Lie algebras associated with labeled directed graphs

TL;DR

This paper introduces a graph-based method to construct and analyze 2-step nilpotent Lie algebras, revealing how graph labels and orientations influence algebraic properties and isomorphisms.

Contribution

It provides a novel construction of Lie algebras from labeled directed graphs and establishes criteria for identifying subalgebras and isomorphisms based on graph features.

Findings

Reversing edge orientation with a unique label yields isomorphic Lie algebras.

Lie algebra structure depends only on the underlying undirected graph when all labels are distinct.

Constructed all 2-step nilpotent Lie algebras of dimension ≤6 from labeled graphs.

Abstract

We present a construction of 2-step nilpotent Lie algebras using labeled directed simple graphs, which allows us to give a criterion to detect certain ideals and subalgebras by finding special subgraphs. We prove that if a label occurs only once, then reversing the orientation of that edge leads to an isomorphic Lie algebra. As a consequence, if every edge is labeled differently, the Lie algebra depends only on the underlying undirected graph. In addition, we construct the labeled directed graphs of all 2-step nilpotent Lie algebras of dimension $\leq6$ and we compute the algebra of strata preserving derivations of the Lie algebra associated with the complete bipartite graph $K_{m,n}$ with two different labelings.