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Explicit quantization of dynamical r-matrices for finite dimensional semisimple Lie algebras

Authors: Pavel Etingof, Travis Schedler and Olivier Schiffmann
Journal: J. Amer. Math. Soc. 13 (2000), 595-609
MSC (2000): Primary 17B37
Published electronically: March 15, 2000
MathSciNet review: 1758755
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Abstract: We provide an explicit quantization of dynamical r-matrices for semisimple Lie algebras, classified earlier by the third author, which includes the Belavin-Drinfeld r-matrices. We do so by constructing an appropriate (dynamical) twist in the tensor square of the Drinfeld-Jimbo quantum group $U_q(\mathfrak g)$, which twists the R-matrix of $U_q(\mathfrak g)$ into the desired quantization. The construction of this twist is based on the method stemming from the work of Jimbo-Konno-Odake-Shiraishi and Arnaudon-Buffenoir-Ragoucy-Roche, i.e. on defining the twist as a unique solution of a suitable difference equation. This yields a simple closed formula for the twist.

This construction allows one to confirm the alternate version of the Gerstenhaber-Giaquinto-Schack conjecture (about quantization of Belavin-Drinfeld r-matrices for $\mathfrak{sl}(n)$ in the vector representation), which was stated earlier by the second author on the basis of computer evidence. It also allows one to define new quantum groups associated to semisimple Lie algebras. We expect them to have a rich structure and interesting representation theory.

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  • [ABRR] Arnaudon, D., Buffenoir, E., Ragoucy, E., and Roche, Ph., Universal Solutions of quantum dynamical Yang-Baxter equations, Lett. Math. Phys. 44 (1998), no. 3, 201-214. MR 2000b:81058
  • [BD] Belavin, A.A., and Drinfeld, V.G., Triangle equations and simple Lie algebras. Soviet Sci. Rev. Sect. C: Math. Phys. Rev. 4 (1984), 93-165. MR 87h:58078
  • [CG] Cremmer, E., and Gervais, J.-L., The quantum group structure associated with non-linearly extended Virasoro algebras, Comm. Math. Phys. 134 (1990), 619-632. MR 92a:81072
  • [CP] Chari, V., and Pressley, A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994. MR 95j:17010; corrected reprint MR 96h:17014
  • [EK] Etingof, P., and Kazhdan, D., Quantization of Lie bialgebras I. Selecta Math. 2 (1996), no. 1, 1-41. MR 97f:17014
  • [ER] Etingof, P., and Retakh, V., Quantum determinants and quasideterminants, Asian Jour. Math. 3 (1999), no. 2, 345-352.
  • [ES] Etingof, P., and Schiffmann, O., Twisted traces of intertwiners for Kac-Moody algebras and classical dynamical r-matrices corresponding to generalized Belavin-Drinfeld triples, Math. Res. Lett. 6 (1999), no. 5-6, 593-613.
  • [EV] Etingof, P., and Varchenko, A., Exchange dynamical quantum groups, Comm. Math. Phys. 205 (1999), no. 1, 19-52. CMP 2000:01
  • [GGS] Gerstenhaber, M., Giaquinto, A., and Schack, S., Construction of quantum groups from Belavin-Drinfeld infinitesimals, in: Quantum deformations of algebras and their representations, Joseph, A., Shnider, S., editors, Israel Math. Conf. Proc. 7 (1993), 45-64. MR 94k:17022
  • [GH] Giaquinto, A., and Hodges, T. J., Nonstandard solutions of the Yang-Baxter equation, Lett. Math. Phys. 44 (1998), 67-75. MR 99h:81086
  • [H1] Hodges, T. J., The Cremmer-Gervais solutions of the Yang-Baxter equation, Proc. Amer. Math. Soc. 127 (1999), no. 6, 1819-1826. MR 99i:81102
  • [H2] Hodges, T. J., Generating functions for the coefficients of the Cremmer-Gervais R-matrices, preprint, 1999.
  • [H3] Hodges, T. J., Nonstandard quantum groups associated to certain Belavin-Drinfeld triples, Perspectives on quantization (South Hadley, MA, 1996), 63-70, Contemp. Math., 214, Amer. Math. Soc., Providence, RI, 1998. MR 98k:17016
  • [JKOS] Jimbo, M., Konno, H., Odake, S., and Shiraishi, J., Quasi-Hopf twistors for elliptic quantum groups, Transform. Groups 4 (1999), no. 4, 303-327. CMP 2000:05
  • [KhT] Khoroshkin, S., and Tolstoy, V., Universal R-matrix for quantized (super)algebras. Comm. Math. Phys. 141 (1991), no. 3, 599-617. MR 93a:16031
  • [S] Schiffmann, O., On classification of dynamical r-matrices, Math. Res. Lett. 5 (1998), 13-30. MR 99j:17026
  • [Sch1] Schedler, T., Verification of the GGS conjecture for $\mathfrak{sl}(n)$, $n \leq 12$. Preprint, math.QA/9901079.
  • [Sch2] Schedler, T., On the GGS conjecture. Preprint, math.QA/9903079.
  • [Xu] Xu, P., Quantum groupoids, math.QA 9905192, (1999).

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Additional Information

Pavel Etingof
Affiliation: Department of Mathematics, Room 2-165, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139

Travis Schedler
Affiliation: 059 Pforzheimer House Mail Center, Cambridge, Massachusetts 02138

Olivier Schiffmann
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Address at time of publication: Department of Mathematics, Yale University, New Haven, Connecticut 06520

Received by editor(s): December 1, 1999
Received by editor(s) in revised form: February 10, 2000
Published electronically: March 15, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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