Invariant Theory of singular Riemannian foliations.

奇异黎曼叶状结构的不变理论。

• 批准号: 318342259
• 负责人: Dr. Ricardo Augusto Mendes, Ph.D.
• 金额: --
• 依托单位: Department of Mathematics
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2016
• 资助国家: 德国
• 起止时间: 2015-12-31 至 2018-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/318342259?language=en
• 关键词: Invariant; Theory; singular; Riemannian; foliations.

The starting point of this project is the recent Algebraicity Theorem due to A. Lytchak and M. Radeschi. It says that every infinitesimal singular Riemannian foliation F on the vector space V with closed leaves is determined by its algebra A of basic polynomials, that is, polynomials on V that are constant on the leaves of F.If F is homogeneous, that is, if it coincides with the orbit decomposition under the linear action of a compact Lie group, then A is known as the algebra of invariants, and is the main object of study of Classical Invariant Theory. For any result in this branch of Mathematics, one may therefore ask whether it holds in the more general setting of (possibly inhomogeneous) singular Riemannian foliations.In collaboration with M. Radeschi I have generalized one such result by describing the algebra of smooth basic functions, which in the homogeneous case is due to G. Schwarz.The goal of the present project (in collaboration with A.Lytchak and M.Radeschi) is to generalize other results from Classical Invariant Theory. More specifically: Classify F for which A is generated in "small" degrees; generalize the notion of real, complex and quaternionic type of a representation; describe the module of vertical vector fields; and find an algorithm that computes a finite set of generators for the algebra A.

这个项目的出发点是最近的代数定理由于A。Lytchak和M.拉德斯基它说，具有闭叶的向量空间V上的每个无穷小奇异黎曼叶化F由其基本多项式代数A确定，即V上的多项式在F的叶上是常数。如果F是齐次的，即如果它与紧李群的线性作用下的轨道分解重合，则A称为不变量代数，是经典不变论的主要研究对象。对于数学的这个分支中的任何结果，人们可能会问它是否在更一般的（可能是非齐次的）奇异黎曼叶理的情况下成立。Radeschi通过描述光滑基函数的代数，推广了一个这样的结果，在齐次情形下，这是由于G。施瓦茨。本项目的目标（与A.Lytchak和M.Radeschi合作）是推广经典不变论的其他结果。更具体地说：对F进行分类，其中A以“小”度生成;推广表示的真实的、复数和四元数类型的概念;描述垂直向量场的模;并找到计算代数A的生成元的有限集的算法。