Minimal algebras and $2-$step nilpotent Lie algebras in dimension 7

TL;DR

This paper classifies 7-dimensional minimal algebras generated in degree 1 with a characteristic filtration of length 2, equivalently classifying 2-step nilpotent Lie algebras in dimension 7 over fields with characteristic not 2.

Contribution

It provides a comprehensive classification of 7-dimensional 2-step nilpotent Lie algebras using minimal algebra methods, extending previous work to arbitrary fields with char not 2.

Findings

Complete classification of 7-dimensional 2-step nilpotent Lie algebras.

Recovery of the real homotopy type of 7-dimensional 2-step nilmanifolds.

Extension of classification methods to fields with char not 2.

Abstract

We use the methods of \cite{BM} to give a classification of $7-$dimensional minimal algebras, generated in degree 1, over any field $\bk$ of characteristic $\textrm{char}(\bk)\neq 2$, whose characteristic filtration has length 2. Equivalently, we classify $2-$step nilpotent Lie algebras in dimension 7. This classification also recovers the real homotopy type of $7-$dimensional $2-$step nilmanifolds.