Dominique Manchon
An extended version of a series of lectures given at Bogota in december 2002. It consists in a presentation of some aspects of Connes' and Kreimer's work on renormalization in the context of general connected Hopf algebras, in particular Birkhoff decomposition and, in the graded case, the scattering-type formula.
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In this review we discuss the relevance of the Hochschild cohomology of renormalization Hopf algebras for local quantum field theories and their equations of motion.
Hopf Algebras, Renormalization and Noncommutative Geometry
我们探讨了与QFT重整化相关联的Hopf代数和与叶理横向指数理论的NCG计算相关联的Hopf代数之间的关系。
Noncommutative renormalization for massless QED
本文从D.克莱默对于QED，我们描述了一个既不是交换的也不是余交换的重整化的Hopf代数。我们得到明确的重整化公式的电子和光子的传播，真空极化和电子的自能，这是相当于Zimmermann的森林公式的总和的所有费曼图在一个给定的相互作用顺序。然后，我们将由理论的耦合常数定义的Connes-Kreimer映射推广到QED，定义了一个非交换的非正规化Hopf代数，然后证明了电荷的重整化定义了这个Hopf代数与QED的重整化Hopf代数之间的同态。最后，我们表明，戴森公式的重整化的电子和光子传播可以给出一个非交换（如矩阵值）的形式。
We show how the Hopf algebra of rooted trees encodes the combinatorics of Epstein-Glaser renormalization and coordinate space renormalization in general. In particular we prove that the Epstein-Glaser time-ordered products can be obtained from the Hopf algebra by suitable Feynman rules, mapping trees to operator-valued distributions. Twisting the antipode with a renormalization map formally solves the Epstein-Glaser recursion and provides local counterterms due to the Hochschild 1-closedness of the grafting operator $B_+$.
This masters thesis reviews the algebraic formulation of renormalization using Hopf algebras as pioneered by Dirk Kreimer and applies it to a toy model of quantum field theory given through iterated insertions of a single primitive divergence into itself. Using this example in a subtraction scheme, we exhibit the renormalized Feynman rules to yield Hopf algebra morphisms into the Hopf algebra of polynomials and as a consequence study the emergence of the renormalization group in connection with combinatorial Dyson-Schwinger equations. In particular we relate the perturbative expansion of the a…
In this paper we describe the right-sided combinatorial Hopf structure of three Hopf algebras appearing in the context of renormalization in quantum field theory: the non-commutative version of the Faà di Bruno Hopf algebra, the non-commutative version of the charge renormalization Hopf algebra on planar binary trees for quantum electrodynamics, and the non-commutative version of the Pinter renormalization Hopf algebra on any bosonic field. We also describe two general ways to define the associative product in such Hopf algebras, the first one by recursion, and the second one by grafting and s…
Renormalization automated by Hopf algebra
最近表明，量子场论的重整化是由装饰根树的Hopf代数组织的，其余积确定需要减法的发散，其对极实现了这一点。我们在几行递归符号代码中自动化了这个过程，它为任何费曼图提供了一个有限的重整化表达式。因此，我们验证了表示的算子乘积展开，推广陈引理的迭代积分。图的森林结构需要一个独特的原始subdivergence的子集提供了一个表示的Hopf代数${\cal H}_R$的未装饰的根树。我们的未修饰的Hopf代数程序被设计用于处理24，213，878 BPHZ对7，813个图的重整化的贡献，最多有12个循环。我们考虑了10个模型，每个模型有9个重整化方案。这两个最简单的模型揭示了Connes和Moscovici的子代数的一个显著特征，对应于同构群的Hopf代数${\cal H}_T$的交换部分：它为费曼图分配了那些从最小减法方案的反项中删除zeta值的权重。我们设计了一个快速算法，这些权重，其平方和置换因子，给合理的counterterterms。
We determine finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero, whose Hopf coradical is isomorphic to a non-pointed basic Hopf algebra of dimension $24$ and the infinitesimal braidings are indecomposable objects. In particular, we obtain families of new finite-dimensional Hopf algebras without the dual Chevalley property.
We classify finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose Hopf coradcial is isomorphic to the smallest non-pointed basic Hopf algebra, under the assumption that the diagrams are strictly graded. In particular, we obtain some new Nichols algebras of non-diagonal type and new finite-dimensional Hopf algebras without the dual Chevalley property.
