Infinite-dimensional Representations of 2-groups

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.