Generators and relations for Lie superalgebras of Cartan type
Journal article, 2019

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.



Lie algebras

generators and relations


Lisa Carbone

Rutgers University

Martin Cederwall

Chalmers, Physics, Theoretical Physics

Jakob Palmkvist

Chalmers, Physics, Theoretical Physics

Journal of Physics A: Mathematical and Theoretical

1751-8113 (ISSN) 1751-8121 (eISSN)

Vol. 52 5 055203

Subject Categories

Algebra and Logic

Other Physics Topics


Basic sciences



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4/6/2022 1