Double extensions of Lie algebras of Kac-Moody type and applications to some hamiltonian systems

We shall describe some Lie algebras of Kac-Moody type, construct their double extensions, central and by derivations; in some cases, the corresponding Lie groups will also be constructed. The case of Lie algebra of unimodular vector fields will be studied in more details. We use the linear Poisson structure on their regular duals to construct generalizations of some in nite dimensional hamiltonian systems, such as magnetohydrodynamics.