Hom-Leibniz Algebras Research Articles

Hom-Leibniz bialgebras and BiHom-Leibniz dendriform algebras

A notion of a Hom-Leibniz bialgebra is introduced and it is shown that matched pairs of Hom-Leibniz algebras, Manin triples of Hom-Leibniz algebras and Hom-Leibniz bialgebras are equivalent in a certain sense. A notion of Hom-Leibniz dendriform algebra is established, their bimodules and matched pairs are defined and their properties and theorems about their interplay and construction are obtained. Furthermore, the concept of BiHom-Leibniz dendriform algebras is introduced and discussed, their bimodules and matched pairs are constructed and properties are described. Finally, the connections between all these algebraic structures using {mathcal {O}}-operators are shown.

Cohomology of nonabelian embedding tensors on Hom-Lie algebras

<abstract><p>In this paper, we generalize known results of nonabelian embedding tensor to the Hom setting. We introduce the concept of Hom-Leibniz-Lie algebra, which is the basic algebraic structure of nonabelian embedded tensors on Hom-Lie algebras and can also be regarded as a nonabelian generalization of Hom-Leibniz algebra. Moreover, we define a cohomology of nonabelian embedding tensors on Hom-Lie algebras with coefficients in a suitable representation. The first cohomology group is used to describe infinitesimal deformations as an application. In addition, Nijenhuis elements are used to describe trivial infinitesimal deformations.</p></abstract>

The Classical Hom–Leibniz Yang–Baxter Equation and Hom–Leibniz Bialgebras

In this paper, we first introduce the notion of Hom–Leibniz bialgebras, which is equivalent to matched pairs of Hom–Leibniz algebras and Manin triples of Hom–Leibniz algebras. Additionally, we extend the notion of relative Rota–Baxter operators to Hom–Leibniz algebras and prove that there is a Hom–pre-Leibniz algebra structure on Hom–Leibniz algebras that have a relative Rota–Baxter operator. Finally, we study the classical Hom–Leibniz Yang–Baxter equation on Hom–Leibniz algebras and present its connection with the relative Rota–Baxter operator.

The structure of split regular Hom-Leibniz–Poisson color algebras

In this paper, we introduce the class of split regular Hom-Leibniz–Poisson color algebras as the natural generalization of split regular Hom-Leibniz algebras, split regular Hom-Poisson algebras and split regular Hom-Leibniz–Poisson superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular Hom-Leibniz–Poisson color algebra [Formula: see text] is of the form [Formula: see text] with [Formula: see text] a subspace of the abelian sub algebra [Formula: see text] and any [Formula: see text] a well-described ideal of [Formula: see text] satisfying [Formula: see text] if [Formula: see text]. Under certain conditions, in the case of [Formula: see text] being of maximal length, the simplicity and the primeness of the algebra is characterized and it is shown that [Formula: see text] is the direct sum of the family of its minimal ideals, each one being a simple split regular Hom-Leibniz–Poisson color algebra.

Hom-Leibniz 2-algebras

We define a Hom-Leibniz 2-algebra, which can be considered as the deformation and categorification of Leibniz algebras. We give the notion of 2-term strongly homotopy (sh) Hom-Leibniz algebras. We prove that the 2-term sh Hom-Leibniz algebras are equivalent to the Hom-Leibniz 2-algebras. We show that there exists a one-to-one correspondence between crossed modules of Hom-Leibniz algebras and 2-term Hom-differential graded Leibniz algebras. We also prove that the skeletal 2-term sh Hom-Leibniz algebras can be classified by using the third cohomology group of Hom-Leibniz algebras. We construct a Hom-Leibniz 2-algebra from an Omni-Hom-Lie 2-algebra. Moreover, we also give the construction of Hom-Leibniz 2-algebras from Hom-associative Rota–Baxter algebras.

Equivariant one-parameter formal deformations of Hom-Leibniz algebras

The aim of this paper is to define a new type of cohomology for multiplicative Hom-Leibniz algebras which control deformations of Hom-Leibniz algebras. The cohomology and the associated deformation theory for Hom-Leibniz algebras as developed here are also extended to equivariant context, under the presence of finite group actions on Hom-Leibniz algebras.

The structure of split regular BiHom-Leibniz color algebras

We introduce the class of split regular BiHom-Leibniz color algebras as the natural generalization of split regular Hom-Leibniz algebras. By developing techniques of connections of roots for this kind of algebra, we show that such a split regular BiHom-Leibniz color algebra ℒ is of the form ℒ=𝒰⊕∑[α]∈Π∕∼I[α], with 𝒰 a subspace of the abelian subalgebra ℋ and any I[α], a well-described ideal of ℒ, satisfying [I[α],I[β]]=0 if [α]≠[β]. Under certain conditions, in the case of ℒ being of maximal length, the simplicity and the primeness of the algebra is characterized.

