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?
Best Answer
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].