Shouchuan Zhang
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.
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In this paper, we obtain a canonical factorization of basis elements in free Rota-Baxter algebras built on bracketed words. This canonical factorization is applied to give a coalgebra structure on the free Rota-Baxter algebras. Together with the Rota-Baxter algebra multiplication, this coproduct gives a bialgebra structure on the free Rota-Baxter algebra of rooted forests. When the weight of the Rota-Baxter algebra is zero, we further obtain a Hopf algebra structure.
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。
Quasi-triangular, triangular, factorizable anti-Leibniz bialgebras and anti-Leibniz Yang-Baxter equation
引入反Leibniz双代数及其与Manin triple和matched pair的等价性，提出反Leibniz Yang-Baxter方程并通过relative Rota-Baxter算子构造对称解，同时研究可分解结构与维数扩展。
Rota-Baxter operators and bialgebras go hand in hand in their applications, such as in the Connes-Kreimer approach to renormalization and the operator approach to the classical Yang-Baxter equation. We establish a bialgebra structure that is compatible with the Rota-Baxter operator, called the Rota-Baxter antisymmetric infinitesimal (ASI) bialgebra. This bialgebra is characterized by generalizations of matched pairs of algebras and double constructions of Frobenius algebras to the context of Rota-Baxter algebras. The study of the coboundary case leads to an enrichment of the associative Yang-B…
Noncommutative pre-Poisson bialgebras and relative Rota-Baxter operators
本文研究了非交换前泊松双代数的结构与性质，引入了相对罗塔-巴克斯特算子，并建立了二次罗塔-巴克斯特非交换前泊松代数与可因子化非交换前泊松双代数的一一对应关系。
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 $dim L \le 3$.
We give a construction of Rota-Baxter coalgebras from Hopf module coalgebras and also derive the structures of the pre-Lie coalgebras via Rota-Baxter coalgebras of different weight. Finally, the notion of Rota-Baxter bialgebra is introduced and some examples are provided.
The aim of this paper is first to introduce and study Rota-Baxter cosystems and bisystems as generalization of Rota-Baxter coalgebras and bialgebras, respectively, with various examples. The second purpose is to provide an alternative definition of covariant bialgebras via coderivations. Furthermore, we consider coquasitriangular covariant bialgebras which are generalization of coquasitriangular infinitesimal bialgebras, coassociative Yang-Baxter pairs, coassociative Yang-Baxter equation, double coalgebras and dendriform coalgebras. We study some of their properties and relationships with Rota…
Quasi-triangular Novikov bialgebras and related bialgebra structures
本文提出拟三角Novikov双代数，由对称部分不变的Novikov Yang-Baxter方程解构造，并证明其与二次Rota-Baxter Novikov代数一一对应，且保持Lie双代数的相应性质。
Zinbiel bialgebras, relative Rota-Baxter operators and the related Yang-Baxter Equation
本文引入Zinbiel双代数概念，证明其与匹配对和Manin三元组等价，并研究余边界情形引出经典Yang-Baxter方程类比，进而定义拟三角和可分解Zinbiel双代数，建立二次Rota-Baxter Zinbiel代数与可分解双代数的一一对应。
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…
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…
Differential perm bialgebras, differential perm Yang-Baxter equation, and their applications in related bialgebra structures
本文引入微分 perm 双代数，通过 Manin 三元组和匹配对刻画，并利用对称解和 O-算子建立三角结构，进而构造微分无穷小双代数和 Novikov 双代数，揭示其与相关双代数结构的联系。
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。
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 $gl_2(\mathbb C)$.
Quasi-triangular dual pre-Poisson bialgebras and its connection with Poisson bialgebras
本文研究了准三角双预李代数与预李代数之间的关系，证明了可因子双预李代数与二次Rota-Baxter李代数的对应关系，并提出了一种构造无限维双预李代数的方法。
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.
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.
We describe $L_\infty$-algebras governing homotopy relative Rota-Baxter Lie algebras and triangular $L_\infty$-bialgebras, and establish a map between them. Our formulas are based on a functorial approach to Voronov's higher derived brackets construction which is of independent interest.
