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 ${\mathrm{HH}}^{\bullet }({\mathsf{A}}_h)$ 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 ${\mathrm{HH}}^{\bullet }({\mathsf{A}}_h)$ 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 ${\mathrm{HH}}^{\bullet }({\mathsf{A}}_h)$ as a module over the Lie algebra ${\mathrm{HH}}^1({\mathsf{A}}_h)$ and find that this action is closely related to the intermediate series modules over the Virasoro algebra. We also determine when ${\mathrm{HH}}^{\bullet }({\mathsf{A}}_h)$ is a semisimple ${\mathrm{HH}}^1({\mathsf{A}}_h)$-module.