[Paper Review] Classical properties of algebras using a new graph association

Rolando Aquino, L.M. Camacho+3 more|arXiv (Cornell University)|Jun 1, 2017

graph theory and CDMA systems2 references3 citations

TL;DR

This paper introduces a novel graph-theoretic framework to analyze algebraic properties of finite-dimensional algebras, particularly Leibniz algebras, by associating weighted directed graphs to algebras via their multiplication tables. The key contribution is a characterization of nilpotency and solvability through structural properties of these graphs—specifically, the absence of oriented cycles and stabilization of a fine sequence—enabling algorithmic verification of solvability and nilpotency in linear time.

ABSTRACT

We study the relation between algebraic structures and Graph Theory. We have defined five different weighted digraphs associated to a finite dimensional algebra over a field in order to tackle important properties of the associated algebras, mainly the nilpotency and solvability in the case of Leibniz algebras.

Motivation & Objective

• To extend graph-theoretic methods beyond Lie algebras to general finite-dimensional algebras over a field.
• To develop a systematic way to associate weighted digraphs to algebras that encode their full multiplication structure.
• To identify algebraic properties—especially nilpotency and solvability—through structural features of the associated digraphs.
• To generalize prior work by Carriazo et al. that was restricted to Lie algebras and relied on the Jacobi identity.
• To establish a computational framework for analyzing derived and lower central series using graph-based algorithms.

Proposed method

• Define five distinct weighted digraphs (e.g., union digraph, fine sequence digraph) based on a fixed basis and multiplication table of a finite-dimensional algebra.
• Use the adjacency structure of the digraphs to represent algebraic products, with edge weights encoding structure constants.
• Introduce the 'fine sequence' of digraphs by iteratively removing vertices with zero indegree, corresponding to elements not in the image of the product map.
• Apply topological sorting to detect acyclic digraphs, which correspond to nilpotent algebras.
• Use the stabilization of the fine sequence to determine solvability: if the sequence stabilizes at a graph with no edges, the derived series reaches zero.
• Leverage algorithmic graph theory (e.g., linear-time cycle detection) to verify properties like acyclicity, which implies nilpotency.

Experimental results

Research questions

• RQ1 Can a graph-theoretic framework be constructed to analyze nilpotency and solvability in general finite-dimensional algebras, not just Lie algebras?
• RQ2 To what extent can the derived series and lower central series of an algebra be reconstructed from the structural properties of its associated digraph?
• RQ3 Is there a graph-based characterization of nilpotency that avoids reliance on the Jacobi identity, thus extending to non-Lie algebras like Leibniz algebras?
• RQ4 Can the fine sequence of a digraph serve as a sufficient or necessary condition for solvability in Leibniz algebras?
• RQ5 How does the graph-based approach compare to traditional algebraic methods in terms of computational efficiency and theoretical insight?

Key findings

• The union digraph of a Leibniz algebra is a directed acyclic graph (DAG) if and only if the algebra is nilpotent.
• If the fine sequence of the digraph stabilizes at a graph with no edges, then the derived series of the algebra stabilizes at zero, implying solvability.
• The method successfully generalizes prior results by Carriazo et al., who were restricted to Lie algebras and the Jacobi identity, by removing this constraint.
• The fine sequence algorithm provides a constructive, graph-based method to test solvability, though it is not a complete characterization, as shown by counterexamples.
• There exist solvable Leibniz algebras whose associated digraphs do not stabilize to an edgeless graph, indicating the fine sequence condition is sufficient but not necessary for solvability.
• The weighted digraphs fully encode the algebraic structure, allowing complete reconstruction of the multiplication table from the graph and edge weights.