A. Eghbali, A. Rezaei-Aghdam
We obtain the classical r-matrices of two and three dimensional Lie super-bialgebras. We thus classify all two and three dimensional coboundary Lie super-bialgebras and their types (triangular, quasi-triangular, or factorable). Using the Sklyanin superbracket, we then obtain the super Poisson structures on the related Poisson-Lie supergroups.
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Using adjoint representation of Lie superalgebras, we obtain the matrix form of super-Jacobi and mixed super-Jacobi identities of Lie superbialgebras. By direct calculations of these identities, and use of automorphism supergroups of two and three dimensional Lie superalgebras, we obtain and classify all two and three dimensional Lie superbialgebras.
Classification of real three-dimensional Lie bialgebras and their Poisson-Lie groups
得到了三维实李双代数的经典r-矩阵。这样，所有的三维实上边界李双代数及其类型（三角形，拟三角形或可因子分解）进行分类。然后利用Sklyanin括号，得到了相关Poisson-Lie群上的Poisson结构.
Classification of Six-dimensional Real Nilpotent Lie Bialgebras of Symplectic Type and their Poisson-Lie Groups
该研究对辛型六维实幂零李双代数进行分类，获取相关泊松-李群的泊松结构，并得到以泊松-李群为相空间、其对偶李群为对称群的新可积哈密顿系统。
Classification of four dimensional real Lie bialgebras of symplectic type and their Poisson-Lie groups
研究分类辛型四维实 Lie 双代数，并给出相应经典 r-矩阵、Poisson-Lie 群的 Poisson 结构及新可积模型。
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 …
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…
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。
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…
Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras
本文将ASI双代数推广至微分代数提出微分ASI双代数，通过双重构造与匹配对进行刻画，引入相干导子作为等价结构，并将其应用于扩展Poisson双代数理论。
The purpose of this paper is to study twistings of Poisson algebras or bialgebras, coPoisson algebras or bialgebras and star-products. We con- sider Hom-algebraic structures generalizing classical algebraic structures by twisting the identities by a linear self map. We summarize the results on Hom-Poisson algebras and introduce Hom-coPoisson algebras and bialge- bras. We show that there exists a duality between Hom-Poisson bialgebras and Hom-coPoisson bialgebras. A relationship between enveloping Hom- algebras endowed with Hom-coPoisson structures and corresponding Hom- Lie bialgebra structure…
(1+1) Schrodinger Lie bialgebras and their Poisson-Lie groups
导出了（1 + 1）维中心扩张Schrodinger代数的所有李双代数结构，并证明了它们都是上边界型的。因此，由于它们都来自一个经典的r-矩阵，薛定谔泊松-李群的完整的家庭可以通过Sklyanin括号推出。研究了谐振子、广义Galilei和gl（2）李双代数在薛定谔分类中的所有可能嵌入。作为应用，构造了Schrodinger代数的新量子（霍普夫代数）变形，包括相应的量子泛R-矩阵.
Poisson bialgebras by deformations-to-quasiclassical limits
本文将泊松代数的形变-拟经典极限过程推广到双代数层面，推导了泊松双代数的对应关系并通过相干导子阐释，是李双代数量子化过程的无穷小版本。
Relative Poisson bialgebras and Frobenius Jacobi algebras
本文通过研究相对泊松代数的Manin三元组和相对泊松双代数，引入相对泊松Yang-Baxter方程及其反对称解的构造方法，并由此给出弗罗贝尼乌斯-雅可比代数的新构造途径。
$D$-bialgebras, dendrification and embeddings into AWB of almost Poisson algebras
