Poisson structures on the cotangent bundle of a Lie group or a principle bundle and their reductions

On a cotangent bundle T*G of a Lie group G one can describe the standard Liouville form θ and the symplectic form dθ in terms of the right Maurer Cartan form and the left moment mapping (of the right action of G on itself), and also in terms of the left Maurer–Cartan form and the right moment mapping, and also the Poisson structure can be written in related quantities. This leads to a wide class of exact symplectic structures on T*G and to Poisson structures by replacing the canonical momenta of the right or left actions of G on itself by arbitrary ones, followed by reduction (to G cross a Weyl-chamber, e.g.). This method also works on principal bundles.