Anisotropic Z n-graded classical r-matrix, deformed A n 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 rJ(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 An Toda models proposed in the work of Skrypnyk [J. Phys. A: Math. Theor. 38, 9665–9680 (2005)] and governed by the r-matrices rJ(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.