We report on the Hopf algebraic description of renormalization theory of quantum electrodynamics. The Ward-Takahashi identities are implemented as linear relations on the (commutative) Hopf algebra of Feynman graphs of QED. Compatibility of these relations with the Hopf algebra structure is the mathematical formulation of the physical fact that WT-identities are compatible with renormalization. As a result, the counterterms and the renormalized Feynman amplitudes automatically satisfy the WT-identities, which leads in particular to the well-known identity $Z_1=Z_2$.
On the Hopf algebra structure of perturbative quantum field theories
我们表明，重整化的过程是一个自然的方式的Hopf代数结构。这揭示了最近提出的结和重整化理论之间的联系。
Let $m$ a positive integer, not divisible by 2,3,5,7. We generalize the classification of basic quasi-Hopf algebras over cyclic groups of prime order given in \cite{EG3} to the case of cyclic groups of order $m$. To this end, we introduce a family of non-semisimple radically graded quasi-Hopf algebras $A(H,s)$, constructed as subalgebras of Hopf algebras twisted by a quasi-Hopf twist, which are not twist equivalent to Hopf algebras. Any basic quasi-Hopf algebra over a cyclic group of order $m$ is either semisimple, or is twist equivalent to a Hopf algebra or a quasi-Hopf algebra of type $A(H,s…
Towards cohomology of renormalization: bigrading the combinatorial Hopf algebra of rooted trees
量子场论的重正化扭曲了根树的非余对易Hopf代数的对极，该代数被无限多的本原发散所修饰。无装饰根树的Hopf代数，${\cal H}_R$，由一个单一的原始分歧产生，解决了Hochschild上同调的一个普遍问题。它有两个非平凡闭Hopf子代数：纯梯形图的余交换子代数${\cal H}_{\rm ladder}$和非交换几何的Connes-Moscovici非余交换子代数${\cal H}_{\rm CM}$。这三个霍普夫代数允许由$n$、节点数和指定连续度的索引$k$进行的二阶化。在每一种情况下，我们使用相关的余积的迭代来计算具有适度值$n$和$k$的子空间的维数，并推断出余数的简单生成过程。${\cal H}_{\rm ladder}$的结果是熟悉的分区理论，而那些为${\cal H}_{\rm CM}$涉及新的分区变换。最美丽的是${\cal H}_R$的双级，它是三个中最大的。多亏了斯隆的{ \tt superseeker}，我们发现它能饱和所有可能的不等式。我们证明了这一点，通过使用通用的Hochschild封闭的一个上循环$B_+$，其中插头一组分歧到另一个，并通过推广的概念，自然增长超出所涉及的康纳斯-莫斯科维奇的情况。我们要强调的是，处理现实量子场论的无限装饰集是一个更大的挑战。
Normal Coordinates and Primitive Elements in the Hopf Algebra of Renormalization
本文在Connes和Kreimer分析有根树的Hopf代数时引入了无穷维群$G$上的法坐标。我们研究了代数的原始元素，并表明它们是由一个简单的应用程序的逆庞加莱引理，给出了一个封闭的左不变1-形式$G$。对于特殊情况下的梯形原语，我们发现第二个描述，将它们与通常的产品的幂级数的Hopf代数泛函。这两种方法都表明梯形基元是由舒尔多项式给出的。本文还讨论了对偶李代数的下中心级数在重整化过程中的相关性，从而引出了$k$ -重复性的自然概念，并证明了它与文献中已有的概念是等价的.
Integrable Renormalization II: the general case
我们推广了我们在早期工作中得到的结果。根梯树的余交换情形被推广到（装饰）根树的满Hopf代数。对于目标空间为Rota-Baxter型的Hopf代数特征标，分别利用Rota-Baxter二重构造和Atkinson定理，导出了重整化理论的Birkhoff分解.我们还概述了通过装饰根树的费曼图的Hopf代数的扩展。
In continuation of the article \cite{L} we classify all radically graded basic Hopf algebras of tame type over an algebraically closed field of characteristic 0.
