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Communications in Mathematical Physics

, Volume 254, Issue 3, pp 603–650 | Cite as

Quantization of Classical Dynamical r-Matrices with Nonabelian Base

  • Benjamin Enriquez
  • Pavel EtingofEmail author
Article

Abstract

We construct some classes of dynamical r-matrices over a nonabelian base, and quantize some of them by constructing dynamical (pseudo)twists in the sense of Xu. This way, we obtain quantizations of r-matrices obtained in earlier work of the second author with Schiffmann and Varchenko. A part of our construction may be viewed as a generalization of the Donin-Mudrov nonabelian fusion construction. We apply these results to the construction of equivariant star-products on Poisson homogeneous spaces, which include some homogeneous spaces introduced by De Concini.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Alekseev, A., Faddeev, L.: (T*G)t: a toy model for conformal field theory. Commun. Math. Phys. 141, no. 2, 413–422 (1991)Google Scholar
  2. 2.
    Alekseev, A., Kosmann-Schwarzbach, Y., Meinrenken, E.: Quasi-Poisson manifolds. Canad. J. Math. 54, no. 1, 3–29 (2002)Google Scholar
  3. 3.
    Alekseev, A., Lachowska, A.: Invariant *-products on coadjoint orbits and the Shapovalov pairing. http://arxiv.org/abs. /math.QA/0308100, 2003.
  4. 4.
    Alekseev, A., Meinrenken, E.: The non-commutative Weil algebra. Invent. Math. 139, no. 1, 135–172 (2000)Google Scholar
  5. 5.
    Alekseev, A., Meinrenken, E.: Clifford algebras and the classical dynamical Yang-Baxter equation. Math. Res. Lett. 10, no. 2–3, 253–268 (2003)Google Scholar
  6. 6.
    Arnaudon, D., Buffenoir, E., Ragoucy, E., and Roche, P.: Universal solutions of quantum dynamical Yang-Baxter equations. Lett. Math. Phys. 44 no. 3, 201–214 (1998)Google Scholar
  7. 7.
    Borel, A., De Siebenthal, J.: Les sous-groupes fermés de rang maximum des groupes de Lie clos. Comment. Math. Helv. 23, 200–221 (1949)Google Scholar
  8. 8.
    Donin, J., Gurevich, D., Shnider, S.: Double quantization on some orbits in the coadjoint representations of simple Lie groups. Commun. Math. Phys. 204, no. 1, 39–60 (1999)Google Scholar
  9. 9.
    Donin, J., Mudrov, A.: Dynamical Yang-Baxter equation and quantum vector bundles. http://arxiv.org/abs/math.QA/0306028, (2003)
  10. 10.
    Drinfeld, V.: On Poisson homogeneous spaces of Poisson-Lie groups. Teoret. Mat. Fiz. 95, no. 2, 226–227 (1993); translation in Theoret. and Math. Phys. 95, no. 2, 524–525 (1993)Google Scholar
  11. 11.
    Drinfeld, V.: Quasi-Hopf algebras. Leningrad Math. J. 1, no. 6, 1419–1457 (1990)Google Scholar
  12. 12.
    Enriquez, B., Etingof, P.: Quantization of Alekseev-Meinrenken dynamical r-matrices (in memory of F.I. Karpelevich). AMS Transl. 210, no. 2, 81–98 (2003)Google Scholar
  13. 13.
    Enriquez, B., Etingof, P., Marshall, I.: Quantization of some Poisson-Lie dynamical r-matrices and Poisson homogeneous spaces. http://arxiv.or/abs/math.QA/0403283, 2004
  14. 14.
    Etingof, P., Frenkel, I., Kirillov, A.: Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations. Providence RI: AMS, 1998Google Scholar
  15. 15.
    Etingof, P., Kazhdan, D.: Quantization of Lie bialgebras, I, II. Selecta Math. (N.S.) 2, no. 1, 1–41 (1996); 4, no. 2, 213–231 (1998)Google Scholar
  16. 16.
    Etingof, P., Schedler, T., Schiffmann, O.: Explicit quantization of dynamical r-matrices. J. Amer. Math. Soc. 13, no. 3, 595–609 (2000)Google Scholar
  17. 17.
    Etingof, P., Schiffmann, O.: Lectures on the Dynamical Yang-Baxter Equations. in: “Quantum groups and Lie theory (Durham, 1999)”, London Math. Soc. Lecture Note Series, 290, Cambridge: Cambridge Univ. Press, 2001, 89–129Google Scholar
  18. 18.
    Etingof, P., Schiffmann, O.: On the moduli space of classical dynamical r-matrices. Math. Res. Lett. 8, no. 1–2, 157–170 (2001)Google Scholar
  19. 19.
    Etingof, P., Varchenko, A.: Geometry and classification of solutions of the classical dynamical Yang-Baxter equation. Commun. Math. Phys. 192, no. 1, 77–120 (1998)Google Scholar
  20. 20.
    Etingof, P., Varchenko, A.: Exchange dynamical quantum groups. Commun. Math. Phys. 205, no. 1, 19–52 (1999)Google Scholar
  21. 21.
    Etingof, P., Varchenko, A.: Dynamical Weyl groups and applications. Adv. Math., 167, no. 1, 74–127 (2002)Google Scholar
  22. 22.
    Faddeev, L.: On the exchange matrix of the WZNW model. Commun. Math. Phys. 132, no. 1, 131–138 (1990)Google Scholar
  23. 23.
    Fehér, L., Gábor, A., Pusztai, P.: On dynamical r-matrices obtained from Dirac reduction and their generalizations to affine Lie algebras. J. Phys. A 34, no. 36, 7335–7348 (2001)Google Scholar
  24. 24.
    Reshetikhin, N., Semenov-Tian-Shansky, M.: Quantum R-matrices and factorization problems. J. Geom. Phys. 5, no. 4, 533–550 (1988)Google Scholar
  25. 25.
    Schiffmann, O.: On classification of dynamical r-matrices. Math. Res. Lett. 5, no. 1–2, 13–30 (1998)Google Scholar
  26. 26.
    Springer, T.: Microlocalisation algébrique. In: Sém. algèbre Dubreil-Malliavin, Lecture Notes in Mathematics 1146, Berlin Heidelberg: Springer-Verlag, 1983, pp. 299–316Google Scholar
  27. 27.
    Xu, P.: Triangular dynamical r-matrices and quantization. Adv. Math. 166, no. 1, 1–49 (2002)Google Scholar
  28. 28.
    Xu, P.: Quantum dynamical Yang-Baxter equation over a nonabelian base. Commun. Math. Phys. 226, no. 3, 475–495 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.IRMA (CNRS)StrasbourgFrance
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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