John Baez

Lecture at the U.C. Irvine Algebra Seminar

May 3, 2005

From Loop Groups to 2-Groups

There is 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. However, for integral k there is an infinite-dimensional Lie 2-group whose Lie 2-algebra is equivalent to gk. The objects of this 2-group are based paths in G, while the automorphisms of any object form the level-k Kac-Moody central extension of the loop group of G. This 2-group is closely related to the kth power of the canonical gerbe over G. Its nerve gives a topological group that is an extension of G by the Eilenberg-MacLane space K(Z,2). When k = ±1, this topological group can also be obtained by killing the third homotopy group of G. Thus, when G = Spin(n), it is none other than String(n).

Click on this to see the transparencies of my talk:

The talk is based on this paper:

For more on this subject, try these:

Also, Urs Schreiber and I will soon be coming out with a related paper. You can see the latest draft here:

In 2004, I gave a related talk at a conference in honor of Larry Breen's birthday.

© 2005 John Baez