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Hochschild homology of Hopf algebras and free Yetter–Drinfeld resolutions of the counit
| Content Provider | Scilit |
|---|---|
| Author | Bichon, Julien |
| Copyright Year | 2012 |
| Description | Journal: Compositio Mathematica 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. |
| Ending Page | 678 |
| Starting Page | 658 |
| ISSN | 00221295 |
| e-ISSN | 15705846 |
| DOI | 10.1112/s0010437x12000656 |
| Journal | Compositio Mathematica |
| Issue Number | 4 |
| Volume Number | 149 |
| Language | English |
| Publisher | Wiley-Blackwell |
| Publisher Date | 2013-04-01 |
| Access Restriction | Open |
| Subject Keyword | Journal: Compositio Mathematica Mathematical Physics Hopf Algebras Hochschild Homology Drinfeld Resolution Free Yetter |
| Content Type | Text |
| Resource Type | Article |
| Subject | Algebra and Number Theory |
