij}. If ⊕_{λ}k_{λ}V_{λ} is a direct sum of irreducible 𝔤_{0}-modules (here k_{λ} is the multiplicity of V_{λ}) with highest weight λ, denote by v_{λ}^{i} the highest weight vectors of the corresponding components: i = 1, …, k_{λ}. We will often represent the elements of 𝔤𝔩(m|n) by the matrices

X = diag(A, D) + antidiag(B, C)

where the dimensions of the matrices A, B, C, and D are m × m, m × n, n × m and n × n, respectively. Denote by A_{i,j} the matrix X whose components B, C, and D are zero and all the entries of A are also zero except for the (i, j)-th. The matrices B_{i,j}, C_{i,j}, and D_{i,j} are defined similarly.

1.1. Theorem. a) If m = 1, n > 1, then 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) = 𝔳𝔢𝔠𝔱(0|n), 𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}) = 𝔰𝔳𝔢𝔠𝔱(0|n);

b) if m, n > 1 and m ≠ n, then 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) = 𝔤, 𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}) = 𝔤_{−1} ⊕ 𝔤̂_{0};

c) if m = n = 2, then 𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}) = 𝔥(0|4), 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) = S^{∗}(𝔤_{−1}^{∗}) ⊂ + 𝔥(0|4);

d) if m = n > 2, then 𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}) = 𝔭𝔰𝔩(n|n), 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) = S^{∗}(𝔤_{−1}^{∗}) ⊂ + 𝔭𝔰𝔩(n|n).

Proof. Consider all cases mentioned in Theorem 1.1.

1.2. m = 1, n ≥ 2. Then 𝔤_{0} = 𝔰𝔩(n) ⊕ ℂ = 𝔤𝔩(n) and 𝔤̂_{0} = 𝔰𝔩(n), where 𝔤_{−1} is the standard 𝔤_{0} (or 𝔤̂_{0}) module. Therefore, 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) = 𝔳𝔢𝔠𝔱(0|n), 𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}) = 𝔰𝔳𝔢𝔠𝔱(0|n).

Notice that if m ≠ n, then

𝔰𝔩(m|n) ⊂ 𝔤_{∗}(𝔤_{−1}, 𝔤_{0})  (1.2.1)

and if m = n, then

𝔭𝔰𝔩(n|n) ⊂ 𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}).  (1.2.2)

Indeed, the Lie superalgebras 𝔰𝔩(m|n), where m ≠ n, and 𝔭𝔰𝔩(n|n) are simple and therefore, they are transitive (i.e., if there exists g ∈ 𝔤_{i} (i ≥ 0) such that [𝔤_{−1}, g] = 0, then g = 0. It follows that 𝔤_{1} is embedded into 𝔤_{0} ⊗ 𝔤_{−1}^{∗} (or 𝔤̂_{0} ⊗ 𝔤_{−1}^{∗}). The Jacobi identity implies 𝔤_{1} ⊂ 𝔤_{−1} ⊗ S^{2}𝔤_{−1}^{∗}.

1.3. Calculation of the first term of the Cartan prolongation for m, n ≥ 2, m ≠ n. Let 𝔤′_{1} be the first term of the Cartan prolongation of the pair (𝔤_{−1}, 𝔤_{0}). Let us show that 𝔤′_{1} = 𝔤_{1}. By definition,

𝔤′_{1} = (𝔤_{0} ⊗ 𝔤_{−1}^{∗}) ∩ (𝔤_{−1} ⊗ S^{2}𝔤_{−1}^{∗}), where, as 𝔤𝔩(m) ⊕ 𝔤𝔩(n)-modules ,
𝔤_{0} ⊗ 𝔤_{−1}^{∗} ≅ [(V ⊗ V^{∗})/ℂ ⊕ (U ⊗ U^{∗})/ℂ ⊕ ℂ] ⊗ (U^{∗} ⊗ V).

Note that if g ∈ 𝔤_{0} ⊗ 𝔤_{−1}^{∗}, then

g ∈ 𝔤_{−1} ⊗ S^{2}𝔤_{−1}^{∗} if and only if g(g_{1})(g_{2}) = −g(g_{2})(g_{1}) for any g_{1}, g_{2} ∈ 𝔤_{−1},

since 𝔤_{−1} is purely odd.

Lemma. The 𝔤𝔩(m) ⊕ 𝔤𝔩(n)-module 𝔤_{0} ⊗ 𝔤_{−1}^{∗} is the direct sum of irreducible submodules whose highest weights and highest vectors are listed in Table 8.

Proof. The proof of the Lemma consists of: a) a verification of the fact that vectors v_{λ} from Table 8 are indeed highest with respect to 𝔤𝔩(m) ⊕ 𝔤𝔩(n), i.e. A_{i,j}v_{λ} = D_{i,j}v_{λ} = 0 for i < j;

b) a calculation of dimension of 𝔤_{0} ⊗𝔤_{−1}^{∗} and of dimensions of the irreducible submodules of 𝔤_{0} ⊗ 𝔤_{−1}^{∗} by the formula from the Appendix.

Let us show that if

λ = 2ε_{1} − ε_{m} − δ_{n}, ε_{1} + δ_{1} − 2δ_{n}, ε_{1} + ε_{2} − ε_{m} − δ_{n} (if m ≥ 3),

or

λ = ε_{1} + δ_{1} − δ_{n−1} − δ_{n} (if n ≥ 3),

then v_{λ} ∉ 𝔤′_{1}. For this it suffices to indicate g_{1}, g_{2} ∈ 𝔤_{−1} such that

v_{λ}(g_{1})(g_{2}) ≠ −v_{λ}(g_{2})(g_{1})  (1.3.1)

or, perhaps, there exists just one g ∈ 𝔤_{−1} such that

v_{λ}(g)(g) ≠ 0.  (1.3.2)

Let λ = 2ε_{1} − ε_{m} − δ_{n}. Then

v_{λ}(f_{n} ⊗ ẽ_{1})(f_{n} ⊗ ẽ_{1}) = A_{1,m}(f_{n} ⊗ ẽ_{1}) = −f_{n} ⊗ ẽ_{m} ≠ 0.

If λ = ε_{1} + δ_{1} − 2δ_{n}, then

v_{λ}(f_{n} ⊗ ẽ_{1})(f_{n} ⊗ ẽ_{1}) = D_{1,n}(f_{n} ⊗ ẽ_{1}) = f_{1} ⊗ ẽ_{1} ≠ 0.

If λ = ε_{1} + ε_{2} − ε_{m} − δ_{n} (for m ≥ 3), then

\begin{matrix}v_{λ}(f_{n} ⊗ ẽ_{2})(f_{n−1} ⊗ ẽ_{1}) = A_{1,m}(f_{n−1} ⊗ ẽ_{1}) = −f_{n−1} ⊗ ẽ_{m}, \\ but v_{λ}(f_{n−1} ⊗ ẽ_{1})(f_{n} ⊗ ẽ_{2}) = 0. \end{matrix}

Finally, if λ = ε_{1} + δ_{1} − δ_{n−1} − δ_{n} (for n ≥ 3), then

\begin{matrix}v_{λ}(f_{n} ⊗ ẽ_{1})(f_{n−1} ⊗ ẽ_{2}) = D_{1,n−1}(f_{n−1} ⊗ ẽ_{2}) = f_{1} ⊗ ẽ_{2}, \\ but v_{λ}(f_{n−1} ⊗ ẽ_{2})(f_{n} ⊗ ẽ_{1}) = 0. \end{matrix}

Now, let us show that if λ = ε_{1} −δ_{n}, then 𝔤′_{1} contains precisely one irreducible 𝔤𝔩(m)⊕𝔤𝔩(n)- module with highest weight λ. Notice that by (1.2.1) 𝔤′_{1} contains at least one such module. Let

v_{λ} = k_{1}v_{λ}^{1} + k_{2}v_{λ}^{2} + k_{3}v_{λ}^{3}, where k_{1}, k_{2}, k_{3} ∈ ℂ,

be a linear combination of highest vectors of weight λ. Then the condition

v_{λ}(f_{n} ⊗ ẽ_{2})(f_{n−1} ⊗ ẽ_{1}) = −v_{λ}(f_{n−1} ⊗ ẽ_{1})(f_{n} ⊗ ẽ_{2})

STANDARD ℤ-GRADING OF 𝔰𝔩(m|n) 37

implies

mk_{1} = nk_{2},  (1.3.3)

whereas the condition

v_{λ}(f_{n} ⊗ ẽ_{1})(f_{n} ⊗ ẽ_{1}) = 0

implies

k_{1}(m − 1) + k_{2}(1 − n) + k_{3}(m − n) = 0.

Hence,

k_{2} = mk_{1}/n and k_{3} = −k_{1}/n.  (1.3.4)

Thus, 𝔤′_{1} = V_{ε_{1}−δ_{n}} and 𝔤′_{1} = 𝔤_{1}. □

1.4. Calculation of the second term of the Cartan prolongation for m, n ≥ 2, m ≠ n. Let 𝔤_{2} be the second term of the Cartan prolongation of (𝔤_{−1}, 𝔤_{0}).

Let us show that 𝔤_{2} = 0. Indeed, by definition, 𝔤_{2} = (𝔤_{1} ⊗ 𝔤_{−1}^{∗}) ∩ (𝔤_{0} ⊗ S^{2}𝔤_{−1}^{∗}). Notice that, as 𝔤_{0}-module,

\begin{matrix}𝔤_{1} ⊗ 𝔤_{−1}^{∗} ≅ (U^{∗} ⊗ V) ⊗ (U^{∗} ⊗ V) = \\ S^{2}U^{∗} ⊗ S^{2}V ⊕ Λ^{2}U^{∗} ⊗ Λ^{2}V ⊕ Λ^{2}U^{∗} ⊗ S^{2}V ⊕ S^{2}U^{∗} ⊗ Λ^{2}V. \end{matrix}

This decomposition and Table 5 of [OV] imply the following

Lemma. The 𝔤𝔩(m) ⊕ 𝔤𝔩(n)-module (U^{∗} ⊗ V) ⊗ (U^{∗} ⊗ V) is the direct sum of irreducible submodules whose highest weights and the corresponding highest vectors are listed in Table 9.

Let us show that v_{λ} ∉ 𝔤_{2}, where v_{λ} is any of the highest vectors listed in Table 9. Let us indicate g_{1}, g_{2} ∈ 𝔤_{−1} for which either (1.3.1) or (1.3.2) holds.

Let λ = 2ε_{1} − 2δ_{n}. Then

v_{λ}(f_{n} ⊗ ẽ_{1})(f_{n} ⊗ ẽ_{2}) = B_{1,n}(f_{n} ⊗ ẽ_{2}) = e_{1} ⊗ ẽ_{2}, but v_{λ}(f_{n} ⊗ ẽ_{2})(f_{n} ⊗ ẽ_{1}) = 0.

If λ = ε_{1} + ε_{2} − 2δ_{n}, then

v_{λ}(f_{n} ⊗ ẽ_{2})(f_{n} ⊗ ẽ_{2}) = B_{1,n}(f_{n} ⊗ ẽ_{2}) = e_{1} ⊗ ẽ_{2} ≠ 0.

If λ = 2ε_{1} − δ_{n−1} − δ_{n}, then

v_{λ}(f_{n−1} ⊗ ẽ_{1})(f_{n} ⊗ ẽ_{2}) = −B_{1,n}(f_{n} ⊗ ẽ_{2}) = −e_{1} ⊗ ẽ_{2}, but v_{λ}(f_{n} ⊗ ẽ_{2})(f_{n−1} ⊗ ẽ_{1}) = 0.

Let λ = ε_{1} + ε_{2} − δ_{n−1} − δ_{n}. Then if n > 2, we have

v_{λ}(f_{n−1} ⊗ ẽ_{2})(f_{1} ⊗ ẽ_{1}) = B_{1,n}(f_{1} ⊗ ẽ_{1}) = f_{1} ⊗ f̃_{n},

but v_{λ}(f_{1} ⊗ ẽ_{1})(f_{n−1} ⊗ ẽ_{2}) = 0. If m > 2, then

v_{λ}(f_{n−1} ⊗ ẽ_{2})(f_{n} ⊗ ẽ_{m}) = B_{1,n}(f_{n} ⊗ ẽ_{m}) = e_{1} ⊗ ẽ_{m},

but v_{λ}(f_{n} ⊗ ẽ_{m})(f_{n−1} ⊗ ẽ_{2}) = 0. Therefore, 𝔤_{2} = 0 and 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) = 𝔤. Note that by (1.3.4) we have 𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}) = 𝔤_{−1} ⊕ 𝔤̂_{0}. This proves part b) of Theorem 1.1.

1.5. m = n. Let m = n = 2 Since 𝔤̂_{0} = 𝔰𝔩(2) ⊕ 𝔰𝔩(2) = 𝔬(4) and 𝔤_{−1} is the standard 𝔬(4)-module (considered as purely odd superspace), then 𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}) = 𝔥(0|4).

Let m = n > 2 and 𝔤′_{1} be the first term of the Cartan prolongation of the pair (𝔤_{−1}, 𝔤̂_{0}). Let us show that 𝔤′_{1} = 𝔤_{1}. Indeed, by (1.2.2) 𝔤_{1} ⊂ 𝔤′_{1}. By sec.1.3 and Table 8 we see that the only highest weights of 𝔤′_{1} are all equal to ε_{1} − δ_{n}. Then formula (1.3.3) implies that the highest vector of such weight in 𝔤′_{1} is precisely one and therefore, 𝔤′_{1} = 𝔤_{1}. By sec.1.4 the second term of the Cartan prolongation of the pair (𝔤_{−1}, 𝔤̂_{0}) is zero. Hence, for m = n > 2 we have 𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}) = 𝔭𝔰𝔩(n|n).

Let m = n > 1 and let 𝔤_{k} be the k-th term of the Cartan prolongation of the pair (𝔤_{−1}, 𝔤_{0}). Recall that 𝔤_{k} = (𝔤_{0} ⊗ S^{k}𝔤_{−1}^{∗}) ∩ (𝔤_{−1} ⊗ S^{k+1}𝔤_{−1}^{∗}) (k ≥ 1). Observe that

𝔤_{0} ⊗ S^{k}𝔤_{−1}^{∗} = (𝔤̂_{0} ⊕ ⟨z⟩) ⊗ S^{k}𝔤_{−1}^{∗}, where z = 1_{2n} is the center of 𝔰𝔩(n|n).

Note that

⟨z⟩ ⊗ S^{k}𝔤_{−1}^{∗} ⊂ 𝔤_{−1} ⊗ S^{k+1}𝔤_{−1}^{∗}.

Therefore,

𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) = S^{∗}(𝔤_{−1}^{∗}) ⊂ + 𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}).
□

Structure functions of Lie superalgebras 𝔳𝔢𝔠𝔱(0|n) and 𝔰𝔳𝔢𝔠𝔱(0|n)

Recall that U is the purely odd standard 𝔤𝔩(n)-module.

2.1. Theorem. If m = 1, n > 1, then

a) H_{𝔤_{0}}^{k,2} = 0 for any k > 0;

b) Hk,2 ; 𝔤̂_{0} = Π^{n}(ℂ)δ_{kn}.

2.2. Proof of part a). Since 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) = 𝔳𝔢𝔠𝔱(0|n) = ∑_{i=−1}^{n−1} 𝔤_{i}, where 𝔤_{0} = 𝔤𝔩(n), and the 𝔤𝔩(n)-module 𝔤_{i} is isomorphic to U ⊗ S^{i+1}U^{∗}, then for k ≥ n + 2 we have H_{𝔤_{0}}^{k,2} = 0, and for k ≤ n + 1 there exist the following Spencer cochain sequences:

\begin{matrix}𝔤𝔩(n) ⊗ U^{∗} ⟶^{∂_{𝔤𝔩(n)}^{2,1}} U ⊗ Λ^{2}U^{∗} ⟶^{∂_{𝔤𝔩(n)}^{1,2}} 0 & (k = 1), \\ C_{𝔤𝔩(n)}^{k+1,1} ⟶^{∂_{𝔤𝔩(n)}^{k+1,1}} C_{𝔤𝔩(n)}^{k,2} ⟶^{∂_{𝔤𝔩(n)}^{k,2}} C_{𝔤𝔩(n)}^{k−1,3} & (2 ≤ k ≤ n), \\ 0 ⟶^{∂_{𝔤𝔩(n)}^{n+2,1}} C_{𝔤𝔩(n)}^{n+1,2} ⟶^{∂_{𝔤𝔩(n)}^{n+1,2}} C_{𝔤𝔩(n)}^{n,3} & (k = n + 1), \end{matrix}

where

C_{𝔤𝔩(n)}^{k+1,1} = 𝔤_{k−1} ⊗ 𝔤_{−1}^{∗} ≅ (U ⊗ S^{k}U^{∗}) ⊗ U^{∗},
C_{𝔤𝔩(n)}^{k,2} = 𝔤_{k−2} ⊗ Λ^{2}𝔤_{−1}^{∗} ≅ (U ⊗ S^{k−1}U^{∗}) ⊗ Λ^{2}U^{∗},
C_{𝔤𝔩(n)}^{k−1,3} = 𝔤_{k−3} ⊗ Λ^{3}𝔤_{−1}^{∗} ≅ (U ⊗ S^{k−2}U^{∗}) ⊗ Λ^{3}U^{∗}.

Recall (see sec.3.1 of Chapter 1) that if 𝔤_{−1} is a faithful module over a Lie superalgebra 𝔤_{0} and

𝔤_{k−1} ⊗ 𝔤_{−1}^{∗} ⟶^{∂_{𝔤_{0}}^{k+1,1}} 𝔤_{k−2} ⊗ Λ^{2}𝔤_{−1}^{∗} ⟶^{∂_{𝔤_{0}}^{k,2}} 𝔤_{k−3} ⊗ Λ^{3}𝔤_{−1}^{∗}  (k ≥ 1)

is the Spencer cochain sequence , which corresponds to the pair (𝔤_{−1}, 𝔤_{0}), then

Im∂_{𝔤_{0}}^{k+1,1} ≅ (𝔤_{k−1} ⊗ 𝔤_{−1}^{∗})/𝔤_{k}.  (2.2.1)

Let us show that H_{𝔤_{0}}^{1,2} = 0. Indeed, by (2.2.1)

Im∂_{𝔤_{0}}^{2,1} ≅ (𝔤𝔩(n) ⊗ U^{∗})/(U ⊗ S^{2}U^{∗}) ≅ U ⊗ Λ^{2}U^{∗} = Ker∂_{𝔤_{0}}^{1,2}.

STRUCTURE FUNCTIONS OF LIE SUPERALGEBRAS 𝔳𝔢𝔠𝔱(0|n) AND 𝔰𝔳𝔢𝔠𝔱(0|n) 39

We will prove that for 2 ≤ k ≤ n + 1 we have H_{𝔤𝔩(n)}^{k,2} = 0 , using the following

Lemma. As 𝔤𝔩(n)-modules C_{𝔤𝔩(n)}^{k,2}, C_{𝔤𝔩(n)}^{k+1,1}, and 𝔤_{k} , where 2 ≤ k ≤ n + 1, are the direct sums of the irreducible submodules whose highest weights and highest vectors are listed in Tables 10, 11, and 12, respectively. [r, s, and t denote the cyclic permutations of (n − k, …, n), (n − k + 1, …, n), and (n − k + 2, …, n), respectively.]

Proof. The proof of the Lemma consists of

a) a verification of the fact that vectors v_{λ_{l}}, v_{β_{l}}, v_{γ_{l}} from the Tables 10, 11, and 12 are indeed highest with respect to 𝔤𝔩(n), i.e., D_{i,j}v_{λ_{l}} = D_{i,j}v_{β_{l}} = D_{i,j}v_{γ_{l}} = 0 for i < j;

b) a calculation of dimensions of given modules and dimensions of their irreducible submodules by the formula from the Appendix.

Notice that D_{i,j}f_{p} = δ_{jp}f_{i}, D_{i,j}f̃_{p} = −δ_{ip}f̃_{j}. According to Table 10,

λ_{1} = δ_{1} − δ_{n−k+2} − … − δ_{n−1} − 3δ_{n}, v_{λ_{1}} = (f_{1} ⊗ f̃_{n−k+2} ∧ f̃_{n−k+3} ∧ … ∧ f̃_{n}) ⊗ f̃_{n}^{2}.

Then for i < n − k + 2 D_{i,j}v_{λ_{1}} = 0. For j > i ≥ n − k + 2

\begin{matrix}D_{i,j}(v_{λ_{1}}) = D_{i,j}(f_{1} ⊗ f̃_{n−k+2} ∧ … ∧ f̃_{i} ∧ … ∧ f̃_{j} ∧ … ∧ f̃_{n}) ⊗ f̃_{n}^{2} = \\ (f_{1} ⊗ f̃_{n−k+2} ∧ … ∧ (−f̃_{j}) ∧ … ∧ f̃_{j} ∧ … ∧ f̃_{n}) ⊗ f̃_{n}^{2} = 0. \end{matrix}

Thus, v_{λ_{1}} is highest vector. The proof of the fact that the other vectors from Tables 10, 11, and 12 are highest with respect to 𝔤𝔩(n) is similar.

