Classification of Minimal Algebras over any Field up to Dimension 6

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.