Multiplier Hopf algebroids are algebraic versions of quantum groupoids that generalize Hopf algebroids to the non-unital case and weak (multiplier) Hopf algebras to non-separable base algebras. The main structure maps of a multiplier Hopf algebroid are a left and a right comultiplication. We show that bijectivity of two associated canonical maps is equivalent to the existence of an antipode, discuss invertibility of the antipode, and present some examples and special cases.
Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Pade-Borel resummation
对无链自能彩虹求和，以获得对反常维数的贡献是很容易的。恢复无彩虹的自我能量链也很容易。驯服原始自能发散的所有可能的嵌套和链式的组合爆炸是一个要求高得多的问题。我们解决它的余积$Δ$，对极S，和未装饰根树的Hopf代数的分次算子Y。重要算子是$S\star Y$，其星形积受$Δ$影响。我们执行30-loop Padé-Borel重建463 020 146 037 416 130 934 BPHZ减法在Yukawa理论中，在时空维度d=4，并在一个三价标量理论，在d=6，遇到残留物的$S\star Y$，涉及素数与多达60位数。即使有一个非常大的汤川耦合，g=30，重新计算的精度是显着的;一个31圈的计算表明，它是$10^{-8}$阶。
We contruct here the Hopf algebra structure underlying the process of renormalization of non-commutative quantum field theory.
We give a natural monomorphism from the necklace Lie coalgebra, defined for any quiver, to Connes and Kreimer's Lie coalgebra of trees, and extend this to a map from a certain quiver-theoretic Hopf algebra to Connes and Kreimer's renormalization Hopf algebra, as well as to pre-Lie versions. These results are direct analogues of Turaev's results in 2004, by replacing algebras of loops on surfaces with algebras of paths on quivers. We also factor the morphism through an algebra of chord diagrams and explain the geometric version. We then explain how all of the Hopf algebras are uniquely determin…
This is a contribution to the classification of finite-dimensional Hopf algebras over an algebraically closed field $\Bbbk$ of characteristic 0. Concretely, we show that a finite-dimensional Hopf algebra whose Hopf coradical is basic is a lifting of a Nichols algebra of a semisimple Yetter-Drinfeld module and we explain how to classify Nichols algebras of this kind. We provide along the way new examples of Nichols algebras and Hopf algebras with finite Gelfand-Kirillov dimension.
It is shown in math.QA/0310253 that a finite dimensional quasi-Hopf algebra over the complex numbers with radical of codimension 2 is twist equivalent to a Nichols Hopf algebra, or to a lifting of one of four special quasi-Hopf algebras of dimensions 2, 8, 8, and 32. The purpose of this paper is to construct new finite dimensional basic quasi-Hopf algebras A(q) of dimension n^3, n>2, parametrized by primitive roots of unity q of order n^2, with radical of codimension n, which generalize the construction of the basic quasi-Hopf algebras of dimension 8 given in math.QA/0310253. These quasi-Hopf …
