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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Some Lie superalgebras associated
to the Weyl algebras

Author: Ian M. Musson
Journal: Proc. Amer. Math. Soc. 127 (1999), 2821-2827
MSC (1991): Primary 17B35; Secondary 16W10
Published electronically: May 4, 1999
MathSciNet review: 1616633
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Abstract: Let ${\mathfrak{g}}$ be the Lie superalgebra $osp(1,2r)$. 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 $A_r$ 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 $[A_r, A_r]$ and $[A_s, A_s]$ are isomorphic as Lie superalgebras, then $r=s$. This answers a question of S. Montgomery.

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Additional Information

Ian M. Musson
Affiliation: Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201

Keywords: Lie superalgebras, Weyl algebras, Joseph ideal
Received by editor(s): February 7, 1997
Received by editor(s) in revised form: December 9, 1997
Published electronically: May 4, 1999
Additional Notes: This research was partially supported by NSF grant DMS 9500486.
Communicated by: Ken Goodearl
Article copyright: © Copyright 1999 American Mathematical Society