Sylvain Attan, Ismail Laraiedh
This paper gives some constructions results and examples of BiHom-Poisson algebras. Next, BiHom-flexible algebras are defined and it is shown that admissible BiHom-Poisson algebras are BiHom-flexible. Furthermore, generalized derivations of Bihom-Poisson algebras are introduced and some their basic properties are given. Finally, BiHom-Poisson modules and several constructions of these notions are obtained.
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The goal of this paper is to introduce BiHom-Poisson color algebras and give various constructions by using some specific maps such as morphisms. We introduce averaging operator and element of centroid for BiHom-Poisson color algebras and point out that BiHom-Poisson color algebras are closed under averaging operators and elements of centroid. We also show that any regular BiHom-associative color algebra leads to BiHom-Poisson color algebra via the commutator bracket. Then we prove that any BiHom-Poisson color algebra together with Rota-Baxter operator or multiplier give rises to another BiHom…
Kupershmidt operators on $3$-BiHom-Poisson color algebras and $3$-BiHom-pre-Poisson color algebras structures
该研究引入非交换3-BiHom-Poisson色代数并研究其表示与Kupershmidt算子，同时提出非交换3-BiHom-预Poisson色代数概念，借助Kupershmidt算子揭示两类代数的关联。
We study BiHom-Novikov-Poisson algebras, which are twisted generalizations of Novikov-Poisson algebras and Hom-Novikov-Poisson algebras, and find that BiHom-Novikov-Poisson algebras are closed under tensor products and several kinds of perturbations. Necessary and sufficient conditions are given under which BiHom-Novikov-Poisson algebras give rise to BiHom-Poisson algebras.
BiHom-Poisson conformal superalgebras arising from related algebraic structures
本文针对BiHom-Poisson共形超代数，给出其构造定理，证明其在扭转变换下封闭，还推导了其与正则BiHom-结合共形代数、Rota-Baxter算子、BiHom-结合超代数的关联及张量积性质，补充了从Hom-post-Poisson共形超代数的构造方法。
In this paper, we introduce the concept of transposed BiHom-Poisson (abbr. TBP) algebras which can be constructed by the BiHom-Novikov-Poisson algebras. Several useful identities for TBP algebras are provided. We also prove that the tensor product of two (T)BP algebras are closed. The notions of BP 3-Lie algebras and TBP 3-Lie algebras are presented and TBP algebras can induce TBP 3-Lie algebras by two approaches. Finally, we give some examples for the TBP algebras of dimension 2.
The paper introduces the class of split regular BiHom-Poisson superalgebras, which is a natural generalization of split regular Hom-Poisson algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular BiHom-Poisson superalgebras $A$ is of the form $A=U+\sum_{\a}I_\a$ with $U$ a subspace of a maximal abelian subalgebra $H$ and any $I_{\a}$, a well described ideal of $A$, satisfying $[I_\a, I_\b]+I_\a I_\b = 0$ if $[\a]\neq [\b]$. Under certain conditions, in the case of $A$ being of maximal leng…
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 purpose of this paper is to introduce the notion of noncommutative BiHom-pre-Poisson algebra. Also we establish the bimodules and matched pairs of noncommutative BiHom-(pre)-Poisson algebras and related relevant properties are also given. Finally, we exploit the notion of $\mathcal{O}$-operator to illustrate the relations existing between noncommutative BiHom-Poisson and noncommutative BiHom pre-Poisson algebras.
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 aim of this paper is to introduce $n$-ary BiHom-algebras, generalizing BiHom-algebras. We introduce an alternative concept of BiHom-Lie algebra called BiHom-Lie-Leibniz algebra and study various type of $n$-ary BiHom-Lie algebras and BiHom-associative algebras. We show that $n$-ary BiHom-Lie-Leibniz algebra can be represented by BiHom-Lie-Leibniz algebra through fundamental objets. Moreover, we provide some key constructions and study $n$-ary BiHom-Lie algebras induced by $(n-1)$-ary BiHom-Lie algebras.
