张量C ~*-范畴的归纳与分类理论
A theory of induction and classification of tensor C*-categories
摘要：
本文讨论了用共轭和不可约张量单位刻画张量C ~*-范畴M的结构问题。没有假设存在一个编织的对称性或顺从。我们的假设的动机的话，这些范畴往往包含非全张量C*-子范畴的共轭和相同的对象允许嵌入到希尔伯特空间。这样的嵌入定义了一个紧凑的量子群的Woronowicz对偶。一个重要的例子是Temperley-Lieb范畴正则地包含在由一个维度大于2的实或伪实对象生成的张量C* 范畴中。相应的量子群是Wang和Van Daele的泛正交量子群。我们的主要结果是，从M到紧量子群的一类Hilbert双模表示存在一个完全的忠实的张量函子.在经典的情形下，这些双模表示归结为紧齐性G-空间上的G-等变埃尔米特丛，其中G是紧群。我们的结构性结果揭示了M在希尔伯特空间中是否存在嵌入函子的问题。我们表明，这是有关的问题，是否一个经典的紧李群可以遍历作用于一个非I型冯诺依曼代数。特别地，将其与Wassermann的结果相结合表明，如果M是由2维的伪实对象生成的，则嵌入存在。
英文摘要：
This paper addresses the problem of describing the structure of tensor C*-categories M with conjugates and irreducible tensor unit. No assumption on the existence of a braided symmetry or on amenability is made. Our assumptions are motivated by the remark that these categories often contain non-full tensor C*-subcategories with conjugates and the same objects admitting an embedding into the Hilbert spaces. Such an embedding defines a compact quantum group by Woronowicz duality. An important example is the Temperley--Lieb category canonically contained in a tensor C*-category generated by a single real or pseudoreal object of dimension bigger than 2. The associated quantum groups are the universal orthogonal quantum groups of Wang and Van Daele. Our main result asserts that there is a full and faithful tensor functor from M to a category of Hilbert bimodule representations of the compact quantum group. In the classical case, these bimodule representations reduce to the G-equivariant Hermitian bundles over compact homogeneous G-spaces, with G a compact group. Our structural results shed light on the problem of whether there is an embedding functor of M into the Hilbert spaces. We show that this is related to the problem of whether a classical compact Lie group can act ergodically on a non-type I von Neumann algebra. In particular, combining this with a result of Wassermann shows that an embedding exists if M is generated by a pseudoreal object of dimension 2.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
