Communications in Mathematical Physics

, Volume 295, Issue 3, pp 731–790

Dirac Operators on Quantum Projective Spaces

Article

Abstract

We construct a family of self-adjoint operators DN, \({N\in{\mathbb Z}}\) , which have compact resolvent and bounded commutators with the coordinate algebra of the quantum projective space \({{\mathbb C}{\rm P}^{\ell}_q}\) , for any  ≥ 2 and 0 < q < 1. They provide 0+-dimensional equivariant even spectral triples. If is odd and \({N=\frac{1}{2}(\ell+1)}\) , the spectral triple is real with KO-dimension 2 mod 8.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ammann B., Bär C.: The Dirac operator on Nilmanifolds and collapsing circle bundles. Ann. Global Anal. Geom. 16(3), 221–253 (1998)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bincer A.M.: Casimir operators for su q(n). J. Phys. A 24(19), L1133–L1138 (1991)MATHCrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Björner A., Brenti F.: Combinatorics of Coxeter Groups. Springer, Berlin-Heidelberg-New York (2005)MATHGoogle Scholar
  4. 4.
    Cahen, M., Franc, A., Gutt, S.: Spectrum of the Dirac operator on complex projective space \({P_{2q-1}(\mathbb{C})}\) . Lett. Math. Phys. 18(2), 165–176 (1989), Erratum in Lett. Math. Phys. 32, 365–368 (1994)Google Scholar
  5. 5.
    Chakrabarti A.: q-analogs of IU(n) and U(n,1). J. Math. Phys. 32(5), 1227–1234 (1991)MATHCrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Chu C.-S., Ho P.-M., Zumino B.: Geometry of the quantum complex projective space CP q (N). Eur. Phys. J. C 72(1), 163–170 (1996)MathSciNetGoogle Scholar
  7. 7.
    Connes A.: Noncommutative Geometry. Academic Press, London-New York (1994)MATHGoogle Scholar
  8. 8.
    Connes A.: Gravity coupled with matter and the foundation of non-commutative geometry. Commun. Math. Phys. 182(1), 155–176 (1996)MATHCrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Dąbrowski, L.: The local index formula for quantum SU(2). In: Traces in Number Theory, Geometry and Quantum Fields, S. Albeverio et al. (eds), Aspects of Mathematics E38, Wiesbaden: Vieweg Verlag, 2008Google Scholar
  10. 10.
    Dąbrowski, L., Sitarz, A.: Dirac operator on the standard Podleś quantum sphere. In: Noncommutative Geometry and Quantum Groups, Vol. 61, Warsaw: Banach Center Publ., 2003, pp. 49–58Google Scholar
  11. 11.
    Dąbrowski L., Sobczyk J.: Left regular representation and contraction of sl q (2) to e q (2). Lett. Math. Phys. 32(3), 249–258 (1994)MATHCrossRefMathSciNetADSGoogle Scholar
  12. 12.
    D’Andrea F., Dąbrowski L., Landi G.: The isospectral Dirac operator on the 4-dimensional orthogonal quantum sphere. Commun. Math. Phys. 279(1), 77–116 (2008)MATHCrossRefADSGoogle Scholar
  13. 13.
    D’Andrea F., Dąbrowski L., Landi G.: The noncommutative geometry of the quantum projective plane. Rev. Math. Phys. 20(8), 979–1006 (2008)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    D’Andrea F., Dąbrowski L., Landi G., Wagner E.: Dirac operators on all Podleś spheres. J. Noncomm. Geom. 1(2), 213–239 (2007)MATHCrossRefGoogle Scholar
  15. 15.
    D’Andrea, F., Landi, G.: Antiself-dual connections on the quantum projective plane: monopoles. http://arXiv.org/abs/0903.3555/v1[math.QA], 2009
  16. 16.
    Dolan B.P., Huet I., Murray S., O’Connor D.: A universal Dirac operator and noncommutative spin bundles over fuzzy complex projective spaces. JHEP 03, 029 (2008)CrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Gelfand, I.M., Tsetlin, M.L.: Finite-dimensional representations of the group of unimodular matrices. Gelfand, I.M.: Collected papers, vol. II, Berlin-Heidelberg-New York: Springer-Verlag, 1988, pp. 653–656, English translation of the paper: Dokl. Akad. Nauk SSSR 71, 825–828 (1950)Google Scholar
  18. 18.
    Heckenberger I., Kolb S.: The locally finite part of the dual coalgebra of quantized irreducible flag manifolds. Proc. London Math. Soc. 89(2), 457–484 (2005)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Heckenberger I., Kolb S.: De Rham complex for quantized irreducible flag manifolds. J. Algebra 305(2), 704–741 (2006)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Itzykson C., Nauenberg M.: Unitary groups: representations and decompositions. Rev. Mod. Phys. 38(1), 95–120 (1966)MATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Klimyk, A., Schmüdgen, K.: Quantum Groups and their Representations. Berlin-Heidelberg-New York: Springer, 1997Google Scholar
  22. 22.
    Krähmer U.: Dirac operators on quantum flag manifolds. Lett. Math. Phys. 67(1), 49–59 (2004)MATHCrossRefMathSciNetADSGoogle Scholar
  23. 23.
    Neshveyev, S., Tuset, L.: The Dirac operator on compact quantum groups. http://arxiv.org/abs/math/0703161v2[math.OA], 2007
  24. 24.
    Reshetikhin, N.Y.: Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links I and II. Preprint LOMI E-4-87 E-17-87, 1987Google Scholar
  25. 25.
    Serre J.-P.: Complex Semisimple Lie Algebras. Springer, Berlin-Heidelberg-New York (2001)MATHGoogle Scholar
  26. 26.
    Schmüdgen K., Wagner E.: Dirac operator and a twisted cyclic cocycle on the standard Podleś quantum sphere. J. Reine Angew. Math. 574, 219–235 (2004)MATHMathSciNetGoogle Scholar
  27. 27.
    Seifarth, S., Semmelmann, U.: The Spectrum of the Dirac Operator on the Odd Dimensional Complex Projective Space \({P^2_{m-1}(C)}\) . SFB 288 Preprint 95, 1993Google Scholar
  28. 28.
    Sitarz, A.: Equivariant spectral triples. In: Noncommutative Geometry and Quantum Groups Vol. 61, Warsaw: Banach Centre Publ., 2003, pp. 231–263Google Scholar
  29. 29.
    Vaksman L., Soibelman Ya.: The algebra of functions on the quantum group SU(n + 1) and odd-dimensional quantum spheres. Leningrad Math. J. 2, 1023–1042 (1991)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Dép. de MathématiqueU.C. LouvainLouvain-La-NeuveBelgique
  2. 2.Scuola Internazionale Superiore di Studi AvanzatiTriesteItalia

Personalised recommendations