Using the formula from the Appendix we find the dimensions of the 𝔤𝔩(n)-modules given in Table 10:

if 2 ≤ k ≤ n, then

dim V_{λ_{1}} = \frac{n(n + 3)(n + 1)!}{2(n − k + 2)(k + 1)(k − 2)!(n − k)!},

if 3 ≤ k = n + 1, then

dim V_{λ_{1}} = \frac{(n − 1)n(n + 2)}{2},

if 2 ≤ k ≤ n + 1, then

dim V_{λ_{2}} = \frac{(n + 1)!}{(k − 2)!((n − k + 1)!k} ,

if k = 2 ≤ n, then

dim V_{λ_{3}} = dim V_{λ_{2}},

if 3 ≤ k ≤ n, then

dim V_{λ_{3}} = \frac{(n + 2)!}{2(k − 3)!(n − k + 2)!k} ,

if 3 ≤ k = n, then

dim V_{λ_{4}} = n^{2} − 1,

if 2 ≤ k ≤ n − 1, then

dim V_{λ_{4}} = \frac{n(n + 2)n!}{(n − k + 1)(k − 1)!(n − k − 1)!(k + 1)},  (2.2.2)

if 2 ≤ k ≤ n, then

dim V_{λ_{5}} = \frac{n!}{k!(n − k)!},  (2.2.3)

if 3 ≤ k ≤ n − 1, then

dim V_{λ_{6}} = dim V_{λ_{2}}.

Therefore, if 2 = k = n, then

dim V_{λ_{1}} + 2dimV_{λ_{2}} + dimV_{λ_{5}} = \frac{n^{3}(n + 1)}{2} = dim(U ⊗ U^{∗}) ⊗ Λ^{2}U^{∗},

if 2 = k ≤ n − 1, then

∑_{l=1}^{5} dim V_{λ_{l}} = \frac{n^{3}(n + 1)}{2} = dim(U ⊗ U^{∗}) ⊗ Λ^{2}U^{∗},

if 3 ≤ k = n + 1, then

∑_{l=1}^{2} dim V_{λ_{l}} = \frac{n^{2}(n + 1)}{2} = dim U ⊗ Λ^{2}U^{∗},

if 3 ≤ k = n, then

∑_{l=1}^{5} dim V_{λ_{l}} = \frac{n^{3}(n + 1)}{2} = dim(U ⊗ S^{n−1}U^{∗}) ⊗ Λ^{2}U^{∗},

if 3 ≤ k ≤ n − 1, then

∑_{l=1}^{6} dim V_{λ_{l}} = \frac{n^{2}(n + 1)!}{2(n − k + 1)!(k − 1)!} = dim(U ⊗ S^{k−1}U^{∗}) ⊗ Λ^{2}U^{∗}.

In order to find the dimensions of the 𝔤𝔩(n)-modules given in Table 11, note that if k ≥ 2, then β_{1} = λ_{4}, β_{2} = β_{4} = λ_{5}, β_{3} = λ_{2}. Using the formula from the Appendix we get

dim V_{β_{5}} = \frac{(n + 1)!}{(n − k)(k + 1)!(n − k − 2)!} for 2 ≤ k ≤ n − 2.  (2.2.4)

Therefore, if 2 ≤ k = n, then

∑_{l=1}^{2} dim V_{β_{l}} = n^{2} = dim U ⊗ U^{∗},

if 2 ≤ k = n − 1, then

∑_{l=1}^{4} dim V_{β_{l}} = n^{3} = dim(U ⊗ S^{n−1}U^{∗}) ⊗ U^{∗},

if 2 ≤ k ≤ n − 2, then

∑_{l=1}^{5} dim V_{β_{l}} = \frac{n^{2}n!}{(n − k)!k!} = dim(U ⊗ S^{k}U^{∗}) ⊗ U^{∗}.

Finally, in order to find the dimensions of the 𝔤𝔩(n)-modules given in Table 12, note that if 2 ≤ k ≤ n − 2, then γ_{1} = β_{5}, and γ_{2} = λ_{5}. Therefore, if 2 ≤ k ≤ n − 2, then

∑_{l=1}^{2} dim V_{γ_{l}} = \frac{nn!}{(n − k − 1)!(k + 1)!} = dim(U ⊗ S^{k+1}U^{∗}) = dim 𝔤_{k},

and if k = n − 1, then dim V_{γ_{1}} = n = dim U = dim 𝔤_{k}. This proves the Lemma. □

Let k ≥ 2 and λ = λ_{1}, λ_{2}, λ_{3}. Then

v_{λ} ∉ Ker∂_{𝔤𝔩(n)}^{k,2}.  (2.2.5)

Indeed,

∂_{𝔤𝔩(n)}^{k,2}v_{λ}(f_{n}, f_{n}, f_{n}) = −3v_{λ}(f_{n}, f_{n})(f_{n}),

and according to Table 10, v_{λ}(f_{n}, f_{n})(f_{n}) ≠ 0. Note that if 2 ≤ k = n, then

Im∂_{𝔤𝔩(n)}^{k+1,1} = V_{λ_{2}} ⊕ V_{λ_{5}}.  (2.2.6)

In fact, according to Table 11,

C_{𝔤𝔩(n)}^{k+1,1} = V_{λ_{2}} ⊕ V_{λ_{5}},

and we get (2.2.6) by (2.2.1), since 𝔤_{k} = 0. Note that if 2 ≤ k ≤ n − 1, then

Im∂_{𝔤𝔩(n)}^{k+1,1} = V_{λ_{2}} ⊕ V_{λ_{4}} ⊕ V_{λ_{5}}.  (2.2.7)

Indeed, according to Table 11, if 2 ≤ k = n − 1, then

C_{𝔤𝔩(n)}^{k+1,1} = V_{λ_{2}} ⊕ V_{λ_{4}} ⊕ 2V_{λ_{5}},  (2.2.8)

and if 2 ≤ k ≤ n − 2, then

C_{𝔤𝔩(n)}^{k+1,1} = V_{λ_{2}} ⊕ V_{λ_{4}} ⊕ 2V_{λ_{5}} ⊕ V_{β_{5}}.  (2.2.9)

Since by Table 12

\begin{matrix}𝔤_{k} = V_{λ_{5}} for 2 ≤ k = n − 1, \\ 𝔤_{k} = V_{λ_{5}} ⊕ V_{β_{5}} for 2 ≤ k ≤ n − 2, \end{matrix}

we get (2.2.7) by (2.2.1).

We will show now that for 2 ≤ k ≤ n + 1

Ker∂_{𝔤𝔩(n)}^{k,2} = Im∂_{𝔤𝔩(n)}^{k+1,1}.  (2.2.10)

Let

Ker∂_{𝔤𝔩(n)}^{k,2} = ⊕_{λ}k_{λ}V_{λ}.  (2.2.11)

Let k = 2. According to Table 10, if n = 2, then

C_{𝔤𝔩(n)}^{k,2} = V_{λ_{1}} ⊕ 2V_{λ_{2}} ⊕ V_{λ_{5}},  (2.2.12)

and if n ≥ 3, then

C_{𝔤𝔩(n)}^{k,2} = V_{λ_{1}} ⊕ 2V_{λ_{2}} ⊕ V_{λ_{4}} ⊕ V_{λ_{5}}.

Therefore, in (2.2.11) k_{λ_{5}} ≤ 1, and by (2.2.5) k_{λ_{1}} = 0, k_{λ_{2}} ≤ 1. Note that if n = 2, then k_{λ_{4}} = 0, and if n ≥ 3, then k_{λ_{4}} ≤ 1. Thus, by (2.2.6) and (2.2.7) we get (2.2.10).

Let k ≥ 3. Then according to Table 10, if k = n + 1, then

C_{𝔤𝔩(n)}^{k,2} = V_{λ_{1}} ⊕ V_{λ_{2}}.

Hence, by (2.2.5) Ker∂_{𝔤𝔩(n)}^{k,2} = 0. If k = n, then

C_{𝔤𝔩(n)}^{k,2} = V_{λ_{1}} ⊕ 2V_{λ_{2}} ⊕ V_{λ_{3}} ⊕ V_{λ_{5}}.  (2.2.13)

Therefore, in (2.2.11) k_{λ_{5}} ≤ 1 and by (2.2.5) k_{λ_{1}} = k_{λ_{3}} = 0, k_{λ_{2}} ≤ 1. So from (2.2.6) we get (2.2.10).

Finally, if k ≤ n − 1, then

C_{𝔤𝔩(n)}^{k,2} = V_{λ_{1}} ⊕ 2V_{λ_{2}} ⊕ V_{λ_{3}} ⊕ V_{λ_{4}} ⊕ V_{λ_{5}}.

Therefore, in (2.2.11) k_{λ_{4}} ≤ 1, k_{λ_{5}} ≤ 1 and by (2.2.5) k_{λ_{1}} = k_{λ_{3}} = 0, k_{λ_{2}} ≤ 1. Thus, by (2.2.7) we get (2.2.10). This proves part a) of Theorem 2.1.

2.3. Proof of part b) of Theorem 2.1. Note that 𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}) = 𝔰𝔳𝔢𝔠𝔱(0|n) = ⊕_{i=−1}^{n−2}𝔤_{i}, where 𝔤̂_{0} = 𝔰𝔩(n), and the 𝔰𝔩(n)-module 𝔤_{i} is isomorphic to U if i = −1 and to V_{δ_{1}−δ_{n−k}−δ_{n−k+1}−...−δ_{n}} if 0 ≤ k ≤ n − 2.

Hence, for k ≥ n + 1 we have H_{𝔰𝔩(n)}^{k,2} = 0 and for 1 ≤ k ≤ n there exist the following Spencer cochain sequences:

\begin{matrix}𝔰𝔩(n) ⊗ U^{∗} ⟶^{∂_{𝔰𝔩(n)}^{2,1}} U ⊗ Λ^{2}U^{∗} ⟶^{∂_{𝔰𝔩(n)}^{1,2}} 0 & (k = 1), \\ C_{𝔰𝔩(n)}^{k+1,1} ⟶^{∂_{𝔰𝔩(n)}^{k+1,1}} C_{𝔰𝔩(n)}^{k,2} ⟶^{∂_{𝔰𝔩(n)}^{k,2}} C_{𝔰𝔩(n)}^{k−1,3} & (2 ≤ k ≤ n − 1), \\ 0 ⟶^{∂_{𝔰𝔩(n)}^{n+1,1}} C_{𝔰𝔩(n)}^{n,2} ⟶^{∂_{𝔰𝔩(n)}^{n,2}} C_{𝔰𝔩(n)}^{n−1,3} & (k = n). \end{matrix}

First, we will show that H_{𝔰𝔩(n)}^{1,2} = 0. In fact, since 𝔤_{1} ≅ V_{δ_{1}−δ_{n−1}−δ_{n}}, then by (2.2.1)

Im∂_{𝔰𝔩(n)}^{2,1} ≅ (𝔰𝔩(n) ⊗ U^{∗})/V_{δ_{1}−δ_{n−1}−δ_{n}}.

Since dim 𝔰𝔩(n) ⊗ U^{∗} = (n^{2} − 1)n and by the formula from the Appendix

dim V_{δ_{1}−δ_{n−1}−δ_{n}} = \frac{n(n + 1)(n − 2)}{2},

then

dim Im∂_{𝔰𝔩(n)}^{2,1} = (n^{2} − 1)n − \frac{n(n + 1)(n − 2)}{2} = \frac{n^{3} + n^{2}}{2} =

dim U ⊗ Λ^{2}U^{∗} = dim Ker∂_{𝔰𝔩(n)}^{1,2}.

Next, we will prove that for 2 ≤ k ≤ n − 1 we have H_{𝔰𝔩(n)}^{k,2} = 0.

Lemma. If 2 ≤ k = n − 1, then

C_{𝔰𝔩(n)}^{k+1,1} = V_{λ_{4}} ⊕ V_{λ_{5}}.  (2.3.1)

If 2 ≤ k ≤ n − 2, then

C_{𝔰𝔩(n)}^{k+1,1} = V_{λ_{4}} ⊕ V_{λ_{5}} ⊕ V_{β_{5}}.  (2.3.2)

Proof. Let c ∈ C_{𝔤𝔩(n)}^{k+1,1}. Then c ∈ C_{𝔰𝔩(n)}^{k+1,1} if and only if

c ∈ (𝔰𝔩(n) ⊗ S^{k−1}U^{∗}) ⊗ U^{∗}.  (2.3.3)

Let

C_{𝔰𝔩(n)}^{k+1,1} = ⊕_{β}k_{β}V_{β}.  (2.3.4)

Using the decomposition of C_{𝔤𝔩(n)}^{k+1,1} into direct sum of irreducible 𝔤𝔩(n)-modules given in (2.2.8) and (2.2.9), we check condition (2.3.3) for the corresponding highest vectors: let β = λ_{4} = β_{1}, then up to a complex constant,

v_{β_{1}}(f_{n})(f_{j_{1}} ∧ f_{j_{2}} ∧ … ∧ f_{j_{k−1}}) = f_{1} ⊗ f̃_{j_{k}}, where 2 ≤ j_{k} ≤ n.

Since f_{1} ⊗ f̃_{j_{k}} ∈ 𝔰𝔩(n), then in (2.3.4) k_{λ_{4}} = 1. Let β = λ_{2} = β_{3}, then up to a nonzero constant,

v_{β_{3}}(f_{n})(f_{n−k+2} ∧ f_{n−k+3} ∧ … ∧ f_{n}) = ∑_{j=1}^{n} f_{j} ⊗ f̃_{j}.

Since ∑_{j=1}^{n} f_{j} ⊗ f̃_{j} ∉ 𝔰𝔩(n), then k_{λ_{2}} = 0. Let β = β_{5}. Then up to a constant,

v_{β_{5}}(f_{j_{1}})(f_{j_{2}} ∧ … ∧ f_{j_{k}}) = f_{1} ⊗ f̃_{j_{k+1}}, where 2 ≤ j_{k+1} ≤ n.

Since f_{1} ⊗ f̃_{j_{k+1}} ∈ 𝔰𝔩(n), then k_{β_{5}} = 1. Let us show that k_{λ_{5}} = 1. Indeed, by the formula from the Appendix

dim V_{δ_{1}−δn−k+1−...−δ_{n}} = \frac{(n + 1)!}{(n − k + 1)k!(n − k − 1)!}.

Therefore,

dim C_{𝔰𝔩(n)}^{k+1,1} = dim(V_{δ_{1}−δn−k+1−...−δ_{n}} ⊗ U^{∗}) = \frac{n(n + 1)!}{(n − k + 1)k!(n − k − 1)!}.

Then by (2.2.2) and (2.2.3) if 2 ≤ k = n − 1, then

dim C_{𝔰𝔩(n)}^{k+1,1} = dim V_{λ_{4}} + dimV_{λ_{5}},

if 2 ≤ k ≤ n − 2, then by (2.2.2), (2.2.3), and (2.2.4)

dim C_{𝔰𝔩(n)}^{k+1,1} = dim V_{λ_{4}} + dimV_{λ_{5}} + dim V_{β_{5}},

Thus, k_{λ_{5}} = 1. This proves the Lemma. □

2.4. Lemma. Ker∂_{𝔰𝔩(n)}^{k,2} doesn’t contain irreducible 𝔰𝔩(n)-submodules with highest weight λ_{2}.

Proof. Let c ∈ C_{𝔤𝔩(n)}^{k,2}. Then c ∈ C_{𝔰𝔩(n)}^{k,2} if and only if

c ∈ (𝔰𝔩(n) ⊗ S^{k−2}U^{∗}) ⊗ λ^{2}U^{∗}.

According to Table 10, each highest vector with weight λ_{2} in C_{𝔤𝔩(n)}^{k,2} is

v = av_{λ_{2}} + bv_{λ_{3}}, if k = 2 ≤ n,
v = av_{λ_{2}} + bv_{λ_{6}}, if 3 ≤ k ≤ n − 1, where a, b ∈ ℂ.

Therefore, up to a nonzero constant,

v(f_{n}, f_{n})(f_{n−k+2} ∧ … ∧ f_{n−1}) = (−1)^{k}af_{n} ⊗ f̃_{n} + b ∑_{j=1}^{n} f_{j} ⊗ f̃_{j}.

Note that

(−1)^{k}af_{n} ⊗ f̃_{n} + b ∑_{j=1}^{n} f_{j} ⊗ f̃_{j} ∈ 𝔰𝔩(n)

if and only if (−1)^{k}a + nb = 0. But in this case v ∉ Ker_{𝔰𝔩(n)}^{k,2}. In fact,

∂_{𝔰𝔩(n)}^{k,2}(f_{n}, f_{n}, f_{n}) = −3v(f_{n}, f_{n})(f_{n}) ≠ 0.

This proves Lemma 2.4. □

In order to prove that for 2 ≤ k ≤ n − 1

Im∂_{𝔰𝔩(n)}^{k+1,1} = Ker∂_{𝔰𝔩(n)}^{k,2},  (2.4.1)

observe that if 2 ≤ k = n − 1, then 𝔤_{k} = 0 and if 2 ≤ k ≤ n − 2, then 𝔤_{k} = V_{β_{5}}. Thus, by (2.2.1), (2.3.1), and (2.3.2)

Im∂_{𝔰𝔩(n)}^{k+1,1} = V_{λ_{4}} ⊕ V_{λ_{5}}.

Therefore, by (2.2.7), (2.2.10), and Lemma 2.4, we get (2.4.1).

Finally, let 2 ≤ k = n. Notice that by (2.2.6) and (2.2.10)

Ker∂_{𝔤𝔩(n)}^{n,2} = V_{λ_{2}} ⊕ V_{λ_{5}},  (2.4.2)

44 2. THE ANALOGUES OF PENROSE’S TENSORS

where V_{λ_{5}} is a trivial 𝔰𝔩(n)-module. By (2.2.12) and (2.2.13) the multiplicity of λ_{5} in C_{𝔤𝔩(n)}^{n,2} is 1. Moreover, this trivial submodule is contained in C_{𝔰𝔩(n)}^{n,2}, because the 𝔰𝔩(n)-module C_{𝔰𝔩(n)}^{n,2} is isomorphic to Λ^{2}U ⊗Λ^{2}U^{∗}, which contains a trivial 𝔰𝔩(n)-submodule (generated by ∑_{i,j} f_{i}f_{j}⊗ f̃_{i}f̃_{j}). Thus, Ker∂_{𝔰𝔩(n)}^{n,2} must contain a trivial submodule. According to (2.4.2) and Lemma 2.4, Ker∂_{𝔰𝔩(n)}^{n,2} is a trivial 𝔰𝔩(n)-submodule, generated by

v_{λ_{5}} = ∑_{j=0}^{n−1}(−1)^{(n−1)j} ∑_{i=1}^{n} f_{i} ⊗ f̃_{s^{j}(1)} ∧ f̃_{s^{j}(2)} ∧ … ∧ f̃_{s^{j}(n−1)} ⊗ f̃_{s^{j}(n)}f̃_{i},

where s is a cyclic permutation of (1, 2, …, n). This proves Theorem 2.1.

Penrose’s tensors

3.1. Theorem. If m, n > 1, then H_{𝔤_{0}}^{k,2} = 0 for k > 2 and the 𝔤_{0}-modules H_{𝔤_{0}}^{1,2} and H_{𝔤_{0}}^{2,2} are the direct sums of irreducible submodules whose highest weights are given in Table 13.

If m = n, then Hk,2 ; 𝔤̂_{0} = H_{𝔤_{0}}^{k,2} for any k and if m ≠ n, then H1,2 ; 𝔤̂_{0} = H_{𝔤_{0}}^{1,2} whereas H2,2 ; 𝔤̂_{0} =

H_{𝔤_{0}}^{2,2} ⊕ V_{ε_{1}+ε_{2}−2δ_{n}} ⊕ V_{2ε_{1}−δ_{n−1}−δ_{n}} if either m = 2 or n = 2;

H2,2 ; 𝔤̂_{0} = H_{𝔤_{0}}^{2,2} ⊕ V_{ε_{1}+ε_{2}−2δ_{n}} ⊕ V_{2ε_{1}−δ_{n−1}−δ_{n}} ⊕ V_{ε_{1}+ε_{2}−δ_{n−1}−δ_{n}} if m, n > 2.

3.2. Calculation of H_{𝔤_{0}}^{1,2} and H1,2 ; 𝔤̂_{0} for m, n ≥ 2, m ≠ n. For k = 1 the Spencer cochain sequence is of the form

𝔤_{0} ⊗ 𝔤_{−1}^{∗} ⟶^{∂_{𝔤_{0}}^{2,1}} 𝔤_{−1} ⊗ Λ^{2}𝔤_{−1}^{∗} ⟶^{∂_{𝔤_{0}}^{1,2}} 0.

Observe that

𝔤_{−1} ⊗ Λ^{2}𝔤_{−1}^{∗} = (U ⊗ V^{∗}) ⊗ Λ^{2}(U^{∗} ⊗ V) ≅
(U ⊗ V^{∗}) ⊗ (Λ^{2}U^{∗} ⊗ S^{2}V ⊕ S^{2}U^{∗} ⊗ Λ^{2}V) ≅
(Λ^{2}U^{∗} ⊗ U) ⊗ (S^{2}V ⊗ V^{∗}) ⊕ (S^{2}U^{∗} ⊗ U) ⊗ (Λ^{2}V ⊗ V^{∗}),
𝔤_{0} ⊗ 𝔤_{−1}^{∗} = (V ⊗ V^{∗}/ℂ ⊕ U ⊗ U^{∗}/ℂ ⊕ ℂ) ⊗ (U^{∗} ⊗ V),
𝔤_{1} = U^{∗} ⊗ V.

Therefore, as 𝔤𝔩(m) ⊕ 𝔤𝔩(n)-modules,

Im∂_{𝔤_{0}}^{2,1} ≅ (V ⊗ V^{∗}/ℂ ⊕ U ⊗ U^{∗}/ℂ) ⊗ (U^{∗} ⊗ V)  (3.2.1)

and

H_{𝔤_{0}}^{1,2} ≅ (Λ^{2}U^{∗} ⊗ U/U^{∗}) ⊗ (S^{2}V ⊗ V^{∗}/V) ⊕ (S^{2}U^{∗} ⊗ U/U^{∗}) ⊗ (Λ^{2}V ⊗ V^{∗}/V). (3.2.2)

Note that

S^{2}V ⊗ V^{∗}/V = V_{2ε_{1}−ε_{m}},
Λ^{2}V ⊗ V^{∗}/V = V_{ε_{1}+ε_{2}−ε_{m}} for m > 2,
Λ^{2}V ⊗ V^{∗}/V = 0 for m = 2.

