A short note on 2-step nilpotent Lie algebras associated with graphs

TL;DR

This paper establishes a precise correspondence between isomorphisms of 2-step nilpotent Lie algebras and the isomorphisms of their associated graphs, providing a clear criterion for algebraic equivalence based on graph structure.

Contribution

It proves that two 2-step nilpotent Lie algebras associated with graphs are isomorphic if and only if their underlying graphs are isomorphic, clarifying the relationship between algebraic and combinatorial structures.

Findings

Isomorphism of Lie algebras corresponds exactly to graph isomorphism.

Provides a criterion to determine Lie algebra isomorphism via graph isomorphism.

Establishes a one-to-one correspondence between algebraic and graph-theoretic structures.

Abstract

In this note we consider 2-step nilpotent Lie algebras associated with graphs. We prove that 2-step nilpotent Lie algebras $\n$ and $\n'$ associated with graphs $(S, E)$ and $(S', E')$ respectively are isomorphic if and only if $(S, E)$ and $(S', E')$ are isomorphic.