Rational solutions of CYBE for simple compact real Lie algebras

JOURNAL OF GEOMETRY AND PHYSICS

卷:57

Article

Pop, Iulia ; Stolin, Alexander

关键词: classical Yang-Baxter equation;    rational solution;    Bruhat-Tits building;    Quasi-Frobenius Lie algebra;    Yangian;   

DOI  :  10.1016/j.geomphys.2006.10.012

来源: Elsevier

【 摘 要 】

In [A.A. Stolin, On rational solutions of Yang-Baxter equation for sl(n), Math. Scand. 69 (1991) 57-80; A.A. Stolin, On rational solutions of Yang-Baxter equation. Maximal orders in loop algebra, Comm. Math. Phys. 141 (1991) 533-548; A. Stolin, A geometrical approach to rational solutions of the classical Yang-Baxter equation. Part I, in: Walter de Gruyter & Co. (Ed.), Symposia Gaussiana, Conf. Alg., Berlin, New York, 1995, pp. 347-357] a theory of rational solutions of the classical Yang-Baxter equation for a simple complex Lie algebra g was presented. We discuss this theory for simple compact real Lie algebras g. We prove that up to gauge equivalence all rational solutions have the form X(u, v) + Omega/u-v + t(1) boolean AND t(2) +(...)+ t(2n-1) boolean AND t(2n), where Omega denotes the quadratic Casimir element of g and {t(i)} are linearly independent elements in a maximal torus t of g. The quantization of these solutions is also emphasized. (c) 2006 Elsevier B.V. All rights reserved.

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