POISSON STRUCTURES ON THE COTANGENT BUNDLE OF A LIE GROUP OR A PRINCIPLE BUNDLE AND THEIR REDUCTIONS

On a cotangent bundle TG of a Lie group G one can describe the Standa rd Liouville form theta and the symplectic form de in terms of the rig ht 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 writt en in related quantities. This leads to a wide class of exact symplect ic structures on TG and to Poisson structures by replacing the, canon ical 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 me thod also works on principal bundles.