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Some Lie superalgebras associated to the Weyl algebras

Authors:	Ian M Musson

Institution:	Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201

Abstract:	Let ${\mathfrak{g}}$ be the Lie superalgebra . We show that there is a surjective homomorphism from $U({\mathfrak{g}})$ to the $r^{th}$ Weyl algebra $A_{r}$ , and we use this to construct an analog of the Joseph ideal. We also obtain a decomposition of the adjoint representation of ${\mathfrak{g}}$ on and use this to show that if $A_{r}$ is made into a Lie superalgebra using its natural ${\mathbb Z}_{2}$ -grading, then $A_{r} = k \oplus A_{r}, A_{r}]$ . In addition, we show that if and are isomorphic as Lie superalgebras, then . This answers a question of S. Montgomery.

Keywords:	Lie superalgebras Weyl algebras Joseph ideal

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