Generalized trigonometric solutions of the classical Yang-Baxter equation

Paper in proceeding, 1998

We consider skew-symmetric solutions of the CYBE of the form ut/upsilon-u + p(u, upsilon), where t epsilon g(x2) is the Casimir element and p(u,upsilon) is a polynomial with coefficients in g(x2) If p(u,upsilon) = const then substituting upsilon/u= e(x) we obtain a trigonometric solution t/1-e(1) + Const in the sense of Ref. 1. We prove that there exists a gauge transformation reducing the polynomial part p(u,upsilon) to a polynomial of degree less than or equal to 1 in u and upsilon. A non-trivial example of a generalized trigonometric solution is constructed.