Lie-Yamaguti Algebras Research Articles

Deformation cohomology of morphisms of Lie-Yamaguti algebras

We study cohomology of morphisms of Lie-Yamaguti algebras. As an application, we establish that this cohomology ‘controls’ the formal deformations. Additionally, we demonstrate its connection to the abelian extensions of morphisms of Lie-Yamaguti algebras.

Para-Kähler and pseudo-Kähler structures on Lie–Yamaguti algebras

For a pre-Lie–Yamaguti algebra [Formula: see text], by using its sub-adjacent Lie–Yamaguti algebra [Formula: see text], we are able to construct a semidirect product Lie–Yamaguti algebra via a representation of [Formula: see text]. The investigation of such semidirect Lie–Yamaguti algebras leads us to the notions of para-Kähler structures and pseudo-Kähler structures on Lie–Yamaguti algebras, and also gives the definition of complex product structures on Lie–Yamaguti algebras. Furthermore, a Levi–Civita product with respect to a pseudo-Riemannian Lie–Yamaguti algebra is introduced and we explore its relation with pre-Lie–Yamaguti algebras.

Cohomology and Deformations of Relative Rota–Baxter Operators on Lie-Yamaguti Algebras

In this paper, we establish the cohomology of relative Rota–Baxter operators on Lie-Yamaguti algebras via the Yamaguti cohomology. Then, we use this type of cohomology to characterize deformations of relative Rota–Baxter operators on Lie-Yamaguti algebras. We show that if two linear or formal deformations of a relative Rota–Baxter operator are equivalent, then their infinitesimals are in the same cohomology class in the first cohomology group. Moreover, an order n deformation of a relative Rota–Baxter operator can be extended to an order n+1 deformation if and only if the obstruction class in the second cohomology group is trivial.

On G-derivations of Lie–Yamaguti algebras

This paper primarily deals with the study of [Formula: see text]-derivations associated with Lie–Yamaguti algebras (LY-algebras). Taking [Formula: see text] as an automorphism group, the concept of [Formula: see text]-derivations, which is a derivation under both the bilinear and trilinear operations is defined for LY-algebras. Then some important properties of [Formula: see text]-derivations are studied along with their relationship with other generalized derivations of LY-algebras.

Automorphisms of extensions of Lie-Yamaguti algebras and inducibility problem

Lie-Yamaguti algebras generalize both the notions of Lie algebras and Lie triple systems. In this paper, we consider the inducibility problem for automorphisms of Lie-Yamaguti algebra extensions. More precisely, given an abelian extension [Display omitted] of a Lie-Yamaguti algebra L, we are interested in finding the pairs (ϕ,ψ)∈Aut(V)×Aut(L), which are inducible by an automorphism in Aut(L˜). We connect the inducibility problem to the (2,3)-cohomology of Lie-Yamaguti algebra. In particular, we show that the obstruction for a pair of automorphisms in Aut(V)×Aut(L) to be inducible lies in the (2,3)-cohomology group H(2,3)(L,V). We develop the Wells exact sequence for Lie-Yamaguti algebra extensions, which relates the space of derivations, automorphism groups, and (2,3)-cohomology groups of Lie-Yamaguti algebras. As an application, we describe certain automorphism groups of semi-direct product Lie-Yamaguti algebras. In a sequel, we apply our results to discuss inducibility problem for nilpotent Lie-Yamaguti algebras of index 2. We give examples of infinite families of such nilpotent Lie-Yamaguti algebras and characterize the inducible pairs of automorphisms for extensions arising from these examples. Finally, we write an algorithm to find out all the inducible pairs of automorphisms for extensions arising from nilpotent Lie-Yamaguti algebras of index 2.

Generalized Reynolds Operators on Lie-Yamaguti Algebras

In this paper, the notion of generalized Reynolds operators on Lie-Yamaguti algebras is introduced, and the cohomology of a generalized Reynolds operator is established. The formal deformations of a generalized Reynolds operator are studied using the first cohomology group. Then, we show that a Nijenhuis operator on a Lie-Yamaguti algebra gives rise to a representation of the deformed Lie-Yamaguti algebra and a 2-cocycle. Consequently, the identity map will be a generalized Reynolds operator on the deformed Lie-Yamaguti algebra. We also introduce the notion of a Reynolds operator on a Lie-Yamaguti algebra, which can serve as a special case of generalized Reynolds operators on Lie-Yamaguti algebras.

Cohomology and relative Rota-Baxter-Nijenhuis structures on [formula omitted] pairs

A LieYRep pair consists of a Lie-Yamaguti algebra and its representation. In this paper, we establish the cohomology theory of LieYRep pairs and characterize their linear deformations by the second cohomology group. Then we introduce the notion of relative Rota-Baxter-Nijenhuis structures on LieYRep pairs, investigate their properties, and prove that a relative Rota-Baxter-Nijenhuis structure gives rise to a pair of compatible relative Rota-Baxter operators under a suitable condition. Finally, we show the equivalence between r-matrix-Nijenhuis structures and Rota-Baxter-Nijenhuis structures on Lie-Yamaguti algebras.

The classification of nilpotent Lie-Yamaguti algebras

In this paper, we consider a generalization of the classical Skjelbred–Sund method, used to classify nilpotent low-dimensional Lie algebras, in order to classify Lie-Yamaguti algebras with non-trivial annihilator. We develop this method with the purpose of classifying nilpotent Lie-Yamaguti algebras, and we obtain from it the algebraic classification of the nilpotent Lie-Yamaguti algebras up to dimension four.

