Comptes Rendus
no. 11
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. Hajac1, 2, 3 ; Rainer Matthes4 ; Wojciech Szymański5
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.

Nous considérons des fibrés en droites non commutatifs associés à la fibration de Hopf quantique de SUq(2) sur toutes les sphères quantiques de Podleś ainsi qu'avec une fibration de Hopf localement triviale de S3pq. 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 SUq(2) et de S3pq. 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 SUq(2) over all Podleś spheres and with a locally trivial Hopf fibration of S3pq. 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 SUq(2) and S3pq. We show that the Chern numbers of these modules coincide with the winding numbers of representations defining them.

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DOI : 10.1016/S1631-073X(03)00190-0
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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     title = {Chern numbers for two families of noncommutative {Hopf} fibrations},
     journal = {Comptes Rendus. Math\'ematique},
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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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