The negative Toda hierarchy and rational Poisson brackets

Abstract

In this paper we extend the usual hierarchies for the finite, nonperiodic Toda lattice for negative values of the index. We define an infinite sequence of rational homogeneous Poisson brackets, master symmetries, invariants and investigate the various relationships between them. All the relations between master symmetries, Poisson tensors and invariants which hold over the positive integers are extended for all integer values. We comment on extensions to other versions of the Toda lattice, i.e. the periodic, infinite and Bogoyavlensky-Toda type systems. © 2002 Elsevier Science B.V. All rights reserved.