Lie structure on the Hochschild cohomology of a family of subalgebras of the Weyl algebra

Abstract

For each nonzero $h\in\mathbb{F}[x]$, where $\mathbb{F}$ is a field, let $\mathsf{A}_h$ be the unital associative algebra generated by elements $x,y$, satisfying the relation $yx-xy=h$. This gives a parametric family of subalgebras of the Weyl algebra $\mathsf{A}_1$, containing many well-known algebras which have previously been studied independently. In this paper, we give a full description of the Hochschild cohomology $\operatorname{\mathsf{HH}}^{\bullet}(\mathsf{A}_h)$ over a field of an arbitrary characteristic. In case $\mathbb{F}$ has a positive characteristic, the center $\mathsf{Z}(\mathsf{A}_{h})$ of $\mathsf{A}_h$ is nontrivial and we describe $\operatorname{\mathsf{HH}}^\bullet(\mathsf{A}_h)$ as a module over $\mathsf{Z}(\mathsf{A}_{h})$. The most interesting results occur when $\mathbb{F}$ has a characteristic $0$. In this case, we describe $\operatorname{\mathsf{HH}}^\bullet(\mathsf{A}_h)$ as a module over the Lie algebra $\operatorname{\mathsf{HH}}^1(\mathsf{A}_h)$ and find that this action is closely related to the intermediate series modules over the Virasoro algebra. We also determine when $\operatorname{\mathsf{HH}}^\bullet(\mathsf{A}_h)$ is a semisimple $\operatorname{\mathsf{HH}}^1(\mathsf{A}_h)$-module.