Defining relations for classical Lie superalgebras without Cartan matrices

Abstract

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.

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