Cohomology of Algebras Over Hopf Algebras

Abstract

We present a cohomology theory for algebras which are modules over a given Hopf algebra.The algebras are commutative, the Hopf algebra cocommutative and under the module action the underlying coalgebra of the Hopf algebra " respects " the multiplication and unit in the algebras.The cohomology is defined by means of an explicit complex.Whenever C is a coalgebra and A an algebra Horn (C, A) has a certain natural algebra structure.The groups in our complex consist of the multiplicative group of invertible elements in Horn (C, A) where C is the underlying coalgebra of the Hopf algebra tensored with itself a number of times.The complex arises as the chain complex associated with a semi-cosimplicial complex whose face operators are induced by maps of the form (g)'l+1 H-+ (g)n H, h0 -• • «"-► h0 ®-• • «i«i + l