The goal of this work is to introduce the notion of $3$-Hom-Lie-dendriform algebras which is the dendriform version of $3$-Hom-Lie-algebras. They can be also regarded as the ternary analogous of Hom-Lie-dendriform algebras. We give the representation of a $3$-Hom-pre-Lie algebra. Moreover, we introduce the notion of Nijenhuis operators on a $3$-Hom-pre-Lie algebra and provide some constructions of $3$-Hom-Lie-dendriform algebras in term of Nijenhuis operators. Parallelly, we introduce the notion of a product and complex structures on a $3$-Hom-Lie-dendriform algebras and there are also four types special integrability conditions.
In this paper we study (associative) Nijenhuis algebras, with emphasis on the relationship between the category of Nijenhuis algebras and the categories of NS algebras. This is in analogy to the well-known theory of the adjoint functor from the category of Lie algebras to that of associative algebras, and the more recent results on the adjoint functor from the categories of dendriform and tridendriform algebras to that of Rota-Baxter algebras. We first give an explicit construction of free Nijenhuis algebras and then apply it to obtain the universal enveloping Nijenhuis algebra of an NS algebr…
Shuffle and quasi-shuffle products are well-known in the mathematics literature. They are intimately related to Loday's dendriform algebras, and were extensively used to give explicit constructions of free commutative Rota-Baxter algebras. In the literature there exist at least two other Rota-Baxter type algebras, namely, the Nijenhuis algebra and the so-called TD-algebra. The explicit construction of the free unital commutative Nijenhuis algebra uses a modified quasi-shuffle product, called the right-shift shuffle. We show that another modification of the quasi-shuffle product, the so-called …
Hom-dendriform algebras are twisted analog of dendriform algebras and are splitting of hom-associative algebras. In this paper, we define a cohomology and deformation for hom-dendriform algebras. We relate this cohomology with the Hochschild-type cohomology of hom-associative algebras. We also describe similar results for the twisted analog of dendriform coalgebras.
The main feature of Hom-algebras is that the identities defining the structures are twisted by linear maps. The purpose of this paper is to introduce and study a Hom-type generalization of pre-Malcev algebras and M-dendriform algebras, called Hom-pre-Malcev algebras and Hom-M-dendriform algebras. We also introduce the notion of $\mathcal{O}$-operators of Hom-Malcev and Hom-pre-Malcev algebras and show the connections between Hom-Malcev, Hom-pre-Malcev and Hom-M-dendriform algebras using $\mathcal{O}$-operators. Hom-pre-Malcev algebras and Hom-M-dendriform algebras generalize Hom-pre-Lie algebr…
In this paper, we introduce the Hom-algebra setting of the notions of matching Rota-Baxter algebras, matching (tri)dendriform algebras and matching (pre)Lie algebras. Moreover, we study the properties and relationships between categories of these matching Hom-algebraic structures.
