Quantum isometries and noncommutative spheres
type: Item
instance of: scholarly article
title: Quantum isometries and noncommutative spheres
zbMATH Open document ID: 1204.58004
author: Teodor Banica
author: Debashish Goswami
published in: Communications in Mathematical Physics
publication date: 2010-10-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/0905.3814
review text: The athors build noncommutative analogs of the usual spheres \(S^{n-1}\subset{\mathbb R}^n\), by following a fashionable approach that aims at considering noncommutative geometric objects, defined by some noncommutative algebras that reduce, in the commutative case, to algebras having as spectra commutative manifolds. So, they axiomatize noncommutative spheres by two noncommutative algebras, denoted \(A^*_n\) and \(A^+_n\), respectively, that in the commutative case reduce to \(A_n=C^*\left(x_1,\dots, x_n \mid x_j=x_j^*, \;x_jx_i=x_ix_j,\;\sum x_i^2=1\right)\), whose spectrum is just \(S^{n-1}\subset{\mathbb R}^{n}\). Furthermore they study the ``differential structure'' of such noncommutative spheres by recognizing there spectral triples in the sense of A. Connes. Then, they obtain ``quantum isometries'' of such noncommutative spheres by using some previous works where quantum isometries are defined in the framework of spectral triples. The paper, after a concise introduction, splits into eight more sections. 1. The usual sphere. 2. Noncommutative spheres. 3. Projective spaces. 4. Probabilistic aspects. 5. Spherical integrals. 6. Spectral triplets. 7. Quantum isometries. 8. Concluding remarks. Remark. At the end of section 7, the authors recognize that ``There are several questions raised by the above results, concerning the axiomatization of the noncommutative spheres. Perhaps the most important one is the following: What conditions on a spherical algebra \(A\) ensure the fact that the corresponding quantum isometry group is ``easy'' in the sense of [6]?''. Really a more natural way to consider noncommutative differential geometry is one in the framework of pseudogroups, as adopted by the reviewer of this paper, in order to consider quantum (super)manifolds and quantum (super) PDEs. See also the review to the paper by \textit{J. Bhowmick} and \textit{D. Goswami} [J. Funct. Anal. 257 No.~8, 2530--2572 (2009; Zbl 1180.58005)]. See also the following paper: \textit{A…
reviewed by: Agostino Prástaro
Mathematics Subject Classification ID: 58B34
Mathematics Subject Classification ID: 46L87
Mathematics Subject Classification ID: 46L89
Mathematics Subject Classification ID: 46L52
Mathematics Subject Classification ID: 46L85
zbMATH DE Number: 5795127
zbMATH Keywords: noncommutative differential geometry
zbMATH Keywords: noncommutative topology
zbMATH Keywords: Connes spectral triples
zbMATH Keywords: quantum isometries.
MaRDI profile type: Publication
OpenAlex ID: W3105845590
arXiv ID: 0905.3814
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DOI: 10.1007/S00220-010-1060-5
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description: scientific article