EVENT DETAILS AND ABSTRACT

Title: From Loop Groups to 2-Groups
Speaker: Alissa Crans
Speaker Info: UC Riverside
Brief Description:
Special Note:
Abstract:

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 g_k 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 g_k 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 g_k. 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. The nerve of this 2-group 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).