Splitting of operations for Hom-diassociative and Hom-triassociative Algebras
University of Carthage（迦太基大学）；Higher Institute of Sciences and Technology of Environment of Borj Cedria (ISTEUB)（布尔吉塞里亚环境科学与技术高等学院）；Université de Gafsa（加夫萨大学）；Facultté des sciences de Gafsa（加夫萨科学学院）；Département de Mathématiques, Faculté des Sciences, Université de Sfax（萨法克斯大学理学院数学系）；IRIMAS - Département de Mathématiques, Faculté des Sciences, Université de Haute Alsace（上阿尔萨斯大学理学院数学系 IRIMAS）
Dendriform algebras are certain splitting of associative algebras and arise naturally from Rota-Baxter operators, shuffle algebras and planar binary trees. In this paper, we first consider involutive dendriform algebras, their cohomology and homotopy analogs. The cohomology of an involutive dendriform algebra splits the Hochschild cohomology of an involutive associative algebra. In the next, we introduce a more general notion of oriented dendriform algebras. We develop a cohomology theory for oriented dendriform algebras that closely related to extensions and governs the simultaneous deformati…
First we use a new approach to give a graded Lie algebra whose Maurer-Cartan elements characterize pre-Lie algebra structures. Then using this graded Lie bracket we define the notion of a Nijenhuis operator on a pre-Lie algebra which generates a trivial deformation of this pre-Lie algebra. There are close relationships between O-operators, Rota-Baxter operators and Nijenhuis operators on a pre-Lie algebra. In particular, a Nijenhuis operator "connects" two O-operators on a pre-Lie algebra whose any linear combination is still an O-operator in certain sense and hence compatible L-dendriform alg…
The aim of this work is to introduce and study the notions of Hom-pre-Jordan algebra and Hom-J-dendriform algebra which generalize Hom-Jordan algebras. Hom-Pre-Jordan algebras are regarded as the underlying algebraic structures of the Hom-Jordan algebras behind the Rota-Baxter operators and $\mathcal{O}$-operators introduced in this paper. Hom-Pre-Jordan algebras are also analogues of Hom-pre-Lie algebras for Hom-Jordan algebras. The anti commutator of a Hom-pre-Jordan algebra is a Hom-Jordan algebra and the left multiplication operator gives a representation of a Hom-Jordan algebra. On the ot…
The aim of this paper is to introduce and study Rota-Baxter Hom-algebras. Moreover we introduce a generalization of the dendriform algebras and tridendriform algebras by twisting the identities by mean of a linear map. Then we explore the connections between these categories of Hom-algebras.
The 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. The 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 connection…
Rota-Baxter systems were introduced by Brzeziński as a generalization of Rota-Baxter operators that are related to dendriform structures, associative Yang-Baxter pairs and covariant bialgebras. In this paper, we define Rota-Baxter systems on Hom-associative algebras and show how they induce Hom-dendriform structures and weak pseudotwistors on the underlying Hom-associative algebra. We introduce Hom-Yang-Baxter pairs and a notion of covariant Hom-bialgebra. Given a Hom-Yang-Baxter pair, we construct a twisted Rota-Baxter system and a (quasitriangular) covariant Hom-bialgebra. Finally, we consid…
The notion of matching Rota-Baxter algebras was recently introduced by Gao, Guo and Zhang [{\em J. Algebra} 552 (2020) 134-170] motivated by the study of algebraic renormalization of regularity structures. The concept of matching Rota-Baxter algebras generalizes multiple integral operators with kernels. The same authors also introduced matching dendriform algebras as the underlying structure of matching Rota-Baxter algebras. In this paper, we introduce matching relative Rota-Baxter algebras that are also related to matching dendriform algebras. We define a matching associative Yang-Baxter equa…
The purpose of this paper is to introduce and study BiHom-NS-algebras, which are a generalization of NS-algebras using two homomorphisms. Moreover, we discuss their relationships with twisted Rota-Baxter operators in a BiHom-associative context. Furthermore, we introduce a generalization of Nijenhuis operators that lead to BiHom-NS-algebras along BiHom-associative algebras
A dendriform algebra is an associative algebra whose product splits into two binary operations and the associativity splits into three new identities. These algebras arise naturally from some combinatorial objects and through Rota-Baxter operators. In this paper, we start by defining cohomology of dendriform algebras with coefficient in a representation. The deformation of a dendriform algebra $A$ is governed by the cohomology of $A$ with coefficient in itself. Next we study $\mathrm{Dend}_\infty$-algebras (dendriform algebras up to homotopy) in which the dendriform identities hold up to certa…
Dendriform coalgebras are the dual notion of dendriform algebras and are splitting of associative coalgebras. In this paper, we define a cohomology theory for dendriform coalgebras based on some combinatorial maps. We show that the cohomology with self coefficients governs the formal deformation of the structure. We also relate this cohomology with the cohomology of dendriform algebras, coHochschild (Cartier) cohomology of associative coalgebras and cohomology of Rota-Baxter coalgebras which we introduce in this paper. Finally, using those combinatorial maps, we introduce homotopy analogue of …
We study Hom-type analogs of Rota-Baxter and dendriform algebras, called Rota-Baxter $G$-Hom-associative algebras and Hom-dendriform algebras. Several construction results are proved. Free algebras for these objects are explicitly constructed. Various functors between these categories, as well as an adjunction between the categories of Rota-Baxter Hom-associative algebras and of Hom-(tri)dendriform algebras, are constructed.