On equivariant Lie–Yamaguti algebras and related structures

In this paper, we first discuss cohomology and a one-parameter formal deformation theory of Lie–Yamaguti algebras. Next, we study finite group actions on Lie–Yamaguti algebras and introduce equivariant cohomology for Lie–Yamaguti algebras equipped with group actions. Finally, we study an equivariant one-parameter formal deformation theory and show that our equivariant cohomology is the suitable deformation cohomology.

Relative Rota-Baxter operators and symplectic structures on Lie-Yamaguti algebras

In this paper, first we show that the invariant bilinear form in a quadratic Lie-Yamaguti algebra induces an isomorphism between the adjoint representation and the coadjoint representation. Then we introduce the notions of relative Rota-Baxter operators on Lie-Yamaguti algebras and pre-Lie-Yamaguti algebras. We prove that a pre-Lie-Yamaguti algebra gives rise to a Lie-Yamaguti algebra naturally and a relative Rota-Baxter operator induces a pre-Lie-Yamaguti algebra. Finally, we study symplectic structures on Lie-Yamaguti algebra, which give rise to relative Rota-Baxter operators as well as pre-Lie-Yamaguti algebras. As applications, we study phase spaces of Lie-Yamaguti algebras, and show that there is a one-to-one correspondence between phase spaces of Lie-Yamaguti algebras and Manin triples of pre-Lie-Yamaguti algebras.

Quasi-derivations of Lie–Yamaguti algebras

The concepts of derivation and centroid for Lie–Yamaguti algebras are generalized in this paper. A quasi-derivation of a LY-algebra can be embedded as a derivation in a larger LY-algebra. The relationship between quasi-derivations and robustness of Lie–Yamaguti algebras has been studied.

Nijenhuis operators, product structures and complex structures on Lie–Yamaguti algebras

In this paper, first, we study linear deformations of a Lie–Yamaguti algebra and introduce the notion of a Nijenhuis operator. Then we introduce the notion of a product structure on a Lie–Yamaguti algebra, which is a Nijenhuis operator [Formula: see text] satisfying [Formula: see text]. There is a product structure on a Lie–Yamaguti algebra if and only if the Lie–Yamaguti algebra is the direct sum of two subalgebras (as vector spaces). There are some special product structures, each of which corresponds to a special decomposition of the original Lie–Yamaguti algebra. In the same way, we introduce the notion of a complex structure on a Lie–Yamaguti algebra. Finally, we add a compatibility condition between a product structure and a complex structure to introduce the notion of a complex product structure on a Lie–Yamaguti algebra.

On Killing Forms and Invariant Forms of Lie-Yamaguti Superalgebras

The notions of the Killing form and invariant form in Lie algebras are extended to the ones in Lie-Yamaguti superalgebras and some of their properties are investigated. These notions are alsoZ2-graded generalizations of the ones in Lie-Yamaguti algebras.

On the deformation of Lie-Yamaguti algebras

The deformation theory of Lie-Yamaguti algebras is developed by choosing a suitable cohomology. The relationship between the deformation and the obstruction of Lie-Yamaguti algebras is obtained.

Deformations and extensions of Lie–Yamaguti algebras

The deformation and extension theory of Lie–Yamaguti algebras is studied. We prove that a 1-parameter infinitesimal deformation of a Lie–Yamaguti algebra corresponds to a Lie–Yamaguti algebra of deformation type and a (2,3)-cocycle of with coefficients in the adjoint representation. The notion of Nijenhuis operators for Lie–Yamaguti algebra is introduced to describe trivial deformations. We also prove that equivalence classes of abelian extensions of Lie–Yamaguti algebras are in one-to-one correspondence to elements of the (2,3)-cohomology group.

One-parameter formal deformations of Hom-Lie-Yamaguti algebras

This paper studies one-parameter formal deformations of Hom-Lie-Yamaguti algebras. The first, second, and third cohomology groups on Hom-Lie-Yamaguti algebras extending ones on Lie-Yamaguti algebras are provided. It is proved that first and second cohomology groups are suitable to the deformation theory involving infinitesimals, equivalent deformations, and rigidity. However, the third cohomology group is not suitable for the obstructions.

Symmetric matrices, orthogonal Lie algebras and Lie-Yamaguti algebras

On the set of symmetric matrices over the field , we can define various binary and ternary products which endow it with the structure of a Jordan algebra or a Lie or Jordan triple system. All these nonassociative structures have the orthogonal Lie algebra as derivation algebra. This gives an embedding for . We obtain a sequence of reductive pairs that provides a family of irreducible Lie-Yamaguti algebras. In this paper, we explain in detail the construction of these Lie-Yamaguti algebras. In the cases , we use computer algebra to determine the polynomial identities of degree ; we also study the identities relating the bilinear Lie-Yamaguti product with the trilinear product obtained from the Jordan triple product.

A polynomial identity for the bilinear operation in Lie–Yamaguti algebras

We use computer algebra to demonstrate the existence of a multilinear polynomial identity of degree 8 satisfied by the bilinear operation in every Lie–Yamaguti algebra. This identity is a consequence of the defining identities for Lie–Yamaguti algebras, but is not a consequence of anticommutativity. We give an explicit form of this identity as an alternating sum over all permutations of the variables in a nonassociative polynomial with 8 terms. Our computations show that no such identities exist in degrees less than 8.

Irreducible Lie–Yamaguti algebras of generic type

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.

Irreducible Lie–Yamaguti algebras

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.