Bin Xin, Guang'ai Song, Yucai Su
We first prove that, for any generalized Hamiltonian type Lie algebra $L$, the first cohomology group $H^1(L,L \otimes L)$ is trivial. We then show that all Lie bialgebra structures on $L$ are triangular.
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Leibniz bialgebras constructed by tensor product from Lie bialgebras and perm bialgebras
本文解决由李双代数和置换双代数构造莱布尼茨双代数的问题，证明二次李代数与置换双代数的张量积具有对应性质的莱布尼茨双代数结构，还构造了无限维莱布尼茨双代数并研究其拟三角、三角性质。
Lie bialgebras were introduced by Drinfeld in studying the solutions to the classical Yang-Baxter equation. The definition of a bialgebra in the sense of Drinfeld (D-bialgebra), related with any variety of algebras, was given by Zhelyabin. In this work, we consider Malcev bialgebras. We describe all structures of a Malcev bialgebra on a simple non-Lie Malcev algebra.
In this paper we investigate Lie bialgebra structures on the Schrödinger-Virasoro algebra $\LL$. Surprisingly, we find out an interesting fact that not all Lie bialgebra structures on the Schrödinger-Virasoro algebra are triangular coboundary, which is different from the related known results of some Lie algebras related to the Virasoro algebra.
Oscillator Lie algebras are the only non commutative solvable Lie algebras which carry a bi-invariant Lorentzian metric. In this paper, we determine all the Poisson structures, and in particular, all symmetric Leibniz algebra structures whose underlying Lie algebra is an oscillator Lie algebra. We give also all the symmetric Leibniz bialgebra structures whose underlying Lie bialgebra structure is a Lie bialgebra structure on an oscillator Lie algebra. We derive some geometric consequences on oscillator Lie groups.
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…
Infinite-dimensional pre-Lie bialgebras induced from Leibniz-dendriform bialgebras and Zinbiel-dendriform bialgebras
通过张量积构造，在Leibniz-dendriform双代数与Zinbiel代数、Zinbiel-dendriform双代数与Leibniz代数上建立完备pre-Lie双代数结构，并利用YBE解构造S-方程的解。
Factorizable Lie conformal bialgebras, quadratic Rota-Baxter Lie conformal algebras and some induced structures
本文引入可分解Lie共形双代数，建立其与非零权二次Rota-Baxter Lie共形代数的对应，并推广至Gel'fand-Dorfman情形，诱导多种共形代数结构。
In the paper we describe structures of quasitriangular Lie bialgebra on $gl_2(\mathbb C)$ using the classification of Rota-Baxter operators of nonzero weight on $gl_2(\mathbb C)$.
Mock-pre-Lie bialgebras
本文系统发展了mock-预李双代数理论，引入相空间、Manin三元组等概念，证明多种结构间的等价性，并研究上边界、拟三角和可分解等特殊类，导出经典Yang-Baxter方程的类似物。
In this paper we investigate Lie bialgebra structures on a twisted Schrödinger-Virasoro type algebra $\LL$. All Lie bialgebra structures on $\LL$ are triangular coboundary, which is different from the relative result on the original Schrödinger-Virasoro type Lie algebra. In particular, we find for this Lie algebra that there are more hidden inner derivations from itself to $\LL\otimes\LL$ and we develop one method to search them.
In this paper we investigate Lie bialgebra structures on the deformative Schrödinger-Virasoro algebras mainly using the techniques introduced recently by Liu, Pei and Zhu, which indicate that all cases considered in this paper except one behave different from their centerless ones.
In this paper, by studying the maximal good subspaces, we determine the dual Lie coalgebras of the centerless twisted Heisenberg-Virasoro algebra. Based on this, we construct the dual Lie bialgebras structures of the twisted Heisenberg-Virasoro type. As by-products, four new infinite dimensional Lie algebras are obtained.
