Irreducible Lie–Yamaguti algebras of generic type

Abstract

a b s t r a c tLie-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.

Irreducible Lie–Yamaguti algebras

“…In fact, most of the irreducible LY-algebras will be shown, here and in the forthcoming paper [3], to appear inside simple Lie algebras as orthogonal complements of subalgebras of derivations of Lie and Jordan algebras, Freudenthal triple systems and Jordan pairs. Let us fix some notation to be used throughout this paper.…”

“…This observation, together with Proposition 1.3, leads to the following definition and structure result: Moreover, the complete classification of the first type is easily obtained as we shall show in what follows. The non-simple type will be studied in Section 4, while the generic type will be the object of a forthcoming paper [3].…”

Symmetric matrices, orthogonal Lie algebras and Lie-Yamaguti algebras

“…The classification of isotropically irreducible reductive homogeneous spaces by Wolf [24,25] was reconsidered in [3,4] using an algebraic approach with reductive pairs related to nonassociative structures, in particular Lie and Jordan algebras and triples. A summary of the close connections between nonassociative structures and isotropically irreducible reductive homogeneous spaces is given in [4,. We emphasize [24, Ch.…”

On Killing Forms and Invariant Forms of Lie-Yamaguti Superalgebras

“…Later on, these algebraic objects were called "Lie triple algebras" [3] and the terminology of "Lie-Yamaguti algebras" is introduced in [4] for these algebras. For further development of the theory of Lie-Yamaguti algebras one may refer, for example, to [5][6][7][8]. From the standard enveloping Lie algebra of a given Lie-Yamaguti algebra, the notions of the Killing-Ricci form and the invariant form of a Lie-Yamaguti algebra are introduced and studied in [9].…”