Hochschild homology of Hopf algebras and free Yetter-Drinfeld resolutions of the counit

Abstract

We show that if A and H are Hopf algebras that have equivalent tensor categories of comodules, then one can transport what we call a free Yetter{Drinfeld resolution of the counit of A to the same kind of resolution for the counit of H, exhibiting in this way strong links between the Hochschild homologies of A and H. This enables us to obtain a finite free resolution of the counit of B(E), the Hopf algebra of the bilinear form associated with an invertible matrix E, generalizing an earlier construction of Collins, Härtel and Thom in the orthogonal case E = In. It follows that B(E) is smooth of dimension 3 and satisfies Poincaré duality. Combining this with results of Vergnioux, it also follows that when E is an antisymmetric matrix, the L2-Betti numbers of the associated discrete quantum group all vanish. We also use our resolution to compute the bialgebra cohomology of B(E) in the cosemisimple case. © Foundation Compositio Mathematica 2013.