Representation theories of conformal Galilei algebras and their applications to orthogonal polynomials and quantum many-body systems

时空对称性是当代物理学中最基本的概念之一。在数学上，对称是由李代数及其表示来描述的。本课题的目的是研究一类特殊的李代数--共形Galilei代数(CGA)及其表示。主要研究结果总结如下。CGA的一些扩展，如超对称性(最广义对称性)和中心(有质量的系统)。扩展代数的最基本表示的分类和其他类型表示的显式构造。作为这些表象的应用，导出了微分方程组和以CGA为对称性的量子多体系统模型。

Symmetries of space-time is one of the most fundamental notions in contemporary physics. Mathematically, symmetries are described by Lie algebras and their representations. The purposes of this project is to investigate the conformal Galilei algebras (CGA), a specific class of Lie algebras, and their representations. Main results are summarized as follows. Some extensions of CGA's such as supersymmetry (widest sense of symmetries) and central (system with mass). Classification of the most fundamental representations of the extended algebras and explicit construction of other types of representations. As applications of the representations, hierarchy of differential equations and a model of quantum many-body systems having CGA as a symmetry are derived.