Hochschild Cohomology and Deformations of Clifford-Weyl Algebras

TL;DR

This paper thoroughly analyzes the Hochschild cohomology and deformation theory of Clifford-Weyl algebras, revealing conditions for rigidity and classifying non-trivial deformations with implications for their representations.

Contribution

It provides a complete Hochschild cohomology computation for Clifford-Weyl algebras and classifies all their non-trivial deformations, especially in the case of odd n and k=1.

Findings

Clifford-Weyl algebra is rigid for even n or k ≠ 1

All non-trivial deformations of ${ m C}(2n+1,2)$ are classified

Deformations have specific representation-theoretic properties

Abstract

We give a complete study of the Clifford-Weyl algebra ${\mathcal C}(n,2k)$ from Bose-Fermi statistics, including Hochschild cohomology (with coefficients in itself). We show that ${\mathcal C}(n,2k)$ is rigid when $n$ is even or when $k \neq 1$. We find all non-trivial deformations of ${\mathcal C}(2n+1,2)$ and study their representations.