Generators and relations for Lie superalgebras of Cartan type

TL;DR

This paper provides a Chevalley-Serre type presentation for certain Lie superalgebras of Cartan type, specifically W(n) and S(n), and extends the framework to a broader class related to Kac-Moody algebras.

Contribution

It introduces a new presentation for W(n) and S(n) superalgebras using Chevalley generators and relations, connecting them to tensor hierarchy algebras and Borcherds-Kac-Moody superalgebras.

Findings

All relations follow from those at level -2 and above.

Definitions are extended to D- and E-series algebras.

Full relations for E-series are conjectured.

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 A_r, D_r or E_r. Then W(A_{n-1}) and S(A_{n-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(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 -2 and higher. 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.