In this paper, Lie super-bialgebra structures on a class of generalized super $W$-algebra $\mathfrak{L}$ are investigated. By proving the first cohomology group of $\mathfrak{L}$ with coefficients in its adjoint tensor module is trivial, namely, $H^1(\mathfrak{L},\mathfrak{L}\otimes {\mathfrak{L}})=0$, we obtain that all Lie super-bialgebra structures on $\mathfrak{L}$ are triangular coboundary.
In this paper we investigate Lie bialgebra structures on the twisted Heisenberg-Virasoro algebra. With the classifications of Lie bialgebra structures on the Virasoro algebra, we determined such structures on the twisted Heisenberg-Virasoro algebra. Moreover, some general and useful results are obtained. With our methods and results we also can easily to determine such structures on some Lie algebras related to the twisted Heisenberg-Virasoro algebra.
Matched pairs and double construction bialgebras of (transposed) Poisson 3-Lie algebras
本文利用匹配对研究 Poisson 3-李代数与转置 Poisson 3-李代数的双构造双代数，证明相关等价关系，并引入容许转置 Poisson 3-李代数以建立其双构造理论。
Verma modules over the $W$-algebra W(2,2) were considered by Zhang and Dong, while the Harish-Chandra modules and irreducible weight modules over the same algebra were classified by Liu and Zhu etc. In the present paper we shall investigate the Lie bialgebra structures on the referred algebra, which are shown to be triangular coboundary.
In this paper, we first introduce the notion of a 3-Hom-Lie bialgebra and prove that it is equivalent to a Manin triple of 3-Hom-Lie algebras. Also, we study the $\mathcal{O}$-operator and construct solutions of the 3-Lie classical Hom-Yang-Baxter equation interms of $\mathcal{O}$-operators and 3-Hom-pre-Lie algebras. Finally, we show that a 3-Hom-Lie algebra has a phase space if and only if it is sub-adjacent to a 3-Hom-pre-Lie algebra.
Motivated by the classical comatrix coalgebra, we introduce the concept of a Newtonian comatrix coalgebra. We construct an infinitesimal unitary bialgebra on a matrix algebra and a weighted infinitesimal unitary bialgebra on a non-commutative polynomial algebra, via two constructions of suitable coproducts. As a consequence, a Newtonian comatrix coalgebra is established. Furthermore, an infinitesimal unitary Hopf algebra, under the view of Aguiar, is constructed on a matrix algebra. By investigating the relationship between weighted infinitesimal bialgebras and pre-Lie algebras, we erect respe…
Reynolds Lie bialgebras
本文为Reynolds Lie代数建立双代数理论，引入二次Reynolds Lie代数、匹配对、Manin三元组及双代数并证明其等价性，通过Reynolds算子与相对Rota-Baxter算子等构造经典Yang-Baxter方程的解以生成Reynolds Lie双代数。
We propose a definition of multiplier infinitesimal bialgebra and a definition of derivator Lie bialgebra. We give some examples of these structures and prove that every bibalanced multiplier infinitesimal bialgebra gives rise to a multiplier Lie bialgebra.
