Question

Does anyone know a simple construction for Affine Kac-Moody groups? There is a book by Kumar ("Kac-Moody groups, their flag varieties, and representation theory") that does the construction for the general Kac-Moody case, but I find the presentation dense. There is also a section that constructs a one-dimensional extension of the loop group by loop rotation, which is a fairly transparent definition. However, I don't know how to add on the final central extension.

Even if the answer to my question is "There is no simpler construction," could someone also tell me about a fruitful way to get my hands on Affine Kac-Moody groups?

Answer 1

I second that central extensions of loop groups over a compact Lie group are treated in Chapter 4 of "Loop Groups" by Pressley and Segal. A completely different, purely algebaric construction (via generators and relations, a la Steinberg) for a (wider? overlapping?, not necessary loop) class of groups is given in J. Tits, Uniqueness and presentation of Kac-Moody groups over fields, J. Algebra 105 (1987), 542-573 [DOI].

Answer 2

I thought the final central extension came by constructing the determinant line bundle over the based loop group and looking at the automorphism group of that.

The place to look for this is the Pressley-Segal classic Loop groups (my copy is at work, hence the hand-waving answer).

Answer 3

The basic examples of affine Kac-Moody groups are the groups SL_n(k[t,t^{-1}]). These groups are Kac Moody groups for the diagram A^~_{n-1} over the field k.