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

Part of: Homological methods Selfadjoint operator algebras Hopf algebras, quantum groups and related topics

Published online by Cambridge University Press:  17 December 2012

Julien Bichon
Julien Bichon*
Affiliation:
Laboratoire de Mathématiques, Université Blaise Pascal, Complexe universitaire des Cézeaux, 63171 Aubière cedex, France (email: julien.bichon@math.univ-bpclermont.fr)
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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 $\mathcal {B}(E)$, 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.

Keywords

Hopf algebraHochschild homologyYetter–Drinfeld module$\uppercase {l}^2$-Betti numbers

MSC classification

Secondary: 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.) 46L89: Other “noncommutative” mathematics based on $C^*$-algebra theory 16T05: Hopf algebras and their applications
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 4 , April 2013 , pp. 658 - 678
Copyright
Copyright © 2012 The Author(s)

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