$ω$-Lie bialgebras and $ω$-Yang-Baxter equation
本文引入了等价于Manin三元组与匹配对的乘法$ω$-李双代数，研究了$ω$-Yang-Baxter方程，并基于由左对称代数构造的$ω$-$\mathcal{O}$-算子给出了该方程的斜对称解。
Rota-Baxter operators of non-scalar weights, connections with coboundary Lie bialgebra structures
本文提出非标量权 Rota-Baxter 算子概念，揭示其与二次有限维 Lie 代数上拟三角 Lie 双代数结构的自然联系，并由其重新导出单 Lie 代数上的经典 Lie 双代数结果。
Primitive elements of a connected free bialgebra
本文利用滤过及 Cartier–Quillen–Milnor–Moore、Shirshov–Witt 定理，证明特征零域上分次连通自由双代数的本原元构成自由李代数。
Infinite-dimensional pre-Lie bialgebras via affinization of pre-Novikov bialgebras
本文通过 pre-Novikov 代数的仿射化构造 pre-Lie 代数，并将其推广到双代数情形，证明 pre-Novikov 双代数的仿射化给出无穷维完备 pre-Lie 双代数，同时由 pre-Novikov Yang-Baxter 方程的对称解构造 S-方程的对称完备解。
We introduce the new concept of braided Hom-Lie bialgebras which is a generalization of Sommerhäuser-Majid's braided Lie bialgebras and Yau's Hom-Lie bialgebras. Using this concept we give the unified product construction for Hom-Lie bialgebras which can be seen as a Hom-Lie version of Bespalov-Drabant's cocycle cross product bialgebras. Some special cases of unified products such as crossed product and matched pair of braided Hom-Lie bialgebras are investigated. As an application, we solve the Agore-Militaru extending problem for Hom-Lie bialgebras by using some non-abelian cohomology theory.…
Biquantization of the necklace Lie bialgebra
本文在Ginzburg等人工作的基础上，进一步构建了项链李双余代数的双量子化，拓展了Schedler之前关于该李双余代数的霍普夫代数构造。
Let $(\mathfrak{g}, [\cdot,\cdot], δ_\mathfrak{g})$ be a fixed Lie bialgebra, $E$ be a vector space containing $\mathfrak{g}$ as a subspace and $V$ be a complement of $\mathfrak{g}$ in $E$. A natural problem is that how to classify all Lie bialgebraic structures on $E$ such that $(\mathfrak{g}, [\cdot,\cdot], δ_\mathfrak{g})$ is a Lie sub-bialgebra up to an isomorphism of Lie bialgebras whose restriction on $\mathfrak{g}$ is the identity map. This problem is called the extending structures problem. In this paper, we introduce a general co-product on $E$, called the unified co-product of $(\mat…
Gel'fand-Dorfman bialgebra, which is both a Lie algebra and a Novikov algebra with some compatibility condition, appears in the study of Hamiltonian pairs in completely integrable systems and a class of special Lie conformal algebras called quadratic Lie conformal algebras. In this paper, we investigate the extending structures problem for Gel'fand-Dorfman bialgebras, which is equivalent to some extending structures problem of quadratic Lie conformal algebras. Explicitly, given a Gel'fand-Dorfman bialgebra $(A, \circ, [\cdot,\cdot])$, this problem asks that how to describe and classify all Gel…
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。
In this paper, we study Lie Rinehart bialgebras, the algebraic generalization of Lie bialgebroids. More precisely, we analyze the structure of Lie Rinehart bialgebras for crossed products induced by actions of Lie algebras on K[t].
In this paper, Lie bialgebra structures on the extended Schrodinger-Virasoro Lie algebra are classified. It is obtained that all the Lie bialgebra structures on L are triangular coboundary. As a by-product, it is derived that the first cohomology group is trivial.
In this paper, we first introduce the notion of Hom-left-symmetric conformal bialgebras and show some nontrivial examples. Also, we present construction methods of matched pairs of Hom-Lie conformal algebras and Hom-left-symmetric conformal algebras. Finally, we prove that a finite Hom-left-symmetric conformal bialgebra is free as a $\mathbb{C}[\partial]$-module is equivalent to a Hom-parakähler Lie conformal algebra. In particular, we investigate the coboundary Hom-left-symmetric conformal bialgebras.