The Hopf algebra structure underlying Feynman diagrams which governs the process of renormalization in perturbative quantum field theory is reviewed. Recent progress is briefly summarized with an emphasis on further directions of research.
Wormhole Renormalization: The gravitational path integral, holography, and a gauge group for topology change
本文借鉴微扰重整化方法自下而上研究因子化悖论，在二维拓扑引力模型中引入“反虫洞”抵消破坏全息对偶因子化的虫洞贡献，并基于对称性Hopf代数提出规范群重组不同拓扑时空贡献。
Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem
本文给出了我们的结果的一个完备的自含证明，证明了量子场论中的重整化是基于Riemann-Hilbert问题的有限值提取的一般数学过程的一个特例.我们将首先证明，对于任何量子场论，费曼图的组合学产生一个Hopf代数$\Hc$，它作为一个代数是交换的。它是一个李代数$\ud G$的赋形代数的对偶Hopf代数，其基由单粒子不可约Feynman图标记.两个这样的图的李括号是从一个图在另一个图中的插入来计算的，反之亦然。相应的李群$G$是$\Hc$的特征标群。然后，我们将证明，使用维数正则化，裸（未重整化）理论产生一个循环$$ \g (z) \in G \qquad z \in C $$，其中$C$是围绕时空的整数维$D$的一个复维数的小圆圈。我们的主要结果是：重整化理论就是$\g$的Birkhoff分解的全纯部分$\g_+$在$z = D$上的求值.我们开始分析组$G$，并表明，它是一个半直接产品的一个容易理解的阿贝尔群由一个高度非平凡的群密切联系在一起的群的同构。后一组的分析以及重整化群和反常维数的解释是我们第二篇论文的内容，总标题相同。
Renormalization in quantum field theory and the Riemann-Hilbert problem II: the $β$-function, diffeomorphisms and the renormalization group
我们在第一部分（hep-th/9912092）中证明了给定QFT中Feynman图的Hopf代数${\cal H}$是复无限维李群$G$上的坐标代数，并且通过在$\ve=0$上计算由维数正则化提供的环$γ(\ve)\in G$的Riemann-Hilbert分解$γ_-(\ve)^{-1}γ_+(\ve)$的全纯部分$γ_+(\ve)$，从未重整化的理论获得重整化理论。本文证明了群$G$自然地作用于理论的无量纲耦合常数的复空间$X$上。更准确地说，当被看作形式幂级数时，有效耦合常数的公式$g_0=gZ_1Z_3^{-3/2}$确实定义了形式同态群上坐标的Hopf代数与Hopf代数${\cal H}$之间的Hopf代数同态。这允许首先直接读出，而不使用组$G$，裸耦合常数和从非重整化有效耦合常数的Riemann-Hilbert分解中的重整化耦合常数，被视为一个正式的环同态。这表明重整化与维数正则化参数$\ve$的黎曼球面上的非线性复丛理论密切相关。它还允许解除这两个重整化群和$β$ -功能的渐近缩放组$G$。这利用了黎曼-希尔伯特分解的全部功率以及质量单位变化下$γ_-(\ve)$的不变性。这不仅从概念上证明了重整化群的存在性，而且给出了在$G$群中关于最小减反项的全高极结构的散射公式.
Basic concepts of ternary Hopf algebras
三元半群、群和代数的理论用抽象箭头语言重新表述。然后利用反向箭头定义了三元余乘、双代数和霍普夫代数，并研究了它们的性质。的主要属性“二进制派生”被认为是详细的。给出了Post定理的共类比形式。结果表明，存在3种类型的三元余结合，3种类型的三元计数器和2种类型的三元对极。文中还列举了一些实例。
QED Hopf algebras on planar binary trees
本文描述了用于重整化量子电动力学Feynman传播子的平面二叉树上的霍普夫代数，以及描述重整化过程的余作用。这两种结构都与霍普夫代数的某些半直余积有关。
We demonstrate that the fundamental algebraic structure underlying the Connes-Kreimer Hopf algebra -- the insertion pre-Lie structure on graphs -- corresponds directly to the canonical pre-Lie structure of polynomial vector fields. Using this fact, we construct a Hopf algebra built from tensors that is isomorphic to a version of the Connes-Kreimer Hopf algebra that first appeared in the perturbative renormalization of quantum field theories.
Non-commutative Hopf algebra of formal diffeomorphisms
本文的研究对象是两个霍普夫代数，它们是两个不同的形式幂级数群的非交换模拟.第一个群是可逆级数的乘法集合，而第二个群是形式反同态的合成集合。引入这些霍普夫代数的动机来自于对非交换系数形式级数的研究。具有非交换系数的可逆级数仍然构成一个群，我们将相应的新的非交换霍普夫代数解释为由群的坐标环给出的自然霍普夫代数的替代，它具有在系数代数中是函子的优点.对于具有非交换系数的形式复同态，这种解释失败了，因为在这种情况下复合不再是结合的。然而，我们证明了对于对偶非交换代数，存在一个自然的余结合余积定义了一个非交换霍普夫代数。此外，我们给出了一个明确的公式的对极，这是一个非交换版本的拉格朗日反演公式，我们表明，其系数有关的平面二叉树。然后我们将这些结果推广到以前的霍普夫代数的半直余积和多元级数。最后，我们还讨论了形式级数的非交换霍普夫代数与由量子场论中的重整化所激发的组合Hopf代数的重整化霍普夫代数代数以及双代数上的双张量代数所给出的重整化函子之间的关系.
