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

TL;DR

This paper extends the theory of finite-dimensional semisimple Lie algebras to a broader class of $bZ$-graded Lie algebras, exploring their structure, representations, and physical applications.

Contribution

It develops the structure and representation theory of finite-dimensional $bZ$-graded Lie algebras, including root systems and modules, expanding the classical Lie algebra framework.

Findings

Classifies finite-dimensional $bZ$-graded Lie algebras.

Analyzes their root systems and modules.

Provides examples from physics such as Schrödinger algebras.

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.