Many features of classical Lie theory generalize to the broader context of algebras over Hopf operads. However, this idea remains largely to be developed systematically. Quasi-shuffle algebras provide for example an interesting illustration of these phenomena, but have not been investigated from this point of view.The notion of quasi-shuffle algebras can be traced back to the beginings of the theory of Rota--Baxter algebras, but was developed systematically only recently, starting essentially with Hoffman's work, that was motivated by multizeta values (MZVs) and featured their bialgebra struct…
In this paper, we study Hom-Lie bialgebras by a new notion of the dual representation of a representation of a Hom-Lie algebra. Motivated by the essential connection between Lie bialgebras and Manin triples, we introduce the notion of a Hom-Lie bialgebra with emphasis on its compatibility with a Manin triple of Hom-Lie algebras associated to a nondegenerate symmetric bilinear form satisfying a new invariance condition. With this notion, coboundary Hom-Lie bialgebras can be studied without a skew-symmetric condition of $r\in\mathfrak{g}\otimes \mathfrak{g}$, naturally leading to the classical H…
We study the Hom-type generalization of infinitesimal bialgebras, called infinitesimal Hom-bialgebras. In particular, we consider infinitesimal Hom-bialgebras arising from quivers, the sub-classes of coboundary and quasi-triangular infinitesimal Hom-bialgebras, the associative Hom-Yang-Baxter equation, and homological perturbation of the comultiplications in infinitesimal Hom-bialgebras. The relationships between infinitesimal Hom-bialgebras, Hom-Lie bialgebras, and the classical Hom-Yang-Baxter equation are also studied.
The aim of this paper is to extend the notion of bialgebra for Leibniz algebras (and Lie algebras) to $3$-Leibniz algebras (and $3$-Lie algebras) by use of the cohomology complex of $3$-Leibniz algebras. Also, some theorems about Leibniz bialgebras are extended and proved in the case of $3$-Leibniz bialgebras ($3$-Lie bialgebras). Moreover, a new theorem on the correspondence between $3$-Leibniz bialgebra and its associated Leibniz bialgebra is proved. Finally, some examples are discussed in detail.
Some constructions of antisymmetric infinitesimal bialgebras and their applications to Lie bialgebras
本文提出利用Zinbiel双代数和双结合双代数构造反对称无穷小双代数及Lie双代数的方法，并证明其保持拟三角、三角和可分解性质。
Affinization of dendriform $\md$-bialgebras, Lie bialgebras and solutions of classical Yang-Baxter equation
本文通过树形$\md$-双代数的仿射化方法，构造了李双代数并研究了其与经典Yang-Baxter方程解的关系。
We introduce the categories of infinitesimal Hopf modules and bimodules over an infinitesimal bialgebra. We show that they correspond to modules and bimodules over the infinitesimal version of the double. We show that there is a natural, but non-obvious way to construct a pre-Lie algebra from an arbitrary infinitesimal bialgebra and a dendriform algebra from a quasitriangular infinitesimal bialgebra. As consequences, we obtain a pre-Lie structure on the space of paths on an arbitrary quiver, and a striking dendriform structure on the space of endomorphisms of an arbitrary infinitesimal bialgeb…
The $n$-Lie bialgebras are studied. In Section 2, the $n$-Lie coalgebra with rank $r$ is defined, and the structure of it is discussed. In Section 3, the $n$-Lie bialgebra is introduced. A triple $(L, μ, Δ)$ is an $n$-Lie bialgebra if and only if $Δ$ is a conformal $1$-cocycle on the $n$-Lie algebra $L$ associated to $L$-modules $(L^{\otimes n}, ρ_s^μ)$, $1\leq s\leq n$, and the structure of $n$-Lie bialgebras is investigated by the structural constants. In Section 4, two-dimensional extension of finite dimensional $n$-Lie bialgebras are studied. For an $m$ dimensional $n$-Lie bialgebra $(L, μ…
We describe the Lie bialgebra structure on the Lie superalgebra sl(2,1) related to an r-matrix that cannot be obtained by a Belavin-Drinfeld type construction. This structure makes sl(2,1) into the Drinfeld double of a four-dimensional subalgebra.
