# Hochschild cohomology of the Weyl algebra and Vasiliev's equations

## TL;DR

This paper introduces an explicit injective resolution for the Hochschild complex of the Weyl algebra, enabling the derivation of explicit cocycles and exploring connections to higher-spin field theory.

## Contribution

It provides a new injective resolution for the Hochschild complex of the Weyl algebra and derives explicit cocycles, linking algebraic structures to higher-spin theories.

## Key findings

- Explicit cocycles for the Weyl algebra are derived.
- A relationship with higher-spin field theory is discussed.
- Resolutions facilitate computations in Hochschild cohomology.

## Abstract

We propose a simple injective resolution for the Hochschild complex of the Weyl algebra. By making use of this resolution, we derive explicit expressions for nontrivial cocycles of the Weyl algebra with coefficients in twisted bimodules as well as for the smash products of the Weyl algebra and a finite group of linear symplectic transformations. A relationship with the higher-spin field theory is briefly discussed.