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

二阶最大超可积系统：半简并性、可分离性和相关的对称代数。

• 批准号: 540196982
• 负责人: Dr. Andreas Vollmer
• 金额: --
• 依托单位: School of Mathematics and Statistics
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/540196982?language=en
• 关键词: Second; order; maximally; superintegrable; systems

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.

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“半退化”系统的几何特征（只有很少是已知的日期），包括其共形变换（承认共形不变的配方）。为此，采用（代数）几何方法，这对于非退化系统已经是成功的。此外，一个（投影）的几何方法是特别适合于半退化的情况。该项目的第二个方面涉及与正交分离坐标系相关的超可积系统。目的是阐明二阶非退化超可积系统的超可积性和可分性之间的相互关系。特别是，该项目旨在澄清是否存在并非来自可分离系统的系统。由双可分系统产生的超可积系统将在上述几何框架内得到表征。在分离坐标理论中已知一个丰富的几何和组合结构，该理论将通过该项目转移到超可积系统（特别是一个运算结构，允许人们从已知的例子构建新的系统）。最后，研究二阶非退化超可积系统的对称代数。为此，所谓的余代数方法，连同提到的几何方法超可积系统。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，以及通过余代数方法从构造中产生哪些几何性质。从长远来看，这对于理解超可积系统中的特殊函数（例如正交超几何多项式）具有重要意义。