Averaging pre-Lie bialgebras
本文研究了平均预李代数的表示及其匹配对、马宁三元组和双代数理论，证明三者在特定条件下等价，并通过引入二次罗塔-巴克斯预李代数上的平均算子构造平均预李双代数，进一步展示相对罗塔-巴克斯算子生成可接受的经典Yang-Baxter方程的对称解。
We introduce the notion of weak Lie 2-bialgebra. Roughly, a weak Lie 2-bialgebra is a pair of compatible 2-term $L_\infty$-algebra structures on a vector space and its dual. The compatibility condition is described in terms of the big bracket. We prove that (strict) Lie 2-bialgebras are in one-one correspondence with crossed modules of Lie bialgebras.
Let $ L $ be a Lie algebra over arbitrary field $ k $ with dim $ L $ =3 and dim $ L' $ =2. All solutions of constant classical Yang- Baxter equation (CYBE) in Lie algebra $ L $ are obtained and the necessary conditions which $ (L,[\ ],Δ_{r}, r) $ is a coboundary (or triangular) Lie bialgebra are given.
We generalize a result on the Heisenberg Lie algebra that gives restrictions to possible Lie bialgebra cobrackets on 2-step nilpotent algebras with some additional properties. For the class of 2-step nilpotent Lie algebras coming from graphs, we describe these extra properties in a very easy graph-combinatorial way. We exhibit applications for $\mathfrak f_n$, the free 2-step nilpotent Lie algebra.
Projective Banach Lie bialgebras, the projective Yang-Baxter equation and projective Poisson-Lie groups
本文引入射影Banach李双代数、射影Yang-Baxter方程及射影Poisson-Lie群，证明拟三角射影r-矩阵产生射影Banach李双代数，射影Poisson-Lie群微分具有该结构，并给出三角射影r-矩阵的等价描述。
In this paper, first we introduce the notion of quadratic Rota-Baxter Lie algebras of arbitrary weight, and show that there is a one-to-one correspondence between factorizable Lie bialgebras and quadratic Rota-Baxter Lie algebras of nonzero weight. Then we introduce the notions of matched pairs, bialgebras and Manin triples of Rota-Baxter Lie algebras of arbitrary weight, and show that Rota-Baxter Lie bialgebras, Manin triples of Rota-Baxter Lie algebras and certain matched pairs of Rota-Baxter Lie algebras are equivalent. The coadjoint representations and quadratic Rota-Baxter Lie algebras pl…
Manin triples associated to $n$-Lie bialgebras
该文围绕$n$-李双代数相关的Manin三元组展开研究，引入operad矩阵概念，通过其特殊情形导出局部上循环$n$-李双代数，还建立了$n$-李双代数偶对与$n$-李代数Manin三元组的一一对应。
All solutions of constant classical Yang-Baxter equation (CYBE) in Lie algebra $L$ with dim $L \le 3$ are obtained and the sufficient and necessary conditions which $(L, \hbox {[ ]}, Δ_r, r)$ is a coboundary (or triangular) Lie bialgebra are given. The strongly symmetric elements in $L\otimes L $ are found and they all are solutions of CYBE in $L$ with $dim L \le 3$.
A bialgebra theory for transposed Poisson algebras via anti-pre-Lie bialgebras and anti-pre-Lie-Poisson bialgebras
本文通过引入反预李双代数和反预李-泊松双代数，利用李代数上交换2-上循环的Manin三元组，为转置泊松代数建立了双代数理论，并研究了共边情形及相关结构。
The notions of transposed Hom-Poisson and Hom-pre-Lie Poisson algebras are introduced. Their bimodules and matched pairs are defined and the relevant properties and theorems are given. The notion of Manin triple of transposed Hom-Poisson algebras is introduced, and its equivalence to the transposed Hom-Poisson bialgebras is investigated. The notion of $\mathcal{O}$-operator is exploited to illustrate the relations existing between transposed Hom-Poisson and Hom-pre-Lie Poisson algebras.
In this paper, we study Lie 2-bialgebras, with special attention to coboundary ones, with the help of the cohomology theory of $L_\infty$-algebras with coefficients in $L_\infty$-modules. We construct examples of strict Lie 2-bialgebras from left-symmetric algebras and symplectic Lie algebras.
