Algebraic Topology: From loop groups to 2-groups
by Alissa Crans (Loyola Marymount University)
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 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).
University of Chicago Algebraic Topology Seminar