From loop groups to 2-groups

Baez, J.C., Stevenson, D., Crans, A.S. and Schreiber, U. (2007) From loop groups to 2-groups. Homology, Homotopy and Applications, 9(2), pp. 101-135.

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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 PkG 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).

Item Type:Articles
Glasgow Author(s) Enlighten ID:Stevenson, Dr Daniel
Authors: Baez, J.C., Stevenson, D., Crans, A.S., and Schreiber, U.
Subjects:Q Science > QA Mathematics
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Homology, Homotopy and Applications
ISSN (Online):1532-0081

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