Sami Mabrouk

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• Professor
• HDR at Faculty of Science of Gafsa

The aim of this paper is to introduce and study the concepts of the Rota-Baxter operator and Reynolds operator within the framework of trusses. Moreover, we introduce and discuss dendriform trusses, tridendriform trusses, and NS-trusses as fundamental algebraic structures underlying these classes of operators. Furthermore, we consider the notions o...

In this paper, we study Hom-mock-Lie algebras as a twisted version of mock-Lie algebras. Also we consider a pair of Hom-mock-Lie algebras structures satisfying that any linear combination of the two Hom-mock-Lie structures is still a Hom-mock-Lie structure called compatible Hom-mock-Lie algebras and exhibit some related results. Next, we introduce...

The main purpose of this paper is to introduce and investigate the notion of Jacobi–Jordan conformal algebras. They are a generalization of Jacobi–Jordan algebras which correspond to the case in which the formal parameter λ equals 0. We consider some related structures such as conformal modules, corresponding representations and O-operators. Theref...

A Malcev-Poisson algebra is a Malcev algebra together with a commutative associative algebra structure related by a Leibniz rule. In this paper, we introduce the notion of Malcev-Poisson bialgebra as an analogue of a Malcev bialgebra and establish the equivalence between matched pairs, Manin triples and Malcev-Poisson bialgebras. Moreover, we intro...

In this paper, we first introduce the notion of twisted $\mathcal O$-operators on a Hom-Lie-Yamaguti algebra by a given $(2,3)$-cocycle with coefficients in a representation. We show that a twisted $\mathcal O$-operator induces a Hom-Lie-Yamaguti structure. We also introduce the notion of a weighted Reynolds operator on a Hom-Lie-Yamaguti algebra,...

A Nijenhuis mock-Lie algebra is a mock-Lie algebra equipped with a Nijenhuis operator. The purpose of this paper is to extend the well-known results about Nijenhuis mock-Lie algebras to the realm of mock-Lie bialgebras. It aims to characterize Nijenhuis mock-Lie bialgebras by generalizing the concepts of matched pairs and Manin triples of mock-Lie...

The aim of this work is to introduce representations of BiHom-left-symmetric algebras and develop its cohomology theory. As applications, we study linear deformations of BiHom-left-symmetric algebras, which are characterized by its second cohomology group with the coefficients in the adjoint representation. The notion of a Nijenhuis operator on a B...

A Lie–Poisson triple system is a Lie triple system together with a commutative associative algebra structure related by a Leibniz rule. In this paper, we study representations theory and 𝒪-operators of Lie–Poisson triple systems. We show that a representation of a Lie–Poisson triple system has a dual representation under an additional condition. Ne...

The purpose of this paper is to study O-(dual-) Nijenhuis structures on a Jordan algebra with a representation. The notion of a (dual-)Nijenhuis pair is introduced and it can generate a trivial deformation of a Jordan algebra with representation. We introduce the notion of a O-(dual-)Nijenhuis structure on a Jordan algebra with representation. Furt...

The objective of this research paper is to broaden the concept of Lie superalgebras into a more generalized framework known as super-Lie superalgebras. It also investigates various super-generalizations of other algebraic structures, including super-associative algebras, left (right) super-Leibniz algebras, and super-left (right)-symmetric superalg...

Leibniz algebras are non skew-symmetric generalization of Lie algebras. In this paper we introduce the notion of anti-Leibniz algebras as a "non commutative version" of mock-Lie algebras. Low dimensional classification of such algebras is given. Then we investigate the notion of averaging operators and more general embedding tensors to build some n...

In this paper, we introduce the notion of averaging Hom-associative algebras wich is a Hom-associative algebra equipped with an averaging operator. We define representation and dual representation without any condition of averaging Hom-associative algebras. Next, we define a cohomology theory of averaging Hom-associative algebras. We also study the...

In this paper, we extend the concept of Lie superalgebras to a more generalized framework called Super-Lie superalgebras. In addition, they seem to be exploring various supergeneralizations of other algebraic structures, such as Super-associative, left (right) Super-Leibniz, and Super-left(right)-symmetric superalgebras, then we give some examples...

We introduce a notion of left-symmetric Rinehart algebras, which is a generalization of the notion of left-symmetric algebras. The left multiplication gives rise to a representation of the corresponding sub-adjacent Lie-Rinehart algebra. We construct left-symmetric Rinehart algebras from O-operators on Lie-Rinehart algebras. We extensively investig...

