Separation of variables for linear Lax algebras and classical r-matrices

Abstract

We consider the problem of separation of variables for the algebraically integrable Hamiltonian systems possessing gl(n)-valued Lax matrices depending on a spectral parameter that satisfy linear Poisson brackets with some 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 polynomials” of Sklyanin [Commun. Math. Phys. 150, 181 (1992)], Scott [J. Math. Phys. 35, 5831 (1994)], Gekhtman [Commun. Math. Phys. 167, 593 (1995)], and Diener and Dubrovin (Algebraic-geometrical Darboux coordinates in R-matrix formalism, SISSA, Preprint Report No. 88-94-FM, 1994) produce a system of canonical variables. We consider two examples of classical r-matrices and separating polynomials. One of these examples, namely, the n-parametric family of non-skew-symmetric non-dynamical classical r-matrices of Skrypnyk [Phys. Lett. A 334, 390 (2005); 347, 266 (2005)] and the corresponding separating polynomials is new. We show that the separating polynomials of Diener and Dubrovin produce in this case a complete set of separated variables for the corresponding generalized Gaudin models with or without external magnetic field.