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Journal Article

#### Infinite-Dimensional Representations of 2-Groups

##### Fulltext (public)

0812.4969v1.pdf

(Preprint), 2MB

##### Supplementary Material (public)

There is no public supplementary material available

##### Citation

Baez, J. C., Baratin, A., Freidel, L., & Wise, D. K. (2012). Infinite-Dimensional
Representations of 2-Groups.* Memoirs of the American Mathematical Society,* *219*.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-4728-E

##### Abstract

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 we continue this work. We begin with a detailed study of measurable categories. Then we give a geometrical description of the measurable representations, intertwiners and 2-intertwiners for any skeletal measurable 2-group. We study tensor products and direct sums for representations, and various concepts of subrepresentation. We describe direct sums of intertwiners, and sub-intertwiners--features not seen in ordinary group representation theory. We classify irreducible and indecomposable representations and intertwiners. We also classify "irretractable" representations--another feature not seen in ordinary group representation theory. Finally, we 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.