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