Second-order maximally superintegrable systems: semi-degeneracy, separability and associated symmetry algebras.

该项目旨在阐明二阶（最大）超可积系统的三个主要方面。第一个方面是所谓的“半简并”系统的几何特征（迄今为止所知甚少），包括它们的共形变换（承认一个共形不变公式）。为此，采用了一种（代数-）几何方法，这种方法对于非退化系统已经取得了成功。此外，还使用了一种（射影-）几何方法，特别适合于半简并情况。该项目的第二个方面涉及与正交分离坐标系统相关的超可积系统。目的是阐明二阶非退化超可积系统的超可积性与可分性的相互关系。特别是，该项目旨在澄清是否存在不是由可分离系统产生的系统。由双可分系统产生的超可积系统将在上述几何框架内进行表征。分离坐标理论中丰富的几何和组合结构是已知的，该项目将其转移到超可积系统中（特别是允许从已知示例构建新系统的操作结构）。最后，研究了二阶非退化超可积系统的对称代数。为了达到这个目的，所谓的协代数方法与前面提到的几何方法一起用于超可积系统。该项目旨在系统地阐明超可积系统如何从代数的实现中产生，以及通过协代数方法构建的几何性质。从长远来看，这对于理解来自超可积系统的特殊函数（例如正交超几何多项式）将是非常重要的。

The project is to shed light onto three major aspects of second-order (maximally) superintegrable systems. The first aspect is the geometric characterisation of so-called "semi-degenerate" systems (on which only little is known to date), including their conformal transformations (admitting a conformally invariant formulation). To this end an (algebraic-)geometric approach is employed, which for non-degenerate systeme has already been successful. In addition, a (projective-)geometric approach is used that is particularly suited for the semi-degenerate case. The second aspect of the project concerns superintegrable systems that are associated with systems of orthogonal separation coordinates. The goal is to shed light on the interrelationship of superintegrability and separability for non-degenerate superintegrable Systems of second order. In particular, the project aims to clarify if systems exist that do not arise from separable systems. Superintegrable systems arising from bi-separable systems will be characterised within the aforementioned geometric framework. A rich geometric and combinatoric structure is known in the theory of separation coordinates, which will be transferred to superintegrable systems by the project (in particular an operad structure which permits one to construct new systems from known examples). Finally, the project investigates symmetry algebras for non-degenerate superintegrable systems of second order. To this end, the so-called coalgebra approach is used, together with the mentioned geometric approach to superintegrable systems. The project aims at clarifying systematically how superintegrable systems arise from realisations of the algebras, and which geometric properties result from the construction via the coalgebra approach. In the long run, this is going to be significant for understanding special functions (e.g. orthogonal hypergeometric polynomials) that arise from superintegrable systems.