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

SIGMA 10 (2014), 088, 20 pages      arXiv:1403.7818      https://doi.org/10.3842/SIGMA.2014.088
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

Piecewise Principal Coactions of Co-Commutative Hopf Algebras

Bartosz Zieliński
Department of Computer Science, Faculty of Physics and Applied Informatics, University of Łódź, Pomorska 149/153 90-236 Łódź, Poland

Received March 31, 2014, in final form August 11, 2014; Published online August 18, 2014

Abstract
Principal comodule algebras can be thought of as objects representing principal bundles in non-commutative geometry. A crucial component of a principal comodule algebra is a strong connection map. For some applications it suffices to prove that such a map exists, but for others, such as computing the associated bundle projectors or Chern-Galois characters, an explicit formula for a strong connection is necessary. It has been known for some time how to construct a strong connection map on a multi-pullback comodule algebra from strong connections on multi-pullback components, but the known explicit general formula is unwieldy. In this paper we derive a much easier to use strong connection formula, which is not, however, completely general, but is applicable only in the case when a Hopf algebra is co-commutative. Because certain linear splittings of projections in multi-pullback comodule algebras play a crucial role in our construction, we also devote a significant part of the paper to the problem of existence and explicit formulas for such splittings. Finally, we show example application of our work.

Key words: strong connections; multi-pullbacks.

pdf (460 kb)   tex (27 kb)

References

  1. Baum P.F., Hajac P.M., Matthes R., Szymański W., The $K$-theory of Heegaard-type quantum 3-spheres, $K$-Theory 35 (2005), 159-186, math.KT/0409573.
  2. Baum P.F., Hajac P.M., Matthes R., Szymański W., Noncommutative geometry approach to principal and associated bundles, in Quantum Symmetry in Noncommutative Geometry, to appear, math.DG/0701033.
  3. Brzeziński T., Fairfax S.A., Quantum teardrops, Comm. Math. Phys. 316 (2012), 151-170, arXiv:1107.1417.
  4. Brzeziński T., Fairfax S.A., Weighted circle actions on the Heegaard quantum sphere, Lett. Math. Phys. 104 (2014), 195-215, arXiv:1305.5942.
  5. Brzeziński T., Hajac P.M., The Chern-Galois character, C. R. Math. Acad. Sci. Paris 338 (2004), 113-116, math.KT/0306436.
  6. Calow D., Matthes R., Covering and gluing of algebras and differential algebras, J. Geom. Phys. 32 (2000), 364-396, math.QA/9910031.
  7. Calow D., Matthes R., Connections on locally trivial quantum principal fibre bundles, J. Geom. Phys. 41 (2002), 114-165, math.QA/0002228.
  8. Cirio L.S., Pagani C., A 4-sphere with non central radius and its instanton sheaf, arXiv:1402.6609.
  9. Dąbrowski L., Grosse H., Hajac P.M., Strong connections and Chern-Connes pairing in the Hopf-Galois theory, Comm. Math. Phys. 220 (2001), 301-331, math.QA/9912239.
  10. Dąbrowski L., Hadfield T., Hajac P.M., Matthes R., Wagner E., Index pairings for pullbacks of $C^*$-algebras, in Operator Algebras and Quantum Groups, Banach Center Publ., Vol. 98, Polish Acad. Sci., Warsaw, 2012, 67-84, math.QA/0702001.
  11. Hajac P.M., Strong connections on quantum principal bundles, Comm. Math. Phys. 182 (1996), 579-617, hep-th/9406129.
  12. Hajac P.M., Krähmer U., Matthes R., Zieliński B., Piecewise principal comodule algebras, J. Noncommut. Geom. 5 (2011), 591-614, arXiv:0707.1344.
  13. Hajac P.M., Matthes R., Szymański W., Chern numbers for two families of noncommutative Hopf fibrations, C. R. Math. Acad. Sci. Paris 336 (2003), 925-930, math.QA/0302256.
  14. Hajac P.M., Matthes R., Szymanski W., A locally trivial quantum Hopf fibration, Algebr. Represent. Theory 9 (2006), 121-146, math.QA/0112317.
  15. 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.
  16. Hajac P.M., Rennie A., Zieliński B., The $K$-theory of Heegaard quantum lens spaces, J. Noncommut. Geom. 7 (2013), 1185-1216, arXiv:1110.5897.
  17. Hajac P.M., Rudnik J., Zieliński B., Reductions of piecewise trivial principal comodule algebras, arXiv:1101.0201.
  18. Hajac P.M., Wagner E., The pullbacks of principal coactions, arXiv:1001.0075.
  19. Hajac P.M., Zieliński B., Cocycle condition for multi-pullbacks of algebras, in Operator Algebras and Quantum Groups, Banach Center Publ., Vol. 98, Polish Acad. Sci., Warsaw, 2012, 239-243, arXiv:1207.0087.
  20. Pedersen G.K., Pullback and pushout constructions in $C^*$-algebra theory, J. Funct. Anal. 167 (1999), 243-344.
  21. Rudnik J., The $K$-theory of the triple-Toeplitz deformation of the complex projective plane, in Operator Algebras and Quantum Groups, Banach Center Publ., Vol. 98, Polish Acad. Sci., Warsaw, 2012, 303-310, arXiv:1207.2066.
  22. Rudnik J., The noncommutative topology of triple-pullback $C^*$-algebra, Ph.D. Thesis, Institute of Mathematics, Polish Academy of Science, Warsaw, 2013.
  23. Vasilevski N.L., Commutative algebras of Toeplitz operators on the Bergman space, Operator Theory: Advances and Applications, Vol. 185, Birkhäuser Verlag, Basel, 2008.

Previous article  Next article   Contents of Volume 10 (2014)