Shouchuan Zhang
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 $dim L \le 3$.
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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。
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.
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。
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。
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。
We introduce the concept of {sigma, tau}-Rota-Baxter operator, as a twisted version of a Rota-Baxter operator of weight zero. We show how to obtain a certain {sigma, tau}-Rota-Baxter operator from a solution of the associative (Bi)Hom-Yang-Baxter equation, and, in a compatible way, a Hom-pre-Lie algebra from an infinitesimal Hom-bialgebra.
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。
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.
We establish a bialgebra structure on Rota-Baxter Lie algebras following the Manin triple approach to Lie bialgebras. Explicitly, Rota-Baxter Lie bialgebras are characterized by generalizing matched pairs of Lie algebras and Manin triples of Lie algebras to the context of Rota-Baxter Lie algebras. The coboundary case leads to the introduction of the admissible classical Yang-Baxter equation (CYBE) in Rota-Baxter Lie algebras, for which the antisymmetric solutions give rise to Rota-Baxter Lie bialgebras. The notions of $\mathcal{O}$-operators on Rota-Baxter Lie algebras and Rota-Baxter pre-Lie …
A generalisation of the notion of a Rota-Baxter operator is proposed. This generalisation consists of two operators acting on an associative algebra and satisfying equations similar to the Rota-Baxter equation. Rota-Baxter operators of any weights and twisted Rota-Baxter operators are solutions of the proposed system. It is shown that dendriform algebra structures of a particular kind are equivalent to Rota-Baxter systems. It is shown further that a Rota-Baxter system induces a weak peudotwistor [F. Panaite & F. Van Oystaeyen, Twisted algebras, twisted bialgebras and Rota-Baxter operators, arX…
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。
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.
In this paper, we introduce 3-Hom-Lie bialgebras whose compatibility conditions between the multiplication and comultiplication are given by local cocycle conditions. We study a twisted 3-ary version of the Yang-Baxter Equation, called the 3-Lie classical HomYang-Baxter Equation (3-Lie CHYBE), which is a general form of 3-Lie classical YangBaxter Equation studied in [2] and prove that the bialgebras induced by the solutions of 3-Lie CHYBE induce the coboundary local cocycle 3-Hom-Lie bialgebras.
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.
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。
A new construction method of $R$-matrix is given. Let $A$ be a C$^{*}$-bialgebra with a comultiplication $Δ$. For two states $ω$ and $ψ$ of $A$ which satisfy certain conditions, we construct a unitary $R$-matrix $R(ω,ψ)$ of the C$^{*}$-bialgebra $(A,Δ)$ on the tensor product of GNS representation spaces associated with $ω$ and $ψ$. The set $\{R(ω,ψ):ω,ψ\}$ satisfies a kind of Yang-Baxter equation. Furthermore, we show a nontrivial example of such $R$-matrices for a non-quasi-cocommutative C$^{*}$-bialgebra.
This paper is devoted to algebro-geometric study of infinite dimensional Lie bialgebras, which arise from solutions of the classical Yang-Baxter equation. We regard trigonometric solutions of this equation as twists of the standard Lie bialgebra cobracket on an appropriate affine Lie algebra and work out the corresponding theory of Manin triples, putting it into an algebro-geometric context. As a consequence of this approach, we prove that any trigonometric solution of the classical Yang-Baxter equation arises from an appropriate algebro-geometric datum. The developed theory is illustrated by …
This paper studies two types of 3-Lie bialgebras whose compatibility conditions between the multiplication and comultiplication are given by local cocycles and double constructions respectively, and are therefore called the local cocycle 3-Lie bialgebra and the double construction 3-Lie bialgebra. They can be regarded as suitable extensions of the well-known Lie bialgebra in the context of 3-Lie algebras, in two different directions. The local cocycle 3-Lie bialgebra is introduced to extend the connection between Lie bialgebras and the classical Yang-Baxter equation. Its relationship with a te…
For a Lie algebra with Lie bracket got by taking commutators in a nonunital associative algebra L, let T(L) be the vector space of tensors over L equipped with the Ito Hopf algebra structure derived from the associative multiplication in L. We show a necessary and sufficient condition that the double product integral satisfy the quantum Yang-Baxter equation over T(L). We construct a quantisation of an arbitrary quasitriangular Lie bialgebra structure on L in the unital associative subalgebra of T(L)[[h]] consisting of formal power series whose zero order coefficient lies in the space S(L) of s…
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…
