Poisson structure and invariant manifolds on Lie groups

For a discrete mechanical system on a Lie group G determined by a (reduced) Lagrangian ℓ we define a Poisson structure via the pull-back of the Lie-Poisson structure on g^∗ by the corresponding Legendre transform. The main result shown in this paper is that this structure coincides with the reduction under the symmetry group G of the canonical discrete Lagrange 2-form ω_L on G × G. Its symplectic leaves then become dynamically invariant manifolds for the reduced discrete system.