A Lie-Poisson triple system is a Lie triple system together with a commutative associative algebra structure related by a Leibniz rule. In this paper, we study representations theory and O-operators of Lie-Poisson triple systems. We show that a representation of a Lie-Poisson triple system has a dual representation under an additional condition. Ne...

The goal of the present work is to introduce the notion of 3-Hom-Lie-Rinehart superalgebras, which is a twisting version of 3-Lie-Rinehart superalgebras, and systematically describe their representations. Furthermore, we study the relationships between a Hom-Lie-Rinehart superalgebra and its induced 3-Hom-Lie-Rinehart superalgebra. Finally, we intr...

The main purpose of this paper is to introduce and investigate the notion of Jacobi-Jordan conformal algebra. They are a generalization of Jacobi-Jordan algebras which correspond to the case in which the formal parameter λ equals 0. We consider some related structures such as conformal modules, corresponding representations and O-operators. Therefo...

In this paper, first we recall the notion of Hom-Jordan superalgebras and study their representations. We define the Yang-Baxter equation in a Hom-Jordan superalgebra. Additionally , we extend the connections between O-operators and skew-symmetric solutions Yang-Baxter equation of Hom-Jordan superalgebras (HJYBE). In which, we prove that a super sk...

The aim of this paper is to introduce the notion of a mock-Lie bialgebra which is equivalent to a Manin triple of mock-Lie algebras. The study of a special case called coboundary mock-Lie bialgebra leads to introducing the mock-Lie Yang-Baxter equation on a mock-Lie algebra which is an analogue of the classical Yang-Baxter equation on a Lie algebra...

Constructions of n-ary bialgebras and n-ary infinitesimal bialgebras of associative type and their hom-analogs, generalizing the hom-bialgebras and infinitesimal hom-bialgebras are investigated. Main algebraic characteristics of n-ary totally, n-ary weak totally, n-ary partially and n-ary alternate partially associative algebras and bialgebras, and...

UDC 512.5 The aim of this paper is to provide a cohomology of n -Hom–Lie color algebras, in particular, a cohomology governing one-parameter formal deformations. Then we also study formal deformations of the n -Hom–Lie color algebras and introduce the notion of Nijenhuis operator on a n -Hom–Lie color algebra, which may give rise to infinitesimally...

The purpose of this paper is to introduce and study the notion of generalized Reynolds operators on Lie triple systems with representations (Abbr. \textsf{L.t.sRep} pairs) as generalization of weighted Reynolds operators on Lie triple systems. First, We construct an $L_{\infty}$-algebra whose Maurer-Cartan elements are generalized Reynolds operator...

We introduce the notion of relative averaging operators on Hom-associative algebras with a representation. Relative averaging operators are twisted generalizations of relative averaging operators on associative algebras. We give two characterizations of relative averaging operators of Hom-associative algebras via graphs and Nijenhuis operators. A (...

In this paper, first, we provide a graded Lie algebra whose Maurer–Cartan elements characterize Lie triple system structures. Then, we use it to study cohomology and deformations of O-operators on Lie triple systems by constructing a Lie 3-algebra whose Maurer–Cartan elements are O-operators. Furthermore, we define a cohomology of an O-operator T a...

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 O-operators introduced in this paper. Hom-pre-Jordan algeb...

The notion of a Dyck m-algebra was introduced recently by Lopez, Preville-Ratelle and Ronco in their work on the splitting of associativity via m-Dyck paths which are generalizations of dendri-form algebras. In this paper, we study this notion of Dyck m-algebras in the Hom-setting. Then we introduce the notion of (λ, µ)-weighted Rota-Baxter operato...

The purpose of this paper is to introduce and study twisted O-operators on 3-Lie algebras. We construct an L ∞-algebra whose Maurer-Cartan elements are twisted O-operators and define a cohomology of a twisted O-operator T as the Chevalley-Eilenberg cohomology of a certain 3-Lie algebra induced by T with coefficients in a suitable representation. Th...

The purpose of this paper is to study pseudo-Euclidean and symplectic Hom-alternative superalgebras and discuss some of their proprieties and provide construction procedures. We also introduce the notion of Rota-Baxter operators of pseudo-Euclidean Hom-alternative superalgebras of any weight and Hom-post-alternative superalgebras. A Hom-post-altern...

The aim of this paper is to deal with BiHom-alternative algebras which are a generalization of alternative and Hom-alternative algebras, their structure is defined with two commuting multiplicative linear maps. We study cohomology and one-parameter formal deformation theory of left BiHom-alternative algebras. Moreover, we study central and $T_\thet...

