Crystals for Demazure Modules of Special Linear Quantum Affine Type

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Title: Crystals for Demazure Modules of Special Linear Quantum Affine Type
Author: Beier, Julie Catherine
Advisors: Tom Lada, Committee Member
Amassa Fauntleroy, Committee Member
Hollylynne Stohl Lee, Committee Member
Ernest Stitzinger, Committee Member
Kailash Misra, Committee Chair
Abstract: Kac-Moody Lie algebras, independently discovered in the 1960's by Victor Kac and Robert Moody, are infinite dimensional analogs of finite dimensional semisimple Lie algebras. Affine Lie algebras form an important class of infinite dimensional Kac-Moody Lie algebras with numerous applications in different areas of mathematics and physics. Quantum groups, discovered by both Drinfeld and Jimbo in the 1980's, 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. For `$q$' generic and $lambda$ a dominant weight there exists a unique (up to isomorphism) irreducible highest weight module $V(lambda)$ for the quantum affine algebra $U_q(g)$. For each $w in mathcal{W}$, the Weyl group of $g$, there is a finite dimensional subspace $V_w(lambda)$ of $V(lambda)$ called a Demazure module generated from the extremal weight vector $u_{wlambda}$ by the positive part of $U_q(g)$. The crystal $B(lambda)$ associated with $V(lambda)$ was introduced by Kashiwara and Lusztig in the 1990's. $B(lambda)$ provides an important tool to study the combinatorics of $V(lambda)$. In 1993, Kashiwara showed that a suitable subset $B_w(lambda)$ of $B(lambda)$ is the crystal for the Demazure module $V_w(lambda)$. In this thesis we give an explicit realization of the Demazure crystals $B_w(lambda)$ for the special linear quantum affine Lie algebra $uqslnh$ where $w=w(k)$, $k >0$, is a suitable linear chain of Weyl group elements. This realization is given in terms of certain combinatorial objects called extended Young diagrams.
Date: 2009-04-23
Degree: PhD
Discipline: Mathematics

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