Communications in Mathematical Physics

, Volume 311, Issue 1, pp 55–96 | Cite as

A Kohno–Drinfeld Theorem for the Monodromy of Cyclotomic KZ Connections

Article

Abstract

We compute explicitly the monodromy representations of “cyclotomic” analogs of the Knizhnik–Zamolodchikov differential system. These are representations of the type B braid group \({B_n^1}\) . We show how the representations of the braid group Bn obtained using quantum groups and universal R-matrices may be enhanced to representations of \({B_n^1}\) using dynamical twists. Then, we show how these “algebraic” representations may be identified with the above “analytic” monodromy representations.

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References

  1. ABRR.
    Arnaudon D., Buffenoir E., Ragoucy E., Roche P.: Universal solutions of quantum dynamical Yang-Baxter equations. Lett. Math. Phys. 44(3), 201–214 (1998)MathSciNetMATHCrossRefGoogle Scholar
  2. Ar.
    Artin E.: Theorie der Zöpfe. Abh. Math. Sem. Univ. Hamburg 4, 47–72 (1925)MATHCrossRefGoogle Scholar
  3. Ba.
    Babelon O.: Universal exchange algebra for Bloch waves and Liouville theory. Commun. Math. Phys. 139(3), 619–643 (1991)MathSciNetADSMATHCrossRefGoogle Scholar
  4. BMR.
    Broué M., Malle G., Rouquier R.: Complex reflection groups, braid groups, Hecke algebras. J. Reine Angew. Math. 500, 127–190 (1998)MathSciNetMATHGoogle Scholar
  5. BR.
    Buffenoir E., Roche P.: Harmonic analysis on the quantum Lorentz group. Commun. Math. Phys. 207(3), 499–555 (1999)MathSciNetADSMATHCrossRefGoogle Scholar
  6. Br.
    Brieskorn E.: Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe. Invent. Math. 12, 57–61 (1971)MathSciNetADSMATHCrossRefGoogle Scholar
  7. Ca.
    Calaque D.: Quantization of formal classical dynamical r-matrices: the reductive case. Adv. Math. 204(1), 84–100 (2006)MathSciNetMATHCrossRefGoogle Scholar
  8. CP.
    Chari V., Pressley A.: A guide to quantum groups. Cambridge University Press, Cambridge (1994)MATHGoogle Scholar
  9. DCK.
    De Concini, C., Kac, V.G.: Representations of quantum groups at roots of 1. In: Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989), Vol. 92 of Progr. Math.. Boston, MA: Birkhäuser Boston, 1990, pp. 471–506Google Scholar
  10. Dr1.
    Drinfeld V.G.: Almost cocommutative Hopf algebras. Algebra i Analiz 1(2), 30–46 (1989)MathSciNetGoogle Scholar
  11. Dr2.
    Drinfeld V.G.: On quasitriangular quasi-Hopf algebras and on a group that is closely connected with \({{\rm Gal}(\overline Q/Q)}\) . Leningrad Math. J. 2(4), 829–860 (1990)MathSciNetGoogle Scholar
  12. Dr3.
    Drinfeld V.G.: Quasi-Hopf algebras. Leningrad Math. J. 1(6), 1419–1457 (1990)MathSciNetGoogle Scholar
  13. EE1.
    Enriquez, B., Etingof, P.: Quantization of Alekseev-Meinrenken dynamical r-matrices. In: Lie groups and symmetric spaces, Vol. 210 of Amer. Math. Soc. Transl. Ser. 2, Providence, RI: Amer. Math. Soc., 2003, pp. 81–98Google Scholar
  14. EE2.
    Enriquez B., Etingof P.: Quantization of classical dynamical r-matrices with nonabelian base. Commun. Math. Phys. 254(3), 603–650 (2005)MathSciNetADSMATHCrossRefGoogle Scholar
  15. EEM.
    Enriquez, B., Etingof, P., Marshall, I.: Quantization of some Poisson-Lie dynamical r-matrices and Poisson homogeneous spaces. In: Quantum groups, Vol. 433 of Contemp. Math., Providence, RI: Amer. Math. Soc. (2007), pp. 135–175Google Scholar
  16. En.
    Enriquez B.: Quasi-reflection algebras and cyclotomic associators. Selecta Mathematica, New Series 13, 391–463 (2008)MathSciNetCrossRefGoogle Scholar
  17. ES1.
    Etingof, P., Schiffmann, O.: Lectures on the dynamical Yang-Baxter equations. In: A. Pressley, editor;. Quantum groups and Lie theory (Durham, 1999), Vol. 290 of London Math. Soc. Lecture Note Ser.. Cambridge: Cambridge Univ. Press, 2001, pp. 89–129, papers from the LMS Symposium on Quantum Groups held at the University of Durham, Durham, July 19–29, 1999Google Scholar
  18. ES2.
    Etingof P., Schiffmann O.: On the moduli space of classical dynamical r-matrices. Math. Res. Lett. 8(1-2), 157–170 (2001)MathSciNetMATHGoogle Scholar
  19. ESS.
    Etingof P., Schedler T., Schiffmann O.: Explicit quantization of dynamical r-matrices for finite dimensional semisimple Lie algebras. J. Amer. Math. Soc. 13(3), 595–609 (2000) (electronic)MathSciNetMATHCrossRefGoogle Scholar
  20. EV.
    Etingof P., Varchenko A.: Geometry and classification of solutions of the classical dynamical Yang-Baxter equation. Commun. Math. Phys. 192(1), 77–120 (1998)MathSciNetADSMATHCrossRefGoogle Scholar
  21. Ga.
    Gavarini F.: The quantum duality principle. Ann. Inst. Fourier (Grenoble) 52(3), 809–834 (2002)MathSciNetMATHCrossRefGoogle Scholar
  22. Ko.
    Kohno T.: Monodromy representations of braid groups and Yang-Baxter equations. Ann. Inst. Fourier (Grenoble) 37(4), 139–160 (1987)MathSciNetMATHCrossRefGoogle Scholar
  23. MR.
    McConnell, J.C., Robson, J.C.: Noncommutative Noetherian rings, Vol. 30 of Graduate Studies in Mathematics Providence, RI: Amer. Math. Soc., 2001, revised ed., with the cooperation of L. W. SmallGoogle Scholar
  24. ST.
    Shephard G.C., Todd J.A.: Finite unitary reflection groups. Canadian J. Math. 6, 274–304 (1954)MathSciNetMATHCrossRefGoogle Scholar
  25. Xu.
    Xu P.: Quantum dynamical Yang-Baxter equation over a nonabelian base. Comm. Math. Phys. 226(3), 475–495 (2002)MathSciNetADSMATHCrossRefGoogle Scholar

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© Springer-Verlag 2012

Authors and Affiliations

  1. 1.IRMA (CNRS), rue René DescartesStrasbourgFrance

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