Infinite-Dimensional Representations of 2-Groups - Couverture souple

Baez, John C.; Baratin, Aristide; Freidel, Laurent; Wise, Derek K.

Couverture souple

ISBN 10 : 0821872842 ISBN 13 : 9780821872840

Editeur : American Mathematical Society, 2012

Synopsis

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.

A propos du livre

À propos de l'auteur

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

Détails de l'édition

• Éditeur: American Mathematical Society
• Date d'édition: 2012
• Langue: anglais
• ISBN 10: 0821872842
• ISBN 13: 9780821872840
• Reliure: Broché
• Nombre de pages: 120
• Coordonnées du fabricant: non disponible
• Personne responsable: non disponible