John C Baez Infinite-Dimensional Representations of 2-Groups

Annotation

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.