  On a cotangent bundle $T\sp*G$ of a Lie group $G$ one can describe the
standard Liouville form $\theta$ and the symplectic form $d \theta$ 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 stuctures on
$T\sp*G$ 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.
