Classical r-Matrices and Compatible Poisson Structures for Lax Equations on Poisson Algebras

Given a classical r-matrix on a Poisson algebra, we show how to construct a natural family of compatible Poisson structures for the Hamiltonian formulation of Lax equations. Examples for which our formalism applies include the Benny hierarchy, the dispersionless Toda lattice hierarchy, the dispersionless KP and modified KP hierarchies, the dispersionless Dym hierarchy, etc. Received: 10 February 1998 / Accepted: 9 December 1998