Averaging antisymmetric infinitesimal bialgebra and perm bialgebras
本文将反反对称无穷小双代数推广至平均代数，提出平均反反对称无穷小双代数理论，通过引入O-算子等工具构造Yang-Baxter方程的解，并建立其与对称平均Frobenius代数的一一对应，最后应用于置换双代数。
$ω$-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。
The notion of a modified Rota-Baxter algebra comes from the combination of those of a Rota-Baxter algebra and a modified Yang-Baxter equation. In this paper, we first construct free modified Rota-Baxter algebras. We then equip a free modified Rota-Baxter algebra with a bialgebra structure by a cocycle construction. Under the assumption that the generating algebra is a connected bialgebra, we further equip the free modified Rota-Baxter algebra with a Hopf algebra structure.
Quasi-triangular, factorizable anti-dendriform bialgebras and relative Rota-Baxter operators
本文引入了准三角形反树形双代数的概念，其可通过对称部分不变的AD-YBE解诱导。可分解的反树形双代数导致底层反树形代数的分解。此外，引入带权重的相对罗塔-巴克斯算子以表征此类解。最后，将可分解的反树形双代数解释为二次罗塔-巴克斯反树形代数。
Anti-pre-Poisson bialgebras and relative Rota-Baxter operators
本文引入反预泊松双代数，证明其与二次反预泊松代数及泊松代数匹配对等价，通过APP-YBE研究余边界情形，并建立二次Rota-Baxter反预泊松代数与可分解反预泊松双代数的一一对应。
Symmetric Rota-Baxter systems and applications
本文提出对称罗塔-巴克斯ASI双系统，通过反称解构造该系统，并推广多种已知结构。
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。
We establish a bialgebra theory for anti-flexible algebras in this paper. We introduce the notion of an anti-flexible bialgebra which is equivalent to a Manin triple of anti-flexible algebras. The study of a special case of anti-flexible bialgebras leads to the introduction of anti-flexible Yang-Baxter equation in an anti-flexible algebra which is an analogue of the classical Yang-Baxter equation in a Lie algebra or the associative Yang-Baxter equation in an associative algebra. It is a unexpected consequence that both the anti-flexible Yang-Baxter equation and the associative Yang-Baxter equa…
We equip a matrix algebra with a weighted infinitesimal unitary bialgebraic structure, via a construction of a suitable coproduct. Furthermore, an infinitesimal unitary Hopf algebra, under the view of Aguiar, is constructed on a matrix algebra. By exploring the relationship between weighted infinitesimal bialgebras and pre-Lie algebras, we construct a pre-Lie algebraic structure and then a new Lie algebraic structure on a matrix algebra. We also introduce the weighted associative Yang-Baxter equations (AYBEs) and obtain the relationship between solutions of weighted AYBEs and weighted infinite…
We study a twisted version of the Yang-Baxter Equation, called the Hom-Yang-Baxter Equation (HYBE), which is motivated by Hom-Lie algebras. Three classes of solutions of the HYBE are constructed, one from Hom-Lie algebras and the others from Drinfeld's (dual) quasi-triangular bialgebras. Each solution of the HYBE can be extended to operators that satisfy the braid relations. Assuming an invertibility condition, these operators give a representation of the braid group.
