Anti-commutative algebras and their groups of automorphisms

TL;DR

This paper classifies four-dimensional anti-commutative algebras over a field of characteristic zero, focusing on their normal forms and automorphism groups, revealing their close relation to binary Lie algebras and extensions of Lie algebras.

Contribution

It provides a detailed classification of these algebras and describes their automorphism groups, linking them to known Lie algebra structures.

Findings

Normal forms of four-dimensional anti-commutative algebras identified

Automorphism groups characterized as extensions of Lie algebra automorphisms

Algebras shown to be extensions of nilpotent and abelian Lie algebras

Abstract

We determine normal forms of the multiplication of four-dimensional anti-commutative algebras over a field $\mathbb K$ of characteristic zero having an analogous family of flags of subalgebras as the four-dimensional non-Lie binary Lie algebras, and hence can be considered as the closest relatives of binary Lie algebras. These algebras are extensions of $\mathbb K$ by the 3-dimensional nilpotent Lie algebra and at the same time extensions of a two-dimensional Lie algebra by a two-dimensional abelian algebra. We describe their groups of automorphisms as extensions of a subgroup of the group of automorphisms of the three-dimensional nilpotent Lie algebra by $\mathbb K$.