本文引入了几乎Poisson Drinfel'd bialgebras（$D$-bialgebras）并建立了匹配对、Manin三元组与该结构的等价性，同时定义了新的代数结构，并展示了如何通过平均算子将几乎Poisson代数嵌入到AWB中。
A new approach to the bialgebra theory for relative Poisson algebras
提出相对 PCA 双代数作为相对 Poisson 代数双代数理论的新途径，引入 RPCA-YBE 及相关算子概念，并证明其可由微分 ASI 双代数自然构造。
Lie bialgebras of complex type and associated Poisson Lie groups
在这项工作中，我们研究了一类特殊的李双代数所产生的埃尔米特结构的李代数，使度量是ad-不变的。我们称它们为复型李双代数。这就产生了泊松李群G，其对应的群G* 是复李群。证明了李代数$\mathfrak{g}$上具有ad-不变度量的厄米特结构在双李代数${\mathcal D}\mathfrak{g}= \mathfrak{g}\oplus\mathfrak{g}^*$上诱导出一个与${\mathcal D}\mathfrak{g}$上中性签名的标准ad-不变度量相同的厄米特结构.我们给出了如何从一个2（n-2）维的李双代数开始构造一个2n维的复型李双代数。这使我们能够确定所有的可解李代数的维数$\leq 6$承认一个埃尔米特结构与ad不变度量。我们展示了一些例子，在4和6维，包括两个单参数的家庭，在那里我们确定相关的单连通李群上的李泊松结构，也得到他们的辛叶。
Reflections on Rota-Baxter Lie algebras, the classical reflection equation and Poisson homogeneous spaces
本文引入权为λ的二次和相对Rota-Baxter李代数上的反射概念，证明其生成经典反射方程的解，并由此导出Poisson李群与Poisson齐次空间。
Quantization of SL(2,R)^* as Bialgebra
利用Kontsevich的积，我们将Poisson-李群SL（2，R）^* 作为双代数.余积是来自群结构的余积的变形。得到的双代数结构同构于量子泛包络代数U_hsl（2，R）.
Balanced infinitesimal bialgebras, double Poisson gebras and pre-Calabi-Yau algebras
本文研究平衡无穷小双代数的 properad，通过 Koszul 对偶与余棒构造揭示其同伦双重 Poisson 代数和预 Calabi-Yau 代数之间的结构关系，并证明弯曲版本等价。
Integrable deformations of Rikitake systems, Lie bialgebras and bi-Hamiltonian structures
本文通过形变Lie-Poisson Hamilton结构并利用Poisson-Lie群与Lie双代数的对应关系，构造了Rikitake系统及其耦合系统的可积形变，并给出保持双Hamilton结构的形变方案，建立了Lie双代数与可积动力系统之间的新联系。
We construct integrable Hamiltonian systems with Lie bialgebras $({\bf g} , {\bf \tilde{g}})$ of the bi-symplectic type for which the Poisson-Lie groups ${\bf G}$ play the role of the phase spaces, and their dual Lie groups ${\bf {\tilde {G}}}$ play the role of the symmetry groups of the systems. We give the new transformations to exchange the role of phase spaces and symmetry groups and obtain the relations between integrals of motions of these integrable systems. Finally, we give some examples of real four-dimensional Lie bialgebras of bi-symplectic type.
Quantum homogeneous spaces are noncommutative spaces with quantum group covariance. Their semiclassical counterparts are Poisson homogeneous spaces, which are quotient manifolds of Lie groups $M=G/H$ equipped with an additional Poisson structure $π$ which is compatible with a Poisson-Lie structure $Π$ on $G$. Since the infinitesimal version of $Π$ defines a unique Lie bialgebra structure $δ$ on the Lie algebra $\frak g=\mbox{Lie}(G)$, we exploit the idea of Lie bialgebra duality in order to study the notion of complementary dual homogeneous space $M^\perp=G^\ast/H^\perp$ of a given homogeneous…
Super Star Products and Quantum Superalgebras
证明了Poisson李超群上的超星积导致了该李超群的李超代数的超量子包络代数上的量子超代数（三角Hopf超代数）的结构，等价的超星积生成同构的量子超代数.
Bidynamical Poisson Groupoids
给出了动力Poisson群胚、经典动力杨-巴克斯特方程和李拟双代数之间的关系。证明了双李拟双代数类与双Poisson广群胚类之间的对应关系。当存在约化分解$\g=ł\oplus\m$时，我们给出了经典动力学杨-巴克斯特方程（经典动力学$\ell$ -矩阵）的一个明确的、解析的和规范的等变解，并证明了任何其它的等变解与规范解形式上是规范等价的.我们还描述了相应的Poisson广群的对偶，得到了动力Poisson广群具有动力对偶的充要条件是存在约化分解$\g=ł\oplus\m$ .
