Double extensions of Lie algebras of Kac–Moody type and applications to some Hamiltonian systems

We describe some Lie algebras of the Kac–Moody type, construct their double extensions, central and by derivations{;} we also construct the corresponding Lie groups in some cases. We study the case of the Lie algebra of unimodular vector fields in more detail and use the linear Poisson structure on their regular duals to construct generalizations of some infinite-dimensional Hamiltonian systems, such as magnetohydrodynamics.