Generators and relations for Lie superalgebras of Cartan type

Abstract

We give an analog of a Chevalley Serre presentation for the Lie superalgebras W(n) and S(n) of Cartan type. These are part of a wider class of Lie superalgebras, the so-called tensor hierarchy algebras, denoted W(g) and S(g), where g denotes the Kac Moody algebra Ar, Dr or Er. Then W(An-1) and S(An-1) are the Lie superalgebras W(n) and S(n). The algebras W(g) and S(g) are constructed from the Dynkin diagram of the Borcherds Kac Moody superalgebras B(g) obtained by adding a single grey node (representing an odd null root) to the Dynkin diagram of g. We redefine the algebras W(Ar) and S(Ar) in terms of Chevalley generators and defining relations. We prove that all relations follow from the defining ones at level -2. The analogous definitions of the algebras in the D- and E-series are given. In the latter case the full set of defining relations is conjectured.