Physica A 161 (1989) 181-220 North-Holland, Amsterdam R-MATRICES AND HIGHER POISSON BRACKETS FOR INTEGRABLE SYSTEMS Walter OEVEL ~ Fachbereich Mathematik, Universitiit Paderborn, Fed. Rep. Germany Orlando RAGNISCO 2 Dipartimento di Fisica, Universit~ degli Studi "" La Sapienza", Roma, Italy and lstituto Nazionale di Fisica Nucleare, Sezione di Roma, Roma, haly Receivcd 24 April 1989 For any Lie algebra g equipped with a second Lie product (engendered by a so-called R-matrix) a natural integrable system arises from the invariant functions on the dual g*. if g has the structure of an associative algebra and admits a trace-form, then in addition to the linear Lie-Poisson structure on g* a quadratic as well as a cubic Poisson bracket arises leading to tri-Hamiitonian formulations of the integrablc equations. The results are applied to PDE's (Korteweg-de Vrics and Boussinesq equation) as well as to lattice systems (classical and relativistic Toda lattice). I. Introduction During the last few decades, the classical subject of integrable systems has been significantly revised, undergoing unexpected and impressive new develop- ments. Crucial steps in this rediscovery have been the celebrated findings by Zabusky and Kruskal [1] on the "recurrence" phenomenon in the Fermi- Pasta-Ulam model, and the Lax formulation [2] of the Korteweg-de Vries equation (KdV) together with the basic results obtained by Gardner, Greene, Kruskal and Miura [3], and Zakharov and Fadeev [4] on the linearization of this equation and its Hamiltonian nature. The phenomenology associated to special solutions of infinite-dimensional integrable systems (nonlinear partial differential equations or difference equa- tions), the so-called solitons, has motivated a large interest in this subject also Work supported in part by the Italian "'lstituto Nazionalc de Fisica Nuclei:re". z Work supported in part by the Italian Ministry of Public Education. 0378-4371/89/$03.50 © Elsevier Science Publishers ,~.V. (North-Holland Physics Publishing Division)