On split regular Hom-Leibniz algebras

We introduce the class of split regular Hom-Leibniz algebras as the natural generalization of split Leibniz algebras and split regular Hom-Lie algebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular Hom-Leibniz algebra [Formula: see text] is of the form [Formula: see text] with [Formula: see text] a subspace of the abelian subalgebra [Formula: see text] and any [Formula: see text], a well described ideal of [Formula: see text], satisfying [Formula: see text] if [Formula: see text]. Under certain conditions, in the case of [Formula: see text] being of maximal length, the simplicity of the algebra is characterized.

Universal α-central extensions of Hom-Leibniz n-algebras

ABSTRACTWe construct homology with trivial coefficients of Hom-Leibniz n-algebras. We introduce and characterize universal ()-central extensions of Hom-Leibniz n-algebras. In particular, we show their interplay with the zero-th and first homology with trivial coefficients. When we recover the corresponding results on universal central extensions of Hom-Leibniz algebras. The notion of non-abelian tensor product of Hom-Leibniz n-algebras is introduced and we establish its relationship with universal central extensions. A generalization of the concept and properties of unicentral Leibniz algebras to the setting of Hom-Leibniz n-algebras is developed.

Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids

In this paper, first we modify the definition of a Hom-Lie algebroid introduced by Laurent-Gengoux and Teles and give its equivalent dual description. Many results that parallel to Lie algebroids are given. In particular, we give the notion of a Hom-Poisson manifold and show that there is a Hom-Lie algebroid structure on the pullback of the cotangent bundle of a Hom-Poisson manifold. Then we give the notion of a Hom-Lie bialgebroid, which is a natural generalization of a purely Hom-Lie bialgebra and a Lie bialgebroid. We show that the base manifold of a Hom-Lie bialgebroid is a Hom-Poisson manifold. Finally, we introduce the notion of a Hom-Courant algebroid and show that the double of a Hom-Lie bialgebroid is a Hom-Courant algebroid. The underlying algebraic structure of a Hom-Courant algebroid is a Hom-Leibniz algebra, or a Hom-Lie 2-algebra.

A non-abelian Hom-Leibniz tensor product and applications

The notion of non-abelian Hom–Leibniz tensor product is introduced and some properties are established. This tensor product is used in the description of the universal (-)central extensions of Hom–Leibniz algebras. We also give its application to the Hochschild homology of Hom-associative algebras.

On the Universal $$\alpha $$ α -Central Extension of the Semi-direct Product of Hom-Leibniz Algebras

We introduce Hom-actions, semidirect product, and establish the equivalence between split extensions and the semi-direct product extension of Hom-Leibniz algebras. We analyze the functorial properties of the universal ( $$\alpha $$ )-central extensions of ( $$\alpha $$ )-perfect Hom-Leibniz algebras. We establish under what conditions an automorphism or a derivation can be lifted in an $$\alpha $$ -cover and we analyze the universal $$\alpha $$ -central extension of the semi-direct product of two $$\alpha $$ -perfect Hom-Leibniz algebras.

On Hom–Lie algebras

In this paper, first we show that is a Hom–Lie algebra if and only if is an differential graded-commutative algebra. Then, we revisit representations of Hom–Lie algebras and show that there are a series of coboundary operators. We also introduce the notion of an omni-Hom–Lie algebra associated to a vector space and an invertible linear map. We show that regular Hom–Lie algebra structures on a vector space can be characterized by Dirac structures in the corresponding omni-Hom–Lie algebra. The underlying algebraic structure of the omni-Hom–Lie algebra is a Hom–Leibniz algebra, or a Hom–Lie 2-algebra.

On universal central extensions of Hom-Leibniz algebras

In the category of Hom-Leibniz algebras we introduce the notion of Hom-co-representation as adequate coefficients to construct the chain complex from which we compute the Leibniz homology of Hom-Leibniz algebras. We study universal central extensions of Hom-Leibniz algebras and generalize some classical results, nevertheless it is necessary to introduce new notions of α-central extension, universal α-central extension and α-perfect Hom-Leibniz algebra due to the fact that the composition of two central extensions of Hom-Leibniz algebras is not central. We also provide the recognition criteria for these kind of universal central extensions. We prove that an α-perfect Hom-Lie algebra admits a universal α-central extension in the categories of Hom-Lie and Hom-Leibniz algebras and we obtain the relationships between both of them. In case α = Id we recover the corresponding results on universal central extensions of Leibniz algebras.

Representations and cohomology of [formula omitted]-ary multiplicative Hom–Nambu–Lie algebras

The aim of this paper is to provide cohomologies of n -ary Hom–Nambu–Lie algebras governing central extensions and one parameter formal deformations. We generalize to n -ary algebras the notions of derivation and representation introduced by Sheng for Hom–Lie algebras. Also we show that a cohomology of n -ary Hom–Nambu–Lie algebras could be derived from the cohomology of Hom–Leibniz algebras.

(Co)Homology and universal central extension of Hom-Leibniz algebras

Hom-Leibniz algebra is a natural generalization of Leibniz algebras and Hom-Lie algebras. In this paper, we develop some structure theory (such as (co)homology groups, universal central extensions) of Hom-Leibniz algebras based on some works of Loday and Pirashvili.