Polynomial realisations of Lie (super)algebras and Bessel operators

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.