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Title: Symmetric algebras and Yang-Baxter equation (English)
Author: Beidar, K.
Author: Fong, Y.
Author: Stolin, A.
Language: English
Journal: Proceedings of the 16th Winter School "Geometry and Physics"
Issue: 1996
Pages: [15]-28
Category: math
Summary: Let $U$ be an open subset of the complex plane, and let $L$ denote a finite-dimensional complex simple Lie algebra. {\it A. A. Belavin} and {\it V. G. Drinfel'd} investigated non-degenerate meromorphic functions from $U\times U$ into $L\otimes L$ which are solutions of the classical Yang-Baxter equation [Funct. Anal. Appl. 16, 159-180 (1983; Zbl 0504.22016)]. They found that (up to equivalence) the solutions depend only on the difference of the two variables and that their set of poles forms a discrete (additive) subgroup $\Gamma$ of the complex numbers (of rank at most 2). If $\Gamma$ is non-trivial, they were able to completely classify all possible solutions. If $\Gamma$ is trivial, the solutions are called rational and for $L= sl_n(\bbfC)$ they were classified by {\it A. Stolin} [in Math. Scand. 69, No. 1, 57-80 (1991; Zbl 0727.17005)]. \par A Lie algebra $L$ is called symmetric if there exists a non-degenerate symmetric invariant bilinear form on $L$. In the ! (English)
MSC: 16W30
MSC: 17B35
MSC: 17B37
MSC: 81R50
idZBL: Zbl 0884.17007
idMR: MR1469019
Date available: 2009-07-13T21:37:23Z
Last updated: 2012-09-18
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