Poisson structures and convexity theorems

Abstract

The proofs of Kostant's nonlinear convexity theorem and Van den Ban's more general convexity theorem remain in the fields of Lie theory and rely on induction. In 1991 Lu and Ratiu discovered an alternative proof to Kostant's nonlinear convexity theorem using a symplectic approach. In 2006 Foth and Otto used a similar symplectic approach for an alternative proof of Van den Ban's convexity theorem. In this thesis we study the prominent part that specific Poisson structures play in these symplectic approaches. For a thorough understanding we include many results concerning Poisson structures on Lie groups, examine the construction of the Lu-Evens Poisson structure and include decompositions of semisimple Lie algebras and groups.