Since U is purely odd, we deduce with the help of Table 5 of [OV] that Λ^{2}U^{∗} ⊗ U/U^{∗} = V_{δ_{1}−2δ_{n}},

S^{2}U^{∗} ⊗ U/U^{∗} = V_{δ_{1}−δ_{n−1}−δ_{n}} for n > 2,
S^{2}U^{∗} ⊗ U/U^{∗} = 0 for n = 2.

Therefore, we have

H_{𝔤_{0}}^{1,2} = V_{2ε_{1}−ε_{m}+δ_{1}−2δ_{n}} if m = 2, n > 2

and

H_{𝔤_{0}}^{1,2} = V_{2ε_{1}−ε_{m}+δ_{1}−2δ_{n}} ⊕ V_{ε_{1}+ε_{2}−ε_{m}+δ_{1}−δ_{n−1}−δ_{n}} if m, n > 2.

By part b) of Theorem 1.1 𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}) = 𝔤_{−1} ⊕ 𝔤̂_{0}. Therefore, by (2.2.1), we have

Im∂2,1 ; 𝔤̂_{0} = 𝔤̂_{0} ⊗ 𝔤_{−1}^{∗} = Im∂_{𝔤_{0}}^{2,1}.

Hence, H1,2 ; 𝔤̂_{0} = H_{𝔤_{0}}^{1,2}.

3.3. Calculation of H1,2 ; 𝔤̂_{0} for m = n > 1. Since by parts c) and d) of Theorem 1.1 the first term of the Cartan prolongation 𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}) is U^{∗} ⊗ V, then by (2.2.1)

Im∂2,1 ; 𝔤̂_{0} = [(V ⊗ V^{∗}/ℂ ⊕ U ⊗ U^{∗}/ℂ) ⊗ (U^{∗} ⊗ V)]/(U^{∗} ⊗ V).

Therefore, by (3.2.1) and (3.2.2),

H1,2 ; 𝔤̂_{0} ≅ (Λ^{2}U^{∗} ⊗ U/U^{∗}) ⊗ (S^{2}V ⊗ V^{∗}/V) ⊕ (S^{2}U^{∗} ⊗ U/U^{∗}) ⊗ (Λ^{2}V ⊗ V^{∗}/V) ⊕ (U^{∗} ⊗ V).

Hence,

H1,2 ; 𝔤̂_{0} = V_{2ε_{1}−ε_{2}+δ_{1}−2δ_{2}} ⊕ V_{ε_{1}−δ_{2}} for n = 2

and

H1,2 ; 𝔤̂_{0} = V_{2ε_{1}−ε_{n}+δ_{1}−2δ_{n}} ⊕ V_{ε_{1}+ε_{2}−ε_{n}+δ_{1}−δ_{n−1}−δ_{n}} ⊕ V_{ε_{1}−δ_{n}} for n > 2.

3.4. Calculation of H_{𝔤_{0}}^{2,2} for m, n > 1, m ≠ n. For k = 2 the Spencer cochain sequence is of the form

𝔤_{1} ⊗ 𝔤_{−1}^{∗} ⟶^{∂_{𝔤_{0}}^{3,1}} 𝔤_{0} ⊗ Λ^{2}𝔤_{−1}^{∗} ⟶^{∂_{𝔤_{0}}^{2,2}} 𝔤_{−1} ⊗ Λ^{3}𝔤_{−1}^{∗}.

Observe that 𝔤_{0} ⊗ Λ^{2}𝔤_{−1}^{∗} = (V ⊗ V^{∗}/ℂ ⊕ U ⊗ U^{∗}/ℂ ⊕ ℂ) ⊗ (Λ^{2}U^{∗} ⊗ S^{2}V ⊕ S^{2}U^{∗} ⊗ Λ^{2}V),

𝔤_{1} ⊗ 𝔤_{−1}^{∗} = (U^{∗} ⊗ V) ⊗ (U^{∗} ⊗ V),
𝔤_{2} = 0.

Lemma. As 𝔤𝔩(m) ⊕ 𝔤𝔩(n)-module, 𝔤_{0} ⊗ Λ^{2}𝔤_{−1}^{∗} is the direct sum of the irreducible submodules whose highest weights and highest vectors are listed in Table 14. [s and t denote the cyclic permutations of (1, 2, 3) and (n − 2, n − 1, n), respectively.]

The proof follows from the formula given in the Appendix.

Let us show that if

\begin{matrix}λ = 3ε_{1} − ε_{m} − 2δ_{n}, 2ε_{1} + ε_{2} − ε_{m} − 2δ_{n} (m > 2), 2ε_{1} + ε_{2} − ε_{m} − δ_{n−1} − δ_{n}, \\ 2ε_{1} + δ_{1} − 3δ_{n}, 2ε_{1} + δ_{1} − δ_{n−1} − 2δ_{n} (n > 2), or ε_{1} + ε_{2} + δ_{1} − δ_{n−1} − 2δ_{n}, \end{matrix}

then v_{λ} ∉ Ker∂_{𝔤_{0}}^{2,2}. Recall that if v ∈ 𝔤_{0} ⊗ Λ^{2}𝔤_{−1}^{∗}, then

∂_{𝔤_{0}}^{2,2}v(g_{1}, g_{2}, g_{3}) = −v(g_{1}, g_{2})g_{3} − v(g_{1}, g_{3})g_{2} − v(g_{2}, g_{3})g_{1}  (3.4.1)

for any g_{1}, g_{2}, g_{3} ∈ 𝔤_{−1}.

Let λ = 3ε_{1} − ε_{m} − 2δ_{n}. Then

\begin{matrix}∂_{𝔤_{0}}^{2,2}v_{λ}(f_{n} ⊗ ẽ_{1}, f_{n} ⊗ ẽ_{1}, f_{n} ⊗ ẽ_{1}) = −3v_{λ}(f_{n} ⊗ ẽ_{1}, f_{n} ⊗ ẽ_{1})(f_{n} ⊗ ẽ_{1}) = \\ = 3A_{1,m}(f_{n} ⊗ ẽ_{1}) = −3f_{n} ⊗ ẽ_{m} ≠ 0. \end{matrix}

Let λ = 2ε_{1} + ε_{2} − ε_{m} − 2δ_{n} (m > 2). Then

\begin{matrix}∂_{𝔤_{0}}^{2,2}v_{λ}(f_{n} ⊗ ẽ_{1}, f_{n} ⊗ ẽ_{1}, f_{n−1} ⊗ ẽ_{2}) = −v_{λ}(f_{n} ⊗ ẽ_{1}, f_{n} ⊗ ẽ_{1})(f_{n−1} ⊗ ẽ_{2}) = \\ = −A_{2,m}(f_{n−1} ⊗ ẽ_{2}) = f_{n−1} ⊗ ẽ_{m} ≠ 0. \end{matrix}

Let λ = 2ε_{1} + ε_{2} − ε_{m} − δ_{n−1} − δ_{n}. Then

\begin{matrix}∂_{𝔤_{0}}^{2,2}v_{λ}(f_{n} ⊗ ẽ_{1}, f_{n} ⊗ ẽ_{1}, f_{n−1} ⊗ ẽ_{2}) = −2v_{λ}(f_{n} ⊗ ẽ_{1}, f_{n−1} ⊗ ẽ_{2})(f_{n} ⊗ ẽ_{1}) = \\ = −A_{1,m}(f_{n} ⊗ ẽ_{1}) = f_{n} ⊗ ẽ_{m} ≠ 0. \end{matrix}

The proof of the fact that v_{λ} ∉ Ker∂_{𝔤_{0}}^{2,2} for λ = 2ε_{1} + δ_{1} − 3δ_{n}, 2ε_{1} + δ_{1} − δ_{n−1} − 2δ_{n} (n > 2), and ε_{1} + ε_{2} + δ_{1} − δ_{n−1} − 2δ_{n} is similar.

Let λ = ε_{1} +ε_{2} +ε_{3} −ε_{m} −δ_{n−1} −δ_{n} (m > 3). Let us show that if n = 2, then v_{λ} ∈ Ker∂_{𝔤_{0}}^{2,2} and if n > 2, then v_{λ} ∉ Ker∂_{𝔤_{0}}^{2,2}. Indeed, if n = 2, then for j = 0, 1, 2 we have

∂_{𝔤_{0}}^{2,2}v_{λ}(f_{1} ⊗ ẽ_{s^{j}(2)}, f_{2} ⊗ ẽ_{s^{j}(3)}, f_{1} ⊗ ẽ_{s^{j}(1)}) =
− v_{λ}(f_{1} ⊗ ẽ_{s^{j}(2)}, f_{2} ⊗ ẽ_{s^{j}(3)})(f_{1} ⊗ ẽ_{s^{j}(1)})
− v_{λ}(f_{2} ⊗ ẽ_{s^{j}(3)}, f_{1} ⊗ ẽ_{s^{j}(1)})(f_{1} ⊗ ẽ_{s^{j}(2)}) =
A_{s^{j}(1),m}(f_{1} ⊗ ẽ_{s^{j}(1)})/2 − A_{s^{j}(2),m}(f_{1} ⊗ ẽ_{s^{j}(2)})/2 =
− f_{1} ⊗ ẽ_{m}/2 + f_{1} ⊗ ẽ_{m}/2 = 0,
∂_{𝔤_{0}}^{2,2}v_{λ}(f_{1} ⊗ ẽ_{s^{j}(2)}, f_{2} ⊗ ẽ_{s^{j}(3)}, f_{2} ⊗ ẽ_{s^{j}(1)}) =
− v_{λ}(f_{1} ⊗ ẽ_{s^{j}(2)}, f_{2} ⊗ ẽ_{s^{j}(3)})(f_{2} ⊗ ẽ_{s^{j}(1)})−
− v_{λ}(f_{2} ⊗ ẽ_{s^{j}(1)}, f_{1} ⊗ ẽ_{s^{j}(2)})(f_{2} ⊗ ẽ_{s^{j}(3)}) =
A_{s^{j}(1),m}(f_{2} ⊗ ẽ_{s^{j}(1)})/2 − A_{s^{j}(3),m}(f_{2} ⊗ ẽ_{s^{j}(3)})/2 =
− f_{2} ⊗ ẽ_{m}/2 + f_{2} ⊗ ẽ_{m}/2 = 0.

Therefore, v_{λ} ∈ Ker∂_{𝔤_{0}}^{2,2}. If n > 2, then

\begin{matrix}∂_{𝔤_{0}}^{2,2}v_{λ}(f_{n−1} ⊗ ẽ_{2}, f_{n} ⊗ ẽ_{3}, f_{1} ⊗ ẽ_{1}) = −v_{λ}(f_{n−1} ⊗ ẽ_{2}, f_{n} ⊗ ẽ_{3})(f_{1} ⊗ ẽ_{1}) = \\ = A_{1,m}(f_{1} ⊗ ẽ_{1})/2 = −f_{1} ⊗ ẽ_{m}/2 ≠ 0. \end{matrix}

The proof of the fact that if λ = ε_{1} + ε_{2} + δ_{1} − δ_{n−2} − δ_{n−1} − δ_{n} (n ≥ 4), then v_{λ} ∈ Ker∂_{𝔤_{0}}^{2,2} for m = 2 and v_{λ} ∉ Ker∂_{𝔤_{0}}^{2,2} for m > 2 is similar.

Finally, let us show that if

λ = 2ε_{1} − 2δ_{n}, ε_{1} + ε_{2} − 2δ_{n}, 2ε_{1} − δ_{n−1} − δ_{n}, or ε_{1} + ε_{2} − δ_{n−1} − δ_{n}

and v_{λ} ∈ Ker∂_{𝔤_{0}}^{2,2}, then v_{λ} ∈ Im∂_{𝔤_{0}}^{3,1}. Note that since 𝔤_{2} = 0, then, as 𝔤𝔩(m) ⊕ 𝔤𝔩(n)-modules,

Im∂_{𝔤_{0}}^{3,1} ≅ 𝔤_{1} ⊗ 𝔤_{−1}^{∗} = (U^{∗} ⊗ V) ⊗ (U^{∗} ⊗ V).

Therefore, by Table 9,

Im∂_{𝔤_{0}}^{3,1} = V_{2ε_{1}−2δ_{n}} ⊕ V_{ε_{1}+ε_{2}−δ_{n−1}−δ_{n}} ⊕ V_{ε_{1}+ε_{2}−2δ_{n}} ⊕ V_{2ε_{1}−δ_{n−1}−δ_{n}}.

Let λ = ε_{1} + ε_{2} − 2δ_{n}. By Table 14 𝔤_{0} ⊗ Λ^{2}𝔤_{−1}^{∗} contains two irreducible components with the indicated highest weight, and one of the corresponding highest vectors is v_{λ}^{1}. Observe that

\begin{matrix}∂_{𝔤_{0}}^{2,2}v_{λ}^{1}(f_{n} ⊗ ẽ_{1}, f_{n} ⊗ ẽ_{1}, f_{n−1} ⊗ ẽ_{2}) = −v_{λ}^{1}(f_{n} ⊗ ẽ_{1}, f_{n} ⊗ ẽ_{1})(f_{n−1} ⊗ ẽ_{2}) = \\ = −A_{2,1}(f_{n−1} ⊗ ẽ_{2}) = f_{n−1} ⊗ ẽ_{1} ≠ 0. \end{matrix}

Therefore, Ker∂_{𝔤_{0}}^{2,2} contains precisely one irreducible submodule with highest weight ε_{1} +ε_{2} − 2δ_{n} and this submodule belongs to Im∂_{𝔤_{0}}^{3,1}. Similarly, 𝔤_{0} ⊗ Λ^{2}𝔤_{−1}^{∗} contains two irreducible submodules with highest weight 2ε_{1}−δ_{n−1}−δ_{n}, one of which belongs to Ker∂_{𝔤_{0}}^{2,2} and, therefore, to Im∂_{𝔤_{0}}^{3,1}.

Let λ = 2ε_{1} − 2δ_{n}. Then by Table 14 any 𝔤𝔩(m) ⊕ 𝔤𝔩(n)-highest vector of weight λ, which belongs to 𝔤_{0} ⊗ Λ^{2}𝔤_{−1}^{∗}, is

v_{λ} = k_{1}v_{λ}^{1} + k_{2}v_{λ}^{2} + k_{3}v_{λ}^{3}, where k_{1}, k_{2}, k_{3} ∈ ℂ.

If v_{λ} ∈ Ker∂_{𝔤_{0}}^{2,2}, then the condition ∂_{𝔤_{0}}^{2,2}v_{λ}(f_{n} ⊗ ẽ_{1}, f_{n} ⊗ ẽ_{1}, f_{n} ⊗ ẽ_{1}) = 0 implies

k_{1}(m − 1) − k_{2}(n − 1) + k_{3}(m − n) = 0,  (3.4.2)

and the condition ∂_{𝔤_{0}}^{2,2}v_{λ}(f_{n} ⊗ ẽ_{2}, f_{n} ⊗ ẽ_{1}, f_{1} ⊗ ẽ_{1}) = 0 implies that

k_{1}m − k_{2}n = 0.  (3.4.3)

Thus, for m ≠ n we have

k_{2} = mk_{1}/n, k_{3} = −k_{1}/n.  (3.4.4)

Therefore, Ker∂_{𝔤_{0}}^{2,2} contains precisely one irreducible submodule with highest weight 2ε_{1}−2δ_{n} and this submodule belongs to Im∂_{𝔤_{0}}^{3,1}.

Finally, let λ = ε_{1} + ε_{2} − δ_{n−1} − δ_{n}. Then by Table 14 any highest vector with weight λ, which belongs to 𝔤_{0} ⊗ Λ^{2}𝔤_{−1}^{∗}, is

v_{λ} = k_{1}v_{λ}^{1} + k_{2}v_{λ}^{2} + k_{3}v_{λ}^{3}, where k_{1}, k_{2}, k_{3} ∈ ℂ,

and if m = 2, then k_{1} = 0. Note that if v_{λ} ∈ Ker∂_{𝔤_{0}}^{2,2}, then

∂_{𝔤_{0}}^{2,2}v_{λ}(f_{n−1} ⊗ ẽ_{1}, f_{n} ⊗ ẽ_{2}, f_{n} ⊗ ẽ_{1}) = 0

implies that

k_{1} + k_{2} + k_{3}(n − m) = 0.  (3.4.5)

Thus, if m = 2, then

k_{2} = (2 − n)k_{3}.  (3.4.6)

If m, n > 2, then the condition

∂_{𝔤_{0}}^{2,2}v_{λ}(f_{n−1} ⊗ ẽ_{m}, f_{n} ⊗ ẽ_{2}, f_{1} ⊗ ẽ_{1}) = 0  (3.4.7)

implies that k_{1} + k_{2} = 0. Hence,

k_{2} = −k_{1}, k_{3} = 0.  (3.4.8)

Therefore, Ker∂_{𝔤_{0}}^{2,2} contains precisely one highest vector of weight ε_{1} + ε_{2} − δ_{n−1} − δ_{n} which belongs to Im∂_{𝔤_{0}}^{3,1}. Thus, we have the description of H_{𝔤_{0}}^{2,2} given in Table 13.

3.5. Calculation of H2,2 ; 𝔤̂_{0} for m, n > 1, m ≠ n. By part b) of Theorem 1.1

𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}) = 𝔤_{−1} ⊕ 𝔤̂_{0}.

Therefore, the Spencer cochain sequence for k = 2 takes the form

0 ⟶^{∂3,1 ; 𝔤̂_{0}} 𝔤̂_{0} ⊗ Λ^{2}𝔤_{−1}^{∗} ⟶^{∂2,2 ; 𝔤̂_{0}} 𝔤_{−1} ⊗ Λ^{3}𝔤_{−1}^{∗}.

Note that since 𝔤_{0} = 𝔤̂_{0} ⊕ ℂ, then

𝔤_{0} ⊗ Λ^{2}𝔤_{−1}^{∗} = 𝔤̂_{0} ⊗ Λ^{2}𝔤_{−1}^{∗} ⊕ V_{2ε_{1}−2δ_{n}} ⊕ V_{ε_{1}+ε_{2}−δ_{n−1}−δ_{n}}.  (3.5.1)

As we have shown in sec.3.4, if λ is one of the weights from Table 14, then an irreducible module with highest weight λ is contained in the decomposition of Ker∂_{𝔤_{0}}^{2,2} into irreducible 𝔤𝔩(m) ⊕ 𝔤𝔩(n)-modules if and only if

\begin{matrix}λ = ε_{1} + ε_{2} + ε_{3} − ε_{m} − δ_{n−1} − δ_{n} (m > 3), \\ ε_{1} + ε_{2} + δ_{1} − δ_{n−2} − δ_{n−1} − δ_{n} (n > 3), \\ ε_{1} + ε_{2} − 2δ_{n}, 2ε_{1} − δ_{n−1} − δ_{n}, 2ε_{1} − 2δ_{n} or ε_{1} + ε_{2} − δ_{n−1} − δ_{n} \end{matrix}  (3.5.2)

48 2. THE ANALOGUES OF PENROSE’S TENSORS

and its multiplicity is 1. Therefore, by (3.5.1), if λ = ε_{1} + ε_{2} + ε_{3} − ε_{m} − δ_{n−1} − δ_{n} (m > 3),

ε_{1} + ε_{2} + δ_{1} − δ_{n−2} − δ_{n−1} − δ_{n} (n > 3),
ε_{1} + ε_{2} − 2δ_{n}, or 2ε_{1} − δ_{n−1} − δ_{n},

then the corresponding submodule is contained in Ker∂2,2 ; 𝔤̂_{0} as well.

Let λ = 2ε_{1}−2δ_{n} and v_{λ} ∈ Ker∂_{𝔤_{0}}^{2,2}. Then (3.4.4) where k_{3} = 0, implies that v_{λ} ∉ Ker∂2,2 ; 𝔤̂_{0}. Let λ = ε_{1} + ε_{2} − δ_{n−1} − δ_{n}, v_{λ} ∈ Ker∂_{𝔤_{0}}^{2,2}. Then (3.4.6) implies that v_{λ} ∉ Ker∂2,2 ; 𝔤̂_{0} for

m = 2, and (3.4.8) implies that v_{λ} ∈ Ker∂2,2 ; 𝔤̂_{0} for m, n > 2. Thus, we have

H2,2 ; 𝔤̂_{0} = H_{𝔤_{0}}^{2,2} ⊕ V_{ε_{1}+ε_{2}−2δ_{n}} ⊕ V_{2ε_{1}−δ_{n−1}−δ_{n}} if either m = 2 or n = 2

and

H2,2 ; 𝔤̂_{0} = H_{𝔤_{0}}^{2,2} ⊕ V_{ε_{1}+ε_{2}−2δ_{n}} ⊕ V_{2ε_{1}−δ_{n−1}−δ_{n}} ⊕ V_{ε_{1}+ε_{2}−δ_{n−1}−δ_{n}} if m, n > 2.

3.6. Calculation of H2,2 ; 𝔤̂_{0} for m = n > 1. By parts c) and d) of Theorem 1.1 the first term of 𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}) is U^{∗} ⊗ V and the second one is ℂ for n = 2 and zero for n > 2. By formula (2.2.1) we have

Im∂3,1 ; 𝔤̂_{0} = (U^{∗} ⊗ V) ⊗ (U^{∗} ⊗ V) for n > 2  (3.6.1)

and

Im∂3,1 ; 𝔤̂_{0} = (U^{∗} ⊗ V) ⊗ (U^{∗} ⊗ V)/ℂ for n = 2.  (3.6.2)

Therefore, by Table 9,

Im∂3,1 ; 𝔤̂_{0} = V_{2ε_{1}−2δ_{n}} ⊕ V_{2ε_{1}−δ_{n−1}−δ_{n}} ⊕ V_{ε_{1}+ε_{2}−2δ_{n}} for n = 2

and

Im∂3,1 ; 𝔤̂_{0} = V_{2ε_{1}−2δ_{n}} ⊕ V_{2ε_{1}−δ_{n−1}−δ_{n}} ⊕ V_{ε_{1}+ε_{2}−2δ_{n}} ⊕ V_{ε_{1}+ε_{2}−δ_{n−1}−δ_{n}} for n > 2.

