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

Sarah Witherspoon and Guodong Zhou

Vol. 283 (2016), No. 1, 223–255

DOI: 10.2140/pjm.2016.283.223

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.

Keywords

Hochschild cohomology, Gerstenhaber bracket, quantum symmetric algebra, skew group algebra

Mathematical Subject Classification 2010

Primary: 16E40

Milestones

Received: 20 July 2015

Accepted: 22 December 2015

Published: 14 June 2016