Classification of $K$-forms in nilpotent Lie algebras associated to graphs

TL;DR

This paper classifies the isomorphism classes of $K$-forms of certain nilpotent Lie algebras derived from graphs, linking algebraic properties to graph automorphisms and field structures.

Contribution

It provides a complete classification of $K$-forms of these Lie algebras based on the graph structure and field, revealing when rational forms are unique or infinite.

Findings

Number of rational forms is either one or infinite.

Unique rational form occurs iff the automorphism group is generated by transpositions.

Classification applies to any subfield of the complex numbers.

Abstract

Given a simple undirected graph, one can construct from it a $c$-step nilpotent Lie algebra for every $c \geq 2$ and over any field $K$, in particular also over the real and complex numbers. These Lie algebras form an important class of examples in geometry and algebra, and it is interesting to link their properties to the defining graph. In this paper, we classify the isomorphism classes of $K$-forms in these real and complex Lie algebras for any subfield $K \subset \mathbb{C}$ from the structure of the graph. As an application, we show that the number of rational forms up to isomorphism is always one or infinite, with the former being true if and only if the group of graph automorphisms is generated by transpositions.