Therefore, by (3.4.6) and (3.5.2),

H2,2 ; 𝔤̂_{0} = 0 for n = 2, 3

and

H2,2 ; 𝔤̂_{0} = V_{ε_{1}+ε_{2}+ε_{3}−ε_{n}−δ_{n−1}−δ_{n}} ⊕ V_{ε_{1}+ε_{2}+δ_{1}−δ_{n−2}−δ_{n−1}−δ_{n}} for n > 3.

3.7. Computation of H_{𝔤_{0}}^{3,2} for m, n > 1, m ≠ n. For k = 3 the Spencer cochain sequence is of the form

𝔤_{2} ⊗ 𝔤_{−1}^{∗} ⟶^{∂_{𝔤_{0}}^{4,1}} 𝔤_{1} ⊗ Λ^{2}𝔤_{−1}^{∗} ⟶^{∂_{𝔤_{0}}^{3,2}} 𝔤_{0} ⊗ Λ^{3}𝔤_{−1}^{∗}.

Observe that

𝔤_{1} ⊗ Λ^{2}𝔤_{−1}^{∗} = (U^{∗} ⊗ V) ⊗ Λ^{2}(U^{∗} ⊗ V) ≅
(Λ^{2}U^{∗} ⊗ U^{∗}) ⊗ (S^{2}V ⊗ V) ⊕ (S^{2}U^{∗} ⊗ U^{∗}) ⊗ (Λ^{2}V ⊗ V),
𝔤_{2} = 0.

By Table 5 from [OV]

S^{2}V ⊗ V = V_{3ε_{1}} ⊕ V_{2ε_{1}+ε_{2}},
Λ^{2}V ⊗ V = V_{2ε_{1}+ε_{2}} ⊕ V_{ε_{1}+ε_{2}+ε_{3}} for m > 2,
Λ^{2}V ⊗ V = V_{2ε_{1}+ε_{2}} for m = 2.

Since U is purely odd,

Λ^{2}U^{∗} ⊗ U^{∗} = V_{−3δ_{n}} ⊕ V_{−δ_{n−1}−2δ_{n}},
S^{2}U^{∗} ⊗ U^{∗} = V_{−δ_{n−1}−2δ_{n}} ⊕ V_{−δ_{n−2}−δ_{n−1}−δ_{n}} for n > 2,
S^{2}U^{∗} ⊗ U^{∗} = V_{−δ_{n−1}−2δ_{n}} for n = 2.

The above decompositions imply the following

Lemma. The 𝔤𝔩(m)⊕𝔤𝔩(n)-module 𝔤_{1} ⊗Λ^{2}𝔤_{−1}^{∗} is the direct sum of irreducible submodules whose highest weights and highest vectors are listed in Table 15. [s and t denote the cyclic permutations of (1, 2, 3) and (n − 2, n − 1, n), respectively.]

Let us show that Ker∂_{𝔤_{0}}^{3,2} = 0. Let λ = 3ε_{1} − 3δ_{n}. Then

∂_{𝔤_{0}}^{3,2}v_{λ}(f_{n} ⊗ ẽ_{1}, f_{n} ⊗ ẽ_{1}, f_{n} ⊗ ẽ_{1}) = 3B_{1,n}(f_{n} ⊗ ẽ_{1}) = 3(e_{1} ⊗ ẽ_{1} + f_{n} ⊗ f̃_{n}) ≠ 0.

Let λ = 2ε_{1} + ε_{2} − 3δ_{n}. Then

∂_{𝔤_{0}}^{3,2}v_{λ}(f_{n} ⊗ ẽ_{1}, f_{n} ⊗ ẽ_{1}, f_{n} ⊗ ẽ_{1}) = −3B_{2,n}(f_{n} ⊗ ẽ_{1}) = −3e_{2} ⊗ ẽ_{1} ≠ 0.

Let λ = 3ε_{1} − δ_{n−1} − 2δ_{n}. Then

∂_{𝔤_{0}}^{3,2}v_{λ}(f_{n} ⊗ ẽ_{1}, f_{n} ⊗ ẽ_{1}, f_{n} ⊗ ẽ_{1}) = 3B_{1,n−1}(f_{n} ⊗ ẽ_{1}) = 3f_{n} ⊗ f̃_{n−1} ≠ 0.

Let λ = 2ε_{1} + ε_{2} − δ_{n−1} − 2δ_{n}. Since by Table 15 𝔤_{1} ⊗ Λ^{2}𝔤_{−1}^{∗} contains two irreducible submodules with highest weight λ, then any highest vector of weight λ in 𝔤_{1} ⊗ Λ^{2}𝔤_{−1}^{∗} is of the form

v_{λ} = k_{1}v_{λ}^{1} + k_{2}v_{λ}^{2}, where k_{1}, k_{2} ∈ ℂ.

Let v_{λ} ∈ Ker∂_{𝔤_{0}}^{3,2}. If m > 2, then

∂_{𝔤_{0}}^{3,2}v_{λ}(f_{n−1} ⊗ ẽ_{2}, f_{n} ⊗ ẽ_{1}, f_{n} ⊗ ẽ_{m}) = −(1/2)k_{2}B_{1,n}(f_{n} ⊗ ẽ_{m}) = −(1/2)k_{2}e_{1} ⊗ ẽ_{m} = 0.

Therefore, k_{2} = 0. Moreover,

∂_{𝔤_{0}}^{3,2}v_{λ}(f_{n} ⊗ ẽ_{1}, f_{n} ⊗ ẽ_{1}, f_{n−1} ⊗ ẽ_{m}) = k_{1}B_{2,n−1}(f_{n−1} ⊗ ẽ_{m}) = k_{1}e_{2} ⊗ ẽ_{m} = 0.

Hence, k_{1} = 0. If n > 2, then

∂_{𝔤_{0}}^{3,2}v_{λ}(f_{n−1} ⊗ ẽ_{2}, f_{n} ⊗ ẽ_{1}, f_{1} ⊗ ẽ_{1}) = −(1/2)k_{2}B_{1,n}(f_{1} ⊗ ẽ_{1}) = −(1/2)k_{2}f_{1} ⊗ f̃_{n} = 0.

Therefore, k_{2} = 0. Moreover,

∂_{𝔤_{0}}^{3,2}v_{λ}(f_{n} ⊗ ẽ_{1}, f_{n} ⊗ ẽ_{1}, f_{1} ⊗ ẽ_{2}) = k_{1}B_{2,n−1}(f_{1} ⊗ ẽ_{2}) = k_{1}f_{1} ⊗ f̃_{n−1} = 0.

Hence, k_{1} = 0.

Let λ = ε_{1} + ε_{2} + ε_{3} − δ_{n−1} − 2δ_{n}. Then

∂_{𝔤_{0}}^{3,2}v_{λ}(f_{n−1} ⊗ ẽ_{2}, f_{n} ⊗ ẽ_{3}, f_{n} ⊗ ẽ_{2}) = (1/2)B_{1,n}(f_{n} ⊗ ẽ_{2}) = (1/2)e_{1} ⊗ ẽ_{2} ≠ 0.

Let λ = 2ε_{1} + ε_{2} − δ_{n−2} − δ_{n−1} − δ_{n}. Then

\begin{matrix}∂_{𝔤_{0}}^{3,2}v_{λ}(f_{n−1} ⊗ ẽ_{2}, f_{n} ⊗ ẽ_{1}, f_{n−1} ⊗ ẽ_{1}) = \\ (1/2)B_{1,n−2}(f_{n−1} ⊗ ẽ_{1}) = (1/2)f_{n−1} ⊗ f̃_{n−2} ≠ 0. \end{matrix}

Finally, let λ = ε_{1} + ε_{2} + ε_{3} − δ_{n−2} − δ_{n−1} − δ_{n}. Then

∂_{𝔤_{0}}^{3,2}v_{λ}(f_{n−1} ⊗ ẽ_{2}, f_{n} ⊗ ẽ_{3}, f_{n−2} ⊗ ẽ_{1}) =
(1/2)(B_{1,n−2}(f_{n−2} ⊗ ẽ_{1}) + B_{3,n}(f_{n} ⊗ ẽ_{3}) + B_{2,n−1}(f_{n−1} ⊗ ẽ_{2})) =
(1/2)(e_{1} ⊗ ẽ_{1} + f_{n−2} ⊗ f̃_{n−2} + e_{3} ⊗ ẽ_{3} + f_{n} ⊗ f̃_{n} + e_{2} ⊗ ẽ_{2} + f_{n−1} ⊗ f̃_{n−1}) ≠ 0.

Thus, H_{𝔤_{0}}^{3,2} = 0.

3.8. Calculation of H3,2 ; 𝔤̂_{0} for m = n > 1. By part d) of Theorem 1.1 for n > 2 the first term of the Cartan prolongation of the pair (𝔤_{−1}, 𝔤̂_{0}) is U^{∗} ⊗ V and the second one is zero. Therefore, by arguments similar to those from sec.3.7 we get H3,2 ; 𝔤̂_{0} = 0.

If n = 2, then by part c) of Theorem 1.1 the first term of 𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}) is U^{∗} ⊗ V, the second one is the 1-dimensional 𝔤𝔩(2) ⊕ 𝔤𝔩(2)-module with highest weight ε_{1} + ε_{2} − δ_{1} − δ_{2}, and the third one is zero. Thus, by (2.2.1),

Im∂4,1 ; 𝔤̂_{0} = V_{2ε_{1}+ε_{2}−δ_{1}−2δ_{2}}.

By Table 15 (U^{∗} ⊗ V) ⊗ Λ^{2}(U^{∗} ⊗ V) contains two irreducible 𝔤𝔩(2) ⊕ 𝔤𝔩(2)-modules with highest weight λ = 2ε_{1} + ε_{2} − δ_{1} − 2δ_{2} and one of the corresponding highest vectors is v_{λ}^{2}. Since

∂3,2 ; 𝔤̂_{0}v_{λ}^{2}(f_{1} ⊗ ẽ_{1}, f_{2} ⊗ ẽ_{2}, f_{2} ⊗ ẽ_{2}) = B_{1,2}(f_{2} ⊗ ẽ_{2}) = e_{1} ⊗ ẽ_{2} ≠ 0,

then Ker∂3,2 ; 𝔤̂_{0} = Im∂4,1 ; 𝔤̂_{0}. Thus, H3,2 ; 𝔤̂_{0} = 0.

3.9. Calculation of H4,2 ; 𝔤̂_{0} for m = n = 2. For k = 4 the Spencer cochain sequence is of the form

𝔤_{3} ⊗ 𝔤_{−1}^{∗} ⟶^{∂5,1 ; 𝔤̂_{0}} 𝔤_{2} ⊗ Λ^{2}𝔤_{−1}^{∗} ⟶^{∂4,2 ; 𝔤̂_{0}} 𝔤_{1} ⊗ Λ^{3}𝔤_{−1}^{∗}.

By part c) of Theorem 1.1 the second term of 𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}) is 𝔤_{2} = V_{ε_{1}+ε_{2}−δ_{1}−δ_{2}} = ⟨g⟩, the 1-dimensional 𝔤𝔩(2) ⊕ 𝔤𝔩(2)-module, and the third one is zero. Since by Table 9

Λ^{2}𝔤_{−1}^{∗} = V_{2ε_{1}−2δ_{2}} ⊕ V_{ε_{1}+ε_{2}−δ_{1}−δ_{2}},

then

𝔤_{2} ⊗ Λ^{2}𝔤_{−1}^{∗} = V_{3ε_{1}+ε_{2}−δ_{1}−3δ_{2}} ⊕ V_{2ε_{1}+2ε_{2}−2δ_{1}−2δ_{2}}.

Let λ = 3ε_{1} + ε_{2} − δ_{1} − 3δ_{2}. Then by Table 9 v_{λ} = g ⊗ (f̃_{2} ⊗ e_{1}) ∧ (f̃_{2} ⊗ e_{1}). Let v be an element from the basis of 𝔤_{−1} such that g(v) ≠ 0. If v = f_{2} ⊗ ẽ_{1}, then

∂4,2 ; 𝔤̂_{0}(f_{2} ⊗ ẽ_{1}, f_{2} ⊗ ẽ_{1}, v) = −3v_{λ}(f_{2} ⊗ ẽ_{1}, f_{2} ⊗ ẽ_{1})(v) = 3g(v) ≠ 0,

and if v ≠ f_{2} ⊗ ẽ_{1}, then

∂4,2 ; 𝔤̂_{0}v_{λ}(f_{2} ⊗ ẽ_{1}, f_{2} ⊗ ẽ_{1}, v) = −v_{λ}(f_{2} ⊗ ẽ_{1}, f_{2} ⊗ ẽ_{1})(v) = g(v) ≠ 0.

Let λ = 2ε_{1} + 2ε_{2} − 2δ_{1} − 2δ_{2}. Then by Table 9

\begin{matrix}v_{λ} = g ⊗ ((f̃_{2} ⊗ e_{1}) ∧ (f̃_{1} ⊗ e_{2}) − (f̃_{2} ⊗ e_{2}) ∧ (f̃_{1} ⊗ e_{1})− \\ − (f̃_{1} ⊗ e_{1}) ∧ (f̃_{2} ⊗ e_{2}) + (f̃_{1} ⊗ e_{2}) ∧ (f̃_{2} ⊗ e_{1})). \end{matrix}

Let v be an element of the basis of 𝔤_{−1} such that g(v) ≠ 0. Then

∂4,2 ; 𝔤̂_{0}v_{λ}(f_{2} ⊗ ẽ_{1}, f_{1} ⊗ ẽ_{2}, v) = −2v_{λ}(f_{2} ⊗ ẽ_{1}, f_{1} ⊗ ẽ_{2})(v) = g(v) ≠ 0

if either v = f_{2} ⊗ ẽ_{1} or v = f_{1} ⊗ ẽ_{2},

and

∂4,2 ; 𝔤̂_{0}v_{λ}(f_{1} ⊗ ẽ_{1}, f_{2} ⊗ ẽ_{2}, v) = −2v_{λ}(f_{1} ⊗ ẽ_{1}, f_{2} ⊗ ẽ_{2})(v) = −g(v) ≠ 0

if either v = f_{2} ⊗ ẽ_{2} or v = f_{1} ⊗ ẽ_{1}.

Therefore, H4,2 ; 𝔤̂_{0} = 0.

3.10. Calculation of H_{𝔤_{0}}^{k,2} for m = n > 1, k > 0.

Lemma. H_{𝔤_{0}}^{k,2} = Hk,2 ; 𝔤̂_{0}.

Proof. Note that if 𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}) = 𝔤_{−1} ⊕(⊕_{k≥0}𝔤̂_{k}) is the Cartan prolongation of the pair (𝔤_{−1}, 𝔤̂_{0}), then, since 𝔤_{k} = 𝔤̂_{k} ⊕S^{k}(𝔤_{−1}^{∗}) (k ≥ 0), the Spencer cochain sequence is of the form

(𝔤̂_{0} ⊕ ℂ) ⊗ 𝔤_{−1}^{∗} ⟶^{∂_{𝔤_{0}}^{2,1}} 𝔤_{−1} ⊗ Λ^{2}𝔤_{−1}^{∗} ⟶^{∂_{𝔤_{0}}^{1,2}} 0  for k = 1,

(𝔤̂_{k−1} ⊕ S^{k−1}(𝔤_{−1}^{∗})) ⊗ 𝔤_{−1}^{∗} ⟶^{∂_{𝔤_{0}}^{k+1,1}} (𝔤̂_{k−2} ⊕ S^{k−2}(𝔤_{−1}^{∗})) ⊗ Λ^{2}𝔤_{−1}^{∗} ⟶^{∂_{𝔤_{0}}^{k,2}}

(𝔤̂_{k−3} ⊕ S^{k−3}(𝔤_{−1}^{∗})) ⊗ Λ^{3}𝔤_{−1}^{∗}  for k > 1.

Note that since 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) = S^{∗}(𝔤_{−1}^{∗}) ⊂ + 𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}), then the sequence

S^{k−1}(𝔤_{−1}^{∗}) ⊗ 𝔤_{−1}^{∗} ⟶^{¯∂_{𝔤_{0}}^{k+1,1}} S^{k−2}(𝔤_{−1}^{∗}) ⊗ Λ^{2}𝔤_{−1}^{∗} ⟶^{¯∂_{𝔤_{0}}^{k,2}} S^{k−3}(𝔤_{−1}^{∗}) ⊗ Λ^{3}𝔤_{−1}^{∗} for k ≥ 1,

where ¯∂_{𝔤_{0}}^{k+1,1} and ¯∂_{𝔤_{0}}^{k,2} are the restrictions of the operators ∂_{𝔤_{0}}^{k+1,1} and ∂_{𝔤_{0}}^{k,2} to S^{k−1}(𝔤_{−1}^{∗}) ⊗ 𝔤_{−1}^{∗} and S^{k−2}(𝔤_{−1}^{∗}) ⊗ Λ^{2}𝔤_{−1}^{∗}, respectively, and S^{k}(𝔤_{−1}^{∗}) = 0 for k < 0, is well-defined. Hence the corresponding cohomology groups

H̄_{𝔤_{0}}^{k,2} = Ker¯∂_{𝔤_{0}}^{k,2}/Im¯∂_{𝔤_{0}}^{k+1,1}

are well-defined and H_{𝔤_{0}}^{k,2} = Hk,2 ; 𝔤̂_{0} ⊕ H̄_{𝔤_{0}}^{k,2}.

Let us show that H̄_{𝔤_{0}}^{k,2} = 0 for k > 0. For k = 1 this is obvious. Let k = 2. Since S^{k−2}(𝔤_{−1}^{∗}) ⊗ Λ^{2}𝔤_{−1}^{∗} = ⟨z⟩ ⊗ Λ^{2}𝔤_{−1}^{∗}, where z is a generator of the center of 𝔤𝔩(n|n), then

Ker¯∂_{𝔤_{0}}^{k,2} ≅ Λ^{2}𝔤_{−1}^{∗}.

By formula (2.2.1)

Im¯∂_{𝔤_{0}}^{k+1,1} ≅ 𝔤_{−1}^{∗} ⊗ 𝔤_{−1}^{∗}/S^{2}𝔤_{−1}^{∗} = Λ^{2}𝔤_{−1}^{∗}.

Therefore, H̄_{𝔤_{0}}^{2,2} = 0. Let k = 3. Observe that

S^{2}(U^{∗} ⊗ V) ⊗ (U^{∗} ⊗ V) = (S^{2}U^{∗} ⊗ U^{∗}) ⊗ (S^{2}V ⊗ V) ⊕ (Λ^{2}U^{∗} ⊗ U^{∗}) ⊗ (Λ^{2}V ⊗ V).

By Table 5 from [OV] we get:

\begin{matrix}S^{2}V ⊗ V = V_{3ε_{1}} ⊕ V_{2ε_{1}+ε_{2}}, Λ^{2}V ⊗ V = V_{2ε_{1}+ε_{2}} if n = 2, \\ Λ^{2}V ⊗ V = V_{2ε_{1}+ε_{2}} ⊕ V_{ε_{1}+ε_{2}+ε_{3}} if n > 2. \end{matrix}

Since U is odd,

Λ^{2}U^{∗} ⊗ U^{∗} = V_{−3δ_{n}} ⊕ V_{−δ_{n−1}−2δ_{n}},
S^{2}U^{∗} ⊗ U^{∗} = V_{−δ_{n−1}−2δ_{n}} if n = 2,
S^{2}U^{∗} ⊗ U^{∗} = V_{−δ_{n−1}−2δ_{n}} ⊕ V_{−δ_{n−2}−δ_{n−1}−δ_{n}} if n > 2.

Therefore,

S^{2}(U^{∗} ⊗ V) ⊗ (U^{∗} ⊗ V) = V_{3ε_{1}−δ_{n−1}−2δ_{n}} ⊕ V_{2ε_{1}+ε_{2}−3δ_{n}} ⊕ 2V_{2ε_{1}+ε_{2}−δ_{n−1}−2δ_{n}} if n = 2

and S^{2}(U^{∗} ⊗ V) ⊗ (U^{∗} ⊗ V) = V_{3ε_{1}−δ_{n−1}−2δ_{n}} ⊕ V_{2ε_{1}+ε_{2}−3δ_{n}} ⊕ 2V_{2ε_{1}+ε_{2}−δ_{n−1}−2δ_{n}} ⊕ V_{3ε_{1}−δ_{n−2}−δ_{n−1}−δ_{n}}⊕

V_{ε_{1}+ε_{2}+ε_{3}−3δ_{n}} ⊕ V_{2ε_{1}+ε_{2}−δ_{n−2}−δ_{n−1}−δ_{n}} ⊕ V_{ε_{1}+ε_{2}+ε_{3}−δ_{n−1}−2δ_{n}}  if n > 2.

