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

Dmitri V. Alekseevsky·Janusz Grabowski·Giuseppe Marmo·Peter W. Michor
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Abstract

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

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Dmitri V. Alekseevsky, Janusz Grabowski, Giuseppe Marmo, Peter W. Michor. 1993-12-01. Poisson structures on the cotangent bundle of a Lie group or a principal bundle and their reductions. https://doi.org/10.1063/1.530822

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