Claude Roger: INFINITE DIMENSIONAL LIE ALGEBRAS AND MAGNETOHYDRODYNAMIC EQUATIONS

Abstract: The interpretation of (incompressible) fluid dynamics equations as a hamiltonian system on the Poisson structure of the dual of the Lie algebra of unimodular vector fields(ie the vector fields with vanishing divergence) is well known since the work of V.I. Arnold in the late sixties;it can be considered as a part of the broad theory of coadjoint orbits of Lie algebras (Kirillov-Kostant-Souriau),here in the infinite dimensional case. We shall describe here some extensions of those results,following the general scheme that algebraic constructions on the Lie algebra (essentially of cohomological kind: semi-direct products, central extensions,derivations,deformations,and superization) induce geometrical modifications of the Poisson structures on the duals,and then change on the corresponding "kinematics" of the Hamiltonian systems. We shall develop mainly the cases of magnetohydrodynamics(MHD) through 'magnetic extensions',and chromohydrodynamics(CHD) in which one has a 'coupling ' between Euler hydrodynamics equation and a non abelian gauge theory,or Euler-Yang-Mills equations.