Lengths of finite dimensional representations of PBW algebras

被引：7

作者：

Constantine, D

Darnall, M

机构：

[1] Eastern Nazarene Coll, Quincy, MA 02170 USA

[2] Humboldt State Univ, Arcata, CA 95521 USA

来源：

LINEAR ALGEBRA AND ITS APPLICATIONS | 2005年 / 395卷

基金：

美国国家科学基金会;

关键词：

representations of lie algebras; representations of quantum groups; PBW sets; lengths of representations;

D O I：

10.1016/j.laa.2004.08.006

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

Let Sigma be a set of n x it matrices with entries from a field, for n > 1, and let c(Sigma) be the maximum length of products in Sigma necessary to linearly span the algebra it generates. Bounds for c(Sigma) have been given by Paz and Pappacena, and Paz conjectures a bound of 2n - 2 for any set of matrices. In this paper we present a proof of Paz's conjecture for sets of matrices obeying a modified Poincare-Birkhoff-Witt (PBW) property, applicable to finite dimensional representations of Lie algebras and quantum groups. A representation of the quantum plane establishes the sharpness of this bound, and we prove a bound of 2n - 3 for sets of matrices with this modified PBW property which do not generate the full algebra of all it x it matrices. This bound of 2n - 3 also holds for representations of Lie algebras, although we do not know whether it is sharp in this case. (C) 2004 Elsevier Inc. All rights reserved.

引用

页码：175 / 181

页数：7