Moreover, we have

\begin{matrix}S^{3}(U^{∗} ⊗ V) = V_{2ε_{1}+ε_{2}−δ_{1}−2δ_{2}} & if n = 2 and \\ S^{3}(U^{∗} ⊗ V) = V_{ε_{1}+ε_{2}+ε_{3}−3δ_{n}} ⊕ V_{3ε_{1}−δ_{n−2}−δ_{n−1}−δ_{n}} ⊕ V_{2ε_{1}+ε_{2}−δ_{n−1}−2δ_{n}} & if n > 2. \end{matrix}

Thus, by (2.2.1)

Im¯∂_{𝔤_{0}}^{4,1} = V_{3ε_{1}−δ_{n−1}−2δ_{n}} ⊕ V_{2ε_{1}+ε_{2}−3δ_{n}} ⊕ V_{2ε_{1}+ε_{2}−δ_{n−1}−2δ_{n}}  if n = 2,
Im¯∂_{𝔤_{0}}^{4,1} = V_{3ε_{1}−δ_{n−1}−2δ_{n}} ⊕ V_{2ε_{1}+ε_{2}−3δ_{n}} ⊕ V_{2ε_{1}+ε_{2}−δ_{n−1}−2δ_{n}}⊕
⊕V_{2ε_{1}+ε_{2}−δ_{n−2}−δ_{n−1}−δ_{n}} ⊕ V_{ε_{1}+ε_{2}+ε_{3}−δ_{n−1}−2δ_{n}}  if n > 2.

Finally, the decomposition of the 𝔤𝔩(n) ⊕ 𝔤𝔩(n)-module 𝔤_{−1}^{∗} ⊗ Λ^{2}𝔤_{−1}^{∗} into the direct sum of irreducible components is given in Table 15. Checking the action of ¯∂_{𝔤_{0}}^{3,2} on the highest vectors we get:

Im¯∂_{𝔤_{0}}^{4,1} = Ker¯∂_{𝔤_{0}}^{3,2}.

Note that for k > 3 the cohomology groups H̄_{𝔤_{0}}^{k,2} coincide with the Spencer cohomology groups H_{𝔬(n^{2})}^{k−2,2} corresponding to the Cartan prolongation 𝔤_{∗}(V (0|n^{2}), 𝔬(n^{2})) = 𝔥(0|n^{2}), where 𝔬(n^{2}) is the orthogonal Lie algebra and V (0|n^{2}) is the standard odd 𝔬(n^{2})-module. These groups are vanishing for k > 3 (see Theorem 1.3 of Chapter 3).

□

CHAPTER 3 The analogues of the Riemann–Weyl tensors for classical superspaces

Recall that ℤ-grading of depth 1 of a Lie (super)algebra 𝔤 is the ℤ-grading of the form 𝔤 = ⊕_{i≥−1}𝔤_{i}. All such ℤ-gradings of simple finite-dimensional complex Lie superalgebras are listed in [S2]. Denote by V_{λ} the irreducible module over a Lie superalgebra with highest weight λ and an even highest vector.

Spencer cohomology of 𝔰𝔩(m|n) and 𝔭𝔰𝔩(n|n)

1.1. Description of the ℤ-gradings of depth 1. Let V (m − p|q) and U(p|n − q) be the standard 𝔰𝔩(m − p|q) and 𝔰𝔩(p|n − q)- modules, respectively.

All ℤ-gradings of depth 1 of 𝔤 = 𝔰𝔩(m|n) and 𝔭𝔰𝔩(n|n) are of the form 𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1}, where 𝔤_{1} = 𝔤_{−1}^{∗} = V (m − p|q) ⊗ U(p|n − q)^{∗}.

A) For 𝔰𝔩(m|n), where m ≠ n, there are the following possible values of 𝔤_{0} for the ℤ-gradings of depth 1:

a) 𝔠(𝔰𝔩(m) ⊕ 𝔰𝔩(n));

b) 𝔠(𝔰𝔩(m|q) ⊕ 𝔰𝔩(n − q)), if p = 0, q ≠ 0, n − q ≠ 0;

c) 𝔠(𝔰𝔩(m − p) ⊕ 𝔰𝔩(p|n)), if q = 0, p ≠ 0, m − p ≠ 0;

d) 𝔠(𝔰𝔩(m − p|q) ⊕ 𝔰𝔩(p|n − q)), if p ≠ 0, q ≠ 0. B) For 𝔰𝔩(n|n) there are the following possible values of 𝔤_{0} for the ℤ-gradings of depth 1:

a) 𝔠(𝔰𝔩(n) ⊕ 𝔰𝔩(n));

b) 𝔠(𝔰𝔩(n|q) ⊕ 𝔰𝔩(n − q)), if p = 0, q ≠ 0, n − q ≠ 0;

c) 𝔠(𝔰𝔩(n − p) ⊕ 𝔰𝔩(p|n)), if q = 0, p ≠ 0, n − p ≠ 0;

d) 𝔠(𝔰𝔩(n − p|q) ⊕ 𝔰𝔩(p|n − q)), if p ≠ 0, q ≠ 0. C) The ℤ-gradings of 𝔭𝔰𝔩(n|n) are similar to those of 𝔰𝔩(n|n), only 𝔤_{0} is centerless.

1.2. Theorem (Cartan prolongations). For the cases of sec.1.1. we have: A) 𝔤 = 𝔰𝔩(m|n), where m ≠ n. Then 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) = 𝔤, except for the following cases:

a) if n = 1, then 𝔤_{∗} = 𝔳𝔢𝔠𝔱(0|m), if m = 1, then 𝔤_{∗} = 𝔳𝔢𝔠𝔱(0|n);

b) if n − q = 1, then 𝔤_{∗} = 𝔳𝔢𝔠𝔱(q|m), if m = 0, q = 1, then 𝔤_{∗} = 𝔳𝔢𝔠𝔱(n − 1|0);

c) if m − p = 1, then 𝔤_{∗} = 𝔳𝔢𝔠𝔱(p|n), if n = 0, p = 1, then 𝔤_{∗} = 𝔳𝔢𝔠𝔱(m − 1|0);

d) if n−q = 0, p = 1, then 𝔤_{∗} = 𝔳𝔢𝔠𝔱(m−1|n), if m−p = 0, q = 1, then 𝔤_{∗} = 𝔳𝔢𝔠𝔱(n−1|m). B) 𝔤 = 𝔰𝔩(n|n). Then 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) = S^{∗}(𝔤_{−1}^{∗}) ⊂ + 𝔭𝔰𝔩(n|n), except for the following cases:

a) if n = 2, then 𝔤_{∗} = S^{∗}(𝔤_{−1}^{∗}) ⊂ + 𝔥(0|4);

b) if n − q = 1, then 𝔤_{∗} = 𝔳𝔢𝔠𝔱(q|n);

c) if n − p = 1, then 𝔤_{∗} = 𝔳𝔢𝔠𝔱(p|n);

d) if n − q = 0, p = 1 or n − p = 0, q = 1 then 𝔤_{∗} = 𝔳𝔢𝔠𝔱(n − 1|n). C) 𝔤 = 𝔭𝔰𝔩(n|n). Then 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) = 𝔤, except for the following cases:

a) if n = 2, then 𝔤_{∗} = 𝔥(0|4);

b) if n − q = 1, then 𝔤_{∗} = 𝔰𝔳𝔢𝔠𝔱(q|n);

c) if n − p = 1, then 𝔤_{∗} = 𝔰𝔳𝔢𝔠𝔱(p|n);

d) if n − q = 0, p = 1 or n − p = 0, q = 1, then 𝔤_{∗} = 𝔰𝔳𝔢𝔠𝔱(n − 1|n).

Let 𝔤 = 𝔰𝔩(m|n), where m ≠ n, or 𝔭𝔰𝔩(n|n). We will describe the Spencer cohomology groups for all ℤ-gradings of depth 1 listed in sec.1.1.

First consider the cases easiest to formulate. Let ⟨π_{i}⟩ be the i-th fundamental weight of 𝔤_{0}.

1.3. Theorem. 1) For 𝔤_{∗} = 𝔳𝔢𝔠𝔱(m|n), 𝔰𝔳𝔢𝔠𝔱(m|n) SFs vanish except for 𝔰𝔳𝔢𝔠𝔱(0|n), when SFs are of order n and constitute the 𝔤_{0}-module Π^{n}(⟨1⟩).

2) For 𝔤_{∗} of series 𝔥(0|n), nonzero SFs are of order 1. For n > 3 SFs constitute 𝔤_{0}-module Π(V_{3π_{1}} ⊕ V_{π_{1}}).

3) For 𝔤_{∗} = 𝔰𝔥(0|n), nonzero SFs are the same as for 𝔥(0|n) and additionally Π^{n−1}(V_{π_{1}}) of order n − 1.

Consider the ℤ-gradings of depth 1 of 𝔤 = 𝔰𝔩(m|n) (m ≠ n) and 𝔭𝔰𝔩(n|n) listed in sec.1.1 for which 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) = 𝔤. Describe the corresponding SFs.

Case a) was discussed in Chapter 2. Consider case b).

1.4. Theorem. The nonzero SFs are of orders 1 and 2. The 𝔤_{0}-module H_{𝔤_{0}}^{2,2} splits into the direct sum of irreducible components whose weights are given in Table 16. Table 16 also contains the highest weights (with respect to the bases ε_{1}, …, ε_{m+q} and δ_{1}, …, δ_{n−q} of the dual spaces to the maximal tori of 𝔰𝔩(m|q) and 𝔰𝔩(n − q), respectively) of irreducible components of H_{𝔤_{0}}^{1,2} for the cases when H_{𝔤_{0}}^{1,2} does split into the direct sum of irreducible 𝔤_{0}-modules.

Exceptional cases are as follows: if m = q − 1, m > 1, n − q ≥ 3, then H_{𝔤_{0}}^{1,2} = V_{ε_{1}+ε_{2}−ε_{m+q}+δ_{1}−δ_{n−q−1}−δ_{n−q}} ⊕X, where X is given by the nonsplit exact sequence of 𝔤_{0}-modules

0 ⟶ V_{2ε_{1}−ε_{m+q}+δ_{1}−2δn−q} ⟶ X ⟶ Π(V_{ε_{1}+δ_{1}−2δ_{n−q}}) ⟶ 0;  (1)

if m = q − 1, m > 1, n − q = 2, then H_{𝔤_{0}}^{1,2} = X, where X is given by (1);

if m = 1, q = 2, n − q ≥ 3, then H_{𝔤_{0}}^{1,2} = Π(V_{ε_{1}+ε_{2}−ε_{3}+δ_{1}−δ_{n−q−1}−δ_{n−q}}) ⊕ X, where X is given by the nonsplit exact sequence of 𝔤_{0}-modules

\begin{matrix}0 ⟶ V_{2ε_{1}−ε_{3}+δ_{1}−2δ_{n−q}} ⊕ Π(V_{−ε_{1}+ε_{2}+ε_{3}+δ_{1}−2δ_{n−q}}) \\ ⟶ X ⟶ Π(V_{ε_{1}+δ_{1}−2δ_{n−q}}) ⟶ 0; \end{matrix}  (2)

if m = 1, q = 2, n − q = 2, then H_{𝔤_{0}}^{1,2} = X, where X is given by (2); if m = q + 1, n − q ≥ 3, then H_{𝔤_{0}}^{1,2} = V_{2ε_{1}−ε_{m+q}+δ_{1}−2δ_{n−q}} ⊕ X, where X is given by the nonsplit exact sequence of 𝔤_{0}-modules

\begin{matrix}0 ⟶ V_{ε_{1}+ε_{2}−ε_{m+q}+δ_{1}−δ_{n−q−1}−δ_{n−q}} ⟶ X \\ ⟶ Π(V_{ε_{1}+δ_{1}−δ_{n−q−1}−δ_{n−q}}) ⟶ 0 & (q ≥ 2), \end{matrix}

\begin{matrix}0 ⟶ V_{ε_{1}−ε_{2}+ε_{3}+δ_{1}−δ_{n−2}−δ_{n−1}} ⊕ V_{ε_{1}+ε_{2}−ε_{3}+δ_{1}−δ_{n−2}−δ_{n−1}} ⟶ X \\ ⟶ Π(V_{ε_{1}+δ_{1}−δ_{n−2}−δ_{n−1}}) ⟶ 0 & (q = 1). \end{matrix}

Case c) is similar to case b). Consider case d).

1.5. Theorem. The nonzero SFs are of orders 1 and 2. The 𝔤_{0}-module H_{𝔤_{0}}^{2,2} splits into the direct sum of irreducible components whose weights are given in Table 17. Table 17 also contains the highest weights (with respect to the bases ε_{1}, …, ε_{m−p+q} and δ_{1}, …, δ_{p+n−q} of the dual spaces to the maximal tori of 𝔰𝔩(m − p|q) and 𝔰𝔩(p|n − q), respectively) of irreducible components of H_{𝔤_{0}}^{1,2} for the cases when H_{𝔤_{0}}^{1,2} does split into the direct sum of irreducible 𝔤_{0}-modules.

Exceptional cases are m = p + q ± 1 and n = p + q ± 1. More precisely: if m = p + q + 1, n ≠ p + q ± 1, q, then H_{𝔤_{0}}^{1,2} = V_{2ε_{1}−ε_{m−p+q}+δ_{1}−2δ_{p+n−q}} ⊕ Y, where Y is given by the nonsplit

SPENCER COHOMOLOGY OF 𝔰𝔩(m|n) AND 𝔭𝔰𝔩(n|n) 55

exact sequence of 𝔤_{0}-modules

\begin{matrix}0 ⟶ V_{ε_{1}+ε_{2}−ε_{3}+δ_{1}−δ_{p+n−2}−δ_{p+n−1}} ⊕ V_{ε_{1}−ε_{2}+ε_{3}+δ_{1}−δ_{p+n−2}−δ_{p+n−1}} ⟶ Y \\ ⟶ V_{ε_{1}+δ_{1}−δ_{p+n−2}−δ_{p+n−1}} ⟶ 0 & (q = 1), \end{matrix}

\begin{matrix}0 ⟶ V_{ε_{1}+ε_{2}−ε_{m−p+q}+δ_{1}−δ_{p+n−q−1}−δ_{p+n−q}} ⟶ Y \\ ⟶ V_{ε_{1}+δ_{1}−δ_{p+n−q−1}−δ_{p+n−q}} ⟶ 0 & (q ≥ 2); \end{matrix}

if m = p + q + 1, n = p + q − 1, then H_{𝔤_{0}}^{1,2} = X ⊕ Y , where X is given by the nonsplit exact sequence of 𝔤_{0}-modules

\begin{matrix}0 ⟶ V_{2ε_{1}−ε_{m−2+q}+δ_{1}−2δ_{3}} ⊕ V_{2ε_{1}−ε_{m−2+q}−δ_{1}−δ_{2}+δ_{3}} ⟶ \\ X ⟶ V_{2ε_{1}−ε_{m−2+q}−δ_{3}} ⟶ 0 & (p = 2), \end{matrix}

\begin{matrix}0 ⟶ V_{2ε_{1}−ε_{m−p+q}+δ_{1}−2δ_{p+n−q}} ⟶ X \\ ⟶ V_{2ε_{1}−ε_{m−p+q}−δ_{p+n−q}} ⟶ 0 & (p ≥ 3), \end{matrix}

and Y is given by the nonsplit exact sequence of 𝔤_{0}-modules

\begin{matrix}0 ⟶ V_{ε_{1}+ε_{2}−ε_{3}+δ_{1}−δ_{p+n−2}−δ_{p+n−1}} ⊕ V_{ε_{1}−ε_{2}+ε_{3}+δ_{1}−δ_{p+n−2}−δ_{p+n−1}} ⟶ \\ Y ⟶ V_{ε_{1}+δ_{1}−δ_{p+n−2}−δ_{p+n−1}} ⟶ 0 & (q = 1), \end{matrix}

\begin{matrix}0 ⟶ V_{ε_{1}+ε_{2}−ε_{m−p+q}+δ_{1}−δ_{p+n−q−1}−δ_{p+n−q}} ⟶ Y \\ ⟶ V_{ε_{1}+δ_{1}−δ_{p+n−q−1}−δ_{p+n−q}} ⟶ 0 & (q ≥ 2); \end{matrix}

if m = p + q + 1, n = q, then H_{𝔤_{0}}^{1,2} = V_{2ε_{1}−ε_{m−p+q}+δ_{1}−δ_{p−1}−δ_{p}} ⊕ Y (p ≥ 3) or H_{𝔤_{0}}^{1,2} = Y (p = 2), where Y is given by the nonsplit exact sequence of 𝔤_{0}-modules

0 ⟶ V_{ε_{1}+ε_{2}−ε_{3}+δ_{1}−2δ_{p}} ⊕ V_{ε_{1}−ε_{2}+ε_{3}+δ_{1}−2δ_{p}} ⟶ Y ⟶ V_{ε_{1}+δ_{1}−2δ_{p}} ⟶ 0  (q = 1),
0 ⟶ V_{ε_{1}+ε_{2}−ε_{m−p+q}+δ_{1}−2δ_{p}} ⟶ Y ⟶ V_{ε_{1}+δ_{1}−2δ_{p}} ⟶ 0  (q ≥ 2);

if n = p + q + 1, m ≠ p + q ± 1, p, then H_{𝔤_{0}}^{1,2} = V_{2ε_{1}−ε_{m−p+q}+δ_{1}−2δ_{p+n−q}} ⊕ Y, where

\begin{matrix}0 ⟶ V_{ε_{1}+ε_{2}−ε_{m−1+q}+δ_{1}−δ_{2}−δ_{3}} ⊕ V_{ε_{1}+ε_{2}−ε_{m−1+q}−δ_{1}+δ_{2}−δ_{3}} ⟶ \\ Y ⟶ V_{ε_{1}+ε_{2}−ε_{m−1+q}−δ_{3}} ⟶ 0 & (p = 1), \end{matrix}

\begin{matrix}0 ⟶ V_{ε_{1}+ε_{2}−ε_{m−p+q}+δ_{1}−δ_{p+n−q−1}−δ_{p+n−q}} ⟶ Y \\ ⟶ V_{ε_{1}+ε_{2}−ε_{m−p+q}−δ_{p+n−q}} ⟶ 0 & (p ≥ 2); \end{matrix}

if n = p + q + 1, m = p + q − 1, then H_{𝔤_{0}}^{1,2} = X ⊕ Y, where

\begin{matrix}0 ⟶ V_{2ε_{1}−ε_{3}+δ_{1}−2δ_{p+n−2}} ⊕ V_{−ε_{1}+ε_{2}+ε_{3}+δ_{1}−2δ_{p+n−2}} ⟶ X \\ ⟶ V_{ε_{1}+δ_{1}−2δ_{p+n−2}} ⟶ 0 & (q = 2), \end{matrix}

0 ⟶ V_{2ε_{1}−ε_{m−p+q}+δ_{1}−2δ_{p+n−q}} ⟶ X ⟶ V_{ε_{1}+δ_{1}−2δ_{p+n−q}} ⟶ 0  (q ≥ 3),

and Y is given by the nonsplit exact sequence of 𝔤_{0}-modules

\begin{matrix}0 ⟶ V_{ε_{1}+ε_{2}−ε_{m−1+q}+δ_{1}−δ_{2}−δ_{3}} ⊕ V_{ε_{1}+ε_{2}−ε_{m−1+q}−δ_{1}+δ_{2}−δ_{3}} \\ ⟶ Y ⟶ V_{ε_{1}+ε_{2}−ε_{m−1+q}−δ_{3}} ⟶ 0 & (p = 1), \end{matrix}

\begin{matrix}0 ⟶ V_{ε_{1}+ε_{2}−ε_{m−p+q}+δ_{1}−δ_{p+n−q−1}−δ_{p+n−q}} ⟶ Y \\ ⟶ V_{ε_{1}+ε_{2}−ε_{m−p+q}−δ_{p+n−q}} ⟶ 0 & (p ≥ 2); \end{matrix}

if n = p + q + 1, m = p, then H_{𝔤_{0}}^{1,2} = V_{ε_{1}+ε_{2}−ε_{q}+δ_{1}−2δ_{p+n−q}} ⊕ Y (q ≥ 3), and H_{𝔤_{0}}^{1,2} = Y (q = 2), where

0 ⟶ V_{2ε_{1}−ε_{q}+δ_{1}−δ_{2}−δ_{3}} ⊕ V_{2ε_{1}−ε_{q}−δ_{1}+δ_{2}−δ_{3}} ⟶ Y ⟶ V_{2ε_{1}−ε_{q}−δ_{3}} ⟶ 0  (p = 1),
0 ⟶ V_{2ε_{1}−ε_{q}+δ_{1}−δ_{p+n−q−1}−δ_{p+n−q}} ⟶ Y ⟶ V_{2ε_{1}−ε_{q}−δ_{p+n−q}} ⟶ 0  (p ≥ 2);

if m = p + q − 1, n ≠ p + q ± 1, q, then H_{𝔤_{0}}^{1,2} = X ⊕ V_{ε_{1}+ε_{2}−ε_{m−p+q}+δ_{1}−δ_{p+n−q−1}−δ_{p+n−q}}, where X is given by the nonsplit exact sequence of 𝔤_{0}-modules

\begin{matrix}0 ⟶ V_{2ε_{1}−ε_{3}+δ_{1}−2δ_{p+n−2}} ⊕ V_{−ε_{1}+ε_{2}+ε_{3}+δ_{1}−2δ_{p+n−2}} ⟶ \\ X ⟶ V_{ε_{1}+δ_{1}−2δ_{p+n−2}} ⟶ 0 & (q = 2), \end{matrix}

0 ⟶ V_{2ε_{1}−ε_{m−p+q}+δ_{1}−2δ_{p+n−q}} ⟶ X ⟶ V_{ε_{1}+δ_{1}−2δ_{p+n−q}} ⟶ 0  (q ≥ 3);

if m = p + q − 1, n = q, then H_{𝔤_{0}}^{1,2} = X ⊕ V_{ε_{1}+ε_{2}−ε_{m−p+q}+δ_{1}−2δ_{p}} (p ≥ 3), where X is given by the nonsplit exact sequence of 𝔤_{0}-modules

\begin{matrix}0 ⟶ V_{2ε_{1}−ε_{3}+δ_{1}−δ_{p−1}−δ_{p}} ⊕ V_{−ε_{1}+ε_{2}+ε_{3}+δ_{1}−δ_{p−1}−δ_{p}} ⟶ \\ X ⟶ V_{ε_{1}+δ_{1}−δ_{p−1}−δ_{p}} ⟶ 0 & (q = 2), \end{matrix}