BiHom-Akivis algebras are introduced. The BiHom-commutator-BiHom-associator algebra of a regular BiHom-algebra is a BiHom-Akivis algebra. It is shown that BiHom-Akivis algebras can be obtained from Akivis algebras by twisting along two algebra endomorphisms. It is pointed out that a BiHom-Akivis algebra associated to a regular BiHom-alternative algebra is a BiHom-Malcev algebra.
The aim of this paper is to introduce and study BiHom-associative dialgebras. We give various constructions and study its connections with BiHom-Poisson dialgebras and BiHom-Leibniz algebras. Next we discuss the central extensions of BiHom-diassociative and we describe the classification of n-dimensional BiHom-diassociative algebras for n less or equal to 4. Finally, we discuss derivations of BiHom-associative dialgebras.
The aim of this paper is to give some construction results and examples of BiHom-alternative and BiHom-Jordan algebras. Next, we define BiHom-alternative and BiHom-Jordan (bi)modules and we prove that from a given BiHom-alternative and BiHom-Jordan bimodules, a sequence of this kind of bimodules can be constructed. Some relations between Bihom-associative, BiHom-alternative and BiHom-Jordan bimodules are given.
The purpose of this paper is to study the construction of $3$-Bihom-Lie algebras. We give some ways of constructing $3$-Bihom-Lie algebras from $3$-Bihom-Lie algebras and $3$-totally Bihom-associative algebras. Furthermore, we introduce $T_θ$-extensions and $T_θ^*$-extensions of $3$-Bihom-Lie algebras and prove the necessary and sufficient conditions for a $2n$-dimensional quadratic $3$-Bihom-Lie algebra to be isomorphic to a $T_θ^*$-extension.
The aim of this paper is to introduce and to give some constructions results of BiHom-X algebras by using Yau's twisting, Rota Baxter and Some elements of centroids. Next, we define the bimodules of BiHom-left symmetric dialgebras, BiHom-associative dialgebras and BiHom-tridendriform algebras. A sequence of this kind of bimodules can be constructed. Their matched pairs are also introduced and related relevant properties are given.
BiHom-Novikov agebra is a generalized Hom-Novikov algebra endowed with two commuting multiplicative linear maps. The main purpose of this paper is to show that two classes of BiHom-Novikov algebras can be constructed from BiHom-commutative algebras together with derivations and BiHom-Novikov algebras with Rota-Baxter operators, respectively. We show that quadratic BiHom-Novikov algebras are associative algebras and the sub-adjacent BiHom-Lie algebras of BiHom-Novikov algebras are 2-step nilpotent. Moreover, we develop the 1-parameter formal deformation theory of BiHom-Novikov algebras.
In this paper, we classify flnite dimensional multiplicative simple BiHom-Lie algebras by investigating the corresponding semisimple Lie algebras and we give a complete classification of the complex 3-dimensional multiplicative BiHom-Lie algebras.
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
In this paper, we introduce the notion of split extension of BiHom- Lie algebra and construct the corresponding cohomology. Also, we establish a one-to-one correspondence between the equivalence classes of extensions of a BiHom-Lie algebra L by an abelian BiHom-Lie algebra V and its second cohomology group.
BiHom-$Ω$-associative algebras and related structures
该研究提出四类BiHom-$\Omega$代数概念，借助权$\lambda$的Rota-Baxter族构造新的BiHom-$\Omega$结合与李代数，并将经典代数间关系推广到BiHom-$\Omega$情形，丰富了同态代数结构理论。
The aim of this paper is introduce and give some constructions results and examples of n-BiHom-Lie color algebras. Next, we introduce the definition of BiHom-modules over n- BiHom-Lie color algebras and we provide some properties. Moreover we investigate generalized derivations of n-BiHom-Lie color algebras and their BiHom-subalgebras.
We introduce the concept of braided BiHom-Frobenius algebras and give the cocycle bicrossproduct construction for BiHom-Frobenius algebras. We find that the extending problem for BiHom-Frobenius algebras can be classified by non-abelian cohomology theory.
This paper defines compatible BiHom-Lie algebras by twisting the compatible Lie algebras by two linear commuting maps. We show the characterization of compatible BiHom-Lie algebra as a Maurer-Cartan element in a suitable bidifferential graded Lie algebra. We also define a cohomology theory for compatible BiHom-Lie algebras.
