Subject: Hochschild cohomology of Hopf algebras Date: Sat, 29 Mar 2003 22:09:01 -0500 From: John Klein Here's a question for the list, which has to do with generalizing the following: Observation: if k[G] is a group ring over a commutative ring k, then there is an an isomorphism (*) Ext^*_k[G](k,k[G^ad]) = THC^*(k[G],k[G]) where: (1) the right side denotes the topological Hochschild cohomology of k[G] with with coefficients itself considered as a bimodule (i.e., Ext_{k[G] tensor k[G]^op}(k[G],k[G])). (2) the left side is the cohomology of the group with coefficients in the **adjoint** module (=: k[G] with conjugation action). To formulate my question, let R denote Hopf algebra over k with antipode R -> R^op. Then the coproduct R -> R tensor R together with antipode assemble to give an algebra map R -> R tensor R^{op}. Since R is a left (R tensor R^{op})-module, restriction of scalars gives rise to an "adjoint" action of R on itself. Let me denote this module structure by R^ad . My Question: When can one conclude that is there an isomorphism Ext^*_R(k,R^ad) = THC^*(R,R) (= Ext^*_{R tensor R^op}(R,R)) ?? Best, john