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

### Samuel A. Lopes

Universidade do Porto, Portugal### Andrea Solotar

Universidad de Buenos Aires, Argentina

## Abstract

For each nonzero $h∈F[x]$, where $F$ is a field, let $A_{h}$ 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 $A_{1}$, containing many well-known algebras which have previously been studied independently. In this paper, we give a full description of the Hochschild cohomology $HH_{∙}(A_{h})$ over a field of an arbitrary characteristic. In case $F$ has a positive characteristic, the center $Z(A_{h})$ of $A_{h}$ is nontrivial and we describe $HH_{∙}(A_{h})$ as a module over $Z(A_{h})$. The most interesting results occur when $F$ has a characteristic $0$. In this case, we describe $HH_{∙}(A_{h})$ as a module over the Lie algebra $HH_{1}(A_{h})$ and find that this action is closely related to the intermediate series modules over the Virasoro algebra. We also determine when $HH_{∙}(A_{h})$ is a semisimple $HH_{1}(A_{h})$-module.

## Cite this article

Samuel A. Lopes, Andrea Solotar, Lie structure on the Hochschild cohomology of a family of subalgebras of the Weyl algebra. J. Noncommut. Geom. 15 (2021), no. 4, pp. 1373–1407

DOI 10.4171/JNCG/439