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。

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.