Welcome to the new version of CaltechAUTHORS. Login is currently restricted to library staff. If you notice any issues, please email coda@library.caltech.edu
Published 2012 | Accepted Version
Journal Article Open

Gerbal representations of double loop groups

Abstract

A crucial role in representation theory of loop groups of reductive Lie groups and their Lie algebras is played by their nontrivial 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∞ of continuous automorphisms of C((t)), and these classes come from a second cohomology class of GL∞. In a similar way, double loop groups embed into a group of automorphisms of C((t))((s)), denoted by GL∞,∞, which has a nontrivial 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∞,∞ (and hence of double loop groups), realizing its nontrivial third cohomology class, on a category of modules over an infinite-dimensional Clifford algebra. This is a two-dimensional analog of the fermionic Fock representations of the ordinary loop groups.

Additional Information

© The Author(s) 2011. Published by Oxford University Press. Received April 26, 2011; Revised May 31, 2011; Accepted July 22, 2011. We thank D. Ben-Zvi, B. Feigin, D. Gaitsgory, V. Kac, D. Kazhdan, and B. Tsygan for useful discussions. This paper was finished while E.F. visited Université Paris VI as Chaire d'Excellence of Fondation Sciences Mathématiques de Paris. He thanks the Foundation for its support and the group "Algebraic Analysis" at Université Paris VI, and especially P. Schapira, for hospitality. Supported by DARPA and AFOSR through the grant FA9550-07-1-0543.

Attached Files

Accepted Version - 0810.1487v3.pdf

Files

0810.1487v3.pdf
Files (658.9 kB)
Name Size Download all
md5:9dd0deeecbd0d018f032b17b9e487ef6
658.9 kB Preview Download

Additional details

Created:
August 19, 2023
Modified:
October 17, 2023