Non-commutative geometry of finite groups
Non-commutative geometry of finite groups
复制标题
有限群的非交换几何
DOI：
--
复制
发表时间：
1995
期刊：
影响因子：
--
通讯作者：
Andrzej Sitarz
中科院分区：
文献类型：
--
作者：
Klaus Bresser;F. Müller;A. Dimakis;Andrzej Sitarz
一个有限集合可以提供一个群结构，然后可以通过左、右和双协方差的概念来选择它上的微分演算（类）。一个相应的框架已经由Woronowicz开发，更一般地用于包含量子群的Hopf代数。微分被认为是最基本的结构需要进一步的几何概念，如线性连接，此外，制定场论和动力学的有限集。有限群上的每一个双共变一阶微分都有一个辫子算子，它在构造特殊的几何结构中起着重要的作用。对于协变演算，有线性连接和张量的不变性的概念。所有这些概念都探讨了有限群，并举例说明。一些结果制定更一般的任意结合（Hopf）代数。特别是，问题的延伸的连接上的双模（在一个结合代数）张量积的调查，导致类的“可扩展连接”。证明了Hopf代数上双模上的可扩联络的不变性可以推广到扩张上。此外，一个联络的不变性也被存在于对偶双模上的一个“对偶联络”所共享（如本文所定义的）。
A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left-, right- and bicovariance. A corresponding framework has been developed by Woronowicz, more generally for Hopf algebras including quantum groups. A differential calculus is regarded as the most basic structure needed for the introduction of further geometric notions like linear connections and, moreover, for the formulation of field theories and dynamics on finite sets. Associated with each bicovariant first-order differential calculus on a finite group is a braid operator which plays an important role for the construction of distinguished geometric structures. For a covariant calculus, there are notions of invariance for linear connections and tensors. All these concepts are explored for finite groups and illustrated with examples. Some results are formulated more generally for arbitrary associative (Hopf) algebras. In particular, the problem of extension of a connection on a bimodule (over an associative algebra) to tensor products is investigated, leading to the class of `extensible connections'. It is shown that invariance properties of an extensible connection on a bimodule over a Hopf algebra are carried over to the extension. Furthermore, an invariance property of a connection is also shared by a `dual connection' which exists on the dual bimodule (as defined in this work).