Gerstenhaber brackets on Hochschild cohomology of quantum symmetric algebras and their group extensions

TL;DR

This paper develops chain maps between resolutions for quantum symmetric algebras to compute Gerstenhaber brackets, extending classical brackets to quantum and group extension contexts.

Contribution

It introduces explicit chain maps for quantum symmetric algebras and computes their Gerstenhaber brackets, including for group extensions, providing a quantum analogue of classical brackets.

Findings

Constructed chain maps between bar and Koszul resolutions.

Computed Gerstenhaber brackets for quantum symmetric algebras.

Extended bracket computations to skew group algebras.

Abstract

We construct chain maps between the bar and Koszul resolutions for a quantum symmetric algebra (skew polynomial ring). This construction uses a recursive technique involving explicit formulae for contracting homotopies. We use these chain maps to compute the Gerstenhaber bracket, obtaining a quantum version of the Schouten-Nijenhuis bracket on a symmetric algebra (polynomial ring). We compute brackets also in some cases for skew group algebras arising as group extensions of quantum symmetric algebras.