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Infinite-Dimensional Representations of 2-Groups
About this Title
John C. Baez, Department of Mathematics, University of California, Riverside, California 92521, Aristide Baratin, Triangle de la Physique (Orsay-Saclay-Ecole Polytechnique), Laboratoire de Physique Théorique, CNRS UMR 8627, Université Paris XI, F-91405 Orsay Cédex, France, Laurent Freidel, Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2J 2Y5, Canada and Derek K. Wise, Department of Mathematics, University of California, Davis, California 95616
Publication: Memoirs of the American Mathematical Society
Publication Year:
2012; Volume 219, Number 1032
ISBNs: 978-0-8218-7284-0 (print); 978-0-8218-9116-2 (online)
DOI: https://doi.org/10.1090/S0065-9266-2012-00652-6
Published electronically: February 28, 2012
Keywords: Representation theory,
2-group,
2-vector space
MSC: Primary 20C35; Secondary 18D05, 22A22
Table of Contents
Chapters
- 1. Introduction
- 2. Representations of 2-Groups
- 3. Measurable Categories
- 4. Representations on Measurable Categories
- 5. Conclusion
- A. Tools from Measure Theory
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.- David L. Armacost, The structure of locally compact abelian groups, Monographs and Textbooks in Pure and Applied Mathematics, vol. 68, Marcel Dekker, Inc., New York, 1981. MR 637201
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