The purpose of this paper is to define an $α$-type cohomology, which we call $α$-type Chevalley-Eilenberg cohomology, for Hom-Lie algebras. We relate it to the known Chevalley-Eilenberg cohomology and provide explicit computations for some examples. Moreover, using this cohomology we study formal deformations of Hom-Lie algebras, where the bracket as well as the structure map $α$ are deformed. Furthermore, we provide a generalization of the Grand Crochet and study, in a particular case, the $α$-type cohomology for Hom-Lie bialgebras.
We approach the classification of Lie bialgebra structures on simple Lie algebras from the viewpoint of descent and non-abelian cohomology. We achieve a description of the problem in terms faithfully flat cohomology over an arbitrary ring over $\mathbb{Q}$, and solve it for Drinfeld-Jimbo Lie bialgebras over fields of characteristic zero. We consider the classification up to isomorphism, as opposed to equivalence, and treat split and non-split Lie algebras alike. We moreover give a new interpretation of scalar multiples of Lie bialgebras hitherto studied using twisted Belavin-Drinfeld cohomolo…
The classical Lie-Yamaguti Yang-Baxter equation and Lie-Yamaguti bialgebras
本文研究 Lie-Yamaguti 双代数，定义经典 Yang-Baxter 方程并建立其与相对 Rota-Baxter 算子的对应，同时证明匹配对、Manin 三元组和双重构造双代数等价。
This paper is devoted to a classification of topological Lie bialgebra structures on the Lie algebra $\mathfrak{g}[\![x]\!]$, where $ \mathfrak{g} $ is a finite-dimensional simple Lie algebra over an algebraically closed field $ F $ of characteristic $ 0 $. We introduce the notion of a topological Manin pair $(L, \mathfrak{g}[\![x]\!])$ and present their classification by relating them to trace extensions of \( F[\![x]\!] \). Then we recall the classification of topological doubles of Lie bialgebra structures on $\mathfrak{g}[\![x]\!]$ and view the latter as a special case of the classificatio…
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 algebra. Note that a skew-symmetric solution of mock-Lie Yang-Baxter equation gives a mock-Lie bialgebra. Finally, the notation of $\mathcal O$-operators are studied to construct skew-symmetric solution of mock-Lie Yang-Baxter equation.
Motivated by recent work on Hom-Lie algebras and the Hom-Yang-Baxter equation, we introduce a twisted generalization of the classical Yang-Baxter equation (CYBE), called the classical Hom-Yang-Baxter equation (CHYBE). We show how an arbitrary solution of the CYBE induces multiple infinite families of solutions of the CHYBE. We also introduce the closely related structure of Hom-Lie bialgebras, which generalize Drinfel'd's Lie bialgebras. In particular, we study the questions of duality and cobracket perturbation and the sub-classes of coboundary and quasi-triangular Hom-Lie bialgebras.
We investigate Lie bialgebra structures on simple Lie algebras of non-split type $A$. It turns out that there are several classes of such Lie bialgebra structures, and it is possible to classify some of them. The classification is obtained using Belavin--Drinfeld cohomology sets, which are introduced in the paper. Our description is particularly detailed over fields of cohomological dimension at most two, and is related to quaternion algebras and the Brauer group. We then extend the results to certain rational function fields over real closed fields via Pfister's theory of quadratic forms and …
In this paper, we study the structure of 3-Lie algebras with involutive derivations. We prove that if $A$ is an $m$-dimensional 3-Lie algebra with an involutive derivation $D$, then there exists a compatible 3-pre-Lie algebra $(A, \{ , , , \}_D)$ such that $A$ is the sub-adjacent 3-Lie algebra, and there is a local cocycle $3$-Lie bialgebraic structure on the $2m$-dimensional semi-direct product 3-Lie algebra $A\ltimes_{ad^*} A^*$, which is associated to the adjoint representation $(A, ad)$. By means of involutive derivations, the skew-symmetric solution of the 3-Lie classical Yang-Baxter equa…
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