Rota-Baxter operators and Loday-type algebras on the BiHom-associative conformal algebras
本文在BiHom-结合共形代数框架下系统研究了(三)树状代数、NS代数与(扭曲)Rota-Baxter算子，揭示了它们之间的内在联系，并建立了BiHom-四元共形代数与Rota-Baxter系统的关联。
In this paper, we introduce the notion of a BiHom-Lie conformal algebra and develop its cohomology theory. Also, we discuss some applications to the study of deformations of regular BiHom-Lie conformal algebras. Finally, we introduce derivations of multiplicative BiHom-Lie conformal algebras and study their properties.
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 BiHom-left-symmetric algebra is introduced. We will prove that it can generate trivial linear deformations of a BiHom-left-symmetric algebra.
In this paper we define and discuss the representations of $n$-BiHom-Lie algebra. We also introduce $T_θ$-extensions and $T_θ^{\ast}$-extensions of $n$-BiHom-Lie algebras and prove the necessary and sufficient conditions for a $2m$-dimensional quadratic $n$-Bihom-Lie algebra to be isomorphic to a $T_θ^{\ast}$-extension. Moreover, we develop the one-parameter formal deformations of $n$-BiHom-Lie algebras, and we proved that the first and second cohomology groups are suitable to the deformation theory involving infinitesimals, equivalent deformations, and rigidity
In this paper, we first introduce the notion of a product structure on a $3$-Bihom-Lie algebra which is a Nijenhuis operator with some conditions. And we provide that a $3$-Bihom-Lie algebra has a product structure if and only if it is the direct sum of two vector spaces which are also Bihom subalgebras. Then we give four special conditions such that each of them can make the $3$-Bihom-Lie algebra has a special decomposition. Similarly, we introduce the definition of a complex structure on a $3$-Bihom-Lie algebra and there are also four types special complex structures. Finally, we show that t…
Sweedler Duality for BiHom-associative Algebras
本文提出BiHom-关联代数的Sweedler对偶构造，定义其对偶结构并证明其煤代数性质，同时扩展到右BiHom模，恢复Hom和经典情况。
Bihom-associative algebras have been recently introduced in the study of group hom-categories. In this paper, we introduce a Hochschild type cohomology for bihom-associative algebras with suitable coefficients. The underlying cochain complex (with coefficients in itself) can be given the structure of an operad with a multiplication. Hence, the cohomology inherits a Gerstenhaber structure. We show that this cohomology also control corresponding formal deformations. Finally, we introduce bihom-associative algebras up to homotopy and show that some particular classes of these homotopy algebras ar…
Cohomology theory of Rota-Baxter family BiHom-$Ω$-associative algebras
本文提出了Rota-Baxter族BiHom-$Ω$-结合代数的概念，构建了其上链复形与权为$λ$的上同调理论，并证明该上同调能控制相应形变及刻画阿贝尔扩张。
Derivations, Cohomology and Deformation of BiHom-Associative Dialgebra
本文定义并分类了2维和3维BiHom-结合双代数的导子、广义导子与拟导子，将BiHom-结合代数的上同调推广到双代数情形，并利用上同调研究了其单参数形式形变。
Classification of Nijenhuis operators on three-dimensional multiplicative simple Bihom-Lie algebras
本文在复数域上完整分类了三维乘法简单BiHom-李代数的Nijenhuis算子，构造了相关上同调与形式形变，并引入BiHom-NS-李代数作为其代数结构。
The aim of this paper is to introduce the notion of BiHom-Lie superalgebras. This class of algebras is a generalization of both BiHom-Lie algebras and Hom-Lie superalgebras. In this article, we first present two ways to construct BiHom-Lie superalgebras from BiHom-associative superalgebras and Hom-Lie superalgebras by Yau's twist principle. Also, we explore some general classes of BiHom-Lie admissible superalgebras and describe all these classes via $G$-BiHom-associative superalgebras, where $G$ is a subgroup of the symmetric group $S_{3}$. Finally, we discuss the concept of $β^{k}$-derivation…
A BiHom-associative algebra is a (nonassociative) algebra $A$ endowed with two commuting multiplicative linear maps $α,β\colon A\rightarrow A$ such that $α(a)(bc)=(ab)β(c)$, for all $a, b, c\in A$. This concept arose in the study of algebras in so-called group Hom-categories. In this paper, we introduce as well BiHom-Lie algebras (also by using the categorical approach) and BiHom-bialgebras. We discuss these new structures by presenting some basic properties and constructions (representations, twisted tensor products, smash products etc).
