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

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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.