The concept of Rota-Baxter family algebra is a generalization of Rota-Baxter algebra. It appears naturally in the algebraic aspects of renormalizations in quantum field theory. Rota-Baxter family algebras are closely related to dendriform family algebras. In this paper, we first construct an $L_\infty$-algebra whose Maurer-Cartan elements correspond to Rota-Baxter family algebra structures. Using this characterization, we define the cohomology of a given Rota-Baxter family algebra. As an application of our cohomology, we study formal and infinitesimal deformations of a given Rota-Baxter family…
The main goal of this work is to introduce the notion of Hom-M-dendriform algebras which are the dendriform version of Hom-Malcev algebras. In fact they are the algebraic structures behind the $\mathcal{O}$-operator of Hom-pre-Malcev algebras. They also fit into a bigger framework as Hom-Malcev algebraic analogues of Hom-L-dendriform algebras. Furthermore, we show a connections between Hom-M-Dendriform algebras and Hom-alternative quadri-algebras.
Rota-Baxter systems of T. Brzeziński are a generalization of Rota-Baxter operators that are related to dendriform structures, associative Yang-Baxter pairs and covariant bialgebras. In this paper, we consider Rota-Baxter systems in the presence of bimodule, which we call generalized Rota-Baxter systems. We define a graded Lie algebra whose Maurer-Cartan elements are generalized Rota-Baxter systems. This allows us to define a cohomology theory for a generalized Rota-Baxter system. Formal one-parameter deformations of generalized Rota-Baxter systems are discussed from cohomological points of vie…
Representations and relative Rota-Baxter operators with respect to representations of Hom-Leibniz Poisson algebras are introduced and studied. Some characterizations of these operators are obtained. The notion of matched pair and Nijenhuis operators of Hom-Leibniz Poisson algebras are given and various relevant constructions of these Hom-algebras are deduced.
Representations and O-operators of Hom-(pre)-Jacobi-Jordan algebras are introduced and studied. The anticommutator of a Hom-pre-Jacobi-Jordan algebra is a Hom-Jacobi-Jordan algebra and the left multiplication operator gives a representation of a Hom-Jacobi-Jordan algebra. The notion of matched pairs and Nijenhuis operators of Hom-(pre)-Jacobi-Jordan algebras are given and various relevant constructions are obtained.
In many previous papers, the authors used an endomorphism of algebra to twist the original algebraic structures in order to produce the corresponding Hom-algebraic structures. In this works, we use these either a bijective linear map, either an element of centroid either an averaging operator either Nijenhuis operator, either a multiplier to produce Hom-Poisson color algebras from given one.
In this paper we introduce the notion of Hom-pre-Lie bialgebra in the general framework of the cohomology theory for Hom-Lie algebras. We show that Hom-pre-Lie bialgebras, standard Manin triples for Hom-pre-Lie algebras and certain matched pairs of Hom-pre-Lie algebras are equivalent. Due to the usage of the cohomology theory, it makes us successfully study the coboundary Hom-pre-Lie bialgebras. The notion of Hom-s-matrix is introduced, by which we can construct Hom-pre-Lie bialgebras naturally. Finally we introduce the notions of Hom-O-operators on Hom-pre-Lie algebras and Hom-L-dendriform al…
In this paper, we introduce the notion of n-Hom-pre-Lie superalgebras. We investigate the representation theory of n-Hom-pre-Lie superalgebras and we give some related results and structures based on Rota-Baxter operators, O-operators and Nijenhuis operators. Moreover, we study relationships between n-Hom-pre-Lie superalgebras and its induced Hom-pre-Lie superalgebras. The same procedure is applied for the representations of n- Hom-pre-Lie superalgebras.
