# Infinite-Dimensional Representations of 2-Groups

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*John C. Baez; Aristide Baratin; Laurent Freidel; Derek Wise*

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 they continue this
work.

They begin with a detailed study of measurable categories. Then they give a
geometrical description of the measurable representations, intertwiners and
\(2\)-intertwiners for any skeletal measurable \(2\)-group. They
study tensor products and direct sums for representations, and various concepts
of subrepresentation. They describe direct sums of intertwiners, and
sub-intertwiners—features not seen in ordinary group representation
theory and study irreducible and indecomposable representations and
intertwiners. They also study “irretractable”
representations—another feature not seen in ordinary group representation
theory. Finally, they 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.