GERSTENHABER BRACKET ON HOPF ALGEBRA COHOMOLOGY
Abstract
M. A. Farinati, A. Solotar, and R. Taillefer showed 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 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.
We also 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 (T_p), most of which are nonquasi-triangular.
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.
Subject
Hochschild cohomologyHopf algebra cohomology
Taft algebra
Gerstenhaber bracket
homotopy lifting
graded Lie bracket
Citation
Karadag, Tekin (2021). GERSTENHABER BRACKET ON HOPF ALGEBRA COHOMOLOGY. Doctoral dissertation, Texas A&M University. Available electronically from https : / /hdl .handle .net /1969 .1 /195063.