Infinite-Dimensional Representations of 2-Groups
Jan 2012 · American Mathematical Soc.
Ebook
120
Pages
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About this ebook
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.
About the author
John C. Baez is at University of California, USA. ||Aristide Baratin is at Max Planck Institute for Gravitational Physics,Germany. |Laurent Freidel is at Perimeter Institute for Theoretical Physics, Canada. |Derek K. Wise is at University of Erlangen-Nurnberg, Germany
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