The purpose of this paper is to study the $\mathcal{O}$-operators on Malcev algebras and discus the solutions of Malcev Yang-Baxter equation by $\mathcal{O}$-operators. Furthermore we introduce the notion of weighted $\mathcal{O}$-operators on Malcev algebras, which can be characterized by graphs of the semi-direct product Malcev algebra . Then we...

We introduce a notion of ternary $F$-manifold algebras which is a generalization of $F$-manifold algebras. We study representation theory of ternary $F$-manifold algebras. In particular, we introduce a notion of dual representation which requires additional conditions similar to the binary case. We then establish a notion of a coherence ternary $F$...

UDC 512.5 We provide and study a Hom-type generalization of Jordan–Malcev–Poisson algebras called Hom–Jordan–Malcev–Poisson algebras. We show that they are closed under twisting by suitable self-maps and give a characterization of admissible Hom–Jordan–Malcev–Poisson algebras. In addition, we introduce the notion of pseudo-Euclidian Hom–Jordan–Malc...

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...

The main goal of this paper is to give some construction results of BiHom-post-Lie algebras which are a generalization of both post-Lie-algebras and Hom-post-Lie algebras. They are the algebraic structures behind the weighted O-operator of BiHom-Lie algebras. They can be also regarded as the splitting into three parts of the structure of a BiHom-Li...

The theory of Hom-conformal algebras, their representations and cohomology has a natural generalization to the case when λ is a vector. This class is called high dimensional Hom-conformal algebras. The main goal of this paper is to introduce the notion of r-dim Hom-Lie conformal algebras, where r be a nonnegative integer. Then we study a special cl...

We introduce a notion of n-Lie–Rinehart algebras as a generalization of Lie–Rinehart algebras to n-ary case. This notion is also an algebraic analogue of n-Lie algebroids. We develop representation theory and describe a cohomology complex of n-Lie–Rinehart algebras. Furthermore, we investigate extension theory of n-Lie–Rinehart algebras by means of...

In this paper, we study Hom-mock-Lie algebras as a twisted version of mock-Lie algebras. Also we consider a pair of Hom-mock-Lie algebras structures satisfying that any linear combination of the two Hom-mock-Lie structures is still a Hom-mock-Lie structure called compatible Hom-mock-Lie algebras and exhibit some related results. Next, we introduce...

The aim of this paper is to introduce the notion of a mock-Lie bialgebra which is equivalent to a Manin triple of mock-Lie algebras. The study of a special case called coboundary mock-Lie bialgebra leads to the introduction the mock-Lie Yang-Baxter equation on a mock-Lie algebra which is an analogue of the classical Yang-Baxter equation on a Lie al...

In this paper, we study the representation of ternary Jordan algebras which allows us to introduce the notion of coherent ternary Jordan algebras. Then the [Formula: see text]-operators of ternary Jordan algebras are introduced and the solutions of ternary Jordan Yang–Baxter equation are discussed involving [Formula: see text]-operators. Moreover,...

In this paper, we give Maurer-Cartan characterizations as well as a cohomology theory for compatible Lie superalgebra and use it to classify infinitesimal deformations of compatible Lie superalgebras. Then we give an interpretation of CYBE in compatible Lie superalgebras.

The aim of this paper is to deal with BiHom-alternative algebras which are a generalization of alternative and Hom-alternative algebras, their structure is defined with two commuting multiplicative linear maps. We study cohomology and one-parameter formal deformation theory of left BiHom-alternative algebras. Moreover, we study central and $T_\thet...

The purpose of this paper is to study the $\mathcal{O}$-operators on Malcev algebras and discuss the solutions of Malcev Yang-Baxter equation by $\mathcal{O}$-operators. Furthermore we introduce the notion of weighted $\mathcal{O}$-operators on Malcev algebras, which can be characterized by graphs of the semi-direct product Malcev algebra. Then we...

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. Furt...

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. Fur...

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...

The purpose of this paper is to study cohomology and deformations of $\mathcal{O}$-operators on Lie triple systems. We define a cohomology of an $\mathcal{O}$-operator $T$ as the Lie-Yamaguti cohomology of a certain Lie triple system induced by $T$ with coefficients in a suitable representation. Then we consider infinitesimal and formal deformation...

In this paper, we introduce the cohomology theory of 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 O-operator and show that they are governed by the abov...