0 ⟶ V_{2ε_{1}−ε_{m−p+q}+δ_{1}−δ_{p−1}−δ_{p}} ⟶ X ⟶ V_{ε_{1}+δ_{1}−δ_{p−1}−δ_{p}} ⟶ 0  (q ≥ 3);

if n = p + q − 1, m ≠ p + q ± 1, p, then H_{𝔤_{0}}^{1,2} = X ⊕ V_{ε_{1}+ε_{2}−ε_{m−p+q}+δ_{1}−δ_{p+n−q−1}−δ_{p+n−q}}, where X is given by the nonsplit exact sequence of 𝔤_{0}-modules

\begin{matrix}0 ⟶ V_{2ε_{1}−ε_{m−2+q}+δ_{1}−2δ_{3}} ⊕ V_{2ε_{1}−ε_{m−2+q}−δ_{1}−δ_{2}+δ_{3}} ⟶ \\ X ⟶ V_{2ε_{1}−ε_{m−2+q}−δ_{3}} ⟶ 0 & (p = 2), \end{matrix}

0 ⟶ V_{2ε_{1}−ε_{m−p+q}+δ_{1}−2δ_{p+n−q}} ⟶ X ⟶ V_{2ε_{1}−ε_{m−p+q}−δ_{p+n−q}} ⟶ 0  (p ≥ 3);

if n = p + q − 1, m = p, then H_{𝔤_{0}}^{1,2} = X ⊕ V_{2ε_{1}−ε_{q}+δ_{1}−δ_{p+n−q−1}−δ_{p+n−q}} (q ≥ 3), where X is given by the nonsplit exact sequence of 𝔤_{0}-modules

\begin{matrix}0 ⟶ V_{ε_{1}+ε_{2}−ε_{q}+δ_{1}−2δ_{3}} ⊕ V_{ε_{1}+ε_{2}−ε_{q}−δ_{1}−δ_{2}+δ_{3}} ⟶ \\ X ⟶ V_{ε_{1}+ε_{2}−ε_{q}−δ_{3}} ⟶ 0 & (p = 2), \end{matrix}

0 ⟶ V_{ε_{1}+ε_{2}−ε_{q}+δ_{1}−2δ_{p+n−q}} ⟶ X ⟶ V_{ε_{1}+ε_{2}−ε_{q}−δ_{p+n−q}} ⟶ 0  (p ≥ 3);

if m = n = p + q + 1, then H_{𝔤_{0}}^{1,2} = X ⊕ V_{2ε_{1}−ε_{m−p+q}+δ_{1}−2δ_{p+n−q}} ⊕ V_{ε_{1}−δ_{p+n−q}}, where X is given by the nonsplit exact sequence of 𝔤_{0}-modules

0 ⟶ Y ⟶ X ⟶ V_{ε_{1}−δ_{p+n−q}} ⟶ 0,

and Y is given by the nonsplit exact sequence of 𝔤_{0}-modules

\begin{matrix}0 ⟶ V_{ε_{1}+ε_{2}−ε_{m−p+q}+δ_{1}−δ_{p+n−q−1}−δp+n−q} ⟶ Y ⟶ V_{ε_{1}+δ_{1}−δ_{p+n−q−1}−δp+n−q}⊕ \\ V_{ε_{1}+ε_{2}−ε_{m−p+q}−δp+n−q} ⟶ 0 & (p ≥ 2, q ≥ 2), \end{matrix}

\begin{matrix}0 ⟶ V_{ε_{1}+ε_{2}−ε_{m−1+q}+δ_{1}−δ_{2}−δ_{3}} ⊕ V_{ε_{1}+ε_{2}−ε_{m−1+q}−δ_{1}+δ_{2}−δ_{3}} ⟶ Y ⟶ V_{ε_{1}+ε_{2}−ε_{m−1+q}−δ_{3}}⊕ \\ V_{ε_{1}+δ_{1}−δ_{2}−δ_{3}} ⊕ V_{ε_{1}−δ_{1}+δ_{2}−δ_{3}} ⟶ 0 & (p = 1, q ≥ 2), \end{matrix}

0 ⟶ V_{ε_{1}+ε_{2}−ε_{3}+δ_{1}−δ_{p+n−2}−δ_{p+n−1}} ⊕ V_{ε_{1}−ε_{2}+ε_{3}+δ_{1}−δ_{p+n−2}−δ_{p+n−1}} ⟶ Y ⟶ V_{ε_{1}+ε_{2}−ε_{3}−δ_{p+n−1}} ⊕ V_{ε_{1}+δ_{1}−δ_{p+n−2}−δ_{p+n−1}} ⊕ V_{ε_{1}−ε_{2}+ε_{3}−δ_{p+n−1}} ⟶ 0

(p ≥ 2, q = 1),

0 ⟶ V_{ε_{1}+ε_{2}−ε_{3}+δ_{1}−δ_{2}−δ_{3}} ⊕ V_{ε_{1}−ε_{2}+ε_{3}+δ_{1}−δ_{2}−δ_{3}} ⊕ V_{ε_{1}+ε_{2}−ε_{3}−δ_{1}+δ_{2}−δ_{3}} ⊕ V_{ε_{1}−ε_{2}+ε_{3}−δ_{1}+δ_{2}−δ_{3}} ⟶ Y ⟶ V_{ε_{1}+ε_{2}−ε_{3}−δ_{3}} ⊕ V_{ε_{1}−ε_{2}+ε_{3}−δ_{3}} ⊕ V_{ε_{1}+δ_{1}−δ_{2}−δ_{3}}⊕

V_{ε_{1}−δ_{1}+δ_{2}−δ_{3}} ⟶ 0  (p = 1, q = 1)

if m = n = p + q − 1, then H_{𝔤_{0}}^{1,2} = X ⊕ V_{ε_{1}+ε_{2}−ε_{m−p+q}+δ_{1}−δ_{p+n−q−1}−δ_{p+n−q}} ⊕ V_{ε_{1}−δ_{p+n−q}}, where X is given by the nonsplit exact sequence of 𝔤_{0}-modules

0 ⟶ Y ⟶ X ⟶ V_{ε_{1}−δ_{p+n−q}} ⟶ 0,

and Y is given by the nonsplit exact sequence of 𝔤_{0}-modules

0 ⟶ V_{2ε_{1}−ε_{m−p+q}+δ_{1}−2δ_{p+n−q}} ⟶ Y
⟶ V_{2ε_{1}−ε_{m−p+q}−δ_{p+n−q}} ⊕ V_{ε_{1}+δ_{1}−2δ_{p+n−q}} ⟶ 0
(p ≥ 3, q ≥ 3),

SPENCER COHOMOLOGY OF 𝔬𝔰𝔭(m|2n) 57

0 ⟶ V_{2ε_{1}−ε_{m−2+q}+δ_{1}−2δ_{3}} ⊕ V_{2ε_{1}−ε_{m−2+q}−δ_{1}−δ_{2}+δ_{3}} ⟶ Y
⟶ V_{2ε_{1}−ε_{m−2+q}−δ_{3}} ⊕ V_{ε_{1}+δ_{1}−2δ_{3}}⊕
V_{ε_{1}−δ_{1}−δ_{2}+δ_{3}} ⟶ 0  (p = 2, q ≥ 3),
0 ⟶ V_{2ε_{1}−ε_{3}+δ_{1}−2δ_{p+n−2}} ⊕ V_{−ε_{1}+ε_{2}+ε_{3}+δ_{1}−2δ_{p+n−2}} ⟶ Y
⟶ V_{ε_{1}+δ_{1}−2δ_{p+n−2}} ⊕ V_{2ε_{1}−ε_{3}−δ_{p+n−2}}⊕
V_{−ε_{1}+ε_{2}+ε_{3}−δ_{p+n−2}} ⟶ 0  (p ≥ 3, q = 2),
0 ⟶ V_{2ε_{1}−ε_{3}+δ_{1}−2δ_{3}} ⊕ V_{−ε_{1}+ε_{2}+ε_{3}+δ_{1}−2δ_{3}} ⊕ V_{2ε_{1}−ε_{3}−δ_{1}−δ_{2}+δ_{3}}⊕
V_{−ε_{1}+ε_{2}+ε_{3}−δ_{1}−δ_{2}+δ_{3}} ⟶ Y ⟶ V_{2ε_{1}−ε_{3}−δ_{3}} ⊕ V_{−ε_{1}+ε_{2}+ε_{3}−δ_{3}}⊕
V_{ε_{1}+δ_{1}−2δ_{3}} ⊕ V_{ε_{1}−δ_{1}−δ_{2}+δ_{3}} ⟶ 0  (p = 2, q = 2).

Remark. The irreducible 𝔤_{0}-modules in the above listed nonsplit exact sequences are given regardless of their parity, which can be easily recovered from the corresponding highest weights.

Spencer cohomology of 𝔭𝔰𝔮(n)

2.1. Definition of 𝔭𝔰𝔮(n). Set 𝔮(n) = {X ∈ 𝔤𝔩(n|n) | [X, J_{2n}] = 0} for an odd nondegenerate form J_{2n}, such that J_{2n}^{2} = −1_{2n}. The usual choice for J_{2n} is J_{2n} = antidiag(1_{n}, −1_{n}). Then we have:

𝔮(n) = {X ∈ 𝔤𝔩(n|n) | X = diag(A, A) + antidiag(B, B), where A, B ∈ 𝔤𝔩(n)}. Let otrX = trB. Set 𝔰𝔮(n) = {X ∈ 𝔮(n) | otrX = 0},

𝔭𝔰𝔮(n) = 𝔰𝔮(n)/⟨1_{2n}⟩,

𝔰(𝔮(p)⊕𝔮(n−p)) = {X ∈ 𝔮(p)⊕𝔮(n−p) | otr𝔮(p)+otr𝔮(n−p) = 0}, 𝔭𝔰(𝔮(p)⊕𝔮(n−p)) = 𝔰(𝔮(p) ⊕ 𝔮(n − p))/⟨1_{p} + 1_{n−p}⟩

2.2. ℤ-gradings of depth 1 of 𝔭𝔰𝔮(n). Let V (n | n) be the standard 𝔮(n)-module. All ℤ-gradings of depth 1 of 𝔤 = 𝔭𝔰𝔮(n) are of the form 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1}, where 𝔤_{0} = 𝔭𝔰(𝔮(p) ⊕ 𝔮(n − p)), p > 0, and as 𝔤_{0}-modules 𝔤_{1} ≅ 𝔤_{−1}^{∗}, where 𝔤_{−1} is either one of the two irreducible 𝔤_{0}-modules in V (p | p)^{∗} ⊗ V (n − p | n − p). Explicitly:

𝔤_{−1} = ⟨(x ± Π(x)) ⊗ (y ± Π(y))⟩, where x ∈ V (p | p)^{∗}, y ∈ V (n − p | n − p).

Let ε_{1}, …, ε_{p} and δ_{1}, …, δ_{n−p} be the standard bases of the dual spaces to the spaces of diagonal matrices in 𝔮(p) and 𝔮(n − p), respectively.

2.3. Theorem. 1) 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) = 𝔤,

2) all SF s are of order 1 and split into the direct sum of two irreducible 𝔤_{0}-submodules with highest weights 2ε_{1} − ε_{p} + δ_{1} − 2δ_{n−p} and ε_{1} − δ_{n−p}.

Spencer cohomology of 𝔬𝔰𝔭(m|2n)

3.1. Definition of 𝔬𝔰𝔭(m|2n). . 𝔬𝔰𝔭(m|2n) is a Lie superalgebra, which preserves a nondegenerate supersymmetric even bilinear form on a superspace V, dim V = (m|2n).

3.2. Consider the ℤ-grading of depth 1 of 𝔤 = 𝔬𝔰𝔭(m|2n), which is defined as follows: 𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1}, where 𝔤_{0} = 𝔠𝔬𝔰𝔭(m − 2|2n) is the central extension of 𝔤̂_{0} = 𝔬𝔰𝔭(m − 2|2n), 𝔤_{1} ≅ 𝔤_{−1} is the standard 𝔤_{0}-module.

Let m = 2r + 2 or m = 2r + 3, n > 0. Let ε_{1}, …, ε_{r} and δ_{1}, …, δ_{n} be the standard bases of the dual spaces to the spaces of diagonal matrices in 𝔬(m − 2) and 𝔰𝔭(n), respectively.

3.3. Theorem. 1) 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) = 𝔤, 𝔤_{∗}(𝔤_{−1}, 𝔤̂_{0}) = 𝔤_{−1} ⊕ 𝔤̂_{0}.

2) If k ≠ 2, then H_{𝔤_{0}}^{k,2} = Hk,2 ; 𝔤̂_{0} = 0. As a 𝔤̂_{0}-module H2,2 ; 𝔤̂_{0} is isomorphic to S^{2}(E^{2}(𝔤_{−1}))/E^{4}(𝔤_{−1}) and splits into the direct sum of three irreducible components (analogues of the Weyl tensor, the traceless Ricci tensor, and the scalar curvature). The highest weights of these components are listed in Table 18. As 𝔤̂_{0}-modules, H2,2 ; 𝔤̂_{0} ≅ H_{𝔤_{0}}^{2,2} ⊕ S^{2}(𝔤_{−1}). The 𝔤_{0}-module H_{𝔤_{0}}^{2,2} is irreducible.

Spencer cohomology of D(α)

4.1. Definition of D(α). D(α), where α ∈ ℂ\{0, −1}, is a one-parameter family consisting of all simple Lie superalgebras for which D(α)_{0} = 𝔰𝔩(2) ⊕ 𝔰𝔩(2) ⊕ 𝔰𝔩(2) and its representation on D(α)_{1} is 𝔰𝔩(2) ⊗ 𝔰𝔩(2) ⊗ 𝔰𝔩(2).

4.2. ℤ-gradings of depth 1 of D(α). Let ε_{1}, ε_{2}, ε_{3} be the standard basis of the dual space to the space of diagonal matrices in 𝔤𝔩(1|2), V_{λ} be the irreducible 𝔰𝔩(1|2)-module with highest weight λ and an even highest vector. All ℤ-gradings of depth 1 of D(α) are of the form 𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1}, where 𝔤_{0} ≅ 𝔤𝔩(1|2). There are the following possible values of 𝔤_{1} and 𝔤_{−1} for the ℤ-gradings of depth 1:

a) 𝔤_{1} = V_{(1+α)ε_{1}}, 𝔤_{−1} = V_{−αε_{1}},

b) 𝔤_{1} = V_{( \frac{1+α}{α})ε_{1}}, 𝔤_{−1} = V_{− \frac{1}{α} ε_{1}},

c) 𝔤_{1} = V_{( \frac{α}{1+α})ε_{1}}, 𝔤_{−1} = V_{\frac{1}{1+α} ε_{1}}.

More explicitly, let e_{1}^{i}, e_{2}^{i} be the basis of the standard 𝔰𝔩(2)_{i}-module V_{i}, where i = 1, 2, 3. Then the ℤ-grading in case a) can be described as follows: 𝔤_{0} = (𝔤_{0})_{0} ⊕ (𝔤_{0})_{1}, where (𝔤_{0})_{0} = 𝔰𝔩(2)_{1} ⊕ ⟨\binom{1 0}{0−1}_{2}⟩ ⊕ ⟨\binom{1 0}{0−1}_{3}⟩, (𝔤_{0})_{1} = V_{1} ⊗ e_{1}^{2} ⊗ e_{2}^{3} ⊕ V_{1} ⊗ e_{2}^{2} ⊗ e_{1}^{3};

𝔤_{1} = (𝔤_{1})_{0} ⊕ (𝔤_{1})_{1}, where (𝔤_{1})_{0} = ⟨\binom{0 1}{0 0}_{2}⟩ ⊕ ⟨\binom{0 1}{0 0}_{3}⟩, (𝔤_{1})_{1} = V_{1} ⊗ e_{1}^{2} ⊗ e_{1}^{3}; 𝔤_{−1} = (𝔤_{−1})_{0} ⊕ (𝔤_{−1})_{1}, where (𝔤_{−1})_{0} = ⟨\binom{0 0}{1 0}_{2}⟩ ⊕ ⟨\binom{0 0}{1 0}_{3}⟩, (𝔤_{−1})_{1} = V_{1} ⊗ e_{2}^{2} ⊗ e_{2}^{3}; The ℤ-gradings in cases b) and c) can be described similarly.

4.3. Theorem. For all ℤ-gradings of depth 1 of 𝔤 = D(α) we have

1) 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) = 𝔤.

2) The nonzero SFs are of order 2, and for the cases considered in sec.4.2, the 𝔤_{0}-module H_{𝔤_{0}}^{2,2} is isomorphic to

a) Π(V_{(2α+1)ε_{1}+ε_{2}}),

b) Π(V_{( \frac{α+2}{α})ε_{1}+ε_{2}}),

c) Π(V_{( \frac{α−1}{α+1} )ε_{1}+ε_{2}}), respectively.

Spencer cohomology of AB_{3}

Definition of AB_{3}. AB_{3} is a simple Lie superalgebra for which (AB_{3})_{0} = 𝔰𝔩(2) ⊕ 𝔬(7) and its representation on (AB_{3})_{1} is 𝔰𝔩(2) ⊗ spin_{7}.

5.2. ℤ-grading of depth 1 of AB_{3}. Let ε_{1}, δ_{1}, δ_{2} be the standard basis of the dual space to the space of diagonal matrices in 𝔬𝔰𝔭(2|4), V_{λ} be an irreducible 𝔬𝔰𝔭(2|4)-module with highest weight λ and an even highest vector.

There is only one ℤ-grading of depth 1 in 𝔤 = AB_{3}, namely, 𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1}, where 𝔤_{−1} = V_{−ε_{1}+δ_{1}+δ_{2}}, 𝔤_{0} = 𝔠𝔬𝔰𝔭(2|4), 𝔤_{1} = V_{3ε_{1}}.

Note that 𝔬(7) = V_{1} ⊕ 𝔬(5) ⊕ ℂ ⊕ V_{2}, where V_{1}, V_{2} are standard 𝔬(5)-modules. The space of the representation spin_{7} after restriction of 𝔬(7) to 𝔬(5) decomposes into the direct sum of two irreducible subspaces, which we denote by U_{1} and U_{2}. Let e_{1}, e_{2} be the basis of the standard 𝔰𝔩(2)-module V. Then 𝔤_{0} = (𝔤_{0})_{0} ⊕ (𝔤_{0})_{1}, where

(𝔤_{0})_{0} = ⟨\binom{1 0}{0−1}⟩ ⊕ 𝔬(5) ⊕ ℂ, (𝔤_{0})_{1} = e_{1} ⊗ U_{1} ⊕ e_{2} ⊗ U_{2};

𝔤_{1} = (𝔤_{1})_{0} ⊕ (𝔤_{1})_{1}, where (𝔤_{1})_{0} = ⟨\binom{0 1}{0 0}⟩ ⊕ V_{2}, (𝔤_{1})_{1} = e_{1} ⊗ U_{2};

𝔤_{−1} = (𝔤_{−1})_{0} ⊕ (𝔤_{−1})_{1}, where (𝔤_{−1})_{0} = ⟨\binom{0 0}{1 0}⟩ ⊕ V_{1}, (𝔤_{−1})_{1} = e_{2} ⊗ U_{1}.

SPENCER COHOMOLOGY OF AB_{3} 59

5.3. Theorem. 1) 𝔤_{∗}(𝔤_{−1}, 𝔤_{0}) = 𝔤.

2) The nonzero SFs are of order 1. The 𝔤_{0}- module H_{𝔤_{0}}^{1,2} is given by the nonsplit exact sequence of 𝔤_{0}- modules

0 ⟶ X ⟶ H_{𝔤_{0}}^{1,2} ⟶ V_{ε_{1}+2δ_{1}} ⟶ 0,

where X is given by the nonsplit exact sequence of 𝔤_{0}-modules

0 ⟶ Π(V_{4ε_{1}+2δ_{1}+δ_{2}}) ⟶ X ⟶ V_{3ε_{1}+2δ_{1}} ⟶ 0.

Appendix. The formula of dimensions of irreducible 𝔰𝔩(n)-modules

Let ε_{1}, …, ε_{n} be the standard basis of the dual space to the space of diagonal matrices in 𝔤𝔩(n), V_{λ} be the irreducible 𝔰𝔩(n)-module with highest weight λ = k_{1}ε_{1} + k_{2}ε_{2} + … + k_{n}ε_{n}, where k_{i} ∈ ℤ. Then

dim V_{λ} = Π_{i=1}^{n−1}Π_{j=1}^{n−i}(1 + \frac{k_{i} − k_{i+j}}{j}).

Proof. A weight λ is the highest weight of an irreducible 𝔰𝔩(n)-module if and only if λ is a dominant integer form, i.e., if

2(λ, α_{i})/(α_{i}, α_{i}) ∈ ℤ_{+}.

It is known [GG] that the inner products of the weights ε_{i} and of weight ρ, where ρ = (∑_{β∈Δ_{+}} β)/2, with fundamental weights α_{j} are:

(ε_{1}, α_{1}) = 1/(2n), (ε_{1}, α_{j}) = 0  for 2 ≤ j ≤ n − 1;

(ε_{i}, α_{i−1}) = −1/(2n), (ε_{i}, α_{i}) = 1/(2n), (ε_{i}, α_{j}) = 0 (j ≠ i − 1, i)

for 2 ≤ i ≤ n − 1;
(ε_{n}, α_{n−1}) = −1/(2n), (ε_{n}, α_{j}) = 0  for 1 ≤ j ≤ n − 2;
(ρ, α_{i}) = 1/(2n)  for 1 ≤ i ≤ n − 1.

