Some Lie superalgebras associated to the Weyl algebras

Musson, Ian

doi:10.1090/S0002-9939-99-04976-X

Some Lie superalgebras associated to the Weyl algebras
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by Ian M. Musson PDF

Proc. Amer. Math. Soc. 127 (1999), 2821-2827 Request permission

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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