Open Access
2007 From loop groups to 2-groups
John C. Baez, Danny Stevenson, Alissa S. Crans, Urs Schreiber
Homology Homotopy Appl. 9(2): 101-135 (2007).

Abstract

We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group String(n). A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the 'Jacobiator.' Similarly, a Lie 2-group is a categorified version of a Lie group. If G is a simply-connected compact simple Lie group, there is a 1-parameter family of Lie 2-algebras gk each having g as its Lie algebra of objects, but with a Jacobiator built from the canonical 3-form on G. There appears to be no Lie 2-group having gk as its Lie 2-algebra, except when k=0. Here, however, we construct for integral k an infinite-dimensional Lie 2-group PkG whose Lie 2-algebra is equivalent to gk. The objects of gk are based paths in G-, while the automorphisms of any object form the level-k Kac-Moody central extension of the loop group ΩG. This 2- group is closely related to the kth power of the canonical gerbe over G. Its nerve gives a topological group |PkG| that is an extension of G by K(Z,2). When k=±1,|PkG| can also be obtained by killing the third homotopy group of G. Thus, when G=Spin(n),|PkG| is none other than String(n).

Citation

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John C. Baez. Danny Stevenson. Alissa S. Crans. Urs Schreiber. "From loop groups to 2-groups." Homology Homotopy Appl. 9 (2) 101 - 135, 2007.

Information

Published: 2007
First available in Project Euclid: 23 January 2008

zbMATH: 1122.22003
MathSciNet: MR2366945

Subjects:
Primary: 22E67

Keywords: 2-group , gerbe , Kac–Moody extension , Lie 2-algebra , Loop group , string group

Rights: Copyright © 2007 International Press of Boston

Vol.9 • No. 2 • 2007
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