Irreducible Lie-Yamaguti algebras of Generic Type

TL;DR

This paper classifies irreducible Lie-Yamaguti algebras of generic type by linking them to various nonassociative algebraic systems, expanding the understanding of their structure and classification.

Contribution

It provides a classification of generic type Lie-Yamaguti algebras by connecting them to Lie, Jordan, and triple systems, complementing previous classifications of other types.

Findings

Classification of generic type Lie-Yamaguti algebras achieved

Connections established with Lie, Jordan, and triple systems

Enhanced understanding of algebraic structures related to homogeneous spaces

Abstract

Lie-Yamaguti algebras (or generalized Lie triple systems) are binary-ternary algebras intimately related to reductive homogeneous spaces. The Lie-Yamaguti algebras which are irreducible as modules over their inner derivation algebras are the algebraic counterpart of the isotropy irreducible homogeneous spaces. These systems splits into three disjoint types: adjoint type, non-simple type and generic type. The systems of the first two types were classified in a previous paper through a generalized Tits Construction of Lie algebras. In this paper, the Lie-Yamaguti algebras of generic type are classified by relating them to several other nonassociative algebraic systems: Lie and Jordan algebras and triple systems, Jordan pairs or Freudenthal triple systems.