## EVENT DETAILS AND ABSTRACT

**Mathematical Physics Seminar**
**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).

**Date:** Wednesday, May 25, 2005

**Time:** 2:00pm

**Where:** Lunt 107

**Contact Person:** Boris Tsygan

**Contact email:** tsygan@math.northwestern.edu

**Contact Phone:** 847-467-6446

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