Given a nonsymmetric operad $\mathcal{O}$, we first construct two new nonsymmetric operads $\mathcal{O}^{\mathrm{comp}}$ and $\mathcal{O}^{\mathrm{Dend}}$. These operads are respectively useful to study compatible and split Loday-algebras. As an application of the operad $\mathcal{O}^{\mathrm{comp}}$, we show that the cohomology of a compatible associative algebra carries a Gerstenhaber structure. We give an application of the operad $\mathcal{O}^\mathrm{Dend}$ to dendriform algebras and find generalizations to other Loday-algebras. In the end, we construct another operad $\mathrm{Fam}(\mathca…
We give the description of homogeneous Rota-Baxter operators, Reynolds operators, Nijenhuis operators, Average operators and differential operator of weight 1 of null-filiform associative algebras of arbitrary dimension.
This paper addresses a Hom-associative algebra built as a direct sum of a given Hom-associative algebra $(\mathcal{A}, \cdot, α)$ and its dual $(\mathcal{A}^{\ast}, \circ, α^{\ast}),$ endowed with a non-degenerate symmetric bilinear form $\mathcal{B},$ where $\cdot$ and $\circ$ are the products defined on $\mathcal{A}$ and $\mathcal{A}^{\ast},$ respectively, and $ α$ and $α^{\ast}$ stand for the corresponding algebra homomorphisms. Such a double construction, also called Hom-Frobenius algebra, is interpreted in terms of an infinitesimal Hom-bialgebra. The same procedure is applied to character…
A relative Rota-Baxter algebra is a generalization of a Rota-Baxter algebra. Relative Rota-Baxter algebras are closely related to dendriform algebras. In this paper, we introduce bimodules over a relative Rota-Baxter algebra that fits with the representations of dendriform algebras. We define the cohomology of a relative Rota-Baxter algebra with coefficients in a bimodule and then study abelian extensions of relative Rota-Baxter algebras in terms of the second cohomology group. Finally, we consider homotopy relative Rota-Baxter algebras and classify skeletal homotopy relative Rota-Baxter algeb…
In this paper, we introduced the notion of Hom-Lie antialgebras. The representations and cohomology theory of Hom-Lie antialgebras are investigated. We prove that the equivalent classes of abelian extensions of Hom-Lie antialgebras are in one-to-one correspondence to elements of the second cohomology group. We also prove that 1-parameter infinitesimal deformation of a Hom-Lie antialgebra are characterized by 2-cocycles of this Hom-Lie antialgebra with adjoint representation in itself. The notion of Nijenhuis operators of Hom-Lie antialgebra is introduced to describe trivial deformations.
The aim of this paper is to provide a cohomology of $n$-Hom-Lie color algebras governing one parameter formal deformations. Then, we study formal deformations of a $n$-Hom-Lie color algebra and introduce the notion of Nijenhuis operator on an $n$-Hom-Lie color algebra, which could give rise to infinitesimally trivial $(n-1)$-order deformations. Furthermore, in connection with Nijenhuis operators we introduce and discuss the notion of a product structure on $n$-Hom-Lie color algebras.
