Unital $3$-dimensional structurable algebras: classification, properties and $\rm{AK}$-construction

TL;DR

This paper classifies complex unital 3-dimensional structurable algebras, describes their properties, and explores the resulting Lie algebras from the Allison-Kantor construction, providing a comprehensive structural analysis.

Contribution

It provides a complete classification of these algebras, detailing their automorphisms, subalgebras, and associated Lie algebra structures, which was previously unknown.

Findings

Five algebras of type (2, 1) identified

Two algebras of type (1, 2) identified

Explicit structure of resulting Lie algebras determined

Abstract

This paper is devoted to the classification and studying properties of complex unital $3$-dimensional structurable algebras. We provide a complete list of non-isomorphic classes, identifying five algebras for type $(2, 1)$ and two algebras for type $(1, 2).$ For each obtained algebra, we describe the derivation algebra, the automorphism group, the lattice of subalgebras and ideals, and functional identities of degree $2$. Furthermore, we investigate the Allison-Kantor construction for the classified algebras. We determine the structure of the resulting $\mathbb{Z}$-graded Lie algebras, providing their dimensions and Levi decompositions.