Gerbal Representations of Double Loop Groups
| dc.creator | Frenkel, Edward | |
| dc.creator | Zhu, Xinwen | |
| dc.date | 2008-10-08 | |
| dc.date | 2008-11-17 | |
| dc.date.accessioned | 2026-07-07T10:18:19Z | |
| dc.date.available | 2026-07-07T10:18:19Z | |
| dc.description | A crucial role in representation theory of loop groups of reductive Lie groups and their Lie algebras is played by their non-trivial second cohomology classes which give rise to their central extensions (the affine Kac-Moody groups and Lie algebras). Loop groups embed into the group GL_\infty of continuous automorphisms of C((t)), and these classes come from a second cohomology class of GL_\infty. In a similar way, double loop groups embed into a group of automorphisms of C((t))((s)), denoted by GL_{\infty,\infty}, which has a non-trivial third cohomology. In this paper we explain how to realize a third cohomology class in representation theory of a group: it naturally arises when we consider representations on categories rather than vector spaces. We call them "gerbal representations." We then construct a gerbal representation of GL_{\infty,\infty} (and hence of double loop groups), realizing its non-trivial third cohomology class, on a category of modules over an infinite-dimensional Clifford algebra. This is a two-dimensional analogue of the fermionic Fock representations of the ordinary loop groups. | |
| dc.description | 60 pages | |
| dc.identifier | https://arxiv.org/abs/0810.1487 | |
| dc.identifier | http://arxiv.org/abs/0810.1487 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/174151 | |
| dc.subject | Representation Theory | |
| dc.subject | High Energy Physics - Theory | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Quantum Algebra | |
| dc.title | Gerbal Representations of Double Loop Groups | |
| dc.type | text |