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

## TL;DR

This paper fully describes the Hochschild cohomology of a family of subalgebras of the Weyl algebra, revealing Lie algebra structures and connections to Virasoro modules, with results varying across different field characteristics.

## Contribution

It provides a comprehensive description of Hochschild cohomology for these subalgebras, including Lie algebra actions and module structures, extending understanding in arbitrary characteristic.

## Key findings

- Hochschild cohomology is described as a module over its center in positive characteristic.
- In characteristic zero, the cohomology acts as a module over a Lie algebra related to Virasoro modules.
- Conditions for semisimplicity of the cohomology as a module are established.

## 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 the Hochschild cohomology $\mathsf{HH}^\bullet(\mathsf{A}_h)$ over a field of arbitrary characteristic. In case $\mathbb{F}$ has positive characteristic, the center of $\mathsf{A}_h$ is nontrivial and we describe $\mathsf{HH}^\bullet(\mathsf{A}_h)$ as a module over its center. The most interesting results occur when $\mathbb{F}$ has characteristic $0$. In this case, we describe $\mathsf{HH}^\bullet(\mathsf{A}_h)$ as a module over the Lie algebra $\mathsf{HH}^1(\mathsf{A}_h)$ and find that this action is closely related to the intermediate series modules over the Virasoro algebra. We also determine when $\mathsf{HH}^\bullet(\mathsf{A}_h)$ is a semisimple $\mathsf{HH}^1(\mathsf{A})$-module.