In this paper, we generalize the results about generalized derivations of Lie algebras to the case of BiHom-Lie algebras. In particular we give the classification of generalized derivations of Heisenberg BiHom-Lie algebras. The definition of the generalized derivation depends on some parameters $ (λ,μ,γ)\in \mathbb{C}^{3}. $ In particular for $(λ,μ,γ)=(1,1,1)$, we obtain classical concept of derivation of BiHom-Lie algebra and for $(λ,μ,γ)=(1,1,0) $ we obtain the centroid of BiHom-Lie algebra. We give classifications of $ 2 $-dimensional BiHom-Lie algebra, centroides and derivations of $ 2 $-d…
The purpose of this paper is to study the structure and the algebraic varieties of BiHom-associative algebras. We provide a classication of n-dimensional BiHom-associative and BiHom-bialgebras and BiHom Hopf algebras for n $\le$ 3.
The first aim of this paper is to introduce and study symmetric (Bi)Hom-Leibniz algebras, which are left and right Leibniz algebras. We discuss $α^kβ^l$-generalized derivations, $α^kβ^l$ -quasi-derivations and $α^kβ^l$-quasi-centroid of (Bi)Hom-Leibniz algebras and colour BiHom-Leibniz algebras. The second aim is to define a new type of BiHom-Lie algebras satisfies the following hierarchy \begin{equation*} \displaystyle \{ \text{ BiHom-Lie type $B_1$} \}\supseteq_{β=id} \{ \text{ Hom-Lie} \}\supseteq_{α=id} \{ \text{ Lie} \}. \end{equation*} Moreover, define representations and a cohomology of…
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 Rota-Baxter operators and O-operators.
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 $\mathcal{O}$-operator of BiHom-Lie algebras. They can be also regarded as the splitting into three parts of the structure of a BiHom-Lie-algebra. Moreover we develop the representation theory of BiHom-post-Lie algebras on a vector space $V$. We show that there is naturally an induced representation of its sub-adjacent Lie algebra.
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 $T_θ$-extensions of BiHom-alternative algebras and their relationship with cohomology. Finally, we investigate generalized derivations and give some relevant results.
In this paper, we introduce the definition of generalized BiHom-Lie algebras and generalized BiHom-Lie admissible algebras in the category ${}_H{\mathcal M}$ of left modules for any quasitriangular Hopf algebra $(H, R) $. Also, we describe the BiHom-Lie ideal structures of the BiHom-associative algebras.
We investigate generalized derivations of $n$-BiHom-Lie algebras. We introduce and study properties of derivations, $( α^{s},β^{r}) $-derivations and generalized derivations. We also study quasiderivations of $n$-BiHom-Lie algebras. Generalized derivations of $(n+1)$-BiHom-Lie algebras induced by $ n $-BiHom-Lie algebras are also considered.
The goal of this paper is to study the structure of split regular BiHom-Leiniz superalgebras, which is a natural generalization of split regular Hom-Leiniz algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular BiHom-Leiniz superalgebras $\mathfrak{L}$ is of the form $\mathfrak{L}=U+\sum_{\a}I_\a$ with $U$ a subspace of a maximal abelian subalgebra $H$ and any $I_{\a}$, a well described ideal of $\mathfrak{L}$, satisfying $[I_\a, I_\b]= 0$ if $[\a]\neq [\b]$. In the case of $\mathfrak{L}$…
BiHom-Lie Colour algebra is a generalized Hom-Lie Colour algebra endowed with two commuting multiplicative linear maps. The main purpose of this paper is to define representations and a cohomology of BiHom-Lie colour algebras and to study some key constructions and properties. Moreover, we discuss $α^{k}β^l$-generalized derivations, $α^{k}β^l$-quasi-derivations and $α^{k}β^l$-quasi-centroid. We provide some properties and their relationships with BiHom-Jordan colour algebra.
