Ling Liu, Abdenacer Makhlouf, Claudia Menini, Florin Panaite
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.
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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.
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…
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.
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。
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 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.
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。
Jacobi algebras and Jacobi Novikov-Poisson algebras
本文研究了Jacobi Novikov-Poisson代数的定义、分类及其与Jacobi代数的关系，提出了Frobenius Jacobi Novikov-Poisson代数的概念，并给出了相关等价刻画和构造方法。
Solvability and nilpotency of transposed Novikov-Poisson algebras
本文发展了转置Novikov-Poisson代数的幂零性与可解性理论，建立了dialgebra幂零的等价条件，证明了可解性与右幂零性等价，并建立了与底代数幂零性的联系。
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。
Classification of Novikov-Poisson Algebras and Their Applications
本文完成3维复诺维科夫-泊松代数的同构分类，以此证明除𝔰𝔩₂(ℂ)外的3维复转置泊松代数均可由其得到，且Sartayev猜想在3维情形成立。
In 1997, X. Xu \cite{Xiaoping Xu Poisson} invented a concept of Novikov-Poisson algebras (we call them Gelfand-Dorfman-Novikov-Poisson (GDN-Poisson) algebras). We construct a linear basis of a free GDN-Poisson algebra. We define a notion of a special GDN-Poisson admissible algebra, based on X. Xu's definition and an S.I. Gelfand's observation (see \cite{Gelfand}). It is a differential algebra with two commutative associative products and some extra identities. We prove that any GDN-Poisson algebra is embeddable into its universal enveloping special GDN-Poisson admissible algebra. Also we prove…
$δ$-Novikov and $δ$-Novikov--Poisson algebras
本文研究 $δ$-Novikov 代数及其 Poisson 推广，刻画交换子结构、嵌入关系与相关构造，并证明对应 operad 对任意 $δ$ 均非 Koszul。
The geometric classification of Novikov-Poisson and Gelfand-Dorfman algebras
本文对复三维诺维科夫-泊松代数、盖尔范德-多夫曼代数及诺维科夫代数开展几何分类，明确了各类代数对应簇的维数、不可约分支数量与刚性代数数量。
$δ$-Mock-Novikov Algebras: Structural Theory, Operad and Poisson-Type Constructions
本文引入参数依赖的 $\delta$-mock-Novikov 代数，研究其结构、operad 性质及 Poisson 型构造，证明有限维情形在特征零下幂零并完成低维分类。
Hom-Novikov-Poisson algebras, which are twisted generalizations of Novikov-Poisson algebras, are studied. Hom-Novikov-Poisson algebras are shown to be closed under tensor products and several kinds of twistings. Necessary and sufficient conditions are given under which Hom-Novikov-Poisson algebras give rise to Hom-Poisson algebras.
The purpose of this paper is to study Hom-Novikov algebras and Hom-Novikov-Poisson algebras, both of which were defined by Yau. In the paper, we give several constructions leading us to some interesting examples of Hom-Novikov algebras and Hom-Novikov-Poisson algebras. Also, we introduce the notion of quadratic Hom-Novikov algebras and provide some properties.
Quantizations of transposed Poisson algebras by Novikov deformations
该研究引入交换结合代数的Novikov形变及经典极限概念，证明其为转置Poisson代数的量子化，推导得Novikov-Poisson型转置Poisson代数均可量子化，并完成二维非阿贝尔复转置Poisson代数量子化的等价分类。
The geometric classification of non-associative algebras: a survey
本文综述了幂零、Lie、Jordan等多种非结合代数簇（含$n$元代数、超代数、Poisson型代数）的几何分类研究，并涵盖了退化水平分类，系统梳理该领域成果。
The main feature of color Hom-algebras is that the identities defining the structures are twisted by even linear maps. The purpose of this paper is to introduce and give some constructions of admissible Hom-Novikov-Poisson color Hom-algebras and Hom-Gelfand-Dorfman color Hom-algebras. Their bimodules and matched pairs are defined and the relevant properties and theorems are given. Also, the connections between Hom-Novikov-Poisson color Hom-algebras and Hom-Gelfand-Dorfman color Hom-algebras is proved. Furthermore, we show that the class of admissible Hom-Novikov-Poisson color Hom-algebras is c…
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。
Transposed Poisson structure on the Witt-type algebra $\mathcal{W}(a,-1)$: Derivations, Automorphisms, and Rota--Baxter operators
分类了Witt型代数W(a,-1)上的导子、自同构和Rota-Baxter算子，证明了非平凡δ-导子仅存在于δ=1和1/2，并建立了刚性结果。
It is shown that the variety of transposed Poisson algebras coincides with the variety of Gelfand-Dorfman algebras in which the Novikov multiplication is commutative. The Gröbner-Shirshov basis for the transposed Poisson operad is calculated up to degree 4. Furthermore, we demonstrate that every transposed Poisson algebra is F-manifold. We verify that the special identities of GD-algebras hold in transposed Poisson algebras. Finally, we propose a conjecture stating that every transposed Poisson algebra is special, i.e., can be embedded into a differential Poisson algebra.
General multi-Novikov algebras, multi-differential algebras and their free constructions
本文提出研究多重Novikov代数和多重微分代数的通用方法，结合泊松代数，扩展了Gelfand构造Novikov代数的方法，并通过有类型装饰根树和非交换多重微分多项式构建自由非交换多重Novikov代数。
In this paper the well-known Dubrovin-Novikov problem posed as long ago as 1984 in connection with the Hamiltonian theory of systems of hydrodynamic type, namely, the classification problem for multidimensional Poisson brackets of hydrodynamic type, is solved. In contrast to the one-dimensional case, in the general case, a nondegenerate multidimensional Poisson bracket of hydrodynamic type cannot be reduced to a constant form by a local change of coordinates. Generally speaking, such a Poisson bracket is generated by a nontrivial special infinite-dimensional Lie algebra. In this paper we obtai…
On the Kantor product, II
本文研究乘法的 Kantor 平方与 Kantor 积，推广已有分类，并给出低维代数示例及构造新 transposed Poisson、Poisson-Novikov 代数和分类相关结构的方法。
On the Dong Property for a binary quadratic operad
本文研究了二元非结合代数满足Dong引理的条件，证明Novikov和Novikov-Poisson代数满足该引理，并通过操作代数语言确定了Dong引理成立的操作代数类型。
Starting from the Virasoro algebra and its relatives the generalization to higher genus compact Riemann surfaces was initiated by Krichever and Novikov. The elements of these algebras are meromorphic objects which are holomorphic outside a finite set of points. A crucial and non-trivial point is to establish an almost-grading replacing the honest grading in the Virasoro case. Such an almost-grading is given by splitting the set of points of possible poles into two non-empty disjoint subsets. Krichever and Novikov considered the two-point case. Schlichenmaier studied the most general multi-poin…
We demonstrate how a simple linear-algebraic technique used earlier to compute low-degree cohomology of current Lie algebras, can be utilized to compute other kinds of structures on such Lie algebras, and discuss further generalizations, applications, and related questions. While doing so, we touch upon such seemingly diverse topics as associative algebras of infinite representation type, Hom-Lie structures, Poisson brackets of hydrodynamic type, Novikov algebras, simple Lie algebras in small characteristics, and Koszul dual operads.
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