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


SIGMA 2 (2006), 089, 9 pages      hep-th/0611025      https://doi.org/10.3842/SIGMA.2006.089
Contribution to the Proceedings of the O'Raifeartaigh Symposium

Einstein-Riemann Gravity on Deformed Spaces

Julius Wess a, b, c
a) Arnold Sommerfeld Center for Theoretical Physics Universität München, Theresienstr. 37, 80333 München, Germany
b) Max-Planck-Institut für Physik, Föhringer Ring 6, 80805 München, Germany
c) Universität Hamburg, II Institut für Theoretische Physik and DESY, Luruper Chaussee 149, 22761 Hamburg, Germany

Received October 27, 2006, in final form November 28, 2006; Published online December 11, 2006

Abstract
A differential calculus, differential geometry and the E-R Gravity theory are studied on noncommutative spaces. Noncommutativity is formulated in the star product formalism. The basis for the gravity theory is the infinitesimal algebra of diffeomorphisms. Considering the corresponding Hopf algebra we find that the deformed gravity is based on a deformation of the Hopf algebra.

Key words: noncommutative spaces; deformed gravity.

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References

  1. Aschieri P., Blohmann C., Dimitrijevic M., Meyer F., Schupp P., Wess J., A gravity theory on noncommutative spaces, Classical Quantum Gravity, 2005, V.22, 3511-3522, hep-th/0504183.
  2. Aschieri P., Dimitrijevic M., Meyer F., Wess J., Noncommutative geometry and gravity, Classical Quantum Gravity, 2006, V.23, 1883-1912, hep-th/0510059.
  3. Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D., Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Physics, 1978, V.111, 61-110.
  4. Sternheimer D., Deformation quantization: twenty years after, AIP Conf. Proc., 1998, V.453, 107-145, math.QA/9809056.
  5. Kontsevich M., Deformation quantization of Poisson manifolds, Lett. Math. Phys., 2003, V.66, 157-216, q-alg/9709040.
  6. Waldman S., An introduction to deformation quantization, Lecture Notes, 2002, see http://idefix.physik.uni-freiburg.de/~stefan/Skripte/intro/index.html.
  7. Weyl H., Quantenmechanik und Gruppentheorie, Zeit. für Phys., 1927, V.46, 1-46.
  8. Moyal J.E., Quantum mechanics as a statistical theory, Proc. Cambridge Phil. Soc., 1949, V.45, 99-124.
  9. Wess J., Zumino B., Covariant differential calculus on the quantum hyperplane, Nuclear Phys. B Proc. Suppl., 1991, V.18, 302-312.
  10. Woronowic S.L., Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys., 1989, V.122, 125-170.
  11. Wess J., Deformed coordinate spaces; derivatives, in Proceedings of the BW2003 Workshop on Mathematical, Theoretical and Phenomenological Challenges Beyond the Standard Model: Perspectives of Balkans Collaboration (2003, Vrnjacka Banja, Serbia), Vrnjacka Banja, 2003, 122-128, hep-th/0408080.
  12. Wess J., Differential calculus and gauge transformations on a deformed space, hep-th/0607251.
  13. Wess J., Deformed gauge theories, hep-th/0608135.
  14. Aschieri P., Dimitrijevic M., Meyer F., Schraml S., Wess J., Twisted gauge theories, Lett. Math. Phys., 2006, V.78, 61-71, hep-th/0603024.
  15. Vassilevich D.V., Twist to close, Modern Phys. Lett. A, 2006, V.21, 1279-1284, hep-th/0602185.
  16. Drinfel'd V.G., On constant quasiclassical solutions of the Yang-Baxter equations, Soviet Math. Dokl., 1983, V.28, 667-671.
  17. Reshetikhin N., Multiparameter quantum groups and twisted quasitriangular Hopf algebras, Lett. Math. Phys., 1990, V.20, 331-335.
  18. Gerstenhaber M., Giaquinto A., Schack S.D., Quantum symmetry, in Quantum Groups, Lecture Notes in Math., V.1510, Editor P.P. Kulish, Berlin, Springer-Verlag, 1992, 9-46.
  19. Kulish P.P., Lyakhovsky V.D., del Olmo M.A., Chains of twists for classical Lie algebras, J. Phys. A: Math. Phys., 1999, V.32, 8671-8684, math.QA/9908061.


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