InfiniteDimensional Representations of 2Groups
Abstract
A "2group" is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, 2groups have representations on "2vector spaces", which are categories analogous to vector spaces. Unfortunately, Lie 2groups typically have few representations on the finitedimensional 2vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinitedimensional 2vector spaces called "measurable categories" (since they are closely related to measurable fields of Hilbert spaces), and used these to study infinitedimensional representations of certain Lie 2groups. 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 2intertwiners for any skeletal measurable 2group. We study tensor products and direct sums for representations, and various concepts of subrepresentation. We describe direct sums of intertwiners, and subintertwiners  features not seen in ordinary group representation theory. We study irreducible and indecomposable representations and intertwiners. We also study "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 2Hilbert spaces", and compare this idea to a tentative definition of 2Hilbert spaces as representation categories of commutative von Neumann algebras.
 Publication:

arXiv eprints
 Pub Date:
 December 2008
 arXiv:
 arXiv:0812.4969
 Bibcode:
 2008arXiv0812.4969B
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematical Physics;
 Mathematics  Category Theory
 EPrint:
 112 pages