Chennai Mathematical Institute


2.00 pm, Seminar Hall
Kac-Moody Algebras and Groups

Rahul Singh
Northeastern University, USA.


Kac-Moody Lie algebra \gl\ were introduced in the mid-1960s independently by V. Kac and R. Moody. These algebras are categorized into three families: finite type, affine type, and indefinite, of which the first two are well understood. Kac-Moody algebras and the associated Kac-Moody groups have a rich structure, generalizing the theory of finite-dimensional semisimple Lie algebras, which in this theory are just the Kac-Moody algebras of finite type. In this talk, we sketch some basic constructions, in particular the Cartan matrix, Weyl group, Bruhat decomposition and Schubert varieties associated to Kac-Moody algebras/groups. Finally, we show how any affine type Kac-Moody algebra can be realized as a two dimensional extension of $\mathfrak g\otimes\mathcal A$, where $\mathfrak g$ is some semisimple Lie algebra $\mathfrak g$ and $\mathcal A=\mathbb C[t,t^{-1}]$. This is a powerful tool in the study of affine Kac-Moody groups, and also opens up avenues to better understand semisimple Lie algebras. The main references are {\em Infinite-Dimensional Lie Algebras}, V. Kac and {\em Kac-Moody Groups, their Flag Varieties and Representation Theory}, S. Kumar.