Invariant symplectic structures and metrics on Lie groups

本课题将重点研究李群上的左不变辛结构和这些群的紧商。与这种流形相关的无穷小对象是辛李代数。对于这些对象，人们希望有一个结构理论，在适当的附加假设下产生明确的分类结果。项目的第一部分处理具有左不变辛结构和双不变伪黎曼度规的群。与之相关的无穷小对象是度量辛李代数。我们将证明这些对象可以通过二次扩展得到。这将导致度量辛李代数的同构类的上同调描述。作为一个应用，我们对具有小度量指标的度量辛李代数进行了分类。此外，我们还确定了那些允许格存在的度量辛李代数，并在一些特殊情况下对格进行了分类。在项目的第二部分，我们将考虑任意辛李代数。我们将尝试修改Medina， Revoy和Dardi 'e开发的双重扩展方法，使其可以应用于这类李代数的分类。此外，我们还尝试将改进的可拓方法应用于外在辛空间。

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.