Irreducible Lie-Yamaguti algebras

TL;DR

This paper classifies irreducible Lie-Yamaguti algebras, revealing their division into three types and connecting some to a generalized Tits construction, advancing understanding of their structure and relation to homogeneous spaces.

Contribution

It introduces a classification of irreducible Lie-Yamaguti algebras into three types and links some to a generalized Tits construction, providing new structural insights.

Findings

Classification into three disjoint types: adjoint, non-simple, and generic.

Most systems of the first two types relate to a generalized Tits construction.

Provides a detailed analysis of irreducible Lie-Yamaguti algebras and their properties.

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 Lie inner derivation algebra are the algebraic counterpart of the isotropy irreducible homogeneous spaces. These systems will be shown to split into three disjoint types: adjoint type, non-simple type and generic type. The systems of the first two types will be classified and most of them will be shown to be related to a Generalized Tits Construction of Lie algebras.