#### Preprint

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.

Andrea Solotar

### Publication

Year of publication: 2019