Categories in Algebra, Geometry and Mathematical Physics
Conference in honour of Ross Street's sixtieth birthday
July 11-16 2005, Macquarie University, Sydney
Fri, 15 July: 15:45 - 16:25
Loop Groups and Lie 2-Algebras
Crans, Alissa (Loyola Marymount University)
A major result in differential geometry is Ado's theorem, which states that every finite-dimensional Lie algebra comes from some Lie group. We seek to prove a higher-dimensional analog of this statement for Lie 2-groups and Lie 2-algebras, which are categorified versions of Lie groups and Lie algebras where we have replaced the associative law and Jacobi identity, respectively, by natural isomorphisms called the `associator' and the `Jacobiator'. While this problem remains open, we describe progress toward a solution by examining the Lie 2-algebras $\mathfrak{g}_k$, each having the simple Lie algebra $\mathfrak{g}$ as its Lie algebra of objects, but with a Jacobiator proportional to a real number $k$ and built from the Killing form. It seems that except for $k=0$, there does not exist a Lie 2-group whose Lie 2-algebra is isomorphic to $\mathfrak{g}_k$. However, we can construct for integral values of $k$ an infinite-dimensional Lie 2-group whose Lie 2-algebra is equivalent to $\mathfrak{g}_k$. Moreover, these Lie 2-groups are closely related to the Kac--Moody central extensions of loop groups!