School of Mathematics Seminars and Lectures

Quantum Groups and Crystal bases -an overview

by Prof. Kailash Misra (North Carolina State University, USA)

Tuesday, December 9, 2014 from to (Asia/Kolkata)
at AG-77
Description Quantum groups are $q$-deformations of universal enveloping algebras of symmetrizable Kac-Moody Lie algebras. The quantum groups associated with affine Lie algebras are called quantum
affine algebras. In 1990, Lusztig (geometric viewpoint) and Kashiwara
(algebraic viewpoint) introduced the theory of crystal bases for
integrable representations of quantum groups. Crystal bases provide a nice tool to study the combinatorics of these representations. In this study explicit realizations of crystal bases are useful. To give explicit realizations of
affine crystals we introduced perfect crystals associated with
certain level
zero representations of quantum affine algebras and realized the
affine crystals as semi-infinite tensor products of perfect
crystals. In these lectures we will take the algebraic approach
of Kashiwara focusing on some of my recent contributions in this
and related directions.