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arXiv · 2507.00384

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

Mark D. Gould·Phillip S. Isaac·Ian Marquette·Jorgen Rasmussen
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Abstract

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.

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BibTeXRIS

Mark D. Gould, Phillip S. Isaac, Ian Marquette, Jorgen Rasmussen. 2025-07-01. Finite-dimensional $\mathbb{Z}$-graded Lie algebras. https://arxiv.org/abs/2507.00384

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