Connections on locally trivial quantum principal fibre bundles
type: Item
instance of: scholarly article
title: Connections on locally trivial quantum principal fibre bundles
zbMATH Open document ID: 1026.46064
DOI: 10.1016/S0393-0440(01)00050-X
author: D. Calow
author: R. Matthes
published in: Journal of Geometry and Physics
publication date: 2002-06-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0002228
review text: A theory of connections in a locally trivial quantum principal bundle is developed as a part of the general theory of noncommutative differential geometry. A locally trivial quantum principal bundle is a noncommutative algebraic version of a principal bundle of the classical Differential Geometry. It is defined in terms of a Hopf algebra \(H\) (which is considered as the ``structure group'' of the bundle), an algebra \(B\) (``the base space'') and a right \(H\) comodule \(P\) with coaction \(\Delta_P\) and an injective homomorphism \(B \to P\) of modules, together with a system of ideals of \(B\) and some additional conditions. The authors define the notion of covariant differential algebras and connections on a quantum principal bundle. They introduce the notion of covariant derivatives, connection forms, curvatures and curvature forms and establish some relations between these notions. They consider as an example a \(U(1)\) quantum principal bundle over a glued quantum sphere and construct a connection in this bundle. The connection may be considered as a \(q\)-version of a Dirac monopole.
Mathematics Subject Classification ID: 46L87
Mathematics Subject Classification ID: 81R50
zbMATH DE Number: 1758081
zbMATH Keywords: quantum principal bundle
zbMATH Keywords: differential structures
zbMATH Keywords: covariant derivative
zbMATH Keywords: connection
zbMATH Keywords: \(q\)-monopole
reviewed by: Dmitri Alekseevsky
MaRDI profile type: Publication
arXiv ID: math/0002228
cites work: Translation map in quantum principal bundles
cites work: Quantum group gauge theory on quantum spaces
cites work: Quantum principal fibre bundles: Topological aspects
cites work: Covering and gluing of algebras and differential algebras
cites work: Q4716420
cites work: The quantum structure of spacetime of the Planck scale and quantum fields
cites work: Geometry of quantum principal bundles. I
cites work: Geometry of Quantum Principal Bundles II
cites work: Q4251396
cites work: Strong connections on quantum principal bundles
cites work: The two-parameter quantum deformation of the unit disc
cites work: Q4375485
cites work: Classifying spaces for quantum principal bundles
cites work: Q4000318
cites work: Quantum groups on fibre bundles
cites work: Q4328430
cites work: Q3136272
cites work: Principal homogeneous spaces for arbitrary Hopf algebras
cites work: Twisted \(\text{SU}(2)\) group. An example of a non-commutative differential calculus
cites work: Differential calculus on compact matrix pseudogroups (quantum groups)
description: scientific article