Poisson homogeneous spaces of Poisson 2-groups
本文研究 2-群的齐性空间，并在 Poisson 2-群背景下推广发展了基于 Lie 双代数 Dirac 结构的 Drinfeld 定理。
A Manin triples (D; g; ~ g) is a bialgebra (g; ~ g which don't intersect each others and a direct sum of this bialgebra D = g ~ g). If the corresponding Lie groups have a Poisson structure, they are called Poisson-Lie groups. A Poisson-Lie sigma models is an action (3.13) calculated by a Poisson vector eld matrix. [3] have deduced the extremal eld which minimize the action of this models, which gives the motion equation (3.19). We calculate here the action and the equations of motion for some 6-dimensionals Manin triples and we give a general formula for each 4-dimensional Manin triples. The 6…
Universal constructions in homotopical algebra
本文利用李图代数有效积分理论，为多种双代数类型的形变与同伦理论提供普适构造，给出Deligne群胚的同伦描述并推广了operadic代数形变公式。
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…
Monomial bialgebras
本文通过单个QYBE解生成无限解族，并研究其在李双代数和霍普夫代数上的准三角结构及相关泊松-李结构。
A systematic computational approach for the explicit construction of any quantum Hopf algebra (U_z(g),Δ_z) starting from the Lie bialgebra (g,δ) that gives the first-order deformation of the coproduct map Δ_z is presented. The procedure is based on the fact that any quantum algebra can be viewed as the quantization of the unique Poisson-Lie structure (G^\ast,Λ_g) on the dual group G^\ast, which is obtained by exponentiating the Lie algebra g^\ast defined by the dual map δ^\ast. From this perspective, the coproduct for U_z(g) is just the pullback of the group law for G^\ast, and the Poisson ana…
Poisson geometrical symmetries associated to non-commutative formal diffeomorphisms
设G是特征为零的域k中所有系数为x的形式幂级数的群，F[G]是它的函数代数. C. Brouder和A. Frabetti通过F[G]的“去阿贝尔化”过程引入了一个非交换、非余交换的分次霍普夫代数H，即取F[G]的类似表示作为代数，但去掉了交换性约束.本文用一般的方法给出了H的四个单参数形变，它们是半经典极限为Poisson几何对称的量子群，如Poisson群或李双代数，即两个量子函数代数和两个量子泛包络代数.特别地，两个Poisson群是G的扩展，同构于原代数Poisson簇，但不同构于原代数群。这种分析很容易扩展到一个类似性质的霍普夫代数的hudge族，从而产生了一种方法，以一种自然的方式将一些经典的几何对称性（如泊松群和李双代数）与这种“广义对称性”联系起来：现在的情况就像是一般情况的一个最简单的玩具模型。
Left-symmetric Bialgebras and An Analogue of the Classical Yang-Baxter Equation
我们引入了左对称双代数的概念，它是李双代数概念的类似物。证明了一个左对称双代数等价于一个辛李代数，该辛李代数分解为两个拉格朗日子代数的基础向量空间的直和。后者在数学物理中被称为parakähler李代数或李代数的相空间。我们引入并研究了余边界左对称双代数，我们的研究导致我们称之为“$S$ -方程”，这是一个类似于经典的杨-巴克斯特方程。从某种意义上说，与左对称代数相联系的$S$ -方程揭示了乘积的左对称性。我们证明了$S$ -方程的对称解给出了一个parakähler李代数。我们还表明，这样的解决方案对应于一个特定的运营商称为“${\cal O}$ -运营商”的对称部分，而反对称解的经典杨-巴克斯特方程对应的反对称部分的${\cal O}$ -运营商。从而提供了一种由${\cal O}$ -算子构造$S$ -方程对称解的方法。此外，通过比较左对称双代数和李双代数，我们观察到它们之间有一个明确的类似物，特别是parakähler李群在这个意义上对应于Poisson-李群。
By direct calculations of matrix form of super Jacobi and mixed super Jacobi identities which are obtained from adjoint representation, and using the automorphism supergroup of the gl(1|1) Lie superalgebra, we determine and classify all gl(1|1) Lie superbialgebras. Then, by calculating their classical r-matrices, the gl(1j1) coboundary Lie superbialgebras and their types (triangular, quasi-triangular or factorizable) are determined, furthermore in this way super Poisson structures on the GL(1|1) Lie supergroup are obtained. Also, we classify Drinfeld superdoubles based on the gl(1|1) as a theo…
All real three dimensional Poisson-Lie groups are explicitly constructed and fully classified under group automorphisms by making use of their one-to-one correspondence with the complete classification of real three-dimensional Lie bialgebras given in [X. Gomez, J. Math. Phys. vol. 41, p. 4939 (2000)]. Many of these 3D Poisson-Lie groups are non-coboundary structures, whose Poisson brackets are given here for the first time. Casimir functions for all three-dimensional PL groups are given, and some features of several PL structures are commented.
