Generalized Trigonometric Solutions of the Classical Yang-Baxter Equation

Research Report MRR98-054

Generalized Trigonometric Solutions of the Classical Yang-Baxter Equation

A. A. Stolin

Abstract: We consider skew-symmetric solutions of the CYBE of the form $\frac{ut}{v-u} +p(u,v)$, where $t\in g^{\otimes 2}$ is the Casimir element and $p(u,v)$ is a polynomial with coefficients in $g^{\otimes 2}$. If $p(u,v)=const$ then substituting $\frac{v}{u}=e^{z}$ we obtain a trigonometric solution $\frac{t}{1-e^z} +const$ in the sense of Ref.~1. We prove that there exists a gauge transformation reducing the polynomial part $p(u,v)$ to a polynomial of degree $\leq 1$ in $u$ and $v$. A non-trivial example of a generalized trigonometric solution is constructed.

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