For a set theoretical solution of the Yang-Baxter equation $(X,σ)$, we define a d.g. bialgebra $B=B(X,σ)$, containing the semigroup algebra $A=k\{X\}/\langle xy=zt : σ(x,y)=(z,t)\rangle$, such that $k\otimes_A B\otimes_Ak$ and $\mathrm{Hom}_{A-A}(B,k)$ are respectively the homology and cohomology complexes computing biquandle homology and cohomology defined in \cite{CJKS} and other generalizations of cohomology of rack-quanlde case (for example defined in \cite{CES}). This algebraic structure allow us to show the existence of an associative product in the cohomology of biquandles, and a compar…
In this paper, we give some low-dimensional examples of local cocycle 3-Lie bialgebras and double construction 3-Lie bialgebras which were introduced in the study of the classical Yang-Baxter equation and Manin triples for 3-Lie algebras. We give an explicit and practical formula to compute the skew-symmetric solutions of the 3-Lie classical Yang-Baxter equation (CYBE). As an illustration, we obtain all skew-symmetric solutions of the 3-Lie CYBE in complex 3-Lie algebras of dimension 3 and 4 and then the induced local cocycle 3-Lie bialgebras. On the other hand, we classify the double construc…
We introduce the notion of a quadri-bialgebra, which gives a bialgebra theory for the quadri-algebra introduced by Aguiar and Loday. We show that a quadri-bialgebra is equivalent to a Manin triple of dendriform algebras associated to a nondegenerate 2-cocycle, and to a Manin triple of quadri-algebras associated to a nondegenerate invariant bilinear form. Quadri-bialgebras also come from a variation of the classical Yang-Baxter equation, called the $Q$-equations. Moreover, quadri-bialgebras fit into the framework of construction of Rota-Baxter operators and Nijenhuis operators on the double spa…
Monoidal structure on Rota-Baxter algebras via augmentation, and free cocycle bialgebras
本文研究增广Rota–Baxter代数的普通张量积上的自然Rota–Baxter算子，构造了非对称幺半范畴，明确其与Rota–Baxter上循环双代数的对应关系，还得到带相容Hopf结构的自由上循环双代数。
Pre-Lie 2-bialgebras and 2-grade classical Yang-Baxter equations
本文建立严格pre-Lie 2-代数的Manin三元组、匹配对与双代数等价理论，引入2阶经典Yang-Baxter方程以构造pre-Lie 2-双代数和仿Kähler严格Lie 2-代数。
In this paper, we introduce the notion of a noncommutative Poisson bialgebra, and establish the equivalence between matched pairs, Manin triples and noncommutative Poisson bialgebras. Using quasi-representations and the corresponding cohomology theory of noncommutative Poisson algebras, we study coboundary noncommutative Poisson bialgebras which leads to the introduction of the Poisson Yang-Baxter equation. A skew-symmetric solution of the Poisson Yang-Baxter equation naturally gives a (coboundary) noncommutative Poisson bialgebra. Rota-Baxter operators, more generally O-operators on noncommut…
We examine classes of quantum algebras emerging from involutive, non-degenerate set-theoretic solutions of the Yang-Baxter equation and their q-analogues. After providing some universal results on quasi-bialgebras and admissible Drinfeld twists we show that the quantum algebras produced from set-theoretic solutions and their q-analogues are in fact quasi-triangular quasi-bialgebras. Specific illustrative examples compatible with our generic findings are worked out. In the q-deformed case of set-theoretic solutions we also construct admissible Drinfeld twists similar to the set-theoretic ones, …
The exotic bialgebra S03, defined by a solution of the Yang-Baxter equation, which is not a deformation of the trivial, is considered. Its FRT dual algebra $s03_F$ is studied. The Baxterisation of the dual algebra is given in two different parametrisations. The finite-dimensional representations of $s03_F$ are considered. Diagonalisations of the braid matrices are used to yield remarkable insights concerning representations of the L-algebra and to formulate the fusion of finite-dimensional representations. Possible applications are considered, in particular, an exotic eight-vertex model and an…
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。
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。
We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie bialgebra. We prove that a left-symmetric bialgebra is equivalent to a symplectic Lie algebra with a decomposition into a direct sum of the underlying vector spaces of two Lagrangian subalgebras. The latter is called a parakähler Lie algebra or a phase space of a Lie algebra in mathematical physics. We introduce and study coboundary left-symmetric bialgebras and our study leads to what we call "$S$-equation", which is an analogue of the classical Yang-Baxter equation. In a certain sense, the $S$-equatio…
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.
An admissible Poisson algebra (or briefly, an adm-Poisson algebra) gives an equivalent presentation with only one operation for a Poisson algebra. We establish a bialgebra theory for adm-Poisson algebras independently and systematically, including but beyond the corresponding results on Poisson bialgebras given in [27]. Explicitly, we introduce the notion of adm-Poisson bialgebras which are equivalent to Manin triples of adm-Poisson algebras as well as Poisson bialgebras. The direct correspondence between adm-Poisson bialgebras with one comultiplication and Poisson bialgebras with one cocommut…
Let $M_{*}({\bf C})$ denote the C$^{*}$-algebra defined as the direct sum of all matrix algebras $\{M_{n}({\bf C}):n\geq 1\}$. It is known that $M_{*}({\bf C})$ has a non-cocommutative comultiplication $Δ_φ$. We show that the C$^{*}$-bialgebra $(M_{*}({\bf C}),Δ_φ)$ has a universal $R$-matrix $R$ such that the quasi-cocommutative C$^{*}$-bialgebra $(M_{*}({\bf C}),Δ_φ,R)$ is triangular.
