24th February – Philip Arathoon (University of Manchester): Flag manifolds

A natural collection of homogeneous symplectic manifolds is given by the orbits of the coadjoint action of a Lie group on the dual of its Lie algebra. Many familiar symplectic manifolds arise in this way, including the complex projective spaces, Grassmannians and more generally, complex flag manifolds. In this talk I will review some essential theory from the classical classification of Lie algebras to illustrate a charming correspondence between such orbit spaces and ‘shaded’ Dynkin diagrams. This very geometric correspondence offers a way to better understand the geometry of these flag manifolds which for the classical Lie algebras I hope to elucidate. If time permits we could also see examples of exceptional flag manifolds arising as coadjoint orbits of the exceptional Lie groups.