Universal R-matrices for quantum groups associated to simple Lie superalgebras
type: Item
instance of: scholarly article
title: Universal R-matrices for quantum groups associated to simple Lie superalgebras
zbMATH Open document ID: 0744.17022
author: Hiroyuki Yamane
published in: Proceedings of the Japan Academy. Series A
publication date: 1992-06-27T00:00:00Z
review text: Let \(G\) be a simple Lie superalgebra of type \(A\)-\(G\). Let \(\Pi\) and \(p:\Pi\to\{0,1\}\) be a set of simple roots and a parity function of \(G\). We assume that \((\Pi,p)\) is of the distinguished type if \(G\) is of type \(F_ 4\) or \(G_ 3\). We define an \(h\)-deformation \(U_ h(\Pi,p)\) of \(U(G)\) by Chevalley generators and defining relations. We prove that \(U_ h(\Pi,p)\) is a topologically free \(Ch\)-Hopf superalgebra which goes to \(U(G)\) when \(h\to 0\). (Moreover, letting \(h\to 0\) in the defining relations, we obtain defining relations of \(U(G)\). This result also seems to be new.) A remarkable fact is that the defining relations are not exhausted by binary relations such as the \(q\)-Serre relations; we also need trinomial and quadrinomial relations. Let \(\sigma\) be the involution of \(U_ h(\Pi,p)\) defined by \(\sigma(x)=x\) (resp. \(-x)\) for an even (resp. odd) element \(x\). We introduce the ``Hopf algebraization'' \(U_ h(\Pi,p)^ \sigma\) of \(U_ h(\Pi,p)\); as an algebra, \(U_ h(\Pi,p)^ \sigma\) is the twisted product of \(U_ h(\Pi,p)\) and \(\langle\sigma\rangle\). Our main result explicitly describes the universal \(R\)-matrices \(\tilde R\) of \(U_ h(\Pi,p)^ \sigma\) by using Lusztig type \(q\)-root vectors and the \(q\)- exponential. If \((\Pi,p)\) is of type \(A\) and \(\pi\) is the fundamental representation of \(U_ h(\Pi,p)^ \sigma\), then \(\pi\otimes\pi(\tilde R)\) is the (constant) \(R\)-matrix of Perk and Schultz.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 17A70
Mathematics Subject Classification ID: 16W55
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 29810
zbMATH Keywords: quantum group
zbMATH Keywords: simple Lie superalgebra
zbMATH Keywords: universal \(R\)-matrices
zbMATH Keywords: defining relations
zbMATH Keywords: Hopf superalgebra
zbMATH Keywords: involution
zbMATH Keywords: Hopf algebraization
zbMATH Keywords: twisted product
MaRDI profile type: Publication
full work available at URL: pjaa.67.108
OpenAlex ID: W2026705944
Wikidata QID: Q115219936
cites work: Q3819193
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DOI: 10.3792/PJAA.67.108
description: scientific article; zbMATH DE number 29810