Separation of variables for quadratic algebras and skew-symmetric classical r-matrices

Abstract

We consider the problem of separation of variables for integrable hamiltonian systems possessing gl(n)-valued Lax matrices satisfying quadratic Poisson brackets with skew-symmetric gl(n) ⊗ gl(n)-valued classical r-matrices. We formulate, in terms of the corresponding r-matrices, a sufficient condition that guarantees that the separating functions A(u), B(u) of Sklyanin [Commun. Math. Phys. 150(1), 181 (1992)], Scott [e-print arXiv:hep-th/9403030 (1994)], Diener and Dubrovin (Preprint Report No. SISSA-88-94-FM, 1994), and Gekhtman [Commun. Math. Phys. 167, 593 (1995)] produce a set of canonical coordinates. We consider an example of gl(n) ⊗ gl(n)-valued skew-symmetric trigonometric r-matrix that satisfies the considered condition and find a class of the Lax operators for it for which the obtained set of canonical coordinates is complete, defining in such a way a separation of variables for the corresponding integrable models. For this purpose, we generalize trigonometric Sklyanin algebra from the n = 2 case onto the case of the arbitrary n.