Six-dimensional nilpotent Lie algebras

TL;DR

This paper provides a comprehensive classification of 6-dimensional nilpotent Lie algebras over arbitrary fields, utilizing automorphism group actions and geometric invariants to distinguish isomorphism types.

Contribution

It offers the first complete classification of 6-dimensional nilpotent Lie algebras over fields that are not algebraically closed or of characteristic 2.

Findings

Classification of all 6-dimensional nilpotent Lie algebras over arbitrary fields

Use of automorphism group action on second cohomology to determine isomorphism classes

Identification of geometric invariants like Gram determinant and Arf invariant for orbit distinction

Abstract

We give a full classification of 6-dimensional nilpotent Lie algebras over an arbitrary field, including fields that are not algebraically closed and fields of characteristic~2. To achieve the classification we use the action of the automorphism group on the second cohomology space, as isomorphism types of nilpotent Lie algebras correspond to orbits of subspaces under this action. In some cases, these orbits are determined using geometric invariants, such as the Gram determinant or the Arf invariant. As a byproduct, we completely determine, for a 4-dimensional vector space $V$, the orbits of $\GL(V)$ on the set of 2-dimensional subspaces of $V\wedge V$.