Polynomial realisations of Lie (super)algebras and Bessel operators

TL;DR

This paper constructs small realizations of Lie superalgebras in Weyl superalgebras, linking them to Bessel operators for Jordan algebras and providing explicit realizations for the exceptional Lie superalgebra D(2,1;a).

Contribution

It introduces new minimal realizations of Lie superalgebras, generalizes Bessel operators for Jordan algebras, and offers explicit realizations for D(2,1;a).

Findings

Realisations of Lie superalgebras in Weyl superalgebras.

Connection between these realizations and Bessel operators.

Explicit realization for the exceptional Lie superalgebra D(2,1;a).

Abstract

We study realisations of Lie (super)algebras in Weyl (super)algebras and connections with minimal representations. The main result is the construction of small realisations of Lie superalgebras, which we apply for two distinct purposes. Firstly it naturally introduces, and generalises, the Bessel operators for Jordan algebras in the study of minimal representations of simple Lie groups. These have already been applied very successfully by several authors, however an easy direct explanation for their relevance seemed still to be missing. Secondly, we work out the theoretical realisation concretely for the exceptional Lie superalgebra D(2,1;a), giving a useful hands-on realisation.