Motivated by the so-called H-cell reduction theorems, we investigate certain classes of bicategories which have only one H-cell apart from possibly the identity. We show that H_0-simple quasi fiab bicategories with unique H-cell H_0 are fusion categories. We further study two classes of non-semisimple quasi-fiab bicategories with a single H-cell apart from the identity. The first is $\cH_A$, indexed by a finite-dimensional radically graded basic Hopf algebra A, and the second is $\cG_A$, consisting of symmetric projective A-A-bimodules. We show that $\cH_A$ can be viewed as a 1-full subbicateg…
Hopf algebras in noncommutative geometry
我们给出了一个介绍性的调查使用的Hopf代数在几个问题的非交换几何。主要的例子，根树的霍普夫代数，是一个分级的，连接的霍普夫代数所产生的普遍建设。我们展示了它的关系代数的横向微分算子引入Connes和Moscovici为了计算一个局部指数公式在循环上同调，和几个Hopf代数定义的Connes和Kreimer简化组合的扰动重整化。我们解释了如何从作用于它的Hopf代数的循环上同调中得到一个Hopf模代数的特征类.最后，我们讨论了非交换球面流形的理论，并说明了它们是如何作为某些紧量子群的齐性空间而产生的.
We introduce and investigate the basic properties of an involutory (dual) quasi-Hopf algebra. We also study the representations of an involutory quasi-Hopf algebra and prove that an involutory dual quasi-Hopf algebra with non-zero integral is cosemisimple.
In this paper we investigate pointed Hopf algebras via quiver methods. We classify all possible Hopf structures arising from minimal Hopf quivers, namely basic cycles and the linear chain. This provides full local structure information for general pointed Hopf algebras.
``Bonsai'' Hopf algebras, introduced here, are generalizations of Connes-Kreimer Hopf algebras, which are motivated by Feynman diagrams and renormalization. We show that we can find operad structure on the set of bonsais. We introduce a new differential on these bonsai Hopf algebras, which is inspired by the tree differential. The cohomologies of these are computed here, and the relationship of this differential with the appending operation $*$ of Connes-Kreimer Hopf algebras is investigated.
Renormalization flows for 1D mixed states and a quantum Goursat lemma
研究非手性二维拓扑有序模型边界态的重整化过程，初始数据经扰动和粗粒化，流可用代数描述，迭代由卷积幂控制，得到新不动点并给出物理解释，还对有限群代数等分类，证明了量子古尔萨引理。
We develop some basic homological theory of hopfological algebra as defined by Khovanov. Several homological properties in hopfological algebra analogous to those of usual homological theory of DG algebras are obtained.
Relating the Connes-Kreimer and Grossman-Larson Hopf algebras built on rooted trees
我们发现了建立在有根树上的两个霍普夫代数之间的一个关系。第一个是Connes-Kreimer Hopf代数H_R，它描述了量子场论中的某种重整化;第二个是Grossman-Larson霍普夫代数A，它是十年前由于某些“微分”和组合的原因而引入的。粗略地说，这两个霍普夫代数之间存在着对偶关系.受Connes和Kreimer为H_R引入的类似算子的启发，我们研究了A上的两个自然算子。
We introduce quasi-Hopf $*$-algebras i.e. quasi-Hopf algebras equipped with a conjugation (star) operation. The definition of quasi-Hopf $*$-algebras proposed ensures that the class of quasi-Hopf $*$-algebras is closed under twisting and additionally, that any Hopf $*$-algebra becomes a quasi-Hopf $*$-algebra via twisting. The basic properties of these algebras are developed. The relationship between the antipode and star structure is investigated. Quasi-triangular quasi-Hopf $*$-algebras are introduced and studied.
A Hopf algebra on nonplanar binary forests
本文研究非平面二叉树生成的多项式代数，通过定义禁止同时切断任意顶点两个子节点的余乘积，构建霍普夫代数结构，并揭示其与布伦德霍普夫代数及预李运算的联系。
Hopf algebras with the dual Chevalley property of finite corepresentation type
本文研究了具有有限共表示类型的Hopf代数，证明其与对偶Chevalley性质、coNakayama性质及链接有向图结构之间的等价关系。
Lessons from Quantum Field Theory - Hopf Algebras and Spacetime Geometries
讨论了Hopf代数在量子场论研究中的重要性。特别是，我们将考虑重整化的霍普夫代数，其对极原来是一个概念理解的减法程序的关键。然后，我们将描述这个或密切相关的Hopf代数在其他数学领域，如叶状，龙格库塔方法，迭代积分和多重zeta值的几个occupational。我们强调的统一作用，布彻组，发现在研究数值积分的常微分方程，在QFT中发挥。
We construct a Hopf algebra structure on the space of specified Feynman graphs of a quantum field theory. We introduce a convolution product and a semigroup of characters of this Hopf algebra with values in some suitable commutative algebra taking momenta into account. We then implement the renormalization described by A. Connes and D. Kreimer and the Birkhoff decomposition for two renormalization schemes: the minimal subtraction scheme and the Taylor expansion scheme.
