Hochschild homology of Hopf algebras and free Yetter–Drinfeld resolutions of the counit
Hochschild homology of Hopf algebras and free Yetter–Drinfeld resolutions of the counit
复制标题
DOI：
10.1112/s0010437x12000656
复制
发表时间：
2012-04
影响因子：
1.8
通讯作者：
J. Bichon
中科院分区：
文献类型：
--
作者：
J. Bichon
摘要：我们证明了如果$A$和$H$是具有等价模张量范畴的Hopf代数，那么我们可以将$A$的自由yeter - drinfeld分辨传递到$H$的自由yeter - drinfeld分辨，从而展示了$A$和$H$的Hochschild同调之间的强联系。这使我们能够获得$\mathcal {B}(E)$的计数的有限自由解析，与可逆矩阵$E$相关的双线性形式的Hopf代数，推广了Collins， Härtel和Thom在正交情况$E=I_n$中的早期构造。由此得出$\mathcal {B}(E)$是3维光滑且满足poincarcarcarr对偶性。结合Vergnioux的结果，也可以得出当$E$是一个反对称矩阵时，相关离散量子群的$L^2$-Betti数全部消失。我们还利用我们的分辨率计算了$\mathcal {B}(E)$在共单情况下的双代数上同调。
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.