Skip to Main Content

Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Explicit quantization of dynamical r-matrices for finite dimensional semisimple Lie algebras
HTML articles powered by AMS MathViewer

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
Similar Articles
  • Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 17B37
  • Retrieve articles in all journals with MSC (2000): 17B37
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