On the structural properties of Lie algebras via associated labeled directed graphs

TL;DR

This paper introduces a graph-theoretic method to analyze finite-dimensional Lie algebras, enabling quick identification of their structural properties and applications in physics, through the association of labeled directed graphs.

Contribution

The paper develops a novel framework linking Lie algebra structures to labeled directed graphs, including criteria and algorithms for key properties, with applications to physical algebras.

Findings

Graph-based criteria for solvability, nilpotency, and simplicity.

Algorithms for constructing graphs from Lie algebras.

Application to physically relevant algebras like Schrödinger and Lorentz.

Abstract

We present a method for associating labeled directed graphs to finite-dimensional Lie algebras, thereby enabling rapid identification of key structural algebraic features. To formalize this approach, we introduce the concept of graph-admissible Lie algebras and analyze properties of valid graphs given the antisymmetry property of the Lie bracket as well as the Jacobi identity. Based on these foundations, we develop graph-theoretic criteria for solvability, nilpotency, presence of ideals, simplicity, semisimplicity, and reductiveness of an algebra. Practical algorithms are provided for constructing such graphs and those associated with the lower central series and derived series via an iterative pruning procedure. This visual framework allows for an intuitive understanding of Lie algebraic structures that goes beyond purely visual advantages, since it enables a simpler and swifter grasping of the algebras of interest beyond computational-heavy approaches. Examples, which include the Schr\"odinger and Lorentz algebra, illustrate the applicability of these tools to physically relevant cases. We further explore applications in physics, where the method facilitates computation of similtude relations essential for determining quantum mechanical time evolution via the Lie algebraic factorization method. Extensions to graded Lie algebras and related conjectures are discussed. Our approach bridges algebraic and combinatorial perspectives, offering both theoretical insights and computational tools into this area of mathematical physics.