Classification of Minimal Algebras over any Field up to Dimension 6

TL;DR

This paper classifies minimal algebras generated in degree 1 over any field up to dimension 6, recovering nilpotent Lie algebra classifications and identifying which nilmanifolds admit symplectic structures.

Contribution

It provides a comprehensive classification of minimal algebras over any field up to dimension 6, extending known classifications of nilpotent Lie algebras and nilmanifolds.

Findings

Classification of minimal algebras over any field up to dimension 6

Recovery of nilpotent Lie algebra classifications up to dimension 6

Identification of nilmanifolds with symplectic structures

Abstract

We give a classification of minimal algebras generated in degree 1, defined over any field $\bk$ of characteristic different from 2, up to dimension 6. This recovers the classification of nilpotent Lie algebras over $\bk$ up to dimension 6. In the case of a field $\bk$ of characteristic zero, we obtain the classification of nilmanifolds of dimension less than or equal to 6, up to $\bk$-homotopy type. Finally, we determine which rational homotopy types of such nilmanifolds carry a symplectic structure.