IJPAM: Volume 25, No. 3 (2005)

LIE-POISSON GROUPS AND THE SOLUTIONS OF
THE CLASSICAL YANG-BAXTER EQUATION

A. Ouadfel
Département de Mathématiques
Ecole Nationale des Sciences Appliquées (ENSAO)
P.O. Box 473, Oujda, 60000, MAROCCO
e-mail: ali_ouadfel@yahoo.fr


Abstract.The solutions of the classical Yang-Baxter equation over an arbitrary Lie group $G$ can be used to assign a Lie-Poisson's structure to $G$. In this paper, the construction of such a structure is explained in the general case and illustrated in the particular case of the Lie group of square matrices. All solutions of the classical Yang-Baxter equation are determined for Lie groups associated with Lie algebras of dimension $\le 3$. For complex Lie algebras of higher dimensions, the existence of solutions to the classical Yang-Baxter equation is proven.

Received: October 18, 2005

AMS Subject Classification: 22E60, 17B45, 14L35

Key Words and Phrases: Lie-Poisson groups, Yang-Baxter equation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 25
Issue: 3