Letters in Mathematical Physics

, Volume 75, Issue 3, pp 235–254

Local Index Formula on the Equatorial Podleś Sphere

Article

Abstract

We discuss spectral properties of the equatorial Podleś sphere Sq2. As a preparation we also study the ‘degenerate’ (i.e. q=0) case (related to the quantum disk). Over Sq2 we consider two different spectral triples:one related to the Fock representation of the Toeplitz algebra and the isopectral one given in [7]. After the identification of the smooth pre-C*-algebra we compute the dimension spectrum and residues. We check the nontriviality of the (noncommutative) Chern character of the associated Fredholm modules by computing the pairing with the fundamental projector of the C*-algebra (the nontrivial generator of the K0-group) as well as the pairing with the q-analogue of the Bott projector. Finally, we show that the local index formula is trivially satisfied.

Mathematics Subject Classifications (2000).

Primary 58B34 Secondary 17B37 

Keywords

Noncommutative geometry spectral triple quantum sphere 

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References

  1. 1.
    Brzezinski T., Majid S. (1998). Line bundles on quantum spheres. AIP Conf. Proc. 453:3–8 math.QA/9807052ADSMathSciNetGoogle Scholar
  2. 2.
    Coburn L. (1967). The C *-algebra generated by an isometry. Bull. Amer. Math. Soc. 73:722–726MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Connes A. (1994). Noncommutative geometry. Academic Press, New YorkMATHGoogle Scholar
  4. 4.
    Connes A. (2004). Cyclic cohomology, quantum group symmetries and the local index formula for SU q(2). J. Inst. Math. Jussieu 3:17–68CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Connes A., Moscovici H. (1995). The local index formula in noncommutative geometry. Geom. Funct. Anal. 5(2):174–243CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Da̧browski, L.: Geometry of quantum spheres. J. Geom. Phys. (in press) math.QA/ 0501240Google Scholar
  7. 7.
    Da̧browski, L., Landi, G., Paschke, M., Sitarz, A.: The spectral geometry of the equatorial Podles sphere. Comptes Rendus Acad. Sci. (in press) math.QA/0408034Google Scholar
  8. 8.
    Da̧browski L., Landi G., Sitarz A., van Suijlekom W., Várilly J.C. (2005). The Dirac operator on SU q(2). Comm. Math. Phys. 259:729–759 math.QA/0411609CrossRefMathSciNetADSGoogle Scholar
  9. 9.
    van Suijlekom, W., Da̧browski, L., Landi, G., Sitarz, A., Varilly, J.C.: The local index formula for SU q(2). K-theory. (in press) math.QA/0501287Google Scholar
  10. 10.
    Gracia-Bondía J.M., Várilly J.C., Figueroa H. (2001). Elements of noncommutative geometry. Birkhäuser, BostonMATHGoogle Scholar
  11. 11.
    Higson, N.: The residue index theorem. Lecture notes for the 2000 Clay Institute symposium on NCG; The local index formula in noncommutative geometry, Lectures given at the School on Algebraic K-Theory and its applications, Trieste 2002. http://www.math.psu.edu/higson/ResearchPapers.htmlGoogle Scholar
  12. 12.
    Masuda T., Nakagami Y., Watanabe J. (1991). Noncommutative differential geometry on the quantum two sphere of Podleś. I: An algebraic viewpoint. K-Theory 5:151–175CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Chakraborty, P.S., Pal, A.: Spectral triples and associated Connes-de Rham complex for the quantum SU(2) and the quantum sphere. math.QA/0210049Google Scholar
  14. 14.
    Podleś P. (1987). Quantum spheres. Lett. Math. Phys. 14:521–531Google Scholar
  15. 15.
    Sheu A.J-L. (1991). Quantization of the Poisson SU(2) and its Poisson homogeneous space – the 2-sphere. Comm. Math. Phys. 135:217–232CrossRefMATHADSMathSciNetGoogle Scholar
  16. 16.
    Wegge-Olsen N.E. (1993). K-theory and C *-algebras: a friendly approach. Oxford University Press, OxfordGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Scuola Internazionale Superiore di Studi AvanzatiTriesteItaly

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