标题： 有限维$\mathbb{Z}$-分次李代数 显示英文标题

标题： Finite-dimensional $\mathbb{Z}$-graded Lie algebras

摘要： 我们研究有限维$\mathbb{Z}$-分次李代数的结构和表示理论，包括相应的根系以及Verma模、不可约模和Harish-Chandra模。 这将有限维半单李代数的熟知理论扩展到更广泛的李代数类，并为依赖于临时方法的领域带来了进展和应用。 物理上有意义的例子由Heisenberg代数和共形Galilei代数提供，包括Schrödinger代数，其$\mathbb{Z}$-分次结构尚未被充分开发。 显示英文摘要

摘要： We investigate the structure and representation theory of finite-dimensional $\mathbb{Z}$-graded Lie algebras, including the corresponding root systems and Verma, irreducible, and Harish-Chandra modules. This extends the familiar theory for finite-dimensional semisimple Lie algebras to a much wider class of Lie algebras, and opens up for advances and applications in areas relying on ad-hoc approaches. Physically relevant examples are afforded by the Heisenberg and conformal Galilei algebras, including the Schr\"odinger algebras, whose $\mathbb{Z}$-graded structures are yet to be fully exploited.