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

Abstract

AbstractWe 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$\mathcal {B}(E)$, 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$E=I_n$. It follows that$\mathcal {B}(E)$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$L^2$-Betti numbers of the associated discrete quantum group all vanish. We also use our resolution to compute the bialgebra cohomology of$\mathcal {B}(E)$in the cosemisimple case.