Leibniz conformal bialgebras and the classical Leibniz conformal Yang-Baxter equation
提出莱布尼茨共形双代数概念，通过匹配对和共形Manin三元组刻画，引入经典莱布尼茨共形杨-巴克斯特方程，其对称解构造双代数，并建立与Novikov双代数的联系。
Quasi-triangular, factorizable Leibniz bialgebras and relative Rota-Baxter operators
本文通过引入拟三角Leibniz双代数，利用带权相对Rota-Baxter算子刻画经典Leibniz Yang-Baxter方程的解，并建立斜对称二次Rota-Baxter Leibniz代数与三角及可分解Leibniz双代数之间的对应关系。
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。
Manin triples, bialgebras and Yang-Baxter equation of $A_3$-associative algebras
本文发展了$A_3$-结合代数的双代数理论，通过引入Manin三元组和双代数并证明其等价性，定义了$A_3$-结合Yang-Baxter方程，并利用相对Rota-Baxter算子构造了该方程的斜对称解。
Bialgebra theory, the Yang-Baxter equation and relative Rota-Baxter operators for diassociative algebras
本文发展了双结合代数的双代数理论，引入双结合Yang-Baxter方程及其对称解与双结合双代数的关系，并利用相对Rota-Baxter算子构造解，进而将双结合代数与Leibniz代数等结构的联系提升至双代数层面。
Leibniz-dendriform bialgebras and relative Rota-Baxter operators
本文引入 Leibniz-dendriform 双代数，建立其与相空间、匹配对的等价关系，并通过 LD-YBE 的解构造拟三角与可分解双代数，最终建立二次 Rota-Baxter 代数与可分解双代数的一一对应。
In this paper, first we introduce the notion of a Leibniz bialgebra and show that matched pairs of Leibniz algebras, Manin triples of Leibniz algebras and Leibniz bialgebras are equivalent. Then we introduce the notion of a (relative) Rota-Baxter operator on a Leibniz algebra and construct the graded Lie algebra that characterizes relative Rota-Baxter operators as Maurer-Cartan elements. By these structures and the twisting theory of twilled Leibniz algebras, we further define the classical Leibniz Yang-Baxter equation, classical Leibniz r-matrices and triangular Leibniz bialgebras. Finally, w…
In this paper, we derive pre-anti-flexible algebras structures in term of zero weight's Rota-Baxter operators defined on anti-flexible algebras, view pre-anti-flexible algebras as a splitting of anti-flexible algebras, introduce the notion of pre-anti-flexible bialgebras and establish equivalences among matched pair of anti-flexible algebras, matched pair of pre-anti-flexible algebras and pre-anti-flexible bialgebras. Investigation on special class of pre-anti-flexible bialgebras leads to the establishment of the pre-anti-flexible Yang-Baxter equation. Both dual bimodules of pre-anti-flexible …
Affinization of Zinbiel bialgebras and pre-Poisson bialgebras, infinite-dimensional Poisson bialgebras
本文通过pre-Poisson代数的affinization构造无限维Poisson bialgebras，并研究了相关代数结构的对应关系。
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 introduce a Hom-type generalization of quantum groups, called quasi-triangular Hom-bialgebras. They are non-associative and non-coassociative analogues of Drinfel'd's quasi-triangular bialgebras, in which the non-(co)associativity is controlled by a twisting map. A family of quasi-triangular Hom-bialgebras can be constructed from any quasi-triangular bialgebra, such as Drinfel'd's quantum enveloping algebras. Each quasi-triangular Hom-bialgebra comes with a solution of the quantum Hom-Yang-Baxter equation, which is a non-associative version of the quantum Yang-Baxter equation. Solutions of …
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。
New types of bialgebras arising from the Hopf equation (pentagonal equation) are introduced and studied. They will play from the Hopf equation the same role as the co-quasitriangular do from the quantum Yang Baxter equation.
