  We present the Hamiltonian (Lie-Poisson) analysis of the Vlasov plasma, and
the dynamics of its kinetic moments, from the matched pair decomposition point
of view. We express these (Lie-Poisson) systems as couplings of
\textit{mutually interacting} (Lie-Poisson) subdynamics. The mutual interaction
is beyond the well-known semi-direct product theory. Accordingly, as the
geometric framework of the present discussion, we address the \textit{matched
pair Lie-Poisson} formulation allowing mutual interactions. Moreover, both for
the kinetic moments and the Vlasov plasma cases, we observe that one of the
constitutive subdynamics is the compressible isentropic fluid flow, and the
other is the dynamics of the kinetic moments of order $\geq 2$. In this regard,
the algebraic/geometric (matched pair) decomposition that we offer, is in
perfect harmony with the physical intuition. To complete the discussion, we
present a momentum formulation of the Vlasov plasma, along with its matched
pair decomposition.
