Laplacians and gauged Laplacians on a quantum Hopf bundle
type: Item
instance of: scholarly article
title: Laplacians and gauged Laplacians on a quantum Hopf bundle
zbMATH Open document ID: 1254.58010
DOI: 10.1007/978-3-8348-9831-9_10
author: Alessandro Zampini
published in: Quantum Groups and Noncommutative Spaces
publication date: 2012-07-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1003.5598
Mathematics Subject Classification ID: 58J42
Mathematics Subject Classification ID: 58B32
Mathematics Subject Classification ID: 81R60
zbMATH DE Number: 6059391
zbMATH Keywords: gauged Laplacian
zbMATH Keywords: quantum Hopf bundle
zbMATH Keywords: Hall Hamiltonian
zbMATH Keywords: quantum sphere
zbMATH Keywords: principal bundle
zbMATH Keywords: gauge connection
zbMATH Keywords: quantum group
MaRDI profile type: Publication
OpenAlex ID: W1482820270
arXiv ID: 1003.5598
review text: The author focusses on the analysis of a class of Hall Hamiltonians in the noncommutative set up. It is intended as a survey of the general formulation of quantum principal bundles, and as a description of a specific procedure to introduce, on both the total space and the base space of a quantum Hopf bundle, a set of Laplacian operators and to couple them with gauge connections. It presents a detailed formulation of the classical Hopf bundle.NEWLINENEWLINEClassical Hall Hamiltonians are gauged Laplace operators acting on the space of sections of the vector bundles associated to the principal bundles \(\pi : G \to G/K\) over homogeneous spaces with \(G\) semisimple and \(K\) compact and can be constructed in terms of the Casimir operators of \(G\) and \(K\). With \((\rho, V)\) a representation of \(K\), one has the identification of sections of the associated vector bundle \(E = G \times_{\rho(K)} V\) with equivariant maps from \(G\) to \(V\), \(\Gamma(G/K, E) \simeq C^{\infty} (G, V)_{\rho(K)} \subset C^{\infty} (G) \otimes V\). Given a connection on \(G\), one has a covariant derivative \(\nabla\) on \(\Gamma(G/K, E)\), so that the gauged Laplacian operator is \(\Delta^E = (\nabla \nabla^* + \nabla^* \nabla )\), where the dual \(\nabla^*\) is defined from the metric induced on the homogeneous space basis \(G/K\) by the Cartan-Killing metric on \(G\).NEWLINENEWLINEThe natural evolution is to develop models of the Hall effect on noncommutative spaces whose symmetries are described in terms of quantum groups. In [\textit{G. Landi, C. Reina} and \textit{A. Zampini}, Commun. Math. Phys. 287, No. 1, 179--209 (2009; Zbl 1180.58004)], the first model of excitations moving on a quantum 2-sphere in the field of a magnetic monopole has been studied. It is described by a quantum principal \(U(1)\)-bundle over a quantum sphere \(S_q^2\) having as a total space the manifold of the quantum group \(SU_q(2)\) [\textit{T. Brzeziński} and \textit{S. Majid}, Commun. Math. Phys. 157, N…
reviewed by: Choonkil Park
description: scientific article; zbMATH DE number 6059391