Home News About Us Contact Contributors Disclaimer Privacy Policy Help FAQ

Home
Search
Quick Search
Advanced
Fulltext
Browse
Collections
Persons
My eDoc
Session History
Login
Name:
Password:
Documentation
Help
Support Wiki
Direct access to
document ID:


          Institute: MPI für Gravitationsphysik     Collection: Quantum Gravity and Unified Theories     Display Documents



  history
ID: 402674.0, MPI für Gravitationsphysik / Quantum Gravity and Unified Theories
Infinite-Dimensional Representations of 2-Groups
Authors:Baez, John C.; Baratin, Aristide; Freidel, Freidel; Wise, Derek K.
Date of Publication (YYYY-MM-DD):2008
Review Status:not specified
Audience:Not Specified
Abstract / Description: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.
External Publication Status:unpublished
Document Type:Article
Communicated by:Hermann Nicolai
Affiliations:MPI für Gravitationsphysik/Quantum Gravity and Unified Theories
Identifiers:LOCALID:arXiv:0812.4969
Full Text:
You have privileges to view the following file(s):
0812.4969v1.pdf  [1,00 Mb] [Comment:arXiv:0812.4969v1 [math.QA]]  
 
The scope and number of records on eDoc is subject to the collection policies defined by each institute - see "info" button in the collection browse view.