Irreducible bimodules associated with crossed product algebras. II
Irreducible bimodules associated with crossed product algebras. II
复制标题
与交叉积代数相关的不可约双模。
DOI：
10.2140/pjm.1995.171.209
复制
发表时间：
1995
影响因子：
0.6
通讯作者：
S. Yamagami
中科院分区：
文献类型：
--
作者：
Tsuyoshi Kajiwara;S. Yamagami
双模理论在近年来算子代数的研究中越来越重要，特别是在Jones指数理论中（如[A]， [CK], [EK],[I1], [12], [LI], [L2], [Ol], [O2]）。在我们之前的论文（[KY]）中，我们介绍了基于有限群的双模的交叉积构造，提供了整数索引的例子以及根据相关有限群的纯有限维表示理论计算其不变量（副群）的方法。对于整数指标的例子，有基于紧群作用不动点代数的Wassermann构造[Wa]。参见[PW]了解该结构的最新状态。本文的目的是将交叉积构造推广到包含紧群的情况，并纯粹从紧群的表示理论出发阐明其范畴结构。作为这一推广的一个意想不到的好处，我们可以将Wassermann的不动点代数构造以oι N x G - N x G双模的形式再现出来。特别是，已知的高相对交换子的描述以及与Wassermann构造相关的图不变量可以从范畴的角度巧妙地处理。我们的构造还提供了具有无限指标的不可约双模的例子，它们仍然是可控的，因为我们可以完全描述它们的融合结构，即张量积的分解方式。至于我们构建的技术部分，我们被迫在相关的自同构行为上强加一个很强的外在性条件。这比从外观上看简约的概念要强得多，但目前还不清楚它们之间是否有任何区别。
The theory of bimodules is becoming more and more important in recent studies of operator algebras, particularly in Jones index theory (see [A], [CK], [EK],[I1], [12], [LI], [L2], [Ol], and [O2] for example). In our previous paper ([KY]), we have introduced crossed product construction of bimodules based on finite groups which provides examples of integer index as well as the way to compute their invariants (paragroups) in terms of purely finite-dimensional representation theory of relevant finite groups. As to examples of integer index, there is the construction of Wassermann based on fixed point algebras of compact group actions ([Wa]). See [PW] for the resent status of this construction. The purpose of the present paper is to generalize the crossed product construction to incorporate the case of compact groups and clarify its categorical structure purely in terms of representation theory of compact groups. As an unexpected bonus of this generalization, we can reproduce the fixed point algebra construction of Wassermann in the form oϊ N x G — N x G bimodules. In particular, the known description of higher relative commutants as well as the associated graph invariants of Wassermann's construction can be neatly handled from the categorical point of view. Our construction also supplies examples of irreducible bimodules with infinite index, which are still controllable in the sense that we can completely describe their fusion structure, i.e., the way of decompositions of tensor products. As to the technical part of our construction, we are forced to impose a strong condition of outerness on relevant automorphic actions. This is much stronger than the notion of minimality from the appearance, but it is not clear at present whether there are any differences between them.
DOI：
10.1007/978-3-642-66243-0
发表时间：
1976
期刊：
Energy Sources, Part B: Economics, Planning, and Policy
影响因子：
--
作者：
A. Kirillov
通讯作者：
A. Kirillov