On crystal bases of the \(q\)-analogue of universal enveloping algebras
type: Item
instance of: scholarly article
title: On crystal bases of the \(q\)-analogue of universal enveloping algebras
zbMATH Open document ID: 0739.17005
DOI: 10.1215/S0012-7094-91-06321-0
published in: Duke Mathematical Journal
publication date: 1992-06-26T00:00:00Z
review text: The aim of this paper is to give a proof of the existence and uniqueness theorem of crystal bases for an arbitrary symmetrizable Kac-Moody Lie algebra. Let \({\mathfrak g}\) be a symmetrizable Kac-Moody Lie algebra and \(U_ q({\mathfrak g})\) be the \(q\)-analogue of the universal enveloping algebra \(U({\mathfrak g})\). For an integrable \(U_ q({\mathfrak g})\)-module \(M\), the endomorphisms \(\tilde e_ i\) and \(\tilde f_ i\) of \(M\) are introduced. Let \(A\) be the subring of \(\mathbb{Q}(q)\) consisting of rational functions regular at \(q=0\). A pair \((L,B)\) is called a (lower) crystal base of \(M\) if it satisfies the conditions: (1) \(L\) is a free sub-A-module of \(M\) such that \(M\simeq\mathbb{Q}(q)\otimes_ AL\); (2) \(B\) is a base of the \(\mathbb{Q}\)-vector space \(L/qL\); (3) \(\tilde e_ iL\subset L\) and \(\tilde f_ iL\subset L\) for any \(i\); (4) \(\tilde e_ iB\subset B\cup\{0\}\) and \(\tilde f_ iB\subset B\cup\{0\}\); (5) \(L=\bigoplus_{\lambda\in P}L_ \lambda\) and \(B=\bigcup_{\lambda\in P}B_ \lambda\), where \(P\) is the weight lattice and \(L_ \lambda=L\cap M_ \lambda\), \(B_ \lambda=B\cap(L_ \lambda/qL_ \lambda)\); (6) For \(b,b'\in B\), \(b'=\tilde f_ ib\) if and only if \(b=\tilde e_ ib'\). For a dominant integral weight \(\lambda\), let \(V(\lambda)\) denote the irreducible \(U_ q({\mathfrak g})\)-module with highest weight \(\lambda\). Let \(u_ \lambda\) be the highest weight vector of \(V(\lambda)\). Let \(L(\lambda)\) be the smallest sub-\(A\)-module of \(V(\lambda)\) that contains \(u_ \lambda\) and that is stable by the actions of \(\tilde f_ i\). Let \(B(\lambda)\) be the subset of \(L(\lambda)/qL(\lambda)\) consisting of the nonzero vectors of the form \(\tilde f_{i_ 1}\dots\tilde f_{i_ k}u_ \lambda\mod qL(\lambda)\). Then \((L(\lambda),B(\lambda))\) is a crystal base of \(V(\lambda)\) (Theorem 2). Let \(M\) be an integrable \(U_ q({\mathfrak g})\)- module such that \(M=\bigoplus_{\lambda\in F-Q_ +}M_ \lambda\) for a finite sub…
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 17B67
zbMATH DE Number: 24068
zbMATH Keywords: irreducible module
zbMATH Keywords: integrable module
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: crystal bases
zbMATH Keywords: Kac-Moody Lie algebra
zbMATH Keywords: \(q\)-analogue of the universal enveloping algebra
zbMATH Keywords: highest weight
zbMATH Keywords: global crystal base
author: Masaki Kashiwara
MaRDI profile type: Publication
cites work: Q4146776
cites work: Q3715293
cites work: A q-difference analogue of \(U({\mathfrak g})\) and the Yang-Baxter equation
cites work: Crystalizing the q-analogue of universal enveloping algebras
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cites work: On quantum groups
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cites work: Crystal base for the basic representation of \(U_ q({\mathfrak sl}^\wedge (n))\)
cites work: Analogues de la forme de Killing et du théorème d'Harish-Chandra pour les groupes quantiques
full work available at URL: s0012-7094-91-06321-0
OpenAlex ID: W1988032111
description: scientific article; zbMATH DE number 24068