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

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- by Pavel Etingof, Travis Schedler and Olivier Schiffmann PDF
- J. Amer. Math. Soc.
**13**(2000), 595-609 Request permission

## 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.## References

- D. Arnaudon, E. Buffenoir, E. Ragoucy, and Ph. Roche,
*Universal solutions of quantum dynamical Yang-Baxter equations*, Lett. Math. Phys.**44**(1998), no. 3, 201–214. MR**1627497**, DOI 10.1023/A:1007498022373 - A. A. Belavin and V. G. Drinfel′d,
*Triangle equations and simple Lie algebras*, Mathematical physics reviews, Vol. 4, Soviet Sci. Rev. Sect. C: Math. Phys. Rev., vol. 4, Harwood Academic Publ., Chur, 1984, pp. 93–165. Translated from the Russian. MR**768939** - Eugène Cremmer and Jean-Loup Gervais,
*The quantum group structure associated with nonlinearly extended Virasoro algebras*, Comm. Math. Phys.**134**(1990), no. 3, 619–632. MR**1086746** - Vyjayanthi Chari and Andrew Pressley,
*A guide to quantum groups*, Cambridge University Press, Cambridge, 1994. MR**1300632** - Pavel Etingof and David Kazhdan,
*Quantization of Lie bialgebras. I*, Selecta Math. (N.S.)**2**(1996), no. 1, 1–41. MR**1403351**, DOI 10.1007/BF01587938 - Etingof, P., and Retakh, V., Quantum determinants and quasideterminants, Asian Jour. Math.
**3**(1999), no. 2, 345–352. - 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. - Etingof, P., and Varchenko, A.,
*Exchange dynamical quantum groups*, Comm. Math. Phys.**205**(1999), no. 1, 19–52. - Murray Gerstenhaber, Anthony Giaquinto, and Samuel D. Schack,
*Construction of quantum groups from Belavin-Drinfel′d infinitesimals*, Quantum deformations of algebras and their representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992) Israel Math. Conf. Proc., vol. 7, Bar-Ilan Univ., Ramat Gan, 1993, pp. 45–64. MR**1261900** - Anthony Giaquinto and Timothy J. Hodges,
*Nonstandard solutions of the Yang-Baxter equation*, Lett. Math. Phys.**44**(1998), no. 1, 67–75. MR**1623770**, DOI 10.1023/A:1007421618929 - Timothy J. Hodges,
*The Cremmer-Gervais solution of the Yang-Baxter equation*, Proc. Amer. Math. Soc.**127**(1999), no. 6, 1819–1826. MR**1621937**, DOI 10.1090/S0002-9939-99-05014-5 - Hodges, T. J., Generating functions for the coefficients of the Cremmer-Gervais R-matrices, preprint, 1999.
- Timothy J. Hodges,
*Nonstandard quantum groups associated to certain Belavin-Drinfeld triples*, Perspectives on quantization (South Hadley, MA, 1996) Contemp. Math., vol. 214, Amer. Math. Soc., Providence, RI, 1998, pp. 63–70. MR**1601221**, DOI 10.1090/conm/214/02905 - Jimbo, M., Konno, H., Odake, S., and Shiraishi, J.,
*Quasi-Hopf twistors for elliptic quantum groups*, Transform. Groups**4**(1999), no. 4, 303–327. - S. M. Khoroshkin and V. N. Tolstoy,
*Universal $R$-matrix for quantized (super)algebras*, Comm. Math. Phys.**141**(1991), no. 3, 599–617. MR**1134942** - Olivier Schiffmann,
*On classification of dynamical r-matrices*, Math. Res. Lett.**5**(1998), no. 1-2, 13–30. MR**1618367**, DOI 10.4310/MRL.1998.v5.n1.a2 - Schedler, T., Verification of the GGS conjecture for $\mathfrak {sl}(n)$, $n \leq 12$. Preprint, math.QA/9901079.
- Schedler, T., On the GGS conjecture. Preprint, math.QA/9903079.
- Xu, P.,
*Quantum groupoids*, math.QA 9905192, (1999).

## Additional Information

**Pavel Etingof**- Affiliation: Department of Mathematics, Room 2-165, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
- MR Author ID: 289118
- Email: etingof@math.harvard.edu
**Travis Schedler**- Affiliation: 059 Pforzheimer House Mail Center, Cambridge, Massachusetts 02138
- Email: schedler@fas.harvard.edu
**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
- Email: schiffma@clipper.ens.fr, schiffma@math.yale.edu
- Received by editor(s): December 1, 1999
- Received by editor(s) in revised form: February 10, 2000
- Published electronically: March 15, 2000
- © Copyright 2000 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**13**(2000), 595-609 - MSC (2000): Primary 17B37
- DOI: https://doi.org/10.1090/S0894-0347-00-00333-7
- MathSciNet review: 1758755