R-matrix theory, formal Casimirs and the periodic Toda lattice

Abstract

The nonunitary r-matrix theory and the associated bi- and triHamiltonian schemes are considered. The language of Poisson pencils and of their formal Casimirs is applied in this framework to characterize the biHamiltonian chains of integrals of motion, pointing out the role of the Schur polynomials in these constructions. This formalism is subsequently applied to the periodic Toda lattice. Some different algebraic settings and Lax formulations proposed in the literature for this system are analyzed in detail, and their full equivalence is exploited. In particular, the equivalence between the loop algebra approach and the method of differential-difference operators is illustrated; moreover, two alternative Lax formulations are considered, and appropriate reduction algorithms are found in both cases, allowing us to derive the multiHamiltonian formalism from r-matrix theory. The systems of integrals for the periodic Toda lattice known after Flaschka and Hénon, and their functional relations, are recovered through systematic application of the previously outlined schemes.