On rational solutions of Yang-Baxter equation for SI(N)

Journal article, 1991

In 1984 Drinfeld conjectured that any rational solution X(u, upsilon) of the classical Yang-Baxter equation (CYBE)' with X taking values in a simple complex Lie algebra g is equivalent to one of the form X(u, upsilon) = C2/(u - upsilon) + r(u, upsilon), where C2 is the quadratic Casimir element, r is a polynmial in u, upsilon, and deg(u)r = deg(upsilon)r less-than-or-equal-to 1. In this paper I will prove this conjecture for g = sl(n) and reduce the problem of listing "nontrivial" (i.e. nonequivalent to C2/(u - upsilon) + const) solutions of CYBE to classification of certain quasi-Frobenius subalgebras of g. There are given all "nontrivial" rational solutions for sl(2), sl(3), sl(4) and several series of examples in general case.