  We investigate some infinite dimensional Lie algebras and their associated
Poisson structures which arise from a Lie group action on a manifold.
  If $G$ is a Lie group, $\g$ its Lie algebra and $M$ is a manifold on which
$G$ acts, then the set of smooth maps from $M$ to $\g$ has at least two Lie
algebra structures, both satisfying the required property to be a Lie
algebroid. We may then apply a {construction} by Marle to obtain a Poisson
bracket on the set of smooth real or complex valued functions on $M\times
\g^*$. In this paper, we investigate these Poisson brackets. We show that the
set of examples include the standard Darboux symplectic structure and the
classical Lie Poisson brackets, but is a strictly larger class of Poisson
brackets than these. Our study includes the associated Hamiltonian flows and
their invariants, canonical maps induced by the Lie group action, and
compatible Poisson structures. Our approach is mainly computational and we
detail numerous examples.
  The Lie brackets from which our results derive, arose from the consideration
of connections on bundles with zero curvature and constant torsion. We give an
alternate derivation of the Lie bracket which will be suited to applications to
Lie group actions for applications not involving a Riemannian metric. We also
begin a study of the infinite dimensional Poisson brackets which may be
obtained by considering a central extension of the Lie algebras.
