  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.
