Mathematical Research Letters

Volume 5 (1998)

Number 1

On classification of dynamical ${\bf r}$-matrices

Pages: 13 – 30

DOI: http://dx.doi.org/10.4310/MRL.1998.v5.n1.a2

Author

Olivier Schiffmann (Harvard University and ENS Paris)

Abstract

Using the gauge transformations of the Classical Dynamical Yang-Baxter Equation introduced by P. Etingof and A. Varchenko in \cite{EV}, we reduce the classification of dynamical r-matrices $r$ on a commutative subalgebra $\frak{l}$ of a Lie algebra $\g$ to a purely algebraic problem, under some assumption on the symmetric part of $r$. We then describe, for a simple complex Lie algebra $\frak{g}$, all non skew-symmetric dynamical r-matrices on a commutative subalgebra $\frak{l} \subset\fral{g}$ which contains a regular semisimple element. This interpolates results of P. Etingof and A. Varchenko (\cite{EV}, when $\frac{l}$ is a Cartan subalgebra) and results of A. Belavin and V. Drinfeld for constant r-matrices (\cite{BD}). This classification is similar, and in some sense simpler than the Belavin-Drinfeld classification.

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