Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies

TL;DR

This paper computes the Gerstenhaber bracket on the Hochschild and Hopf algebra cohomologies of the Taft algebra, revealing it is zero in this nonquasi-triangular case and providing a general formula for such brackets.

Contribution

It presents the first known computation of the Gerstenhaber bracket for a nonquasi-triangular Hopf algebra and derives a general formula for brackets on Hopf algebra cohomology.

Findings

Gerstenhaber bracket is zero on Hopf algebra cohomology of Taft algebra

Provides a general formula for brackets on Hopf algebra cohomology with bijective antipode

First example of bracket computation for a nonquasi-triangular algebra

Abstract

We calculate the Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies of the Taft algebra $T_p$ for any integer $p>2$ which is a nonquasi-triangular Hopf algebra. We show that the bracket is indeed zero on Hopf algebra cohomology of $T_p$, as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi-triangular algebra. Also, we find a general formula for the bracket on Hopf algebra cohomology of any Hopf algebra with bijective antipode on the bar resolution that is reminiscent of Gerstenhaber's original formula for Hochschild cohomology.