8-dimensional 2-step nilpotent Lie algebras over algebraically closed fields of char $\ne 2, 3$

TL;DR

This work presents a complete classification of $8$-dimensional $2$-step nilpotent Lie algebras over algebraically closed fields of characteristic not $2$ or $3$ by translating Lie algebra structures into minimal CDGAs generated in degree $1$ and analyzing the differential via its image in $igwedge^2 W_0$. The authors reduce the problem to a GL$(W_0)$-orbit classification on subspaces of $igwedge^2 W_0$ and perform a detailed stratification by bivector rank through the Grassmannian $ ext{Gr}(2,6)$ and the Pfaffian hypersurface, handling the cases $(6,2)$, $(5,3)$, and $(4,4)$ with explicit normal forms. The result is a self-contained catalog of $35$ isomorphism classes of $8$-dimensional minimal algebras (and thus $8$-dimensional $2$-step nilpotent Lie algebras) with explicit models; many irreducible cases recover known complex classifications, while the method applies uniformly over any algebraically closed field of char $ eq 2,3$. The work provides practical tools for constructing and identifying algebras in geometric contexts (e.g., nilmanifolds) and suggests avenues for extending the approach to $ ext{R}$, leveraging algebro-geometric techniques for Lie-theoretic classification.

Abstract

We provide a self contained, elementary, and geometrically-flavored classification of $8$-dimensional $2$-step nilpotent Lie algebras over algebraically closed fields of characteristic $\ne 2,3$, using the algebro-geometric arguments from \cite{B} and elementary linear algebra.

Theorems & Definitions (40)

• Theorem 1
• Corollary 2
• Lemma 3
• Remark 4
• Definition 5
• Proposition 6
• proof
• Proposition 7
• proof
• Remark 8