Thus,

(λ, α_{i}) = \frac{k_{i} − k_{i+1}}{2n} and k_{i} ≥ k_{i+1}.

By Weyl’s character formula [GG]

dim V_{λ} = Π_{β∈Δ_{+}}(1 + \frac{(λ, β)}{(ρ, β)}).

For 𝔰𝔩(n) we have Δ_{+} = {α_{i} + α_{i+1} + … + α_{j}, where 1 ≤ i ≤ n − 1, j ≥ i}. Since

(λ, α_{i} + … + α_{j}) = 1/(2n)((k_{i} − k_{i+1}) + (k_{i+1} − k_{i+2}) + … + (k_{j} − k_{j+1})) = \frac{k_{i} − k_{j+1}}{2n} ,

we have

\begin{matrix}Π_{β∈Δ_{+}}(1 + \frac{(λ, β)}{(ρ, β)}) = Π_{i=1}^{n−1}Π_{j=i}^{n−1}(1 + \frac{(\frac{k_{i}−k_{j+1}}{2n})}{(\frac{j−i+1}{2n})}) = \\ = Π_{i=1}^{n−1}Π_{j=1}^{n−i}(1 + \frac{k_{i} − k_{i+j}}{j}). \end{matrix}

□

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Tables

Table 1. Irreducible 𝔤𝔩(n)-submodules of 𝔠𝔭𝔢(n) ⊗ V^{∗}

𝔤𝔩(n)-submodule Highest weight  Highest vector
E^{2}V_{0}^{∗} ⊗ V_{0} ε_{1} − ε_{n−1} − ε_{n} f_{n−1} ∧ f̃_{n} ⊗ ẽ_{1}
−ε_{n}  ∑_{i=1}^{n} f_{n} ∧ f̃_{i} ⊗ ẽ_{i}
E^{2}V_{0}^{∗} ⊗ V_{0}^{∗} −ε_{n−1} − 2ε_{n} f_{n−1} ∧ f̃_{n} ⊗ f̃_{n}
−ε_{n−2} − ε_{n−1} − ε_{n} f_{n−2} ∧ f̃_{n−1} ⊗ f̃_{n} + f_{n−1} ∧ f̃_{n} ⊗ f̃_{n−2}

+f_{n} ∧ f̃_{n−2} ⊗ f̃_{n−1}

V_{0}^{∗} ∧ V_{0} ⊗ V_{0} 2ε_{1} − ε_{n} f_{n} ∧ ẽ_{1} ⊗ ẽ_{1}
ε_{1} + ε_{2} − ε_{n}  f_{n} ∧ ẽ_{1} ⊗ ẽ_{2} − f_{n} ∧ ẽ_{2} ⊗ e_{1}
ε_{1}  ∑_{i=1}^{n} f_{i} ∧ ẽ_{i} ⊗ ẽ_{1}
ε_{1}  ∑_{i=1}^{n} f_{i} ∧ ẽ_{1} ⊗ ẽ_{i}
V_{0}^{∗} ∧ V_{0} ⊗ V_{0}^{∗} ε_{1} − 2ε_{n} f_{n} ∧ ẽ_{1} ⊗ f̃_{n}
ε_{1} − ε_{n−1} − ε_{n}  f_{n−1} ∧ ẽ_{1} ⊗ f̃_{n} − f_{n} ∧ ẽ_{1} ⊗ f̃_{n−1}
−ε_{n}  ∑_{i=1}^{n} f_{i} ∧ ẽ_{i} ⊗ f̃_{n}
−ε_{n}  ∑_{i=1}^{n} f_{n} ∧ ẽ_{i} ⊗ f̃_{i}
S^{2}V_{0} ⊗ V_{0} 3ε_{1} e_{1}ẽ_{1} ⊗ ẽ_{1}
2ε_{1} + ε_{2}  e_{1}ẽ_{2} ⊗ ẽ_{1} − e_{1}ẽ_{1} ⊗ ẽ_{2}
S^{2}V_{0} ⊗ V_{0}^{∗} 2ε_{1} − ε_{n} e_{1}ẽ_{1} ⊗ f̃_{n}
ε_{1}  ∑_{i=1}^{n} e_{1}ẽ_{i} ⊗ f̃_{i}
z ⊗ V_{0}  ε_{1}  ∑_{i=1}^{n} e_{i}f̃_{i} ⊗ ẽ_{1}
z ⊗ V_{0}^{∗}  −ε_{n}  ∑_{i=1}^{n} e_{i}f̃_{i} ⊗ f̃_{n}

Table 2. Irreducible 𝔤𝔩(n)-submodules of H^{q}(V_{0}, 𝔤_{∗})

q Highest weight  Highest vector
0 −ε_{n} f_{n}
2ε_{1}  e_{1}ẽ_{1}
ε_{1}  e_{1}
0  τ − z
1 −2ε_{n} f_{n} ⊗ f̃_{n}
2ε_{1} − ε_{n}  (e_{1}ẽ_{1}) ⊗ f̃_{n}
ε_{1} − 2ε_{n}  (e_{1} ∧ f̃_{n}) ⊗ f̃_{n}
−ε_{n}  (τ − z) ⊗ f̃_{n}
2 −2ε_{n−1} − 2ε_{n} (f_{n−1} ∧ f̃_{n}) ⊗ f̃_{n−1} ∧ f̃_{n}
2ε_{1} − ε_{n−1} − ε_{n}  e_{1}ẽ_{1} ⊗ f̃_{n−1} ∧ f̃_{n}
ε_{1} − ε_{n−1} − 2ε_{n} (e_{1} ∧ f̃_{n}) ⊗ f̃_{n−1} ∧ f̃_{n}
−ε_{n−1} − ε_{n}  (τ − z) ⊗ f̃_{n−1} ∧ f̃_{n}

66 TABLES

Table 3. Irreducible 𝔤𝔩(n)-submodules of E_{1}^{p,0}

weight E_{1}^{1,0} = ⊕_{λ}V_{λ} ⊗ V_{0} E_{1}^{2,0} = ⊕_{λ}V_{λ} ⊗ S^{2}V_{0} E_{1}^{3,0} = ⊕_{λ}V_{λ} ⊗ S^{3}V_{0}
λ  V_{λ} ⊗ V_{0}  V_{λ} ⊗ S^{2}V_{0}  V_{λ} ⊗ S^{3}V_{0}
−ε_{n}  ε_{1} − ε_{n}  2ε_{1} − ε_{n}  3ε_{1} − ε_{n}
0  ε_{1}  2ε_{1}
2ε_{1}  3ε_{1}  4ε_{1}  5ε_{1}
2ε_{1} + ε_{2}  2ε_{1} + 2ε_{2}  4ε_{1} + ε_{2}
3ε_{1} + ε_{2}  3ε_{1} + 2ε_{2}
ε_{1}  2ε_{1}  3ε_{1}  4ε_{1}
ε_{1} + ε_{2}  2ε_{1} + ε_{2}  3ε_{1} + ε_{2}
0  ε_{1}  2ε_{1}  3ε_{1}

Table 4. Irreducible 𝔤𝔩(n)-submodules of E_{1}^{p,1}

weight E_{1}^{0,1} = ⊕_{λ}V_{λ} E_{1}^{1,1} = ⊕_{λ}V_{λ} ⊗ V_{0} E_{1}^{2,1} = ⊕_{λ}V_{λ} ⊗ S^{2}V_{0}
λ  V_{λ}  V_{λ} ⊗ V_{0}  V_{λ} ⊗ S^{2}V_{0}
−2ε_{n}  −2ε_{n}  ε_{1} − 2ε_{n}  2ε_{1} − 2ε_{n}
−ε_{n}  ε_{1} − ε_{n}

0

2ε_{1} − ε_{n} 2ε_{1} − ε_{n} 3ε_{1} − ε_{n} 4ε_{1} − ε_{n}
2ε_{1}  3ε_{1}
2ε_{1} + ε_{2} − ε_{n}  3ε_{1} + ε_{2} − ε_{n}
2ε_{1} + 2ε_{2} − ε_{n}

2ε_{1} + ε_{2}

ε_{1} − 2ε_{n} ε_{1} − 2ε_{n} 2ε_{1} − 2ε_{n} 3ε_{1} − 2ε_{n}
ε_{1} + ε_{2} − 2ε_{n}  2ε_{1} − ε_{n}
ε_{1} − ε_{n}  ε_{1}
2ε_{1} + ε_{2} − 2ε_{n}
ε_{1} + ε_{2} − ε_{n}
−ε_{n} −ε_{n} ε_{1} − ε_{n} 2ε_{1} − ε_{n}
0  ε_{1}

Table 5. Irreducible 𝔤𝔩(n)-submodules of E_{1}^{p,2}

weight  E_{1}^{0,2} = ⊕_{λ}V_{λ} E_{1}^{1,2} = ⊕_{λ}V_{λ} ⊗ V_{0}
λ V_{λ} V_{λ} ⊗ V_{0}
−2ε_{n−1} − 2ε_{n} −2ε_{n−1} − 2ε_{n} ε_{1} − 2ε_{n−1} − 2ε_{n}
−ε_{n−1} − 2ε_{n}
2ε_{1} − ε_{n−1} − ε_{n} 2ε_{1} − ε_{n−1} − ε_{n} 3ε_{1} − ε_{n−1} − ε_{n}
2ε_{1} − ε_{n}
ε_{1} − ε_{n−1} − 2ε_{n} ε_{1} − ε_{n−1} − 2ε_{n} 2ε_{1} − ε_{n−1} − 2ε_{n}
ε_{1} − 2ε_{n}
ε_{1} − ε_{n−1} − ε_{n}
−ε_{n−1} − ε_{n} −ε_{n−1} − ε_{n} ε_{1} − ε_{n−1} − ε_{n}
−ε_{n}

TABLE 8. IRREDUCIBLE 𝔤𝔩(m) ⊕ 𝔤𝔩(n)-SUBMODULES OF 𝔤_{0} ⊗ 𝔤_{−1}^{∗} 67

Table 6. Irreducible 𝔤𝔩(n)-submodules of E_{0}^{p,0}

E_{0}^{1,0} = ⊕_{U}U ⊗ V_{0} E_{0}^{2,0} = ⊕_{U}U ⊗ S^{2}V_{0} E_{0}^{3,0} = ⊕_{U}U ⊗ S^{3}V_{0}
U  U ⊗ V_{0}  U ⊗ S^{2}V_{0}  U ⊗ S^{3}V_{0}
V_{0}(mult2)  2ε_{1}  3ε_{1}  4ε_{1}
ε_{1} + ε_{2}  2ε_{1} + ε_{2}  3ε_{1} + ε_{2}
V_{0}^{∗}(mult2)  ε_{1} − ε_{n}  2ε_{1} − ε_{n}  3ε_{1} − ε_{n}
0  ε_{1}  2ε_{1}
S^{2}V_{0}  3ε_{1}  4ε_{1}  5ε_{1}
2ε_{1} + ε_{2}  3ε_{1} + ε_{2}  4ε_{1} + ε_{2}
2ε_{1} + 2ε_{2}  3ε_{1} + 2ε_{2}
E^{2}V_{0}^{∗}  ε_{1} − ε_{n−1} − ε_{n}  2ε_{1} − ε_{n−1} − ε_{n}  3ε_{1} − ε_{n−1} − ε_{n}
−ε_{n}  ε_{1} − ε_{n}  2ε_{1} − ε_{n}
𝔰𝔩(n)  2ε_{1} − ε_{n}  3ε_{1} − ε_{n}  4ε_{1} − ε_{n}
ε_{1}  2ε_{1}  3ε_{1} + ε_{2} − ε_{n}
ε_{1} + ε_{2} − ε_{n}  2ε_{1} + ε_{2} − ε_{n}  3ε_{1}
ε_{1} + ε_{2}  2ε_{1} + ε_{2}
ℂ(mult3)  ε_{1}  2ε_{1}  3ε_{1}

Table 7. Irreducible 𝔤𝔩(n)-submodules of E^{3}V

Space Highest vectors Highest weights
E^{3}V_{0}  e_{1} ∧ e_{2} ∧ e_{3}  0
(E^{2}V_{0})(V_{0}^{∗})  (e_{1} ∧ e_{2})f_{3}  −2ε_{3}
∑_{i=1}^{3}(e_{1} ∧ e_{i})f_{i}  ε_{1}
V_{0}(S^{2}V_{0}^{∗}) e_{1}f_{3}^{2} ε_{1} − 2ε_{3}
∑_{i=1}^{3} e_{i}f_{i}f_{3}  −ε_{3}
S^{3}V_{0}^{∗}  f_{3}^{3}  −3ε_{3}

Table 8. Irreducible 𝔤𝔩(m) ⊕ 𝔤𝔩(n)-submodules of 𝔤_{0} ⊗ 𝔤_{−1}^{∗}
𝔤𝔩(m) ⊕ 𝔤𝔩(n)-module Highest weight  Highest vector
(V ⊗ V^{∗})/ℂ ⊗ (U^{∗} ⊗ V) 2ε_{1} − ε_{m} − δ_{n}  (e_{1} ⊗ ẽ_{m}) ⊗ (f̃_{n} ⊗ e_{1})
ε_{1} − δ_{n}  v_{λ}^{1} = ∑_{i=1}^{m}(e_{i} ⊗ ẽ_{i}) ⊗ (f̃_{n} ⊗ e_{1})−

−m ∑_{i=1}^{m}(e_{1} ⊗ ẽ_{i}) ⊗ (f̃_{n} ⊗ e_{i})

ε_{1} + ε_{2} − ε_{m} − δ_{n}  (e_{1} ⊗ ẽ_{m}) ⊗ (f̃_{n} ⊗ e_{2})−
(if m ≥ 3)  −(e_{2} ⊗ ẽ_{m}) ⊗ (f̃_{n} ⊗ e_{1})
(U ⊗ U^{∗})/ℂ ⊗ (U^{∗} ⊗ V) ε_{1} + δ_{1} − 2δ_{n} (f_{1} ⊗ f̃_{n}) ⊗ (f̃_{n} ⊗ e_{1})
ε_{1} − δ_{n}  v_{λ}^{2} = ∑_{i=1}^{n}(f_{i} ⊗ f̃_{i}) ⊗ (f̃_{n} ⊗ e_{1})−

−n ∑_{i=1}^{n}(f_{i} ⊗ f̃_{n}) ⊗ (f̃_{i} ⊗ e_{1})

ε_{1} + δ_{1} − δ_{n−1} − δ_{n}  (f_{1} ⊗ f̃_{n−1}) ⊗ (f̃_{n} ⊗ e_{1})−
(if n ≥ 3)  −(f_{1} ⊗ f̃_{n}) ⊗ (f̃_{n−1} ⊗ e_{1})
ℂ ⊗ (U^{∗} ⊗ V) ε_{1} − δ_{n} v_{λ}^{3} = (n ∑_{i=1}^{m} e_{i} ⊗ ẽ_{i}+

+m ∑_{i=1}^{n} f_{i} ⊗ f̃_{i}) ⊗ (f̃_{n} ⊗ e_{1})

Table 9. Irreducible 𝔤𝔩(m) ⊕ 𝔤𝔩(n)-submodules of (U^{∗} ⊗ V) ⊗ (U^{∗} ⊗ V)

𝔤𝔩(m) ⊕ 𝔤𝔩(n)-module Highest weight  Highest vector
Λ^{2}U^{∗} ⊗ S^{2}V 2ε_{1} − 2δ_{n} (f̃_{n} ⊗ e_{1}) ⊗ (f̃_{n} ⊗ e_{1})
S^{2}U^{∗} ⊗ Λ^{2}V  ε_{1} + ε_{2} − δ_{n−1} − δ_{n} (f̃_{n} ⊗ e_{1}) ⊗ (f̃_{n−1} ⊗ e_{2})−
−(f̃_{n} ⊗ e_{2}) ⊗ (f̃_{n−1} ⊗ e_{1})−
−(f̃_{n−1} ⊗ e_{1}) ⊗ (f̃_{n} ⊗ e_{2})+
+(f̃_{n−1} ⊗ e_{2}) ⊗ (f̃_{n} ⊗ e_{1})
Λ^{2}U^{∗} ⊗ Λ^{2}V ε_{1} + ε_{2} − 2δ_{n} (f̃_{n} ⊗ e_{1}) ⊗ (f̃_{n} ⊗ e_{2})−
−(f̃_{n} ⊗ e_{2}) ⊗ (f̃_{n} ⊗ e_{1})
S^{2}U^{∗} ⊗ S^{2}V 2ε_{1} − δ_{n−1} − δ_{n} (f̃_{n−1} ⊗ e_{1}) ⊗ (f̃_{n} ⊗ e_{1})−
−(f̃_{n} ⊗ e_{1}) ⊗ (f̃_{n−1} ⊗ e_{1})

Table 10. Irreducible 𝔤𝔩(n)-submodules of C_{𝔤𝔩(n)}^{k,2}

k  Highest weight  Highest vector
2  λ_{1} = δ_{1} − 3δ_{n}  v_{λ_{1}} = (f_{1} ⊗ f̃_{n}) ⊗ f̃_{n}^{2}
λ_{2} = −2δ_{n}  v_{λ_{2}} = ∑_{j=1}^{n}(f_{j} ⊗ f̃_{n}) ⊗ f̃_{j} f̃_{n}
λ_{3} = −2δ_{n}  v_{λ_{3}} = ∑_{j=1}^{n}(f_{j} ⊗ f̃_{j}) ⊗ f̃_{n}^{2}
λ_{4} = δ_{1} − δ_{n−1} − 2δ_{n}( if n ≥ 3)  v_{λ_{4}} = (f_{1} ⊗ f̃_{n−1}) ⊗ f̃_{n}^{2} − (f_{1} ⊗ f̃_{n}) ⊗ f̃_{n−1}f̃_{n}
λ_{5} = −δ_{n−1} − δ_{n}  v_{λ_{5}} = ∑_{j=1}^{n}((f_{j} ⊗ f̃_{n−1}) ⊗ f̃_{j} f̃_{n} − (f_{j} ⊗ f̃_{n}) ⊗ f̃_{j} f̃_{n−1})
3 ≤ k ≤ n + 1 λ_{1} = δ_{1} − δ_{n−k+2} − … − δ_{n−1} − 3δ_{n}  v_{λ_{1}} = (f_{1} ⊗ f̃_{n−k+2} ∧ … ∧ f̃_{n}) ⊗ f̃_{n}^{2}
λ_{2} = −δ_{n−k+2} − … − δ_{n−1} − 2δ_{n}  v_{λ_{2}} = ∑_{j=1}^{n}(f_{j} ⊗ f̃_{n−k+2} ∧ … ∧ f̃_{n}) ⊗ f̃_{j} f̃_{n}
3 ≤ k ≤ n λ_{3} = −δ_{n−k+3} − … − δ_{n} − 2δ_{n} v_{λ_{3}} = ∑_{j=1}^{n}(f_{j} ⊗ f̃_{j} ∧ f̃_{n−k+3} ∧ … ∧ f̃_{n}) ⊗ f̃_{n}^{2}
λ_{4} = δ_{1} − δ_{n−k+1} − … − δ_{n−1} − 2δ_{n}  v_{λ_{4}} = ∑_{j=0}^{k−1}(−1)^{(k−1)j}(f_{1} ⊗ f̃_{s^{j} (n−k+1)} ∧ …
∧f̃_{s^{j} (n−1)}) ⊗ f̃_{s^{j} (n)}f̃_{n}
λ_{5} = −δ_{n−k+1} − δ_{n−k+2} − … − δ_{n}  v_{λ_{5}} = ∑_{j=0}^{k−1}(−1)^{(k−1)j} ∑_{i=1}^{n}(f_{i} ⊗ f̃_{s^{j} (n−k+1)} ∧ …
∧f̃_{s^{j} (n−1)}) ⊗ f̃_{i}f̃_{s^{j} (n)}
3 ≤ k ≤ n − 1 λ_{6} = −δ_{n−k+2} − … − δ_{n−1} − 2δ_{n} v_{λ_{6}} = ∑_{j=0}^{k−2}(−1)^{(k−2)j} ∑_{i=1}^{n}(f_{i} ⊗ f̃_{i} ∧ f̃_{t^{j} (n−k+2)} ∧ …
∧f̃_{t^{j} (n−1)}) ⊗ f̃_{t^{j} (n)}f̃_{n}

Table 11. Irreducible 𝔤𝔩(n)-submodules of C_{𝔤𝔩(n)}^{k+1,1}

k  Highest weight  Highest vector
2 ≤ k ≤ n β_{1} = δ_{1} − δ_{n−k+1} − … − δ_{n−1} − 2δ_{n}  v_{β_{1}} = (f_{1} ⊗ f̃_{n−k+1} ∧ … ∧ f̃_{n−1} ∧ f̃_{n}) ⊗ f̃_{n}
β_{2} = −δ_{n−k+1} − … − δ_{n−1} − δ_{n}  v_{β_{2}} = ∑_{j=1}^{n} f_{j} ⊗ f̃_{n−k+1} ∧ … ∧ f̃_{n−1} ∧ f̃_{n}) ⊗ f̃_{j}
2 ≤ k ≤ n − 1 β_{3} = −δ_{n−k+2} − … − δ_{n−1} − 2δ_{n}  v_{β_{3}} = ∑_{j=6}^{n}(f_{j} ⊗ f̃_{j} ∧ f̃_{n−k+2} ∧ … ∧ f̃_{n−1} ∧ f̃_{n}) ⊗ f̃_{n}
β_{4} = −δ_{n−k+1} − … − δ_{n−1} − δ_{n}  v_{β_{4}} = ∑_{j=0}^{k−1}(−1)^{(k−3)j} ∑_{i=1}^{n}(f_{i} ⊗ f̃_{i} ∧ f̃_{s^{j} (n−k+1)} ∧ …
∧f_{s^{j} (n−1)}) ⊗ f̃_{s^{j} (n)}
2 ≤ k ≤ n − 2 β_{5} = δ_{1} − δ_{n−k} − … − δ_{n−1} − δ_{n}  v_{β_{5}} = ∑_{j=0}^{k}(−1)^{kj}(f_{1} ⊗ f̃_{r^{j} (n−k)} ∧ …
∧f̃_{r^{j} (n−1)}) ⊗ f̃_{r^{j} (n)}

