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

TL;DR

This work develops a framework that associates finite-dimensional Lie algebras with labeled directed graphs to reveal key structural features. By introducing graph-admissible (including minimal and redundant) algebras, the authors define constructive procedures to attach graphs, derive graph-based criteria for solvability and nilpotency, and identify central elements and ideals directly from the graph. They demonstrate the approach on prominent examples such as the Schrödinger and Lorentz algebras, illustrating Levi–Mal’tsev decompositions and semisimple components through graphical means, and discuss extensions to graded structures and quantum-physics applications. The methods yield practical algorithms for the derived and lower central series, as well as for detecting simplicity, semisimplicity, and reductiveness, offering a bridge between combinatorics and algebra with potential impact on Lie-algebra classification and quantum dynamics analyses.

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 Schrödinger 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.

Figures (33)

• Figure 1: Schematic overview of the workflow presented in this paper. We consider finite-dimensional Lie algebras and begin by introducing the notion of graph-admissibility. Based on this concept, we construct labeled directed graphs associated with these algebras. Subsequently, we develop graph-theoretical tools, such as the lower central series and the derived series for graphs, in order to analyze structural properties. These tools enable the formulation of classification criteria for Lie algebras using graph-theoretical methods. Finally, we consider some specific examples and discuss possible extensions of these methods.
• Figure 2: Depiction of the graphs associated with the Lie algebras $\mathfrak{su}(2)$ and $\sl{2}{\mathbb{R}}$ from Examples \ref{['exa:first:example:graphs:assocaited:with:algebras']} and \ref{['exa:second:example:graphs:assocaited:with:algebras']} respectively.
• Figure 3: Depiction of the graphs from Example \ref{['exa:different:L:alpha:3:graphs']}. Graph (a) is associated with the Lie algebra $L_0^3$ with $v_1=x_1$, $v_2=x_2$, and $v_3=x_3$. Graph (b) is associated with the Lie algebra $L_{1}^3$ with $v_1=x_1+\varphi x_2$, $v_2=x_1+\vartheta x_2$, and $v_3=x_3$. Graph (c) is associated with the Lie algebra $L_{-1}^3$ with $v_1=x_1$, $v_2=x_2$, $v_3=x_1-x_2$, and $v_4=x_3$.
• Figure 4: Depiction of the graph $G(V,E)$ associated with the non-abelian Lie algebra from the proof of Proposition \ref{['prop:tight:bounds']}, for the case $n=3$. In (a) visualized by drawing every edge in $E$ explicitly, in (b) the graph is drawn using the convention of collapsing multiple edges with the same starting and end vertices into a single edge and annotating them with all relevant labeling vertices.
• Figure 5: Illustration of an improper labeled directed graph that satisfies all conditions from Theorem \ref{['thm:neccersary:conditions:proper:graph']}.

Theorems & Definitions (200)

• Definition 1: Lie algebra
• Definition 2: Center
• Definition 3: Derived series
• Definition 4: Lower central series
• Definition 5: Solvable and Nilpotent algebra
• Definition 6: Ideal
• Definition 7: Normalizer
• Definition 8: Radical
• Definition 9: Simple algebra, Semisimple algebra
• Definition 10: Reductive algebra