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

Etingof, Pavel; Schedler, Travis; Schiffmann, Olivier

doi:10.1090/S0894-0347-00-00333-7

Explicit quantization of dynamical r-matrices for finite dimensional semisimple Lie algebras
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by Pavel Etingof, Travis Schedler and Olivier Schiffmann

J. Amer. Math. Soc. 13 (2000), 595-609

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

Published electronically: March 15, 2000

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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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