Differential operators on quantum spaces for \(GL_ q(n)\) and \(SO_ q(n)\)
type: Item
instance of: scholarly article
title: Differential operators on quantum spaces for \(GL_ q(n)\) and \(SO_ q(n)\)
zbMATH Open document ID: 0755.17012
DOI: 10.1007/BF00402900
published in: Letters in Mathematical Physics
publication date: 1993-01-16T00:00:00Z
review text: This paper is concerned with rings of \(q\)-differential operators \(\text{Diff}_ G(n)\) (as defined by Wess and Zumino) on the `quantum planes' associated with the quantum groups \(G=GL_ q(n)\) and \(G=SO_ q(n)\). In both of these cases the author explicitly constructs normal elements of \(\text{Diff}_ G(n)\) and is thus able to show that a suitable extension ring of \(\text{Diff}_ G(n)\) contains a tensor product of \(n\) copies of \(\text{Diff}_ \alpha(1)\), for suitable \(\alpha\). Here \(\text{Diff}_ \alpha(1)=\mathbb{C}[x][\partial]\) where \(\partial\) is the \(\alpha\)-derivation with \(\partial(x)=1\), so that \(\partial x-\alpha x\partial=1\). The author interprets this as an `isomorphism' between the classical and quantized rings of differential operators for generic values of the parameter \(q\). D. A. Jordan has pointed out that, in the \(GL_ q(n)\) case, replacing \(q^ 2\) by \(q\) in the relations allows one to obtain the required tensor product simply by inverting normal elements.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 81R50
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 81R05
zbMATH DE Number: 89336
zbMATH Keywords: rings of \(q\)-differential operators
zbMATH Keywords: quantum groups
zbMATH Keywords: normal elements
author: O. V. Ogievetsky
reviewed by: Martin P. Holland
MaRDI profile type: Publication
cites work: Bicovariant differential calculus on quantum groups \(SU_ q(N)\) and \(SO_ q(N)\)
description: scientific article; zbMATH DE number 89336