半单Lie代数相关的若干经典和量子可积系统的代数和几何性质

This project aims to study integrable systems from the aspect of representation theory. To be more precise, the project is about construction and algebraic and geometric properties of classical and quantum integrable systems associated with semi-simple Lie algebras. The following concrete topics are covered:..(1) A number of new completely and super integrable Hamiltonian systems will be constructed, based on the algebraic extensions of semi-simple Lie algebras. Simultaneously, we also discuss the related problems such as Poisson algebra structure, dimensional reductions, linearization and symmetry classification for the resulting super integrable systems...(2) Quantum integrable systems associated with semi-simple Lie algebras will be considered. This includes quantization of classical integrable systems, construction of operator algebras for quantum super integrable systems, and their connections with Poisson algebras...(3) The Krall-Sheffer operators will be studied, especially from the viewpoint of their algebraic and geometric properties, which can further develop the theory of orthogonal polynomials and help us to find novel eigenfunctions, recurrence relations and ladder operators...The research in this project enables us to construct new integrable systems and establish a unified framework for Hamiltonian systems associated with semi-simple Lie algebras. The studies in exact solutions of these models may also have impacts on other topics in mathematics and physics.

代数表示论与可积系统密切相关，本项目主要研究半单Lie代数相关的经典和量子可积系统的构造及其代数与几何性质。首先，基于半单Lie代数的代数扩张构造一系列完全可积和超可积经典Hamilton系统，讨论超可积系统的Poisson代数、降维、线性化和对称群分类。其次，研究半单Lie代数相关的量子可积系统，讨论经典可积系统的量子化、量子超可积系统算子代数的构造及其与相应Possion代数的关系。最后，讨论三维Krall-Sheffer型算子的可积性与可解性，重点研究超可积性与（准）精确可解性的关系。通过对三维Krall-Sheffer型算子超可积系统的研究建立相应的正交多项式理论，包括多项式本征函数的形式、迭代关系和阶梯算子等。本项目的研究可系统构造新的Hamilton系统，建立半单Lie代数相关的Hamilton系统的统一框架，其中对精确可解系统的研究还可以推动数学物理领域其他理论的发展。

代数表示论与可积系统密切相关，本项目主要研究了半单Lie代数相关的经典可积系统的构造及其代数与几何性质。主要取得如下成果：. 1．建立了基于共形代数构造完全可积和超可积经典Hamilton系统的一般方法，并将其应用于三个自由度的欧式度量的共形代数上。得到了三个自由度的非常曲率空间上的共形平坦度量（广义Darboux-Koenigs度量）和相应的二次Killing张量，讨论了三维广义Darboux-Koenigs度量的Poisson代数和约化降维；构造了容许不超过四维的低维对称代数的超可积系统及其Poisson代数和一致约化，建立了它们与 二维Darboux-Koenigs系统的联系，并对相应系统添加势函数，得到超可积的自然Hamilton系统。. 2. 研究了平坦和共形平坦空间上Calogero-Moser 系统的可积和超可积推广。. 3. 完善和补充了非线性偏微分方程的对称、解结构和守恒律以及无穷维Lie代数的微分算子向量场实现。. 本项目的研究可系统构造新的Hamilton系统，建立以共形代数为基础的Hamilton系统的统一框架。