As a natural basis of the Hopf algebra of quasisymmetric functions, monomial quasisymmetric functions are formal power series defined from compositions. The same definition applies to left weak compositions, while leads to divergence for other weak compositions. We adapt the method of renormalization in quantum field theory, in the framework of Connes and Kreimer, to deal with such divergency. This approach defines monomial quasisymmetric functions for any weak composition as power series while extending the quasi-shuffle (stuffle) relation satisfied by the usual quasisymmetric functions. The …
Differential Geometry on Hopf Algebras and Quantum Groups (Ph.D. Thesis)
利用Hopf代数的基本公理和非交换微分几何，构造了Hopf代数上的微分几何。利用Smash积给出了函数的Hopf代数上的广义导子空间，并利用它定义和讨论了量子李代数及其性质。对于任意的Hopf代数的泛微分和一般微分，我们都得到了外导数、李导数和内导数的Cartan演算，并且通过限制到拟三角的情形并利用数值R-矩阵形式，确定了量子群的上述结构.
Semiquasitriangular Hopf algebras
我们说一个霍普夫代数H是半交换的，如果右伴随余作用通过H与H的中心的张量积分解.例如，交换和上交换的霍普夫代数都是半交换的.拟三角霍普夫代数推广了余交换霍普夫代数.在本文中，我们引入并开始研究半交换的一个类似的推广。这些代数，我们称之为半拟三角霍普夫代数，具有拟三角代数的许多基本性质。特别是，它们以自然的方式将编织的表征类别关联起来。
In this paper, we present a general method for constructing finite-dimensional quasi-Hopf algebras from finite abelian groups and braided vector spaces of Cartan type. The study of such quasi-Hopf algebras leads to the classification of finite-dimensional radically graded basic quasi-Hopf algebras over abelian groups with dimensions not divisible by $2,3,5,7$ and associators given by abelian $3$-cocycles. As special cases , the small quasi-quantum groups are introduced and studied. Many new explicit examples of finite-dimensional genuine quasi-Hopf algebras are obtained.
We study the basic monoidal properties of the category of Hopf modules for a coquasi Hopf algebra. In particular we discuss the so called fundamental theorem that establishes a monoidal equivalence between the category of comodules and the category of Hopf modules. We present a categorical proof of Radford's $S^4$ formula for the case of a finite dimensional coquasi Hopf algebra, by establishing a monoidal isomorphism between certain double dual functors.
Killing Form on Quasitriangular Hopf Algebras and Quantum Lie Algebras
简要回顾了拟三角霍普夫代数和量子李代数的基础知识，并证明了它们的性质允许引入Killing形式。对于量子李代数，这导致了Killing度量和二次卡西米尔的定义。具体情况下的$\uea{su}{N}$进行了详细的检查，它表明，许多经典的结果再现，并明确计算来说明的结论，提出了$\uea{su}{2}$。
Hopf Images of Hopf algebra Coactions
本文定义了 Hopf 代数余作用的 Hopf 像为最小可分解子代数，证明其内忠实性，并针对 Nichols 代数给出检测结果及玻色化实例。
We develop intrinsic tools for computing the periodic Hopf cyclic cohomology of Hopf algebras related to transverse symmetry in codimension 1. Besides the Hopf algebra found by Connes and the first author in their work on the local index formula for transversely hypoelliptic operators on foliations, this family includes its `Schwarzian' quotient, on which the Rankin-Cohen universal deformation formula is based, the extended Connes-Kreimer Hopf algebra related to renormalization of divergences in QFT, as well as a series of cyclic coverings of these Hopf algebras, motivated by the treatment of …
Combinatorics of Rooted Trees and Hopf Algebras
我们首先考虑分次向量空间，其基由有根树组成，通过非根顶点的计数进行分次。我们定义了两个线性算子在这个向量空间，增长和修剪运营商，分别提高和降低等级;他们的交换机是操作员，乘以有根树的顶点数。我们定义了一个内积，生长算子和剪枝算子在这个内积上是伴随的，并得到了关于每个算子的重数的几个结果。有根树的向量空间上的对称代数（在一个度移动之后）可以被赋予一个余积以构成一个霍普夫代数;这是由Kreimer在与重整化有关的方面定义的。我们将增长和修剪算子以及上面提到的内积推广到Kreimer的霍普夫代数。另一方面，有根树的向量空间本身可以被赋予一个非交换乘法：与一个适当的余积，这给出了Grossman和Larson的霍普夫代数。证明了有根树上的内积导致Grossman-Larson霍普夫代数与Kreimer的霍普夫代数的分次对偶的同构，修正了Panaite的一个较早的结果.
