Infinite-Dimensional Representations of 2-Groups

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 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 2-Hilbert spaces", and compare this idea to a tentative definition of 2-Hilbert spaces as representation categories of commutative von Neumann algebras.

112 pages

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Cited by in corpus (20)

• An Invitation to Higher Gauge Theory
• Non-invertible Symmetries and Higher Representation Theory I
• Higher categorical symmetries and gauging in two-dimensional spin systems
• Teleparallel Gravity as a Higher Gauge Theory
• A Prehistory of n-Categorical Physics
• On unitary 2-representations of finite groups and topological quantum field theory
• Transport of structure in higher homological algebra
• Poincare 2-group and quantum gravity
• Orbifolds by 2-groups and decomposition
• Quantum geometry from higher gauge theory
• A 2-categorical state sum model
• Fluxes, bundle gerbes and 2-Hilbert spaces
• Hamiltonian analysis of the BFCG theory for the Poincare 2-group
• 2-Group Representations for Spin Foams
• Causal Dynamical Triangulations in the Spincube Model of Quantum Gravity
• Two-dimensional algebra in lattice gauge theory
• (2-)Drinfel'd Double and (2-)BF Theory
• 2-Drinfel'd double symmetry of the 4d Kitaev model
• Symmetric self-adjoint Hopf categories and a categorical Heisenberg double
• On the regular representation of an (essentially) finite 2-group