Invariant Theory of singular Riemannian foliations.

本计画的起点是最近由A. Lytchak和M. Radeschi所提出的代数定理。它说向量空间V上具有闭叶的每一个无穷小奇异黎曼叶化F都是由它的基本多项式代数A决定的，即V上的多项式在F的叶上是常数。如果F是齐次的，即它与紧李群线性作用下的轨道分解重合，则A被称为不变量代数，是经典不变理论的主要研究对象。因此，对于这个数学分支的任何结果，人们可能会问，它是否在更一般的（可能是非齐次的）奇异黎曼叶的设定中成立。在与Radeschi的合作中，我通过描述光滑基本函数的代数推广了这样一个结果，在齐次情况下，这是由于G. Schwarz。本项目的目标（与A.Lytchak和M.Radeschi合作）是推广经典不变理论的其他结果。更具体地说：将A以“小”度生成的F分类；推广了实数型、复数型和四元数型的概念；描述垂直矢量场模块；并找到一种算法来计算代数a的有限生成器集。

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.