Quasi-Lie bialgebroids and twisted Poisson manifolds
我们开发了一个理论的准李双代数胚使用同调的方法。这个概念是拟李双代数的推广，以及最近出现在弦理论背景下的具有3-形式背景的扭曲泊松结构，并且由Ševera和Weinstein使用不同的方法研究。
The chiral WZNW phase space as a quasi-Poisson space
它被解释为手征WZNW相空间是一个准泊松空间相对于'规范'李准双代数，这是经典的极限Drinfeld的准Hopf变形的通用包络代数。这证明了Alekseev和Kosmann-Schwarzbach最近引入的准Poisson-Lie对称性的概念，并且还允许我们推广以前在这个例子中考虑的某些动力学扭曲。
From entangled codipterous coalgebras to coassociative manifolds
我们从余结合余代数、双代数、霍普夫代数、Poisson代数、Loday和M. Ronco和探索的概念coassociative流形。
Using technique of wheeled props we establish a correspondence between the homotopy theory of unimodular Lie 1-bialgebras and the famous Batalin-Vilkovisky formalism. Solutions of the so called quantum master equation satisfying certain boundary conditions are proven to be in 1-1 correspondence with representations of a wheeled dg prop which, on the one hand, is isomorphic to the cobar construction of the prop of unimodular Lie 1-bialgebras and, on the other hand, is quasi-isomorphic to the dg wheeled prop of unimodular Poisson structures. These results allow us to apply properadic methods for…
We study {\em right-invariant (resp., left-invariant) Poisson quasi-Nijenhuis structures} on a Lie group $G$ and introduce their infinitesimal counterpart, the so-called {\em r-qn structures} on the corresponding Lie algebra $\mathfrak g$. We investigate the procedure of the classification of such structures on the Lie algebras and then for clarity of our results we classify, up to a natural equivalence, all $r$-$qn$ structures on two types of four-dimensional real Lie algebras. We mention some remarks on the relation between $r$-$qn$ structures and the generalized complex structures on the Li…
A general approach to noncommutative spaces from Poisson homogeneous spaces: Applications to (A)dS and Poincaré
本文提出了一般方法，通过将余边界李双代数导出的各向同性泊松齐性空间量子化，构造具有量子群不变性的非交换空间，并应用于(3+1)维(A)dS和庞加莱，得到κ-Minkowski、κ-(A)dS时空及6维类时κ-Poincaré空间。
Poisson Diffeomorphism Groups
我们在${\Bbb R}^n$的形式同态群上构造了一类共边界Poisson-Lie结构.等价地，这些结果给出了${\Bbb R}^n$上形式向量场的李代数$W_n$上的一类共边界三角李双代数结构。我们猜想，这类占所有这样的共边界结构。所构造的Poisson-Lie同构群的自然作用在${\Bbb R}^n$上诱导出大量相容的Poisson结构，从而使其成为Poisson齐性空间.此外，Poisson-李群$FDiff({\Bbb R}^m)\times FDiff({\Bbb R}^n)$的左右作用在光滑映射的无限喷流空间$J^{\infty}({\Bbb R}^m,{\Bbb R}^n)$ ${\Bbb R}^m\to {\Bbb R}^n$上诱导出相容Poisson结构类，这使得它对于该作用也是Poisson齐性空间.对这些结构进行分类的初步步骤。
Using the concept of Jacobi-Lie group and Jacobi-Lie bialgebra, we generalize the definition of Poisson-Lie symmetry to Jacobi-Lie symmetry. In this regard, we generalize the concept of Poisson-Lie T-duality to Jacobi-Lie T-duality and present Jacobi-Lie T-dual sigma models on Lie groups, which have Jacobi-Lie symmetry. Using this symmetry, new cases of duality appear and some examples are given. This generalization may provide insights to understand the quantum features of Poisson-Lie T-duality, in a more satisfactory way.
