Chern numbers for two families of noncommutative Hopf fibrations

Piotr M. Hajac; Rainer Matthes; Wojciech Szymański

doi:10.1016/S1631-073X(03)00190-0

Functional Analysis

Chern numbers for two families of noncommutative Hopf fibrations
[Nombres de Chern pour deux familles de fibrations de Hopf non commutatives]

Présenté par : Alain Connes

Piotr M. Hajac ^1,^2,³ ; Rainer Matthes ⁴ ; Wojciech Szymański ⁵

¹ Mathematisches Institut, Universität München, Theresienstr. 39, 80333 München, Germany
² Instytut Matematyczny, Polska Akademia Nauk, ul. Śniadeckich 8, Warszawa 00-956, Poland
³ Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski, ul. Hoża 74, Warszawa 00-682, Poland
⁴ Fachbereich 2, TU Clausthal, Leibnizstr. 4, 38678 Clausthal-Zellerfeld, Germany
⁵ School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia

Comptes Rendus. Mathématique, Volume 336 (2003) no. 11, pp. 925-930.

Résumé
Abstract

Nous considérons des fibrés en droites non commutatifs associés à la fibration de Hopf quantique de SU_q(2) sur toutes les sphères quantiques de Podleś ainsi qu'avec une fibration de Hopf localement triviale de S³_pq. Ces fibrés sont construits comme des modules projectifs associés aux représentations de dimension 1 de U(1) avec des extensions galoisiennes relatives aux fibrés principaux de SU_q(2) et de S³_pq. Nous montrons que les nombres de Chern de ces fibrés coı̈ncident avec les degrés des représentations qui les définissent.

We consider noncommutative line bundles associated with the Hopf fibrations of SU_q(2) over all Podleś spheres and with a locally trivial Hopf fibration of S³_pq. These bundles are given as finitely generated projective modules associated via 1-dimensional representations of U(1) with Galois-type extensions encoding the principal fibrations of SU_q(2) and S³_pq. We show that the Chern numbers of these modules coincide with the winding numbers of representations defining them.

Reçu le : 2003-03-20
Accepté le : 2003-04-01
Publié le : 2003-06-01

DOI : 10.1016/S1631-073X(03)00190-0

Affiliations des auteurs :

Piotr M. Hajac ^{1, 2, 3} ; Rainer Matthes ⁴ ; Wojciech Szymański ⁵

¹ Mathematisches Institut, Universität München, Theresienstr. 39, 80333 München, Germany
² Instytut Matematyczny, Polska Akademia Nauk, ul. Śniadeckich 8, Warszawa 00-956, Poland
³ Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski, ul. Hoża 74, Warszawa 00-682, Poland
⁴ Fachbereich 2, TU Clausthal, Leibnizstr. 4, 38678 Clausthal-Zellerfeld, Germany
⁵ School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia

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     author = {Piotr M. Hajac and Rainer Matthes and Wojciech Szyma\'nski},
     title = {Chern numbers for two families of noncommutative {Hopf} fibrations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {925--930},
     publisher = {Elsevier},
     volume = {336},
     number = {11},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00190-0},
     language = {en},
}

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Piotr M. Hajac; Rainer Matthes; Wojciech Szymański. Chern numbers for two families of noncommutative Hopf fibrations. Comptes Rendus. Mathématique, Volume 336 (2003) no. 11, pp. 925-930. doi : 10.1016/S1631-073X(03)00190-0. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00190-0/

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