Invariant Theory of singular Riemannian foliations.

这个项目的起点是A.Lytchak和M.Radeschi最近提出的代数性定理。它指出，具有闭叶的向量空间V上的每个无穷小奇异黎曼叶F由它的基本多项式的代数A决定，即V上的多项式在F的叶上不变。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，是经典不变理论的主要研究对象。因此，对于这一数学分支中的任何结果，人们可能会问，它是否适用于更一般的(可能是非齐次的)奇异黎曼叶。与M.Radeschi合作，我通过描述光滑基本函数的代数来推广一个这样的结果，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，对A以“小”度生成的F进行分类；推广表示的实数、复数和四元数类型的概念；描述垂直向量场的模；找到计算代数A的有限生成元集的算法。

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.