The equations of motion of a super non-Abelian T-dual sigma model on the Lie supergroup $(C^1_1+A)$ in the curved background are explicitly solved by the super Poisson-Lie T-duality. To find the solution of the flat model we use the transformation of supercoordinates, transforming the metric into a constant one, which is shown to be a supercanonical transformation. Then, using the super Poisson-Lie T-duality transformations and the dual decomposition of elements of Drinfel'd superdouble, the solution of the equations of motion for the dual sigma model is obtained. The general form of the dilat…
All possible Poisson-Lie (PL) structures on the 3D real Lie group generated by a dilation and two commuting translations are obtained. Its classification is fully performed by relating these PL groups with the corresponding Lie bialgebra structures on the corresponding "book" Lie algebra. By construction, all these Poisson structures are quadratic Poisson-Hopf algebras for which the group multiplication is a Poisson map. In contrast to the case of simple Lie groups, it turns out that most of the PL structures on the book group are non-coboundary ones. Moreover, from the viewpoint of Poisson dy…
Universal solutions to deformation quantization problems can be conveniently classified by the cohomology of suitable graph complexes. In particular, the deformation quantizations of (finite-dimensional) Poisson manifolds and Lie bialgebras are characterised by an action of the Grothendieck-Teichmüller group via one-colored directed and oriented graphs, respectively. In this note, we study the action of multi-oriented graph complexes on Lie bialgebroids and their "quasi" generalisations. Using results due to T. Willwacher and M. Zivković on the cohomology of (multi)-oriented graphs, we show th…
Super Chern Simons Theory and Flat Super Connections on a Torus
研究拓扑为${\bf R}\times §$的流形上的超Chern-Simons理论的模空间，其中$§$是紧致曲面.模空间是$§$上的平坦超联络模规范变换的模空间，并详细研究了$§$是超环且超群是$OSp(m|2n)$的情形.玻色子模空间由最大玻色子子群$O(m)\times Sp(2n)$的平坦联络确定，而费米子模只出现在玻色子模空间的特殊部分，这是由与holonomy的玻色子部分相关联的矩阵的消失行列式确定的。如果CS超群是超李代数的指数，则费米子模只出现在超伴随表示的费米子-费米子块中的生成元为零的玻色子完整性中。由超Chern-Simons理论导出了模空间上的一个自然辛结构，它由完整性的Poisson括号代数决定.我们表明，齐次连接的辛结构是有用的理解的模空间和holonomy代数的性质，我们说明了这一点上的例子$OSp(1|2)$超群。
Shifted Poisson structures on higher Chevalley-Eilenberg algebras
本文研究了在更高Chevalley-Eilenberg代数上构造n-移位泊松结构的方法，并将其与准李双代数和不变对称张量联系起来，揭示了其与更高量子群的半经典数据关系。
The first part of this paper is devoted to the theory of Poisson-Lie groups in the Banach setting. Our starting point is the straightforward adaptation of the notion of Manin triples to the Banach context. The difference with the finite-dimensional case lies in the fact that a duality pairing between two non-reflexive Banach spaces is necessary weak (as opposed to a strong pairing where one Banach space can be identified with the dual space of the other). The notion of generalized Banach Poisson manifolds introduced in this paper is compatible with weak duality pairings between the tangent spa…
In this paper, first we modify the definition of a Hom-Lie algebroid introduced by Laurent-Gengoux and Teles and give its equivalent dual description. Many results that parallel to Lie algebroids are given. In particular, we give the notion of a Hom-Poisson manifold and show that there is a Hom-Lie algebroid structure on the pullback of the cotangent bundle of a Hom-Poisson manifold. Then we give the notion of a Hom-Lie bialgebroid, which is a natural generalization of a purely Hom-Lie bialgebra and a Lie bialgebroid. We show that the base manifold of a Hom-Lie bialgebroid is a Hom-Poisson man…
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