Lie brackets on Hopf algebra cohomology

TL;DR

This paper investigates the Lie algebra structure of Hopf algebra cohomology, demonstrating that it is abelian in positive degrees for quantum elementary abelian groups, extending known results beyond quasi-triangular cases.

Contribution

It generalizes the understanding of Gerstenhaber brackets on Hopf algebra cohomology to nonquasi-triangular cases using homotopy liftings and monoidal functors.

Findings

Lie brackets on Hopf cohomology can be expressed via projective resolutions.

The Lie structure is abelian in positive degrees for quantum elementary abelian groups.

The result extends the abelian property beyond quasi-triangular Hopf algebras.

Abstract

By work of Farinati, Solberg, 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.