Invariant symplectic structures and metrics on Lie groups

李群上的不变辛结构和度量

• 批准号: 229604565
• 负责人: Professorin Dr. Ines Kath
• 金额: --
• 依托单位: Institut für Mathematik und Informatik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2012
• 资助国家: 德国
• 起止时间: 2011-12-31 至 2015-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/229604565?language=en
• 关键词: Invariant; symplectic; structures; metrics; Lie

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.

该项目将集中在李群上的左不变辛结构和这些群的紧对偶上。与这种流形相关联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，产生明确的分类结果下适当的额外的assumption.The第一部分的项目涉及的群体，赋予了一个leftinvariant辛结构，此外，与一个bivariant伪黎曼度量。相关的无穷小对象是度量辛李代数。我们将表明，这些对象可以得到二次扩张。这将导致度量辛李代数的同构类的上同调描述。作为应用，我们对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。在第二部分中，我们将考虑任意辛李代数。我们将尝试修改由Medina，Revoy和Dardi\'e开发的双重扩张方法，使其可以应用于此类李代数的分类。此外，我们将尝试将改进的扩张方法应用于外辛空间。