Generators and relations for Lie superalgebras of Cartan type

We give an analog of a Chevalley–Serre presentation for the Lie superalgebras and of Cartan type. These are part of a wider class of Lie superalgebras, the so-called tensor hierarchy algebras, denoted and , where denotes the Kac–Moody algebra A r , D r or E r . Then W ( A n −1 ) and S ( A n −1 ) are the Lie superalgebras and . The algebras and are constructed from the Dynkin diagram of the Borcherds–Kac–Moody superalgebras obtained by adding a single grey node (representing an odd null root) to the Dynkin diagram of . We redefine the algebras W ( A r ) and S ( A r ) in terms of Chevalley generators and defining relations. We prove that all relations follow from the defining ones at level . 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.