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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
DOI: https://doi.org/10.1090/S0894-0347-00-00333-7
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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Additional Information

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

DOI: https://doi.org/10.1090/S0894-0347-00-00333-7
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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