On rational solutions of Yang-Baxter equations. Maximal orders in loop algebra

Abstract

In 1982 Belavin and Drinfeld listed all elliptic and trigonometric solutionsX(u, v) of the classical Yang-Baxter equation (CYBE), whereX takes values in a simple complex Lie algebrag, and left the classification problem of the rational one open. In 1984 Drinfeld conjectured that if a rational solution is equivalent to a solution of the formX(u,v)=C ₂/(u−v)+r(u,v), whereC ₂ is the quadratic Casimir element andr is a polynomial inu,v, then deg _u r=deg _v r≦1. In another paper I proved this conjecture forg=sl (n) and reduced the problem of listing “nontrivial” (i.e. nonequivalent toC ₂/(u−v)) solutions of CYBE to classification of quasi-Frobenius subalgebras of g. They, in turn, are related with the so-called maximal orders in the loop algebra of g corresponding to the vertices of the extended Dynkin diagramD ^e(g). In this paper I give an algorithm which enables one to list all solutions and illustrate it with solutions corresponding to vertices ofD ^e(g) with coefficient 2 or 3. In particular I will find all solutions forg=o=(5) and some solutions forg=o(7),o(10),o(14) andg ₂.