In this paper, first we give the cohomologies of an $n$-Hom-Lie algebra and introduce the notion of a derivation of an $n$-Hom-Lie algebra. We show that a derivation of an $n$-Hom-Lie algebra is a $1$-cocycle with the coefficient in the adjoint representation. We also give the formula of the dual representation of a representation of an $n$-Hom-Lie algebra. Then, we study $(n-1)$-order deformation of an $n$-Hom-Lie algebra. We introduce the notion of a Hom-Nijenhuis operator, which could generate a trivial $(n-1)$-order deformation of an $n$-Hom-Lie algebra. Finally, we introduce the notion of…
We define a graded twisted-coassociative coproduct on the tensor algebra $TW$ of any $\Z^n$-graded vector space $W$. If $W$ is the desuspension space $\da V$ of a graded vector space $V$, the coderivations (resp. quadratic ``degree 1'' codifferentials, arbitrary odd codifferentials) of this coalgebra are 1-to-1 with sequences $\zp_s$, $s\ge 1$, of $s$-linear maps on $V$ (resp. $\Z^n$-graded Loday structures on $V$, sequences that we call Loday infinity structures on $V$). We prove a minimal model theorem for Loday infinity algebras, investigate Loday infinity morphisms, and observe that the $\…
In this paper, first we study dual representations and tensor representations of Hom-pre-Lie algebras. Then we develop the cohomology theory of Hom-pre-Lie algebras in term of the cohomology theory of Hom-Lie algebras. As applications, we study linear deformations of Hom-pre-Lie algebras, which are characterized by the second cohomology groups of Hom-pre-Lie algebras with the coefficients in the regular representation. The notion of a Nijenhuis operator on a Hom-pre-Lie algebra is introduced which can generate trivial linear deformations of a Hom-pre-Lie algebra. Finally, we introduce the noti…
The aim of this paper is to extend to Hom-algebra structures the theory of formal deformations of algebras which was introduced by Gerstenhaber for associative algebras and extended to Lie algebras by Nijenhuis-Richardson. We deal with Hom-associative and Hom-Lie algebras. We construct the first groups of a deformation cohomology and give several examples of deformations. We provide families of Hom-Lie algebras deforming Lie algebra $sl_2$ and describe as formal deformations the q-deformed Witt algebra and Jackson $sl_2$.
In this paper, first we discuss Hom-pre-Poisson algebras and their relationships with Hom-Poisson algebra. Then we introduce the notion of a Hom-pre-Gerstenhaber algebra and show that a Hom-pre-Gerstenhaber algebra gives rise to a Hom-Gerstenhaber algebra. Moreover, we consider Hom-dendriform formal deformations of Hom-zinbiel algebras and show that Hom-pre-Poisson algebras are the corresponding semi-classical limits. Furthermore, we consider Hom-O-operators on Hom-Poisson algebras and study their relationships with Hom-pre-Poisson algebras. Finally, we define the notion of dual-Hom-pre-Poisso…
In this paper, we introduce the cohomology theory of $\mathcal{O}$-operators on Hom-associative algebras. This cohomology can also be viewed as the Hochschild cohomology of a certain Hom-associative algebra with coefficients in a suitable bimodule. Next, we study infinitesimal and formal deformations of an $\mathcal{O}$-operator and show that they are governed by the above-defined cohomology. Furthermore, the notion of Nijenhuis elements associated with an $\mathcal{O}$-operator is introduced to characterize trivial infinitesimal deformations.
The purpose of this paper is to define cohomology structures on Hom-associative algebras and Hom-Lie algebras. The first and second coboundary maps were introduced by Makhlouf and Silvestrov in the study of one-parameter formal deformations theory. Among the relevant formulas for a generalization of Hochschild cohomology for Hom-associative algebras and a Chevalley-Eilenberg cohomology for Hom-Lie algebras, we define Gerstenhaber bracket on the space of multilinear mappings of Hom-associative algebras and Nijenhuis-Richardson bracket on the space of multilinear mappings of Hom-Lie algebras. Al…
The main feature of Hom-algebras is that the identities defining the structures are twisted by linear maps. The purpose of this paper is to introduce and study a Hom-type generalization of pre-Malcev algebras, called Hom-pre-Malcev algebras. We also introduce the notion of Kupershmidt operators of Hom-Malcev and Hom-pre-Malcev algebras and show the connections between Hom-Malcev and Hom-pre-Malcev algebras using Kupershmidt operators. Hom-pre-Malcev algebras generalize Hom-pre-Lie algebras to the Hom-alternative setting and fit into a bigger framework with a close relationship with Hom-pre-alt…