The notion of a F -manifold algebras is an algebraic description of a F -manifold. In this paper, we introduce the notion of Hom-F -manifold algebras which is generalisation of F -manifold algebras and Hom-Poisson algebras. We develop the representation theory of Hom-F -manifold algebras and generalize the notion of Hom-pre-Poisson algebras by intr...

The aim of this paper is to study infinitesimal deformations of a Malcev algebra with a representation and introduce the notion of Nijenhuis pair, which gives a trivial deformation of a Malcev algebra with a representation. We introduce the notion of Kupershmidt-(dual- )Nijenhuis structure on a Malcev algebra with a representation. Furthermore, we...

The main purpose of this paper is to introduce the notion of $n$-L-dendriform algebra which can be seen as a dendrification of $n$-pre-Lie algebras by means of $\mathcal{O}$-operators. We investigate the representation theory of $n$-pre-Lie algebras and provide some related constructions. Furthermore, we introduce the notion of phase space of a $n$...

We introduce a concept of 3-Lie-Rinehart superalgebra and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we study the relationships between a Lie-Rinehart superalgebra and its induced 3-Lie-Rinehart superalgebra. The deformations of 3-Lie-Rinehart superalgebra are considered via a cohomology theory.

The purpose of this paper is to introduce twisted $\mathcal{O}$-operators on $3$-Lie algebras. We define a cohomology of a twisted $\mathcal{O}$-operator $T$ as the Chevalley-Eilenberg cohomology of a certain $3$-Lie algebra induced by $T$ with coefficients in a suitable representation. Then we consider infinitesimal and formal deformations of twis...

In this paper, we define and study (co)homology theories of a compatible associative algebra $A$. At first, we construct a new graded Lie algebra whose Maurer-Cartan elements are given by compatible associative structures. Then we define the cohomology of a compatible associative algebra $A$ and as applications, we study extensions, deformations an...

The goal of this work is to introduce the notion of 3-Hom-L-dendriform algebras which is the dendriform version of 3-Hom-Lie-algebras. They can be also regarded as the ternary analogous of Hom-L-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 alg...

The aim of this paper is to generalise the construction of n -ary Hom-Lie bracket by means of an $$(n-2)$$ ( n - 2 ) -cochain of given Hom-Lie algebra to super case inducing n -Hom-Lie superalgebras. We study the notion of generalized derivations and Rota-Baxter operators of n -ary Hom-Nambu and n -Hom-Lie superalgebras and their relation with gene...

In this paper we introduce the notion pre-Lie triple systems. Pre-Lie triple systems algebras are regarded as the underlying algebraic structures of Lie triple systems. They are the algebraic structures behind the Rota-Baxter operators and O-operators of triple systems introduced in this paper. We also study the notion of generalized derivations of...

The purpose of this paper is to introduce and study 3-Hom-Lie bialgebras, which are a ternary version of Hom-Lie bialgebras introduced by Yau (2015). We provide their properties, some key constructions and their 3-dimensional classification. Moreover we discuss their representation theory and their generalized derivations and coderivations. Further...

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 $\mathcal{O}$-operat...

The aim of this paper is to study infinitesimal deformations of a Malcev algebrawith a representation and introduce the notion of a Nijenhuis pair, which gives a trivial deformation of a Malcev algebra with a representation. We introduce the notion ofa Kupershmidt-(dual-)Nijenhuis structure on a Malcev algebra with a representation.Furthermore, we...

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...

We introduce a notion of $n$-Lie Rinehart algebras as a generalization of Lie Rinehart algebras to $n$-ary case. This notion is also an algebraic analogue of $n$-Lie algebroids. We develop representation theory and describe a cohomology complex of $n$-Lie Rinehart algebras. Furthermore, we investigate extension theory of $n$-Lie Rinehart algebras b...

The notion of a F-manifold algebras is an algebraic description of a F-manifold. In this paper, we introduce the notion of Hom-F-manifold algebras which is generalisation of F-manifold algebras and Hom-Poisson algebras. We develop the representation theory of Hom-F-manifold algebras and generalize the notion of Hom-pre-Poisson algebras by introduci...

The main goal of this paper is to introduce the notion of 3-L-dendriform algebras which are the dendriform version of 3-pre-Lie algebras. In fact they are the algebraic structures behind the O-operator of 3-pre-Lie algebras. They can be also regarded as the ternary analogous of L-dendriform algebras. Moreover, we study the generalized derivations o...

