Geometry of Vlasov kinetic moments: A bosonic Fock space for the symmetric Schouten bracket

The dynamics of Vlasov kinetic moments is shown to be Lie–Poisson on the dual Lie algebra of symmetric contravariant tensor fields. The corresponding Lie bracket is identified with the symmetric Schouten bracket and the moment Lie algebra is related with a bundle of bosonic Fock spaces, where creation and annihilation operators are used to construct the cold plasma closure. Kinetic moments are also shown to define a momentum map, which is infinitesimally equivariant. This momentum map is the dual of a Lie algebra homomorphism, defined through the Schouten bracket. Finally the moment Lie–Poisson bracket is extended to anisotropic interactions.