  Let G be a Lie group acting by diffeomorphisms on a manifold M and consider
the image of T[1]G and T[1]M, of G and M respectively, in the category of
differential graded manifolds. We show that the obstruction to lift the action
of T[1]G on T[1]M to an action on a R[n]-bundle over T[1]M is measured by the G
equivariant cohomology of M. We explicitly calculate the differential graded
Lie algebra of the symmetries of the R[n]-bundle over T[1]M and we use this
differential graded Lie algebra to understand which actions are hamiltonian.
  We show how split Exact Courant algebroids could be obtained as the derived
Leibniz algebra of the symmetries of R[2]-bundles over T[1]M, and we use this
construction to propose that the infinitesimal symmetries of a split Exact
Courant algebroid should be encoded in the differential graded Lie algebra of
symmetries of a R[2]-bundle over T[1]M. With this setup at hand, we propose a
definition for an action of a Lie group on an Exact Courant algebroid and we
propose conditions for the action to be hamiltonian.
