Nilpotent Lie algebras of nilpotence class at most three via graphs

Abstract

In this work, we associate with each finite simple graph $G$ a nilpotent Lie algebra $\mathfrak{n}_G$ of class at most three, whose underlying vector space has a basis indexed by the vertices, edges, and paths of length two of $G$. Its nilpotency class is exactly three if and only if $G$ contains a path of length two. We compute the dimensions of its lower central series and describe its center. Over fields of characteristic different from two, we prove that two such Lie algebras are isomorphic if and only if their underlying graphs are isomorphic. We also give an algebraic description using quotients involving exterior and symmetric squares. Finally, we study balanced blow-ups and prove that $\mathfrak{n}_G$ is a retract of $\mathfrak{n}_{G^{(m)}}$.