Skip Navigation

International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn005, 25 pages, doi:10.1093/imrn/rnn005 published on April 15, 2008
This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Right arrow How to cite this article
Google Scholar
Right arrow Articles by Peña, J. L.
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

A Connection for Product Manifolds in Noncommutative Geometry

Javier López Peña

Max-Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany

Correspondence: Correspondence to be sent to: jlopez{at}mpim-bonn.mpg.de

Motivated by some results in classical differential geometry, we give a constructive procedure for building up a connection over a (twisted) tensor product of two algebras, starting from connections defined on the factors. The curvature for the product connection is explicitly calculated, and shown to be independent of the choice of the twisting map and the module twisting map used to define the product connection. As a consequence, we obtain that a product of two flat connections is again a flat connection. We show that our constructions also behave well with respect to bimodule structures, namely being the product of two bimodule connections again a bimodule connection. As an application of our theory, all the product connections on the quantum plane are computed.



References

  1. Beck J. Distributive laws. In: Seminar on Triples and Categorical Homolgy Theory—Eckman B., ed. (1969) Berlin: Springer. 119–40. Lecture Notes in Mathematics 80.
  2. Beggs E. Braiding and exponentiating noncommutative vector fields. (2003) preprint arXiv:math.QA/0306094.
  3. Beggs E. J., Brzezinski T. The Serre spectral sequence of a noncommutative fibration for de Rham cohomology. Acta Mathematica (2005) 195(2):155–96.[CrossRef][ISI]
  4. Borowiec A. Cartan pairs. Czechoslovak Journal of Physics (1996) 46(12):1197–202.[CrossRef]
  5. Brzezinski T., Majid S. Coalgebra bundles. Communications in Mathematical Physics (1998) 191(2):467–92.[CrossRef][ISI]
  6. Brzezinski T., Majid S. Quantum geometry of algebra factorisations and coalgebra bundles. Communications in Mathematical Physics (2000) 213(3):491–521.[CrossRef][ISI]
  7. Cap A., Schichl H., Vanzura J. On twisted tensor products of algebras. Communications in Algebra (1995) 23(12):4701–35.[CrossRef][ISI]
  8. Connes A. Non-commutative differential geometry. Publications Mathematiques. Institut de Hautes Etudes Scientifiques (1986) 62:44–144.
  9. Cuntz J., Quillen D. Algebra extensions and nonsingularity. Journal of the American Mathematical Society (1995) 8:251–89.[CrossRef][ISI]
  10. Dabrowski L., Hajac P. M., Landi G., Siniscalco P. Metrics and pairs of left and right connections on bimodules. Journal of Mathematical Physics (1996) 37(9):4635–46.[CrossRef][ISI]
  11. Dubois-Violette M. Lectures on graded differential algebras and noncommutative geometry. In: Noncommutative Differential Geometry and its Applications to Physics (2001) Dordrecht, The Netherlands: Kluwer Academic Press. 245–306.
  12. Dubois-Violette M., Madore J., Masson T., Mourad J. Linear connections on the quantum plane. Letters in Mathematical Physics (1995) 35(4):351–9.[CrossRef][ISI]
  13. Dubois-Violette M., Masson T. On the first-order operators in bimodules. Letters in Mathematical Physics (1996) 37(4):467–74.[CrossRef][ISI]
  14. Gracia-Bondía J. M., Varilly J. C., Figueroa H. Elements of Noncommutative Geometry (2001) 1st ed. Berlin: Birkhäuser.
  15. Jara P., Llena D. Lie bracket of vector fields on noncommutative geometry. Czechoslovak Journal of Physics (2003) 53(9):743–58.[CrossRef][ISI]
  16. Jara Martínez P., Peña J. López, Panaite F., Van Oystaeyen F. On iterated twisted tensor products of algebras. International Journal of Mathematics. forthcoming.
  17. Koszul J. L. Fibre Bundles and Differential Geometry. Lecture notes, Tata Institute of Fundamental Research, Bombay, 1960.
  18. Landi G. Noncommutative Spaces and Their Geometry (1997) Berlin: Springer. Lecture Notes in Physics 51.
  19. López Peña J., Panaite F., Van Oystaeyen F. General twisting of algebras. Advances in Mathematics (2007) 212(1):315–37.[CrossRef][ISI]
  20. Madore J. Noncommutative Differential Geometry and Its Physical Applications (1995) Cambridge, United Kingdom: Cambridge University Press.
  21. Majid S. Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction. Journal of Algebra (1990) 130:17–64.[CrossRef][ISI]
  22. Majid S. Diagrammatics of braided group gauge theory. Journal of Knot Theory and its Ramifications (1999) 8:731–71.[CrossRef][ISI]
  23. Mourad J. Linear connections in non-commutative geometry. Classical and Quantum Gravity (1995) 12:965–74.[CrossRef][ISI]
  24. Nuss P. Noncommutative descent and nonabelian cohomology. K-Theory (1997) 12:23–74.
  25. Tambara D. The coendomorphism bialgebra of an algebra. Journal of The Faculty of Science, The University of Tokyo, Section IA, Mathematics (1990) 37:425–56.
  26. Van Daele A., Van Keer S. The Yang–Baxter and Pentagon equation. Compositio Mathematica (1994) 91:201–21.[ISI]
  27. Woronowicz S. L. An example of a braided locally compact group. In: Proceedings of the 9th Max Born Symposium, Karpacz (1996) Warsaw, Poland: Polish Scientific Publishers.

Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?



This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Right arrow How to cite this article
Google Scholar
Right arrow Articles by Peña, J. L.
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?