Kac-Moody Groups, Their Flag Varieties & Representation Theory

by Shrawan Kumar

Kac-Moody Lie algebras 9 were introduced in the mid-1960s independently by V. Kac and R. Moody, generalizing the finite-dimensional semisimple Lie alge­ bras which we refer to as the finite case. The theory has undergone tremendous developments in various directions and connections with diverse areas abound, including mathematical physics, so much so that this theory has become a stan­ dard tool in mathematics. A detailed treatment of the Lie algebra aspect of the theory can be found in V. Kac's book [Kac-90l This self-contained work treats the algebro-geometric and the topological aspects of Kac-Moody theory from scratch. The emphasis is on the study of the Kac-Moody groups 9 and their flag varieties XY, including their detailed construction, and their applications to the representation theory of g. In the finite case, 9 is nothing but a semisimple Y simply-connected algebraic group and X is the flag variety 9 /Py for a parabolic subgroup p y C g.

624 pages, Hardcover

First published January 1, 2001

Reviews

[Andrew]

I'd heard of Kac-Moody algebras before, but I didn't know very much about the parallel theory of their groups and flag varieties. The approach to the subject is fairly algebraic / combinatorial approach to the subject, although the fact that we are working with groups and their flag varieties gives it a more geometric flavor as well. One thing that surprised me was how the generalization from the finite-dimensional case usually involved simply coming up with a framework to express the objects in question as ind-varieties / pro-varieties; the generalizations were almost straightforward once the right framework is there. Also, this book is not too difficult if you are familiar with representation theory, say, on the level of Jantzen's Representations of Algebraic Groups.