Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 027, 11 pages      arXiv:1403.5626
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

The Structure of Line Bundles over Quantum Teardrops

Albert Jeu-Liang Sheu
Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA

Received October 07, 2013, in final form March 15, 2014; Published online March 22, 2014

Over the quantum weighted 1-dimensional complex projective spaces, called quantum teardrops, the quantum line bundles associated with the quantum principal U(1)-bundles introduced and studied by Brzezinski and Fairfax are explicitly identified among the finitely generated projective modules which are classified up to isomorphism. The quantum lens space in which these quantum line bundles are embedded is realized as a concrete groupoid C*-algebra.

Key words: quantum line bundle; quantum principal bundle; quantum teardrop; quantum lens space; groupoid C*-algebra; finitely generated projective module; quantum group.

pdf (370 kb)   tex (18 kb)


  1. Bach K.A., A cancellation problem for quantum spheres, Ph.D. Thesis, University of Kansas, Lawrence, 2003.
  2. Blackadar B., K-theory for operator algebras, Mathematical Sciences Research Institute Publications, Vol. 5, 2nd ed., Cambridge University Press, Cambridge, 1998.
  3. Brzeziński T., Fairfax S.A., Quantum teardrops, Comm. Math. Phys. 316 (2012), 151-170, arXiv:1107.1417.
  4. Brzeziński T., Hajac P.M., The Chern-Galois character, C. R. Math. Acad. Sci. Paris 338 (2004), 113-116, math.KT/0306436.
  5. Brzeziński T., Majid S., Quantum group gauge theory on quantum spaces, Comm. Math. Phys. 157 (1993), 591-638, hep-th/9208007.
  6. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
  7. Curto R.E., Muhly P.S., C*-algebras of multiplication operators on Bergman spaces, J. Funct. Anal. 64 (1985), 315-329.
  8. Gootman E.C., Rosenberg J., The structure of crossed product C*-algebras: a proof of the generalized Effros-Hahn conjecture, Invent. Math. 52 (1979), 283-298.
  9. Hajac P.M., Strong connections on quantum principal bundles, Comm. Math. Phys. 182 (1996), 579-617, hep-th/9406129.
  10. Hajac P.M., Private communication.
  11. Hong J.H., Szymański W., Quantum lens spaces and graph algebras, Pacific J. Math. 211 (2003), 249-263.
  12. Husemoller D., Fibre bundles, McGraw-Hill Book Co., New York - London - Sydney, 1966.
  13. Kumjian A., Pask D., Raeburn I., Renault J., Graphs, groupoids, and Cuntz-Krieger algebras, J. Funct. Anal. 144 (1997), 505-541.
  14. Muhly P.S., Renault J.N., C*-algebras of multivariable Wiener-Hopf operators, Trans. Amer. Math. Soc. 274 (1982), 1-44.
  15. Paterson A.L.T., Graph inverse semigroups, groupoids and their C*-algebras, J. Operator Theory 48 (2002), 645-662, math.OA/0304355.
  16. Peterka M.A., Finitely-generated projective modules over the θ-deformed 4-sphere, Comm. Math. Phys. 321 (2013), 577-603, arXiv:1203.6441.
  17. Renault J., A groupoid approach to C*-algebras, Lecture Notes in Mathematics, Vol. 793, Springer, Berlin, 1980.
  18. Rieffel M.A., Dimension and stable rank in the K-theory of C*-algebras, Proc. London Math. Soc. 46 (1983), 301-333.
  19. Rieffel M.A., The cancellation theorem for projective modules over irrational rotation C*-algebras, Proc. London Math. Soc. 47 (1983), 285-302.
  20. Rieffel M.A., Projective modules over higher-dimensional noncommutative tori, Canad. J. Math. 40 (1988), 257-338.
  21. Salinas N., Sheu A.J.L., Upmeier H., Toeplitz operators on pseudoconvex domains and foliation C*-algebras, Ann. of Math. 130 (1989), 531-565.
  22. Sheu A.J.L., A cancellation theorem for modules over the group C*-algebras of certain nilpotent Lie groups, Canad. J. Math. 39 (1987), 365-427.
  23. Sheu A.J.L., Compact quantum groups and groupoid C*-algebras, J. Funct. Anal. 144 (1997), 371-393.
  24. Sheu A.J.L., The structure of quantum spheres, Proc. Amer. Math. Soc. 129 (2001), 3307-3311.
  25. Swan R.G., Vector bundles and projective modules, Trans. Amer. Math. Soc. 105 (1962), 264-277.
  26. Vaksman L.L., Soibelman Ya.S., Algebra of functions on the quantum group SU(n+1), and odd-dimensional quantum spheres, Leningrad Math. J. 2 (1991), 1023-1042.
  27. Woronowicz S.L., Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613-665.
  28. Woronowicz S.L., Compact quantum groups, in Symétries Quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, 845-884.

Previous article  Next article   Contents of Volume 10 (2014)