The objects of study in this paper are Hopf algebras $H$ which are finitely generated algebras over an algebraically closed field and are extensions of a commutative Hopf algebra by a finite dimensional Hopf algebra. Basic structural and homological properties are recalled and classes of examples are listed. Bounds are obtained on the dimensions of simple $H$-modules, and the structure of $H$ is shown to be severely constrained when the finite dimensional extension is semisimple and cosemisimple.
On a noncommutative deformation of the Connes-Kreimer algebra
本文研究了有根树交换霍普夫代数的一种非对易变形，这种非对易变形与量子场论中重整化的数学结构有关.非对易变形的对极存在的要求导致代数的自然扩张。Connes-Kreimer代数的非对易变形可能与非对易空间上域论的重整化有关，并且有迹象表明，在这种情况下，代数的扩展可能与红外和紫外发散的混合有关。我们也给出了一个论点，对于某一类非对易的量子场论重整化应该联系到一个非对易和非余对易的自对偶霍普夫代数，它可以被看作是Grothendieck-Teichmueller群的一个非对易对应。
Nakayama functor for coalgebras and a categorical perspective of the integral theory for Hopf algebras
本研究回顾余代数的Nakayama函子的基本性质并介绍其在张量范畴的应用，同时给出带预对极的余拟双代数存在非零余积分的等价条件。
We extend the notion of Poisson-Lie groups and Lie bialgebras from Poisson to g-quasi-Poisson geometry and provide a quantization to braided Hopf algebras in the corresponding Drinfeld category. The basic examples of these g-quasi-Poisson Lie groups are nilpotent radicals of parabolic subgroups. We also provide examples of moment maps in this new context coming from moduli spaces of flat connections on surfaces.
In this paper, we introduce the concept of a Rota-Baxter paired module to study Rota-Baxter modules without necessarily a Rota-Baxter operator. We obtain two characterizations of Rota-Baxter paired modules, and give some basic properties of Rota-Baxter paired modules. Beginning with the connection between the notion of integrals in the representations of Hopf algebra and of the notion of an integral algebra as a motivation and special case of Rota-Baxter algebra of weight zero, we obtain a large number of Rota-Baxter paired modules from Hopf related algebras and modules, including semisimple H…
We survey variety theory for modules of finite dimensional Hopf algebras, recalling some definitions and basic properties of support and rank varieties where they are known. We focus specifically on properties known for classes of examples such as finite group algebras and finite group schemes. We list open questions about tensor products of modules and projectivity, where varieties may play a role in finding answers.
Algebras and Hopf Algebras in Braided Categories
这是一个介绍代数理论的代数和霍普夫代数在编织类别。这样的对象推广了超代数和超Hopf代数，以及色李代数。回顾了辫范畴C的基本事实，研究了这类范畴中Hopf代数的模和余模，回顾了辫-交换或辫-余交换的Hopf代数（辫群）的概念，并给出了辫群aut（C）的重构定理的一个完全图解证明.该理论对拟三角Hopf代数（量子群）理论有重要的意义。它还包括重要的例子，如退化Sklyanin代数和量子平面。
We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field by using the braiding structures of A. We investigate basic properties of the polynomial invariants including stability under extension of the base field. Furthermore, we show that our polynomial invariants are indeed tensor invariants of the representation category of A, and recognize the difference of the representation category and the representation ring of A. Actually, by computing and comparing polynomial invariants, we find new examples of pairs of Hopf algebras whose r…
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