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Gerbal representations of double loop groups

Frenkel, Edward and Zhu, Xinwen (2012) Gerbal representations of double loop groups. International Mathematics Research Notices, 2012 (17). pp. 3929-4013. ISSN 1073-7928. doi:10.1093/imrn/rnr159.

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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.

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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.
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Fondation Sciences Mathématiques de ParisUNSPECIFIED
Defense Advanced Research Projects Agency (DARPA)UNSPECIFIED
Air Force Office of Scientific Research (AFOSR)FA9550-07-1-0543
Issue or Number:17
Record Number:CaltechAUTHORS:20140919-152325076
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:49870
Deposited By: George Porter
Deposited On:24 Sep 2014 20:06
Last Modified:10 Nov 2021 18:49

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