On a class of gl(n) ⊗ gl(n)-valued classical r-matrices and separation of variables

Abstract

We consider a problem of separation of variables for the Lax-integrable Hamiltonian systems governed by gl(n) ⊗ gl(n)-valued classical r-matrices r(u, v). We report on a class of classical non-skew-symmetric non-dynamical gl(n) ⊗ gl(n)-valued r-matrices r_J(u, v) labeled by arbitrary anisoropy matrix J ∈ gl(n) for which the “magic recipe” of Sklyanin [Prog. Theor. Phys., 118, 35 (1995)] in the theory of variable separation is applicable. An example of n = 3 corresponding to gl(3) ⊗ gl(3)-valued r-matrices is elaborated in detail. For the case of the r-matrices r_J(u, v) and n = 3, the coordinates of separation, the reconstruction formulas, and the Abel-type equations are explicitly written for the different types of matrices J.