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

Kobiljon Abdurasulov, Maqpal Eraliyeva, Ivan Kaygorodov

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.

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

Abstract

This paper is devoted to the classification and studying properties of complex unital -dimensional structurable algebras. We provide a complete list of non-isomorphic classes, identifying five algebras for type and two algebras for type For each obtained algebra, we describe the derivation algebra, the automorphism group, the lattice of subalgebras and ideals, and functional identities of degree . Furthermore, we investigate the Allison-Kantor construction for the classified algebras. We determine the structure of the resulting -graded Lie algebras, providing their dimensions and Levi decompositions.
Paper Structure (20 sections, 66 theorems, 26 equations)

This paper contains 20 sections, 66 theorems, 26 equations.

Table of Contents

  1. Classification and properties
  2. Preliminaries
  3. Classification of $3$-dimensional structurable algebras
  4. Type $(2,1)$
  5. Type $(1,2)$
  6. Properties of unital $3$-dimensional structurable algebras
  7. Derivations
  8. Automorphisms
  9. Subalgebras of unital $3$-dimensional structurable algebras
  10. Identities of unital $3$-dimensional structurable algebras
  11. Allison---Hein construction
  12. Allison---Kantor construction
  13. The Allison---Kantor structure algebra
  14. $\rm{AK}$ construction for ${\rm A}_1$
  15. $\rm{AK}$ construction for ${\rm A}_2$
  16. ...and 5 more sections

Key Result

Lemma 6

Let $\mathcal{A}$ be a unital algebra and let $\varphi \in {\rm Aut} (\mathcal{A}),$ then $\varphi(e_1)=e_1.$

Theorems & Definitions (110)

  • Definition 1
  • Definition 2: see A78
  • Definition 3
  • Definition 4
  • Remark 5
  • Lemma 6
  • Theorem 7
  • proof
  • Theorem 8
  • proof
  • ...and 100 more