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

SIGMA 10 (2014), 036, 15 pages      arXiv:1309.7096

A Note on Gluing Dirac Type Operators on a Mirror Quantum Two-Sphere

Slawomir Klimek a and Matt McBride b
a) Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford St., Indianapolis, IN 46202, USA
b) Department of Mathematics, University of Oklahoma, 601 Elm St., Norman, OK 73019, USA

Received September 30, 2013, in final form March 25, 2014; Published online March 29, 2014

The goal of this paper is to introduce a class of operators, which we call quantum Dirac type operators on a noncommutative sphere, by a gluing construction from copies of noncommutative disks, subject to an appropriate local boundary condition. We show that the resulting operators have compact resolvents, and so they are elliptic operators.

Key words: Dirac type operator; quantum space, C*-algebra.

pdf (343 kb)   tex (18 kb)


  1. Ahlfors L.V., Complex analysis. An introduction to the theory of analytic functions of one complex variable, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978.
  2. Brain S., Landi G., The 3D spin geometry of the quantum two-sphere, Rev. Math. Phys. 22 (2010), 963-993, arXiv:1003.2150.
  3. Carey A.L., Klimek S., Wojciechowski K.P., A Dirac type operator on the non-commutative disk, Lett. Math. Phys. 93 (2010), 107-125.
  4. Dąbrowski L., Landi G., Paschke M., Sitarz A., The spectral geometry of the equatorial Podleś sphere, C. R. Math. Acad. Sci. Paris 340 (2005), 819-822, math.QA/0408034.
  5. Dąbrowski L., Sitarz A., Dirac operator on the standard Podleś quantum sphere, in Noncommutative Geometry and Quantum Groups (Warsaw, 2001), Banach Center Publ., Vol. 61, Polish Acad. Sci., Warsaw, 2003, 49-58, math.QA/0209048.
  6. D'Andrea F., Dąbrowski L., Landi G., Wagner E., Dirac operators on all Podleś quantum spheres, J. Noncommut. Geom. 1 (2007), 213-239, math.QA/0606480.
  7. Hajac P.M., Matthes R., Szymański W., Noncommutative index theory for mirror quantum spheres, C. R. Math. Acad. Sci. Paris 343 (2006), 731-736, math.KT/0511309.
  8. Klimek S., Lesniewski A., Quantum Riemann surfaces. I. The unit disc, Comm. Math. Phys. 146 (1992), 103-122.
  9. Klimek S., Lesniewski A., A two-parameter quantum deformation of the unit disc, J. Funct. Anal. 115 (1993), 1-23.
  10. Klimek S., McBride M., d-bar operators on quantum domains, Math. Phys. Anal. Geom. 13 (2010), 357-390, arXiv:1001.2216.
  11. Klimek S., McBride M., A note on Dirac operators on the quantum punctured disk, SIGMA 6 (2010), 056, 12 pages, arXiv:1003.5618.
  12. Klimek S., McBride M., Global boundary conditions for a Dirac operator on the solid torus, J. Math. Phys. 52 (2011), 063518, 14 pages, arXiv:1103.4569.
  13. Klimek S., McBride M., Dirac type operators on the quantum solid torus with global boundary conditions, in preparation.
  14. Wagner E., Fibre product approach to index pairings for the generic Hopf fibration of SUq(2), J. K-Theory 7 (2011), 1-17, arXiv:0902.3777.

Previous article  Next article   Contents of Volume 10 (2014)