Truncated quantum Drinfeld Hecke algebras and Hochschild cohomology

TL;DR

This paper studies a special class of algebra deformations called truncated quantum Drinfeld Hecke algebras, establishing conditions for their existence and exploring their Hochschild cohomology to understand their structure.

Contribution

It introduces truncated quantum Drinfeld Hecke algebras, provides necessary and sufficient conditions for their occurrence, and connects their structure to Hochschild cohomology.

Findings

Derived conditions using Bergman's Diamond Lemma

Computed Hochschild cohomology for these algebras

Presented classical and novel examples of such algebras

Abstract

We consider deformations of quantum exterior algebras extended by finite groups. Among these deformations are a class of algebras which we call truncated quantum Drinfeld Hecke algebras in view of their relation to classical Drinfeld Hecke algebras. We give the necessary and sufficient conditions for which these algebras occur, using Bergman's Diamond Lemma. We compute the relevant Hochschild cohomology to make explicit the connection between Hochschild cohomology and truncated quantum Drinfeld Hecke algebras. To demonstrate the variance of the allowed algebras, we compute both classical type examples and demonstrate an example that does not arise as a factor algebra of a quantum Drinfeld Hecke algebra.