Abstract:

The Lie groups were discovered by Sophus Lie around 1880 while searching for a framework to analyze the continuous symmetries of differential equations in much the same way as permutation groups are used in Galois theory for analylyzing the discrete symmetries of algebraic equations. One of the important idea in the theory of Lie groups is to replace the gobal object, the group, with its local or linearized version which Lie called its “infinitesimal group”, now known as its Lie algebra. Around 1940, after Elie Cartan’s beautiful classification of finite dimensional semisimple Lie algebras, Lie algebras emerged as an independent branch of Algebra. An important class of infinite dimensional Lie algebras generalizing the finite dimensional semisimple Lie algebras were discovered independently by Victor Kac and Robert Moody in 1968. An important family of these infinite Lie algebras is known as affine Lie algebras. These Lie algebras have proved to be very important with interactions in many areas of mathematics and physics. One such interaction is with number theory, particularly combinatorial identities. In this talk I will give an overview of some applications in this direction.