The purpose of this paper is to study the relationships between a Hom-Jordan algebras and its induced ternary Hom-Jordan algebras. We give some properties of the α k -generalized derivation algebra G D e r ( J ) of a ternary Hom-Jordan algebras. In particular, we prove that G D e r ( J ) = Q D e r ( J ) + Q Γ ( J ) , the sum of the α k -quasideriva...

The purpose of this paper is to provide and study a Hom-type generalization of Jordan-Malcev-Poisson algebras, called Hom-Jordan-Malcev-Poisson algebras. We show that they are closed under twisting by suitable self-maps and give a characterization of admissible Hom-Jordan-Malcev-Poisson algebras. In addition, we introduce the notion of pseudo-Eucli...

We introduce the concept of 3-Lie-Rinehart superalgebra and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we study the relationships between a Lie-Rinehart superalgebra and its induced 3-Lie-Rinehart superalgebra. The deformations of 3-Lie-Rinehart superalgebra are considered via the cohomology theory...

We introduce a notion of left-symmetric Rinehart algebras, which is a generalization of a left-symmetric algebras. The left multiplication gives rise to a representation of the corresponding sub-adjacent Lie-Rinehart algebra. We construct left-symmetric Rinehart algebra from $\mathcal O$-operators on Lie-Rinehart algebra. We extensively investigate...

The aim of this paper is to introduce and study BiHom-Poisson algebras, in particular Non-BiHom-Commutative BiHom-Poisson algebras. We discuss their representation theory and Semi-direct product. Furthermore, we characterize admissible BiHom-Poisson algebras. Finally, we establish the classification of 2-dimensional BiHom-Poisson algebras.

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...

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-...

The purpose of this paper is to study the relationships between a Hom-Jordan algebras and its induced ternary Hom-Jordan algebras. We give some properties of the α k-generalized derivation algebra GDer(J) of a ternary Hom-Jordan algebras. In particular , we prove that GDer(J) = QDer(J) + QΓ(J), the sum of the α k-quasiderivation algebra and the α k...

The propose of this paper is to extend generalized representations of 3-Lie algebras to Hom-type algebras. We introduce the concept of generalized representation of multiplicative 3-Hom-Lie algebras, develop the corresponding cohomology theory and study semi-direct products. We provide a key construction, various examples and computation of 2-cocyc...

The main goal of this paper is to introduce the notion of $3$-L-dendriform algebras which are the dendriform version of $3$-pre-Lie algebras. In fact they are the algebraic structures behind the $\mathcal{O}$-operator of $3$-pre-Lie algebras. They can be also regarded as the ternary analogous of L-dendriform algebras. Moreover, we study the general...

The aim of this paper is to generalise the construction of $n$-ary Hom-Lie bracket by means of an $(n-2)$-cochain of given Hom-Lie algebra to super case inducing a $n$-Hom-Lie superalgebras. We study the notion of generalized derivation and Rota-Baxter operators of $n$-ary Hom-Nambu and $n$-Hom-Lie superalgebras and their relation with generalized...

The purpose of this paper is to introduce and study BiHom-alternative algebras and BiHom-Malcev algebras. It is shown that BiHom-alternative algebras are BiHom-Malcev admissible and BiHom-Jordan admissible. Moreover, BiHom-type generalizations of some well known identities in alternative algebras, including the Moufang identities, are obtained.

The purpose of this paper is to study the relationships between a BiHom-Lie superalgebras and its induced 3-BiHom-Lie superalgebras. We introduce the notion of (αs,βr)-derivation, (αs,βr)-quasiderivation and generalized (αs,βr)-derivation of 3-BiHom-Lie superalgebras and their relation with derivation of BiHom-Lie superalgebras. We introduce also t...

The main goal of this paper is to introduce the notion of BiHom-post-Lie-algebras. They are a generalization of both post-Lie-algebras and Hom-post-Lie algebras. They are the algebraic structures behind the $\mathcal{O}$-operator of BiHom-Lie algebras. They can be also regarded as the splitting into three parts of the structure of a BiHom-Lie-algeb...

The purpose of this paper is to study quadratic color Hom-Lie algebras. We present some constructions of quadratic color Hom-Lie algebras which we use to provide several examples. We describe \(T^*\)-extensions and central extensions of color Hom-Lie algebras and establish some cohomological characterizations.

The purpose of this paper is to introduce and study the notion of BiHom-pre-alternative algebra which may be viewed as a BiHom-alternative algebra whose product can be decomposed into two compatible pieces. Furthermore, we introduce the notion of BiHom-alternative quadri-algebra and show the connections between all these algebraic structures using...

