Infinite-Dimensional Representations of 2-Groups
Infinite-Dimensional Representations of 2-Groups
复制标题
2 群的无限维表示
DOI：
10.1090/s0065-9266-2012-00652-6
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发表时间：
2008
期刊：
影响因子：
--
通讯作者：
D. Wise
中科院分区：
文献类型：
--
作者：
J. Baez;A. Baratin;L. Freidel;D. Wise
一个“$2$-群”是一个配备有乘法满足像群一样的定律的范畴。就像群在向量空间上有表示一样，$2$-群在“$2$-向量空间”上有表示，这是类似于向量空间的范畴。不幸的是，Lie $2$-群在Kapranov和Voevodsky引入的有限维$2$-向量空间上通常只有很少的表示。由于这个原因，起重机，Sheppeard和Yetter引入了某些称为“可测范畴”的无限维2 $-向量空间（因为它们与希尔伯特空间的可测场密切相关），并使用它们来研究某些Lie 2 $-群的无限维表示。他们在这里继续这项工作。他们开始详细研究可测量的类别。然后，他们给出了几何描述的可测表示，缠绕和$2$-缠绕的任何骨骼可测$2$-组。他们研究张量积和直和的陈述，以及各种概念的子代表。他们描述的直接总和的缠绕，和子缠绕-功能没有看到在普通的群表示理论和研究不可约和不可分解的表示和缠绕。他们还研究“不可伸缩”的表示-另一个在普通群表示理论中看不到的特征。最后，他们认为，配备了一些额外的结构的可测范畴值得考虑的“可分离的2 $-希尔伯特空间”，并比较这个想法的一个试探性的定义2 $-希尔伯特空间作为代表类别的交换冯诺依曼代数。
A "$2$-group" is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, $2$-groups have representations on "$2$-vector spaces", which are categories analogous to vector spaces. Unfortunately, Lie $2$-groups typically have few representations on the finite-dimensional $2$-vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinite-dimensional $2$-vector spaces called "measurable categories" (since they are closely related to measurable fields of Hilbert spaces), and used these to study infinite-dimensional representations of certain Lie $2$-groups. Here they continue this work. They begin with a detailed study of measurable categories. Then they give a geometrical description of the measurable representations, intertwiners and $2$-intertwiners for any skeletal measurable $2$-group. They study tensor products and direct sums for representations, and various concepts of subrepresentation. They describe direct sums of intertwiners, and sub-intertwiners--features not seen in ordinary group representation theory and study irreducible and indecomposable representations and intertwiners. They also study "irretractable" representations--another feature not seen in ordinary group representation theory. Finally, they argue that measurable categories equipped with some extra structure deserve to be considered "separable $2$-Hilbert spaces", and compare this idea to a tentative definition of $2$-Hilbert spaces as representation categories of commutative von Neumann algebras.