Irreducible Lie–Yamaguti algebras

Abstract

Communicated by C.A. Weibel MSC:Primary: 17A30 17B60 a b s t r a c t 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.

Symmetric matrices, orthogonal Lie algebras and Lie-Yamaguti algebras

“…• For [1,1], the projection onto [1,1] gives the Lie bracket on the adjoint module, so this is a Lie-Yamaguti algebra of adjoint type: This produces an LY algebra that appears in references [3] and [4].…”

“…Following [3,22], we note that two elements x, y in an LY algebra m define a linear map d x,y : m → m, z → x, y, z , which by (LY5-6) is a derivation of both the binary and ternary products. In this case, we call (g, h) a reductive pair.…”

“…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,.…”

Representations and cohomologies of Hom-Lie–Yamaguti algebras with applications

“…In [8], the authors introduced the concept of Hom-Lie-Yamaguti algebras. It is a Hom-type generalization of a Lie-Yamaguti algebra in [11,4], a general Lie triple system in [16,17] and a Lie triple algebra in [10]. In [12], the authors studied the formal deformations of Hom-Lie-Yamaguti algebras, where only low dimensional deformation cohomology were defined without the help of any representation.…”

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].…”