We show that given a hom–Lie algebra one can construct the n-ary hom–Lie bracket by means of an \((n-2)\)-cochain of the given hom–Lie algebra and find the conditions under which this n-ary bracket satisfies the Filippov–Jacobi identity, thereby inducing the structure of n-hom–Lie algebra. We introduce the notion of a hom–Lie n-tuple system which i...

The aim of this work is to introduce representations of BiHom-left-symmetric algebras. and develop its cohomology theory. As applications, we study linear deformations of BiHom-left-symmetric algebras, which are characterized by its second cohomology group with the coefficients in the adjoint representation. The notion of a Nijenhuis operator on a...

The propose of this paper is to extend generalized representations of 3-Lie algebras to Hom-type algebras. We introduce the concept of generalized representation of multiplicative 3-Hom-Lie algebras, develop the corresponding cohomology theory and study semi-direct products. We provide a key construction, various examples and computation of 2-cocyc...

The purpose of this paper is to study the relationships between a BiHom-Lie superalgebras and its induced 3-BiHom-Lie superalgebras. We introduce the notion of $(\alpha^s,\beta^r)$-derivation, $(\alpha^s,\beta^r)$-quasiderivation and generalized $(\alpha^s,\beta^r)$-derivation of 3-BiHom-Lie superalgebras, and their relation with derivation of BiHo...

The purpose of this paper is to introduce and study the notion of BiHom-pre-alternative algebra which may be viewed as a BiHom-alternative algebra whose product can be decomposed into two compatible pieces. Furthermore, we introduce the notion of BiHom-alternative quadri-algebra and show the connections between all these algebraic structures using...

The purpose of this paper is to introduce and study the notion of BiHom-pre-alternative algebra which may be viewed as a BiHom-alternative algebra whose product can be decomposed into two compatible pieces. Furthermore, we introduce the notion of BiHom-alternative quadri-algebra and show the connections between all these algebraic structures using...

The purpose of this paper is to introduce and study BiHom-alternative algebras and BiHom-Malcev algebras. It is shown that BiHom-alternative algebras are BiHom-Malcev admissible and BiHom-Jordan admissible. Moreover, BiHom-type generalizations of some well known identities in alternative algebras, including the Moufang identities, are obtained.

We show that given a Hom-Lie algebra one can construct the n-ary Hom-Lie bracket by means of an (n-2)-cochain of given Hom-Lie algebra and find the conditions under which this n-ary bracket satisfies the Filippov-Jacobi identity, there by inducing the structure of n-Hom-Lie algebra. We introduce the notion of Hom-Lie $n$-uplet system which is the g...

The purpose of this paper is to generalize to ℤ2-graded case the study of Hom-Lie bialgebras which were discussed first by D. Yau, then by Y. Sheng and C. Bai. We first provide various constructions of Hom-Lie superbialgebras and a classification of 3-dimensional Hom-Lie superbialgebras with 2-dimensional even part. Moreover, we study coboundary an...

The purpose of this paper is to generalize to $\mathbb{Z}_2$-graded case the study of Hom-Lie bialgebras which were discussed first by D. Yau, then by C. Bai and Y. Sheng. We provide different ways for constructing Hom-Lie superbialgebras. Also we define Matched pairs, Manin supertriples and discuss their relationships. Moreover, we study coboundar...

In this paper, we study Rota-Baxter operators and super $\mathcal{O}$-operator of associative superalgebras, Lie superalgebras, pre-Lie superalgebras and $L$-dendriform superalgebras. Then we give some properties of pre-Lie superalgebras constructed from associative superalgebras, Lie superalgebras and $L$-dendriform superalgebras. Moreover, we pro...

The aim of this paper is provide a survey on n-ary Hom-Nambu algebras and study quadratic n-ary Hom-Nambu algebras, which are n-ary Hom-Nambu algebras with an invariant, nondegenerate and symmetric bilinear forms that are also α-symmetric and β-invariant where α and β are twisting maps. We provide various constructions of quadratic n-ary Hom-Nambu...

The aim of this paper is to study the cohomology of Hom-Leibniz superalgebras. We construct the $q$-deformed Heisenberg-Virasoro superalgebra of Hom-type and provide as application the computations of the derivations and second cohomology group. Moreover, we extend to graded case the Takhtajan's construction of a cohomology of $n$-ary Hom-Nambu-Lie...