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.
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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- ID: 9780821872840
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