Representations of Extended Affine Lie Algebras Coordinatized by Certain Quantum Tori

An irreducible representation of the extended affine Lie algebra of type A _n-1 coordinatized by a quantum torus of variables is constructed by using the Fock space for the principal vertex operator realization of the affine Lie algebra .     

Crystal Bases for Quantum Generalized Kac-Moody Algebras

In this paper, we develop the crystal basis theory for quantumgeneralized Kac–Moody algebras. For a quantum generalizedKac–Moody algebra U_q(g), we first introduce the categoryO_int of U_q(g)-modules and prove its semisimplicity. Next, wedefine the notion of crystal bases for U_q(g)-modules in thecategory O_int and for the subalgebra . We then prove the tensor product rule and the existence theoremfor crystal bases. Finally, we construct the global bases forU_q(g)-modules in the category O_int and for the subalgebra . 2000 Mathematics Subject Classification17B37, 17B67.     

Characters of Fundamental Representations of Quantum Affine Algebras

We give closed formulae for the q-characters of the fundamental representations of the quantum loop algebra of a classical Lie algebra, in terms of a family of partitions satisfying some simple properties. We also give the multiplicities of the eigenvalues of the imaginary subalgebra in terms of these partitions.     

Strip Bundles and the Highest Weight Crystals Over Quantum Infinite Rank Affine Algebras

We give new realizations of the highest weight crystals B(λ) over the quantum infinite rank affine algebras U _q (A _∞), U _q (B _∞), U _q (C _∞), and U _q (D _∞) using strip bundles, which are related with Nakajima monomials.     

Verma-Type Modules for Quantum Affine Lie Algebras

Let be an untwisted affine Kac–Moody algebra and M_J() a Verma-type module for with J-highest weight P. We construct quantum Verma-type modules M_J^q() over the quantum group , investigate their properties and show that M_J^q() is a true quantum deformation of M_J() in the sense that the weight structure is preserved under the deformation. We also analyze the submodule structure of quantum Verma-type modules. Presented by A. VerschorenMathematics Subject Classifications (2000) 17B37, 17B67, 81R50.The first author is a Regular Associate of the ICTP. The third author was supported in part by a Faculty Research Grant from St. Lawrence University.     

胞腔代数和仿射胞腔代数简介

表示论中一个最基本的问题是确定不可约表示的参数集,这个问题至今没有完全解决.对于Graham和Lehrer引入的有限维胞腔代数,这个问题得到了完满解答,并被成功地应用于数学和物理中出现的许多代数.近来,人们引入仿射胞腔代数,将Graham和Lehrer有限维胞腔代数的表示理论框架推广到一类无限维代数上.仿射胞腔代数不仅包括有限维胞腔代数,也包括无限维的仿射Temperley-Lieb代数和Lusztig的A-型仿射Hecke代数.本文将对胞腔代数的发展历史和主要研究成果做一些综述,同时,对新引入的仿射胞腔代数及其最新成果做一点简介.     

Drinfeld Realization of Twisted Quantum Affine Algebras

The quantum affine algebra has two realizations, the usual Drinfeld–Jimbo definition and a new Drinfeld realization given by Drinfeld. In this article, we use the adjoint action to prove that these two realizations are isomorphic for the twisted quantum affine algebra.     

On the Quantum KPRV Determinants for Semisimple and Affine Lie Algebras

Let g be a semisimple or affine Lie algebra and U _q (g) its quantized enveloping algebra. Extending earlier work, the KPRV determinant for an admissible integrable U _q (g) module V relative to a parabolic subalgebra pg is defined and shown to be nonzero. These determinants had previously been evaluated for g semisimple and p a Borel subalgebra. The present results can be used to extend this to g affine as will be shown in a subsequent publication.For a parabolic subalgebra the evaluation of these determinants is much more difficult. For appropriate overalgebras of the primitive quotients of the enveloping algebra U(g) defined by one-dimensional representations of p, these determinants had been calculated for g semisimple. However the quantum case is interesting because it is unnecessary to pass to overalgebras and besides for U(g):g affine, it is not even clear how these determinants should be defined. Here for g semisimple, the degrees of the determinants are computed and shown to depend on being the same type of functions as in the enveloping algebra case; yet in a different fashion. Some special cases (in type A ₄) are computed explicity. Here, as in the Borel case, the determinants take a remarkably simple form and notably can be expressed as a product of linear factors. However compared to the enveloping algebra case one finds additional factors corresponding to what are called quantum zeros and whose origin remains unknown.