ON RATIONAL SOLUTIONS OF YANG-BAXTER EQUATIONS - MAXIMAL-ORDERS IN LOOP ALGEBRA

In 1982 Belavin and Drinfeld listed all elliptic and trigonometric solutions X(u, upsilon) of the classical Yang-Baxter equation (CYBE), where X takes values in a simple complex Lie algebra g, 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 form X(u, upsilon) = C2/(u - upsilon) + r(u, upsilon), where C2 is the quadratic Casimir element and r is a polynomial in u, upsilon, then deg(u) r = deg(upsilon) r less-than-or-equal-to 1. In another paper I proved this conjecture for g = sI(n) and reduced the problem of listing "nontrivial" (i.e. nonequivalent to C2/(u - upsilon)) 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 diagram D(e)(g). In this paper I give an algorithm which enables one to list all solutions and illustrate it with solutions corresponding to vertices of D(e)(g) with coefficient 2 or 3. In particular I will find all solutions for g = D(5) and some solutions for g = D(7), D(10), D(14) and g2.