# Hochschild cohomology of group extensions of quantum symmetric algebras

## TL;DR

This paper computes the Hochschild cohomology ring structure of quantum symmetric algebras extended by finite groups, providing explicit descriptions of their algebraic properties in a noncommutative setting.

## Contribution

It explicitly determines the multiplicative structure of Hochschild cohomology for group extensions of quantum symmetric algebras, a novel result in noncommutative algebra.

## Key findings

- Explicit graded vector space structure for Hochschild cohomology
- Complete description of the Hochschild cohomology ring of skew group algebras
- New insights into the algebraic structure of quantum symmetric algebra extensions

## Abstract

Quantum symmetric algebras (or noncommutative polynomial rings) arise in many places in mathematics. In this article we find the multiplicative structure of their Hochschild cohomology when the coefficients are in an arbitrary bimodule algebra. When this bimodule algebra is a finite group extension (under a diagonal action) of a quantum symmetric algebra, we give explicitly the graded vector space structure. This yields a complete description of the Hochschild cohomology ring of the corresponding skew group algebra.