Table 12. Irreducible 𝔤𝔩(n)-submodules of 𝔤_{k}

k  Highest weight  Highest vector
2 ≤ k ≤ n − 1 γ_{1} = δ_{1} − δ_{n−k} − δ_{n−k+1} − … − δ_{n}  v_{γ_{1}} = f_{1} ⊗ f̃_{n−k} ∧ … ∧ f̃_{n}
2 ≤ k ≤ n − 2 γ_{2} = −δ_{n−k+1} − δ_{n−k+2} − … − δ_{n} v_{γ_{2}} = ∑_{j=1}^{n} f_{j} ⊗ f̃_{j} ∧ f̃_{n−k+1} ∧ … ∧ f̃_{n}

TABLE 14. IRREDUCIBLE 𝔤𝔩(m) ⊕ 𝔤𝔩(n)SUBMODULES OF 𝔤_{0} ⊗ Λ^{2}𝔤_{−1}^{∗} 69

Table 13. Structure functions of 𝔰𝔩(m|n) endowed with the standard ℤ-grading

m n  H_{𝔤_{0}}^{1,2}  H_{𝔤_{0}}^{2,2}
3 2  2ε_{1} − ε_{3} + δ_{1} − 2δ_{2}  ——
2 3  2ε_{1} − ε_{2} + δ_{1} − 2δ_{3}  ——
≥ 4 2  2ε_{1} − ε_{m} + δ_{1} − 2δ_{2}  ε_{1} + ε_{2} + ε_{3} − ε_{m} − δ_{1} − δ_{2}
2 ≥ 4  2ε_{1} − ε_{2} + δ_{1} − 2δ_{n}  ε_{1} + ε_{2} + δ_{1} − δ_{n−2} − δ_{n−1} − δ_{n}

2ε_{1} − ε_{m} + δ_{1} − 2δ_{n}

≥ 3 ≥ 3 ε_{1} + ε_{2} − ε_{m} + δ_{1} − δ_{n−1} − δ_{n}  ——

ε_{1} − δ_{n} (if m = n)

2 2  2ε_{1} − ε_{2} + δ_{1} − 2δ_{2}

——

ε_{1} − δ_{2}

Table 14. Irreducible 𝔤𝔩(m) ⊕ 𝔤𝔩(n)submodules of 𝔤_{0} ⊗ Λ^{2}𝔤_{−1}^{∗}

𝔤𝔩(m) ⊕ 𝔤𝔩(n)-module  Highest weight  Highest vector
V ⊗ V^{∗}/ℂ ⊗ Λ^{2}U^{∗} ⊗ S^{2}V  3ε_{1} − ε_{m} − 2δ_{n}  (e_{1} ⊗ ẽ_{m}) ⊗ (f̃_{n} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{1})
2ε_{1} − 2δ_{n}  v_{λ}^{1} = ∑_{i=1}^{m}(e_{i} ⊗ ẽ_{i}) ⊗ (f̃_{n} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{1})−

m ∑_{i=1}^{m}(e_{1} ⊗ ẽ_{i}) ⊗ (f̃_{n} ⊗ e_{i}) ∧ (f̃_{n} ⊗ e_{1})

ε_{1} + ε_{2} − 2δ_{n}  v_{λ}^{1} = ∑_{i=1}^{m}((e_{1} ⊗ ẽ_{i}) ⊗ (f̃_{n} ⊗ e_{i}) ∧ (f̃_{n} ⊗ e_{2})−

(e_{2} ⊗ ẽ_{i}) ⊗ (f̃_{n} ⊗ e_{i}) ∧ (f̃_{n} ⊗ e_{1}))

2ε_{1} + ε_{2} − ε_{m} − 2δ_{n} (m ≥ 3)  (e_{1} ⊗ ẽ_{m}) ⊗ (f̃_{n} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{2})−

(e_{2} ⊗ ẽ_{m}) ⊗ (f̃_{n} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{1})

V ⊗ V^{∗}/ℂ ⊗ S^{2}U^{∗} ⊗ Λ^{2}V 2ε_{1} + ε_{2} − ε_{m} − δ_{n−1} − δ_{n} (e_{1} ⊗ ẽ_{m}) ⊗ (f̃_{n−1} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{2})−

(e_{1} ⊗ ẽ_{m}) ⊗ (f̃_{n} ⊗ e_{1}) ∧ (f̃_{n−1} ⊗ e_{2})

2ε_{1} − δ_{n−1} − δ_{n} (m ≥ 3)  v_{λ}^{1} = ∑_{i=1}^{m}(e_{1} ⊗ ẽ_{i}) ⊗ (f̃_{n−1} ⊗ e_{i}) ∧ (f̃_{n} ⊗ e_{1})−

(e_{1} ⊗ ẽ_{i}) ⊗ (f̃_{n} ⊗ e_{i}) ∧ (f̃_{n−1} ⊗ e_{1})

ε_{1} + ε_{2} − δ_{n−1} − δ_{n}  v_{λ}^{1} = ∑_{i=1}^{m}((e_{1} ⊗ ẽ_{i}) ⊗ (f̃_{n−1} ⊗ e_{i}) ∧ (f̃_{n} ⊗ e_{2})−
(m ≥ 3)  (e_{1} ⊗ ẽ_{i}) ⊗ (f̃_{n} ⊗ e_{i}) ∧ (f̃_{n−1} ⊗ e_{2})−

(e_{2} ⊗ ẽ_{i}) ⊗ (f̃_{n−1} ⊗ e_{i}) ∧ (f̃_{n} ⊗ e_{1})+ (e_{2} ⊗ ẽ_{i}) ⊗ (f̃_{n} ⊗ e_{i}) ∧ (f̃_{n−1} ⊗ e_{1}))

ε_{1} + ε_{2} + ε_{3} − ε_{m} − δ_{n−1} − δ_{n} ∑_{j=0}^{2}((e_{s^{j} (1)} ⊗ ẽ_{m}) ⊗ (f̃_{n−1} ⊗ e_{s^{j} (2)}) ∧ (f̃_{n} ⊗ e_{s^{j} (3)})−
(m ≥ 4)  (e_{s^{j} (1)} ⊗ ẽ_{m}) ⊗ (f̃_{n} ⊗ e_{s^{j} (2)}) ∧ (f̃_{n−1} ⊗ e_{s^{j} (3)}))
U ⊗ U^{∗}/ℂ ⊗ Λ^{2}U^{∗} ⊗ S^{2}V 2ε_{1} + δ_{1} − 3δ_{n} (f_{1} ⊗ f̃_{n}) ⊗ (f̃_{n} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{1})
2ε_{1} − 2δ_{n}  v_{λ}^{2} = ∑_{i=1}^{n}(f_{i} ⊗ f̃_{i}) ⊗ (f̃_{n} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{1})−

n ∑_{i=1}^{n}(f_{i} ⊗ f̃_{n}) ⊗ (f̃_{i} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{1})

2ε_{1} − δ_{n−1} − δ_{n}  v_{λ}^{2} = ∑_{i=1}^{n}((f_{i} ⊗ f̃_{n−1}) ⊗ (f̃_{i} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{1})−

(f_{i} ⊗ f̃_{n}) ⊗ (f̃_{i} ⊗ e_{1}) ∧ (f̃_{n−1} ⊗ e_{1}))

2ε_{1} + δ_{1} − δ_{n−1} − 2δ_{n}  (f_{1} ⊗ f̃_{n}) ⊗ (f̃_{n−1} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{1})−
(n ≥ 3)  (f_{1} ⊗ f̃_{n−1}) ⊗ (f̃_{n} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{1})
U ⊗ U^{∗}/ℂ ⊗ S^{2}U^{∗} ⊗ Λ^{2}V ε_{1} + ε_{2} + δ_{1} − δ_{n−1} − 2δ_{n} (f_{1} ⊗ f̃_{n}) ⊗ (f̃_{n−1} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{2})−

(f_{1} ⊗ f̃_{n}) ⊗ (f̃_{n} ⊗ e_{1}) ∧ (f̃_{n−1} ⊗ e_{2})

ε_{1} + ε_{2} − 2δ_{n} (n ≥ 3)  v_{λ}^{2} = ∑_{i=1}^{n}((f_{i} ⊗ f̃_{n}) ⊗ (f̃_{i} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{2})−

(f_{i} ⊗ f̃_{n}) ⊗ (f̃_{n} ⊗ e_{1}) ∧ (f̃_{i} ⊗ e_{2}))

ε_{1} + ε_{2} − δ_{n−1} − δ_{n} (n ≥ 3)  v_{λ}^{2} = ∑_{i=1}^{n}((f_{i} ⊗ f̃_{n}) ⊗ (f̃_{i} ⊗ e_{1}) ∧ (f̃_{n−1} ⊗ e_{2})−

(f_{i} ⊗ f̃_{n}) ⊗ (f̃_{i} ⊗ e_{2}) ∧ (f̃_{n−1} ⊗ e_{1})− (f_{i} ⊗ f̃_{n−1}) ⊗ (f̃_{i} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{2})+ (f_{i} ⊗ f̃_{n−1}) ⊗ (f̃_{i} ⊗ e_{2}) ∧ (f̃_{n} ⊗ e_{1}))

ε_{1} + ε_{2} + δ_{1} − δ_{n−2} − δ_{n−1} − δ_{n} ∑_{j=0}^{2}((f_{1} ⊗ f̃_{t^{j} (n−2)}) ⊗ (f̃_{t^{j} (n−1)} ⊗ e_{1}) ∧ (f̃_{t^{j} (n)} ⊗ e_{2})−
(n ≥ 4)  (f_{1} ⊗ f̃_{t^{j} (n−2)}) ⊗ (f̃_{t^{j} (n−1)} ⊗ e_{2}) ∧ (f̃_{t^{j} (n)} ⊗ e_{1}))
ℂ ⊗ Λ^{2}U^{∗} ⊗ S^{2}V 2ε_{1} − 2δ_{n} v_{λ}^{3} = (n ∑_{i=1}^{m} e_{i} ⊗ ẽ_{i}+

+m ∑_{i=1}^{n} f_{i} ⊗ f̃_{i}) ⊗ (f̃_{n} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{1})

ℂ ⊗ S^{2}U^{∗} ⊗ Λ^{2}V ε_{1} + ε_{2} − δ_{n−1} − δ_{n} v_{λ}^{3} = (n ∑_{i=1}^{m} e_{i} ⊗ ẽ_{i}+

+m ∑_{i=1}^{n} f_{i} ⊗ f̃_{i}) ⊗ ((f̃_{n−1} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{2})− (f̃_{n} ⊗ e_{1}) ∧ (f̃_{n−1} ⊗ e_{2}))

Table 15. Irreducible 𝔤𝔩(m) ⊕ 𝔤𝔩(n)-submodules of 𝔤_{1} ⊗ Λ^{2}𝔤_{−1}^{∗}

𝔤𝔩(m) ⊕ 𝔤𝔩(n)-module  Highest weight  Highest vector
(Λ^{2}U^{∗} ⊗ U^{∗}) ⊗ (S^{2}V ⊗ V)  3ε_{1} − 3δ_{n}  (f̃_{n} ⊗ e_{1}) ⊗ (f̃_{n} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{1})
2ε_{1} + ε_{2} − 3δ_{n}  (f̃_{n} ⊗ e_{1}) ⊗ (f̃_{n} ⊗ e_{2}) ∧ (f̃_{n} ⊗ e_{1})−

(f̃_{n} ⊗ e_{2}) ⊗ (f̃_{n} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{1})

3ε_{1} − δ_{n−1} − 2δ_{n}  (f̃_{n−1} ⊗ e_{1}) ⊗ (f̃_{n} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{1})−

(f̃_{n} ⊗ e_{1}) ⊗ (f̃_{n−1} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{1})

2ε_{1} + ε_{2} − δ_{n−1} − 2δ_{n}  v_{λ}^{1} = (f̃_{n} ⊗ e_{1}) ⊗ (f̃_{n−1} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{2})−

(f̃_{n} ⊗ e_{2}) ⊗ (f̃_{n−1} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{1})− (f̃_{n−1} ⊗ e_{1}) ⊗ (f̃_{n} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{2})+ (f̃_{n−1} ⊗ e_{2}) ⊗ (f̃_{n} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{1})

(S^{2}U^{∗} ⊗ U^{∗}) ⊗ (Λ^{2}V ⊗ V) 2ε_{2} + ε_{2} − δ_{n−1} − 2δ_{n} v_{λ}^{2} = (f̃_{n} ⊗ e_{1}) ⊗ (f̃_{n−1} ⊗ e_{1}) ∧ (f̃_{n} ⊗ e_{2})−

(f̃_{n} ⊗ e_{1}) ⊗ (f̃_{n−1} ⊗ e_{2}) ∧ (f̃_{n} ⊗ e_{1})

ε_{1} + ε_{2} + ε_{3} − δ_{n−1} − 2δ_{n}  ∑_{j=0}^{2}((f̃_{n} ⊗ e_{s^{j} (1)}) ⊗ (f̃_{n−1} ⊗ e_{s^{j} (2)}) ∧ (f̃_{n} ⊗ e_{s^{j} (3)})−
(m ≥ 3)  (f̃_{n} ⊗ e_{s^{j} (1)}) ⊗ (f̃_{n} ⊗ e_{s^{j} (2)}) ∧ (f̃_{n−1} ⊗ e_{s^{j} (3)})
2ε_{1} + ε_{2} − δ_{n−2} − δ_{n−1} − δ_{n}  ∑_{j=0}^{2}((f̃_{t^{j} (n−2)} ⊗ e_{1}) ⊗ (f̃_{t^{j} (n−1)} ⊗ e_{2}) ∧ (f̃_{t^{j} (n)} ⊗ e_{1})−
(n ≥ 3)  (f̃_{t^{j} (n−2)} ⊗ e_{1}) ⊗ (f̃_{t^{j} (n−1)} ⊗ e_{1}) ∧ (f̃_{t^{j} (n)} ⊗ e_{2}))
ε_{1} + ε_{2} + ε_{3} − δ_{n−2} − δ_{n−1} − δ_{n} ∑_{i=0}^{2} ∑_{j=0}^{2}((f̃_{t^{j} (n−2)} ⊗ e_{s^{i}(1)}) ⊗ (f̃_{t^{j} (n−1)} ⊗ e_{s^{i}(2)})∧
(m, n ≥ 3)  (f̃_{t^{j} (n)} ⊗ e_{s^{i}(3)}) − (f̃_{t^{j} (n−2)} ⊗ e_{s^{i}(1)})⊗

(f̃_{t^{j} (n)} ⊗ e_{s^{i}(2)}) ∧ (f̃_{t^{j} (n−1)} ⊗ e_{s^{i}(3)}))

Table 16. Spencer cohomology of 𝔰𝔩(m|n) endowed with a ℤ-grading, where 𝔤_{0} = 𝔠(𝔰𝔩(m|q) ⊕ 𝔰𝔩(n − q))

m q n − q  H_{𝔤_{0}}^{1,2}  H_{𝔤_{0}}^{2,2}
≥ 2 ≥ 1 ≥ 3  m ≠ q ± 1

2ε_{1} − ε_{m+q} + δ_{1} − 2δ_{n−q}

——

ε_{1} + ε_{2} − ε_{m+q} + δ_{1} − δ_{n−q−1} − δ_{n−q} ε_{1} − δ_{n−q} (if m = n)

≥ 3 ≥ 1 2  m ≠ q − 1
2ε_{1} − ε_{m+q} + δ_{1} − 2δ_{2}  ε_{1} + ε_{2} + ε_{3} − ε_{m+q} − δ_{1} − δ_{2}

ε_{1} − δ_{2} (if m = n)

2 ≥ 2 2 q ≠ 3 ε_{1} + ε_{2} + ε_{3} − ε_{q+2} − δ_{1} − δ_{2}

2ε_{1} − ε_{q+2} + δ_{1} − 2δ_{2}

2 1 2 2ε_{1} − ε_{3} + δ_{1} − 2δ_{2} ε_{1} + ε_{3} − δ_{1} − δ_{2}
1 ≥ 1 ≥ 3  q ≠ 2
2ε_{1} − ε_{q+1} + δ_{1} − 2δ_{n−q}  ——

ε_{1} + ε_{2} − ε_{q+1} + δ_{1} − δ_{n−q−1} − δ_{n−q}

1 ≥ 1 2 q ≠ 2 ε_{1} + 2ε_{2} − ε_{q+1} − δ_{1} − δ_{2} (q ≠ 1)
2ε_{1} − ε_{q+1} + δ_{1} − 2δ_{2}  2ε_{2} − δ_{1} − δ_{2} (q = 1)
0 2 2 3ε_{1} − ε_{2} − δ_{1} − δ_{2}

——

ε_{1} + ε_{2} + δ_{1} − 3δ_{2}
0 2 ≥ 3  2ε_{1} − ε_{2} + δ_{1} − δ_{n−3} − δ_{n−2}  ε_{1} + ε_{2} + δ_{1} − 3δ_{n−2}
0 ≥ 3 2  ε_{1} + ε_{2} − ε_{q} + δ_{1} − 2δ_{2}  3ε_{1} − ε_{q} − δ_{1} − δ_{2}
0 ≥ 3 ≥ 3  2ε_{1} − ε_{q} + δ_{1} − δ_{n−q−1} − δ_{n−q}

——

ε_{1} + ε_{2} − ε_{q} + δ_{1} − 2δ_{n−q}

TABLE 18. SPENCER COHOMOLOGY OF 𝔬𝔰𝔭(m|2n) 71

Table 17. Spencer cohomology of 𝔰𝔩(m|n) endowed with a ℤ-grading, where

𝔤_{0} = 𝔠(𝔰𝔩(m − p|q) ⊕ 𝔰𝔩(p|n − q))

m − p q  p n − q  H_{𝔤_{0}}^{1,2}  H_{𝔤_{0}}^{2,2}
0  2  2  0  2ε_{1} − ε_{2} + δ_{1} − 2δ_{2}

——

ε_{1} − δ_{2}

0  2  3  0  2ε_{1} − ε_{2} + δ_{1} − 2δ_{3}  ——
0  3  2  0  2ε_{1} − ε_{3} + δ_{1} − 2δ_{2}  ——
0  2 ≥ 4  0  2ε_{1} − ε_{2} + δ_{1} − 2δ_{p}  ε_{1} + ε_{2} + δ_{1} − δ_{p−2} − δ_{p−1} − δ_{p}
0  ≥ 4 2  0  2ε_{1} − ε_{q} + δ_{1} − 2δ_{2}  ε_{1} + ε_{2} + ε_{3} − ε_{q} − δ_{1} − δ_{2}
0  ≥ 3 ≥ 3  0  2ε_{1} − ε_{q} + δ_{1} − 2δ_{p}
ε_{1} + ε_{2} − ε_{q} + δ_{1} − δ_{p−1} − δ_{p}  ——
ε_{1} − δ_{p} (m = n)
0 2 ≥ 1 ≥ 1 n ≠ p + q + 1
2ε_{1} − ε_{2} + δ_{1} − δ_{p+n−3} − δ_{p+n−2}  ε_{1} + ε_{2} + δ_{1} − 3δ_{p+n−2}
ε_{1} − δ_{p+n−2} (m = n)
≥ 1 ≥ 1 2 0 m ≠ p + q + 1
ε_{1} + ε_{2} − ε_{m−p+q} + δ_{1} − 2δ_{2}  3ε_{1} − ε_{m−p+q} − δ_{1} − δ_{2}
ε_{1} − δ_{2} (m = n)
0 ≥ 3 ≥ 1 ≥ 1 n ≠ p + q ± 1
ε_{1} + ε_{2} − ε_{q} + δ_{1} − 2δ_{p+n−q}

——

2ε_{1} − ε_{q} + δ_{1} − δ_{p+n−q−1} − δ_{p+n−q}

ε_{1} − δ_{p+n−q} (m = n)
≥ 1 ≥ 1 ≥ 3 0 m ≠ p + q ± 1
2ε_{1} − ε_{m−p+q} + δ_{1} − δ_{p−1} − δ_{p}

——

ε_{1} + ε_{2} − ε_{m−p+q} + δ_{1} − 2δ_{p}

ε_{1} − δ_{p} (m = n)
≥ 1 ≥ 1 ≥ 1 ≥ 1  m, n ≠ p + q ± 1
2ε_{1} − ε_{m−p+q} + δ_{1} − 2δ_{p+n−q}

——

ε_{1} + ε_{2} − ε_{m−p+q} + δ_{1} − δ_{p+n−q−1} − δ_{p+n−q}

ε_{1} − δ_{p+n−q} (m = n ≥ 3)

Table 18. Spencer cohomology of 𝔬𝔰𝔭(m|2n)

r n H_{𝔤_{0}}^{2,2} S^{2}(𝔤_{−1})
0 (if m = 2)
0 1  ——
0, δ_{1} (if m = 3)
0 ≥ 2 2δ_{1} + 2δ_{2}  δ_{1} + δ_{2}, 0
1 1 ε_{1} + δ_{1} 2ε_{1}, 0
1 ≥ 2 2ε_{1} + δ_{1} + δ_{2}  2ε_{1}, 0
≥ 2 ≥ 1 2ε_{1} + 2ε_{2}  2ε_{1}, 0