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

Abstract:We present a method for associating labeled directed graphs to finite-dimensional Lie algebras that are $graph$-$admissible$. The proposed framework enables visual identification of key structural algebraic features. We first formalize this approach and establish several classes of Lie algebras that admit the categorization as graph-admissible Lie algebras, and then investigate suitable valid graphs for this approach by translating the antisymmetry of the Lie bracket and the Jacobi identity into rules that such graphs must satisfy. Furthermore, based on these foundations, we provide graph-theoretic criteria for solvability, nilpotency, presence of ideals, simplicity, semisimplicity, and reductiveness of an algebra. We also present practical algorithms for constructing such graphs and those associated with the lower central series and derived series via iterative pruning procedures. We further cover specific examples, such as the Schrödinger and Lorentz algebras, which illustrate the applicability of the developed tools to physically relevant cases. Moreover, we explore applications in physics, where the method facilitates computation of similarity transformations essential for determining quantum mechanical time evolution via the Lie-algebraic factorization method. We also discuss a possible generalization of the framework based on magma-gradings of Lie algebras and formulate related open questions. Our approach bridges algebraic and combinatorial perspectives, offering theoretical tools for this area of mathematical physics.

We have significantly overhauled the manuscript. In particular, we have withdrawn Conjectures 80, 82, and 83 of the original version. The appendix has also been updated. 95 pages (66+29), 32 figures (24+8)

arXiv:2601.16161 [math-ph]

(or arXiv:2601.16161v2 [math-ph] for this version)