The notion of defining relations is well-defined for a nilpotent Lie (super)algebra. One of the ways to present a simple Lie algebra is, therefore, by splitting it into the direct sum of a maximal diagonalizing (commutative) subalgeba and 2 nilpotent subalgebras (positive and negative). The relations obtained for finite dimensional Lie algebras are neat; they are called Serre relations and can be encoded via an integer symmetrizable matrix, the Cartan matrix, which, in turn, can be encoded by means of a graph, the Dynkin diagram. The complete set of relations for Lie algebras with an arbitrary Cartan matrix is unknown.We completely describe presentations of Lie superalgebras with Cartan matrix if they are simple Zgraded of polynomial growth. Such matrices can be neither integer nor symmetrizable. There are non-Serre relations encountered. In certain cases there are infinitely many relations.Our results are applicable to the Lie algebras with the same Cartan matrices as the Lie superalgebras considered.1991 Mathematics Subject Classification. 17A70, 17B01, 17B70.

Super)conformal many-body quantum mechanics with extended supersymmetry Equivalence between chain categories of representations of affine sl(2) and N=2 superconformal algebras

Key words and phrases. Supermanifolds, Riemann tensor, penrose tensor, structure functions.I warmly thank IAS (1989), SFB-170 (1990), and MPIM, Bonn during my several visits in the past decades 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 Z-graded 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. 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 REDUCED STRUCTURES 9Goncharov has shown that a manifold M with generalized conformal structure of type X is a manifold with aG-structure, whereG 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 CThe 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,

The Einstein equations (EE) are certain conditions on the Riemann tensor on the real Minkowski space M . In the twistor picture, after complexification and compactification M becomes the Grassmannian Gr 4 2 of 2-dimensional subspaces in the 4-dimensional complex one. Here we answer for which of the classical domains considered as manifolds with G-structure it is possible to impose conditions similar in some sense to EE. The above investigation has its counterpart on superdomains: an analog of the Riemann tensor is defined for any supermanifold with G-structure with any Lie supergroup G. We also derive similar analogues of EE on supermanifolds. Our analogs of EE are not what physicists consider as SUGRA (supergravity), for SUGRA see [16,34].

No abstract