Modified induction and a torsion-theoretic equivalence for relative BiHom-Hopf modules
本文针对相对BiHom-Hopf模建立诱导理论，利用Haar积分构造遗传挠理论，通过修正诱导函子实现右B-BiHom模与特定无挠相对BiHom-Hopf模的等价，推广了Hom及经典相对Hopf模的相关结果。
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…
Nijenhuis BiHom-Lie bialgebras and differential Lie bialgebras
引入 Nijenhuis BiHom-Lie 代数概念，建立其 Manin 三元组、双代数与匹配对之间的等价关系，并推广至微分 Lie 双代数。
Matrices as graded BiHom-algebras and decompositions
本文通过引入连接概念构造典范分次理想，证明在Σ-乘性等条件下矩阵正则BiHom-结合代数可分解为分次单理想的直和，并推广至一般情形及经典分次。
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…
Infinitesimal (BiHom-)bialgebras of any weight (II): Representations
本文研究任意权无穷小（双Hom-）双代数的表示理论，提出双积双代数，揭示其与Rota-Baxter、树状及pre-Lie双模的联系，并给出两类Gelfand-Dorfman定理。
Infinitesimal (BiHom-)bialgebras of any weight (I): Basic definitions and properties
本文引入任意权重的无穷小BiHom-双代数，推广了多种已知无穷小双代数，给出其构造、Hopf模结构及与Yang-Baxter方程的联系，并提供了构造BiHom-预Lie代数的方法。
A new class of Poisson algebras, the class of {\em generalized Weyl Poisson algebras}, is introduced. It can be seen as Poisson algebra analogue of generalized Weyl algebras or as giving a Poisson structure to (certain) generalized Weyl algebras. A Poisson simplicity criterion is given for generalized Weyl Poisson algebras and explicit descriptions of the Poisson centre and the absolute Poisson centre are obtained. Many examples are considered.
On the generalized Poisson and transposed Poisson algebras
本文研究了广义Poisson代数与转置Poisson代数，给出了同时属于两者的代数的多项式恒等式，通过单一运算建立广义Poisson代数的定义恒等式，并证明Ito定理对其成立。
For a generalized Weyl Poisson algebra $A$, explicit sets of generators and defining relations are presented for its Poisson enveloping algebra $\CU (A)$. Simplicity criteria are given for the algebra $\CU (A)$ and algebra of Poisson differential operators $P\CD (A)$ on $A$. The Gelfand-Kirillov dimensions of the algebras $\CU (A)$ and $P\CD (A)$ are calculated. It is proven that the algebra $\CU (A)$ is a domain provided that the coefficient ring $D$ of the generalized Weyl Poisson algebra $A$ is a domain of essentially finite type over a perfect field. For the algebra $A$, the set of its min…
Transposed Novikov-Poisson algebras
本文研究了转置Novikov-泊松代数的定义、结构及其与Novikov代数的关系，展示了其在张量积和导子中的应用，并探讨了其在单代数中的性质。
We introduce the notion of weakly associative algebra and its relations with the notion of nonassociative Poisson algebras.
Kähler-Poisson algebras were introduced as algebraic analogues of function algebras on Kähler manifolds, and it turns out that one can develop geometry for these algebras in a purely algebraic way. A Kähler-Poisson algebra consists of a Poisson algebra together with the choice of a metric structure, and a natural question arises: For a given Poisson algebra, how many different metric structures are there, such that the resulting Kähler-Poisson algebras are non-isomorphic? In this paper we initiate a study of such moduli spaces of Kähler-Poisson algebras defined over rational functions in two v…
Degenerations of Poisson-type algebras
本文研究了二维情形下多种Poisson型代数的退化问题，给出了代数与几何分类，并构建了完整的退化图及轨道闭包描述。
On transposed Poisson conformal algebras
本文引入非交换转置Poisson共形代数的概念，探讨其与Hom-Lie共形代数的关系，并给出其构造与分类。
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