Infinite-Dimensional Representations of 2-Groups
Memoirs of the American Mathematical Society(2012)
摘要
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.
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关键词
representation theory,tensor product,von neumann algebra,hilbert space,vector space,category theory,group representation theory,satisfiability,quantum algebra
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