Some Lie superalgebras associated to the Weyl algebras

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.