Abstract:

Quantum groups, introduced by physicists to model observables for the quantum scattering problem, have a variety of mathematical applications. Among these, the best known examples are to link invariants and to the representation theory of semisimple Lie groups and Lie algebras.

In this lecture I trace one such development due to G. Lusztig---the introduction of a basis for the universal enveloping algebra of a Borel subalgebra U(b), which when projected to finite dimensional quotient modules remains a basis.

As time permits, I will discuss the extension of the theory to affine Lie algebras and the important connection to the combinatorics of symmetric functions.