Vol. 316, No. 2, 2022

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Lie brackets on Hopf algebra cohomology

Tekin Karadağ and Sarah Witherspoon

Vol. 316 (2022), No. 2, 395–407
Abstract

By work of Farinati, Solotar, and Taillefer, it is known that the Hopf algebra cohomology of a quasi-triangular Hopf algebra, as a graded Lie algebra under the Gerstenhaber bracket, is abelian. Motivated by the question of whether this holds for nonquasi-triangular Hopf algebras, we show that Gerstenhaber brackets on Hopf algebra cohomology can be expressed via an arbitrary projective resolution using Volkov’s homotopy liftings as generalized to some exact monoidal categories. This is a special case of our more general result that a bracket operation on cohomology is preserved under exact monoidal functors — one such functor is an embedding of Hopf algebra cohomology into Hochschild cohomology. As a consequence, we show that this Lie structure on Hopf algebra cohomology is abelian in positive degrees for all quantum elementary abelian groups, most of which are nonquasi-triangular.

Keywords
Hochschild cohomology, Hopf algebra cohomology, Gerstenhaber brackets, homotopy lifting
Mathematical Subject Classification
Primary: 17B55, 17B56, 17B70, 18M15, 18M20
Milestones
Received: 3 October 2021
Revised: 20 November 2021
Accepted: 27 November 2021
Published: 6 April 2022
Authors
Tekin Karadağ
Department of Mathematics
College of Charleston
Charleston, SC
United States
Sarah Witherspoon
Department of Mathematics
Texas A&M University
College Station, TX
United States