On quadratic Poisson brackets

In this paper, we present a method for constructing large families of quadratic Poisson brackets on a manifold using more elementary brackets on a different manifold. The method is then applied to several examples of completely integrable systems. One can recover several known brackets for systems such as the Toda lattice or the open discrete KP hierarchy. New brackets for a doubly periodic discrete KP hierarchy are also constructed. (C) 2005 American Institute of Physics.