Anisotropic Zn-graded classical r-matrix, deformed An Toda- and Gaudin-type models, and separation of variables

Abstract

We consider a problem of separation of variables for Lax-integrable Hamiltonian systems governed by gl( n) ⨂ gl( n)-valued classical r-matrices r( u, v). We find a new class of classical non-skew-symmetric non-dynamical gl( n) ⨂ gl( n)-valued r-matrices r J( u, v) for which the “magic recipe” of Sklyanin [Prog. Theor. Phys. Suppl. 118, 35 (1995)] in the theory of variable separation is applicable, i.e., for which standard separating functions A( u) and B( u) of Gekhtman [Commun. Math. Phys. 167, 593 (1995)] and Scott [“Classical functional Bethe ansatz for SL( N): Separation of variables for the magnetic chain,” arXiv:hep-th 940303] produce a complete set of canonical coordinates satisfying the equations of separation. We illustrate the corresponding separation of variable theory by the example of the anisotropically deformed A n Toda models proposed in the work of Skrypnyk [J. Phys. A: Math. Theor. 38, 9665–9680 (2005)] and governed by the r-matrices r J( u, v) and by the generalized Gaudin models [T. Skrypnyk, Phys. Lett. A 334(5–6), 390 (2005)] governed by the same classical r-matrices. The n = 2 and n = 3 cases are considered in detail.