Comptes Rendus
Functional Analysis
Chern numbers for two families of noncommutative Hopf fibrations
[Nombres de Chern pour deux familles de fibrations de Hopf non commutatives]
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.

Reçu le :
Accepté le :
Publié le :
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
@article{CRMATH_2003__336_11_925_0,
     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},
}
TY  - JOUR
AU  - Piotr M. Hajac
AU  - Rainer Matthes
AU  - Wojciech Szymański
TI  - Chern numbers for two families of noncommutative Hopf fibrations
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 925
EP  - 930
VL  - 336
IS  - 11
PB  - Elsevier
DO  - 10.1016/S1631-073X(03)00190-0
LA  - en
ID  - CRMATH_2003__336_11_925_0
ER  - 
%0 Journal Article
%A Piotr M. Hajac
%A Rainer Matthes
%A Wojciech Szymański
%T Chern numbers for two families of noncommutative Hopf fibrations
%J Comptes Rendus. Mathématique
%D 2003
%P 925-930
%V 336
%N 11
%I Elsevier
%R 10.1016/S1631-073X(03)00190-0
%G en
%F CRMATH_2003__336_11_925_0
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/

[1] T. Brzeziński Quantum homogeneous spaces as quantum quotient spaces, J. Math. Phys., Volume 37 (1996), pp. 2388-2399

[2] T. Brzeziński, P.M. Hajac, The Chern–Galois character, joint project. See http://www.fuw.edu.pl/~pmh for a preliminary version

[3] T. Brzeziński; S. Majid Quantum geometry of algebra factorisations and coalgebra bundles, Comm. Math. Phys., Volume 213 (2000), pp. 491-521

[4] D. Calow; R. Matthes Covering and gluing of algebras and differential algebras, J. Geom. Phys., Volume 32 (2000), pp. 364-396

[5] D. Calow; R. Matthes Connections on locally trivial quantum principal fibre bundles, J. Geom. Phys., Volume 41 (2002), pp. 114-165

[6] A. Connes Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math., Volume 62 (1985), pp. 257-360

[7] P.M. Hajac Bundles over quantum sphere and noncommutative index theorem, K-Theory, Volume 21 (2000), pp. 141-150

[8] P.M. Hajac, R. Matthes, W. Szymański, A locally trivial quantum Hopf fibration, to appear in Algebr. Represent. Theory, | arXiv

[9] S. Klimek; A. Lesniewski A two-parameter quantum deformation of the unit disc, J. Funct. Anal., Volume 115 (1993), pp. 1-23

[10] J.-L. Loday Cyclic Homology, Springer-Verlag, Berlin, 1998

[11] T. Masuda; Y. Nakagami; J. Watanabe Noncommutative differential geometry on the quantum two sphere of Podleś. I: An algebraic viewpoint, K-Theory, Volume 5 (1991), pp. 151-175

[12] E.F. Müller; H.-J. Schneider Quantum homogeneous spaces with faithfully flat module structures, Israel J. Math., Volume 111 (1999), pp. 157-190

[13] P. Podleś Quantum spheres, Lett. Math. Phys., Volume 14 (1987), pp. 193-202

[14] S.L. Woronowicz Twisted SU(2) group. An example of a non-commutative differential calculus, Publ. Res. Inst. Math. Sci., Volume 23 (1987), pp. 117-181

Cité par Sources :

Commentaires - Politique