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.
Nijenhuis operators and mock-Lie bialgebras
本文通过推广匹配对、Manin三元组等概念，刻画了Nijenhuis mock-Lie双代数，并研究了其形式形变、阿贝尔扩张及与mock-Lie-Yang-Baxter方程的联系。
Yang-Baxter bialgebras, as previously introduced by the authors, are shown to arise from a double crossproduct construction applied to the bialgebra R T T = T T R, E T = T E R, Δ(T) = T \hat{\otimes} T, Δ(E) = E \hat{\otimes} T + 1 \hat{\otimes} E and its skew dual, with R being a numerical matrix solution of the Yang-Baxter equation. It is further shown that a set of relations generalizing q-Serre ones in the Drinfeld-Jimbo algebras U_q(g) can be naturally imposed on Yang-Baxter algebras from the requirement of non-degeneracy of the pairing.
We introduce a new type of categorical object called a \emph{hom-tensor category} and show that it provides the appropriate setting for modules over an arbitrary hom-bialgebra. Next we introduce the notion of \emph{hom-braided category} and show that this is the right setting for modules over quasitriangular hom-bialgebras. We also show how the hom-Yang-Baxter equation fits into this framework and how the category of Yetter-Drinfeld modules over a hom-bialgebra with bijective structure map can be organized as a hom-braided category. Finally we prove that, under certain conditions, one can obta…
For an Yang Baxter operator we show that a bialgebra homomorphism from a free braided tensor bialgebra to a cofree braided shuffle bialgebra is the Woronowicz braided antisymmetrizer. A cofree braided shuffle bialgebra is a braided generalization of a cofree shuffle bialgebra introduced by Sweedler. Its graded dual bialgebra is a free braided tensor bialgebra.
It was proved by Montaner and Zelmanov that up to classical twisting Lie bialgebra structures on $\mathfrak{g}[u]$ fall into four classes. Here $\mathfrak{g}$ is a simple complex finite-dimensional Lie algebra. It turns out that classical twists within one of these four classes are in a one-to-one correspondence with the so-called quasi-trigonometric solutions of the classical Yang-Baxter equation. In this paper we give a complete list of the quasi-trigonometric solutions in terms of sub-diagrams of the certain Dynkin diagrams related to $\mathfrak{g}$. We also explain how to quantize the corr…
A homotopy analogue of the notion of a triangular Lie bialgebra is proposed with a goal of extending the basic notions of theory of quantum groups to the context of homotopy algebras and, in particular, introducing a homotopical generalization of the notion of a quantum group, or quantum$_\infty$-group.
In the classification of solutions of the Yang--Baxter equation, there are solutions that are not deformations of the trivial solution (essentially the identity). We consider the algebras defined by these solutions, and the corresponding dual algebras. We then study the representations of the latter. We are also interested in the Baxterisation of these $R$-matrices and in the corresponding quantum planes.
Conformal Yang-Baxter equation on $\mathrm{Cur}(\mathrm{sl}_2(\mathbb{C}))$
针对J. Liberati 2008年提出的共形经典杨-巴克斯特方程（CCYBE），本文给出当前李共形代数$\text{Cur}(\text{sl}_2(\text{C}))$上CCYBE及其弱版本的所有解，为共形李双代数结构研究提供基础。
Sufficient conditions for an invertible two-tensor $F$ to relate two solutions to the Yang-Baxter equation via the transformation $R\to F^{-1}_{21} R F$ are formulated. Those conditions include relations arising from twisting of certain quasitriangular bialgebras.
The quantum bialgebra related to the Baxter's eight-vertex R-matrix is found as a quantum deformation of the Lie algebra of sl(2)-valued automorphic functions on a complex torus.
In this paper, we introduce a new definition of a hom-Lie bialgebra, which is equivalent to a Manin triple of hom-Lie algebras. We also introduce a notion of an $\mathcal O$-operator and then construct solutions of the classical hom-Yang-Baxter equation in terms of $\mathcal O$-operators and hom-left-symmetric algebras.
Gaussian solutions to the Yang--Baxter equation and their twists
研究常数量子杨 - 巴克斯特方程的高斯解，用FRT构造生成双代数，通过张扭转等对高斯解扭转，得到不一定是高斯解的杨 - 巴克斯特方程的解。
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