Twisted \(\text{SU}(2)\) group. An example of a non-commutative differential calculus
type: Item
instance of: scholarly article
title: Twisted \(\text{SU}(2)\) group. An example of a non-commutative differential calculus
zbMATH Open document ID: 0676.46050
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1987-01-01T00:00:00Z
review text: Summary: For any number \(\mu\) in the interval \([-1,1]\) a \(C^*\)-algebra \(A\) generated by two elements \(\alpha\) and \(\gamma\) satisfying a simple (depending on \(\mu)\) commutation relation is introduced and investigated. If \(\mu =1\) then the algebra coincides with the algebra of all continuous functions on the group \(\text{SU}(2)\). Therefore one can introduce many notions related to the fact that \(\text{SU}(2)\) is a Lie group. In particular, one can speak about convolution products, Haar measure, differential structure, cotangent bundle, left invariant differential forms, Lie brackets, infinitesimal shifts and Cartan-Maurer formulae. One can also consider representations of \(\text{SU}(2)\). For \(\mu <1\) the algebra \(A\) is no longer commutative, however the notions listed above are meaningful. Therefore \(A\) can be considered as the algebra of all ``continuous functions'' on a ``pseudospace \(\text{S}_{\mu}\text{U}(2)\)'' and this pseudospace is endowed with a Lie group structure. Potential applications to quantum physics are indicated.
Mathematics Subject Classification ID: 46L87
Mathematics Subject Classification ID: 46L65
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 4107479
zbMATH Keywords: twisted SU(2) group
zbMATH Keywords: non-commutative differential calculus
zbMATH Keywords: \(C^*\)- algebra
zbMATH Keywords: Lie group
zbMATH Keywords: convolution products
zbMATH Keywords: Haar measure
zbMATH Keywords: differential structure
zbMATH Keywords: cotangent bundle
zbMATH Keywords: left invariant differential forms
zbMATH Keywords: Lie brackets
zbMATH Keywords: infinitesimal shifts
zbMATH Keywords: Cartan Maurer formulae
author: Stanisław Lech Woronowicz
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195176848
OpenAlex ID: W2110612849
cites work: Compact matrix pseudogroups
cites work: Q3965125
cites work: Q3658578
DOI: 10.2977/PRIMS/1195176848
description: scientific article; zbMATH DE number 4107479