Comptes Rendus
Functional Analysis
Noncommutative index theory for mirror quantum spheres
[Théorie de l'indice non commutative pour des sphères quantiques miroirs]
Comptes Rendus. Mathématique, Volume 343 (2006) no. 11-12, pp. 731-736.

Nous introduisons et analysons un nouveau type de 2-sphères quantiques. Nous appliquons la théorie de l'indice pour les fibrés en droites non commutatifs sur ces sphères afin de déduire que les espaces lenticulaires quantiques sont des exemples d'extensions principales de C-algèbres qui ne sont pas des produits croisés.

We introduce and analyse a new type of quantum 2-spheres. Then we apply index theory for noncommutative line bundles over these spheres to conclude that quantum lens spaces are non-crossed-product examples of principal extensions of C-algebras.

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DOI : 10.1016/j.crma.2006.09.021

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

1 Instytut Matematyczny, Polska Akademia Nauk, ul. Śniadeckich 8, Warszawa, 00-956 Poland
2 Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski, ul. Hoża 74, Warszawa, 00-682 Poland
3 School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia
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Piotr M. Hajac; Rainer Matthes; Wojciech Szymański. Noncommutative index theory for mirror quantum spheres. Comptes Rendus. Mathématique, Volume 343 (2006) no. 11-12, pp. 731-736. doi : 10.1016/j.crma.2006.09.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.09.021/

[1] P.F. Baum; P.M. Hajac; R. Matthes; W. Szymański The K-theory of Heegaard-type quantum 3-spheres, K-Theory, Volume 35 (2005), pp. 159-186

[2] T. Brzeziński; P.M. Hajac The Chern–Galois character, C. R. Acad. Sci. Paris, Ser. I, Volume 338 (2004), pp. 113-116

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

[4] L. Da̧browski The garden of quantum spheres, Warsaw, 2001 (Banach Center Publ.), Volume vol. 61, Polish Acad. Sci., Warsaw (2003), pp. 37-48

[5] L. Da̧browski; G. Landi; M. Paschke; A. Sitarz The spectral geometry of the equatorial Podleś sphere, C. R. Math. Acad. Sci. Paris, Ser. I, Volume 340 (2005), pp. 819-822

[6] K.R. Davidson C-Algebras by Example, Fields Institute Monographs, vol. 6, Amer. Math. Soc., Providence, RI, 1996

[7] D.A. Ellwood A new characterisation of principal actions, J. Funct. Anal., Volume 173 (2000), pp. 49-60

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

[9] P.M. Hajac; S. Majid Projective module description of the q-monopole, Comm. Math. Phys., Volume 206 (1999), pp. 247-264

[10] P.M. Hajac; R. Matthes; W. Szymański Chern numbers for two families of noncommutative Hopf fibrations, C. R. Acad. Sci. Paris, Ser. I, Volume 336 (2003), pp. 925-930

[11] P.M. Hajac; R. Matthes; W. Szymański A locally trivial quantum Hopf fibration, Algebras Rep. Theory, Volume 9 (2006), pp. 121-146

[12] J.H. Hong; W. Szymański Quantum spheres and projective spaces as graph algebras, Comm. Math. Phys., Volume 232 (2002), pp. 157-188

[13] J.H. Hong; W. Szymański Quantum lens spaces and graph algebras, Pacific J. Math., Volume 211 (2003), pp. 249-263

[14] J.H. Hong; W. Szymański The primitive ideal space of the C-algebras of infinite graphs, J. Math. Soc. Japan, Volume 56 (2004), pp. 45-64

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

[16] 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

[17] K. Matsumoto; J. Tomiyama Noncommutative lens spaces, J. Math. Soc. Japan, Volume 44 (1992), pp. 13-41

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

[19] A.J.-L. Sheu Quantization of the Poisson SU(2) and its Poisson homogeneous space—the 2-sphere, Comm. Math. Phys., Volume 135 (1991), pp. 217-232

[20] W. Szymański Quantum lens spaces and principal actions on graph C-algebras, Warsaw, 2001 (Banach Center Publ.), Volume vol. 61, Polish Acad. Sci., Warsaw (2003), pp. 299-304

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