Invariant symplectic structures and metrics on Lie groups

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The project will focus on leftinvariant symplectic structures on Lie groups and on compact quotients of these groups. The infinitesimal objects associated with such manifolds are symplectic Lie algebras. For these objects, one wishes to have a structure theory which yields explicit classification results under suitable additional assumptions.The first part of the project deals with groups that are endowed with a leftinvariant symplectic structure and, in addition, with a biinvariant pseudo-Riemannian metric. The associated infinitesimal objects are metric symplectic Lie-Algebras. We will show that these objects can be obtained by quadratic extensions. This will lead to a cohomological description of the isomorphism classes of metric symplectic Lie algebras. As an application we classify metric symplectic Lie algebras with small index of the metric. Moreover, we determine those metric symplectic Lie algebras that admit a lattice and we classify lattices in some special cases.In the second part of the project we will consider arbitrary symplectic Lie algebras. We will try to modify the double extension method developed by Medina, Revoy and Dardi\'e in such a way that it can be applied to the classification of such Lie algebras. Moreover we will try to apply the modified extension method to extrinsic symplectic spaces.