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.
