Irreducible Lie–Yamaguti algebras of generic type

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 counterparts 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.