A class of non-associative and non-coassociative generalizations of cobraided bialgebras, called cobraided Hom-bialgebras, is introduced. The non-(co)associativity in a cobraided Hom-bialgebra is controlled by a twisting map. Several methods for constructing cobraided Hom-bialgebras are given. In particular, Hom-type generalizations of FRT quantum groups, including quantum matrices and related quantum groups, are obtained. Each cobraided Hom-bialgebra comes with solutions of the operator quantum Hom-Yang-Baxter equations, which are twisted analogues of the operator form of the quantum Yang-Bax…
Bialgebra theory for nearly associative algebras and $LR$-algebras: equivalence, characterization, and $LR$-Yang-Baxter Equation
该研究构建近结合代数与$LR$-代数的双代数理论，证明二者双代数等价，引入近结合$L$-代数并刻画相关双代数，还提出余边界近结合双代数作为研究YBE的框架。
Quasi-triangular and factorizable dendriform D-bialgebras
本文提出拟三角与可因子化树形D-双代数，证明树形双重具有可因子化结构，并建立其与二次Rota-Baxter树形代数的一一对应及表示同构。
Quasi-triangular and factorizable Poisson bialgebras
本文引入拟三角和可分解泊松双代数的概念，证明泊松双代数的Drinfeld经典对偶自然具有可分解结构，并建立可分解泊松双代数与非零权二次Rota-Baxter泊松代数的一一对应。
Species of Rota-Baxter algebras by rooted trees, twisted bialgebras and Fock functors
本文引入简单角度装饰森林构造自由Rota-Baxter物种，建立其上的扭曲双代数结构，并通过Fock函子给出自由Rota-Baxter代数双代数结构的新证明。
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。
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。
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。
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.
Skew-symmetric Frobenius dialgebras, diassociative bialgebras and Yang-Baxter equations
本文系统建立了双结合双代数理论，引入双结合双代数概念，证明其等价于Frobenius双代数的双重构造和双代数的匹配对，并给出斜对称Rota-Baxter Frobenius双代数与三角/可分解双代数的联系。
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 introduce the notion of associative (BiHom-)Yang-Baxter pair of weight $(λ,γ)$ which can provide the solution to the double curved Rota-Baxter (BiHom-)system. Equivalent characterizations of (quasitriangular) covariant BiHom-bialgebra are given. We also prove that associative BiHom-Yang-Baxter equation of weight $-1$ can be obtained by the unitary quasitriangular covariant BiHom-bialgebra. At last, we present two approaches to construct (BiHom-)pre-Lie modules from Rota-Baxter (BiHom-)paired modules.
Braided cohomology of quasi-triangular bialgebras and braided Morita invariance
引入线性幺半范畴中编织余代数的编织上链复形和编织上同调，证明在有限维条件下拟三角双代数的编织上链复形是编织Morita不变量。
The binary product in the 2-category of triangular bialgebras and twisted morphisms
本文在2-范畴框架下，通过引入适当的二元积概念并利用张量积双代数上的扭曲描述，将双代数张量积的二元积性质推广到三角双代数及其扭曲态射，并应用于解释三角双代数的扭曲张量积。
In this paper, we first introduce the notion of a phase space of a Poisson algebra, and show that a Poisson algebra has a phase space if and only if it is sub-adjacent to a pre-Poisson algebra. Moreover, we introduce the notion of Manin triples of pre-Poisson algebras and show that there is a one-to-one correspondence between Manin triples of pre-Poisson algebras and phase spaces of Poisson algebras. Then we introduce the notion of pre-Poisson bialgebras, which is equivalent to Manin triples of pre-Poisson algebras. We study coboundary pre-Poisson bialgebras, which leads to an analogue of the …
Anti-pre-Novikov algebras, quasi-triangular and factorizable anti-pre-Novikov bialgebras
本文提出反 pre-Novikov 代数与双代数框架，通过反 O-算子、APN-YBE 及其斜对称解构造余边界双代数，并刻画拟三角与可因子化情形，建立二次 Rota-Baxter 代数与可因子化双代数的一一对应。
For any algebra two families of coloured Yang-Baxter operators are constructed, thus producing solutions to the two-parameter quantum Yang-Baxter equation. An open problem about a system of functional equations is stated. The matrix forms of these operators for two and three dimensional algebras are computed. A FRT bialgebra for one of these families is presented. Solutions for the one-parameter quantum Yang-Baxter equation are derived and a Yang-Baxter system constructed.
Construction of diassociative bialgebras from antisymmetric infinitesimal bialgebras and related algebra structures
本文证明反对称无穷小双代数与二次 perm 代数的张量积具有双结合双代数结构，且其结构性质与原反对称无穷小双代数对应，还明确了几类双代数及辛代数间的密切关系。
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