Dualising complexes and twisted Hochschild (co)homology for noetherian Hopf algebras
Dualising complexes and twisted Hochschild (co)homology for noetherian Hopf algebras
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DOI：
10.1016/j.jalgebra.2007.03.050
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发表时间：
2006-03
影响因子：
0.9
通讯作者：
K. Brown;James J. Zhang
中科院分区：
文献类型：
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作者：
K. Brown;James J. Zhang
我们证明了许多Noether Hopf代数A都有一个刚性对偶复形R，R ∈ A1ν[d].这里，d是代数的内射维数，ν是A的某个k-代数自同构，直到内自同构为止是唯一的。为了纪念在此推广的有限维理论，我们称ν为A的Nakayama自同构。证明了ν=S2 <$，其中S是A的对极，<$是由A的左积分确定的A的左缠绕自同构.当A是一个整体维数为d的Artin-Schelter正则Noether Hopf代数时，证明了系数在适当扭曲自由双模中的Hochschild同调群和上同调群在顶维d中是非零的.（扭曲）庞加莱对偶在这种情况下成立，这是从货车登伯格定理推导出来的。我们还利用相反的余代数结构计算A的ν，确定了S 4的一个公式，推广了有限维A的拉德福1976年的一个公式。给出了这些结果在A为PI、包络代数、量子群、量子化函数代数和群代数等情形中的应用。
We show that many noetherian Hopf algebras A have a rigid dualising complex R with R≅A1ν[d]. Here, d is the injective dimension of the algebra and ν is a certain k-algebra automorphism of A, unique up to an inner automorphism. In honour of the finite-dimensional theory which is hereby generalised we call ν the Nakayama automorphism of A. We prove that ν=S2ξ, where S is the antipode of A and ξ is the left winding automorphism of A determined by the left integral of A. The Hochschild homology and cohomology groups with coefficients in a suitably twisted free bimodule are shown to be non-zero in the top dimension d, when A is an Artin–Schelter regular noetherian Hopf algebra of global dimension d. (Twisted) Poincaré duality holds in this setting, as is deduced from a theorem of Van den Bergh. Calculating ν for A using also the opposite coalgebra structure, we determine a formula for S4generalising a 1976 formula of Radford for A finite-dimensional. Applications of the results to the cases where A is PI, an enveloping algebra, a quantum group, a quantised function algebra and a group algebra are outlined.