Algorithms to obtain the canonical basis in some fundamental modules of quantum groups
type: Item
instance of: scholarly article
title: Algorithms to obtain the canonical basis in some fundamental modules of quantum groups
zbMATH Open document ID: 0874.17011
published in: Journal of Algebra
publication date: 1997-11-11T00:00:00Z
review text: Let \(U\) be the quantized universal enveloping algebra of a finite-dimensional complex simple Lie algebra. \textit{M. Kashiwara} [Duke Math. J. 63, 465-516 (1991; Zbl 0739.17005)] proved the existence and uniqueness of a crystal basis for every finite-dimensional simple \(U\)-module using certain operators and defines its crystal graph which encodes how these operators act on the crystal basis. \textit{M. Kashiwara} and \textit{T. Nakashima} [J. Algebra 165, 295-345 (1994; Zbl 0808.17005)] gave explicit descriptions of these crystal graphs in the case that \(U\) is of type \(A, B, C\), or \(D\). On the other hand, \textit{G. Lusztig} [J. Am. Math. Soc. 3, 447-498 (1990; Zbl 0703.17008)] showed the existence of a canonical basis for every finite-dimensional simple \(U\)-module. The author of the paper under review observes that if for each element of the crystal basis a certain sequence of Kashiwara's operators can be found, then the canonical basis can be written in a very nice way in terms of these sequences. Then he presents algorithms which generate sequences with the required properties for most of the fundamental modules in types \(A, B, C\), or \(D\). Moreover, he shows that with his choice of sequences there is a close link between the crystal and canonical bases of the fundamental modules in type \(C\) with the corresponding bases in type \(A\) and a similar connection also occurs for type \(D\) and type \(B\). By a different method the author gives in [Commun. Algebra 25, 1365-1386 (1997); see the review Zbl 0874.17013 below] an explicit description of the canonical bases for the fundamental \(U\)-modules when \(U\) is of type \(A\).
reviewed by: Jörg Feldvoss
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 81R50
Mathematics Subject Classification ID: 17B35
zbMATH DE Number: 1013956
zbMATH Keywords: quantized universal enveloping algebra
zbMATH Keywords: crystal basis
zbMATH Keywords: canonical basis
zbMATH Keywords: sequence of Kashiwara's operators
zbMATH Keywords: algorithms
author: Bethany R. Marsh
Wikidata QID: Q56285895
MaRDI profile type: Publication
full work available at URL: jabr.1996.0398
OpenAlex ID: W2096412442
DOI: 10.1006/JABR.1996.0398
description: scientific article; zbMATH DE number 1013956