School of Mathematics Seminars and Lectures

Crystal bases and Imaginary Verma module $U_q(\widehat{sl}(2))$.

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

Friday, December 12, 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.