Héctor Figueroa, José M. Gracia-Bondía, Joseph C. Várilly
This is a short review on the Faà di Bruno formulas, and some Hopf algebras associated to them. These allow, among several other things, a short proof of the Lie-Scheffers theorem, and relating the Lagrange inversion formulas with antipodes. The link of Faà di Bruno formulas with the theory of set partitions is developed in some detail.
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In this paper we describe the Hopf algebra approach to the center problem for the differential equation $\frac{dv}{dx}=\sum_{i=1}^{\infty}a_{i}(x)v^{i+1}$, $x\in [0,T]$, and study some combinatorial properties of the first return map of this equation. The paper summarizes and extends previously developed approaches to the center problem due to Devlin and the author.
Center problem, Abel equation and the Faa di Bruno Hopf algebra for output feedback
本文给出了非自治微分方程经典Poincare中心焦点问题的Devlin字问题的组合解释。从连通分次Hopf代数的观点出发，证明了Devlin的正则多项式与应用于Fferera的形式幂级数的霍普夫代数的对极的分次分量密切相关.这种联系是通过控制理论建立的，因为描述中心的阿贝尔方程等价于输出反馈方程，而输出反馈的霍普夫代数是由迭代积分的合成而不是迭代积分的乘积导出的，后者产生了洗牌代数。这意味着Devlin方法中起作用的主要代数结构实际上不是shuffle代数，而是Faa di Bruno类型的霍普夫代数，它是根据shuffle积定义的，但却是一个不同的代数结构。
Combinatorial Hopf algebras in quantum field theory I
这份手稿站在组合霍普夫代数理论和重整化理论之间的接口。它的计划如下：第一部分是介绍，并包含一个基本的邀请主题。第一部分的其余部分，包括第2-6节，致力于霍普夫代数理论和例子的基础知识，在复杂性的上升水平。第二部分围绕所有重要的Faa di Bruno Hopf代数。第七节包含了第一个直接的方法，第八节给出了Faa di Bruno代数在量子场论和拉格朗日反演中的应用。第九节重新推导了相关的Connes-Moscovici代数。在第三部分中，我们转向费曼图的Connes-Kreimer Hopf代数，更一般地，关联双代数。在第10节中，我们描述了第一个。然后在第11节中，我们给出了齐默尔曼的无消去法的简单推导（正确组合部分），以其原始的图解形式。在第12节中引入了一般的关联代数，并将Faa di Bruno双代数描述为关联双代数。在第13节中，关于罗塔关联代数的更深层次的知识使我们能够用分配格来重新解释Connes-Kreimer代数。其次，确定了对映体的无消去公式的一般代数组合证明;这是本文的核心。交换Hopf代数的结构结果在第14节和第15节中找到。展望部分非常简要地回顾了量子化的共代数方面和重整化中的Rota-Baxter映射。
Zimmermann Type Cancellation in the Free Faa di Bruno Algebra
由Brouder，Fabretti，and Krattenaler（BFK）在非交换Lagrange反演的上下文中引入的N-变量霍普夫代数可以与N-着色区间划分的关联代数的逆相一致。（BFK）对极及其反射决定了具有N个非交换变量的非交换系数幂级数的（通常不同的）左逆和右逆。在Faa di Bruno霍普夫代数的情况下，有一个类似的齐默尔曼取消公式。（BFK）对极的和数可以用压缩平面树上顶点的深度优先排序来索引，这同样适用于区间划分对极。这两个也可以索引的宽度第一顺序的顶点在无序的可收缩平面树，其中恰好一个非退化顶点出现在每一个级别。
Five interpretations of Faà di Bruno's formula
在这些讲座中，我们提出了五种解释法'迪布鲁诺公式计算的n阶导数的组成两个职能的一个变量：在方面的团体，李代数和霍普夫代数，在组合和运算。
The general goal of this paper is to identify a transformation group that can be used to describe a class of feedback interconnections involving subsystems which are modeled solely in terms of Chen-Fliess functional expansions or Fliess operators and are independent of the existence of any state space models. This interconnection, called an output affine feedback connection, is distinguished from conventional output feedback by the presence of a multiplier in an outer loop. Once this transformation group is established, three basic questions are addressed. How can this transformation group be …
A combinatorial Hopf algebra for nonlinear output feedback control systems
在这项工作中，提供了一个组合描述的Faa di Bruno型霍普夫代数自然出现在非线性反馈控制理论中的Fliess算子的上下文中。它是一个定义在根圆树上的连通分次交换和非余交换的霍普夫代数。给出了其对极的无消去森林公式。
For any coloured operad R, we prove a Faà di Bruno formula for the `connected Green function' in the incidence bialgebra of R. This generalises on one hand the classical Faà di Bruno formula (dual to composition of power series), corresponding to the case where R is the terminal reduced operad, and on the other hand the Faa di Bruno formula for P-trees of Gálvez--Kock--Tonks (P a finitary polynomial endofunctor), which corresponds to the case where R is the free operad on P. Following Gálvez--Kock--Tonks, we work at the objective level of groupoid slices, hence all proofs are `bijective': the …
We introduce a general construction that takes as input a so-called stiff PRO and that outputs a Hopf algebra. Stiff PROs are particular PROs that can be described by generators and relations with precise conditions. Our construction generalizes the classical construction from operads to Hopf algebras of van der Laan. We study some of its properties and review some examples of application. We get in particular Hopf algebras on heaps of pieces and retrieve some deformed versions of the noncommutative Faà di Bruno algebra introduced by Foissy.
Discrete Faà di Bruno via Möbius Inversion
该研究从莫比乌斯反演角度探讨离散和微分法阿·迪·布鲁诺公式，通过组合泰勒展开与反演得到离散公式，经分组、迭代得出相关版本及公式，还展示了覆盖与划分关系，最后将代数恒等式提升到分析设置，恢复并扩展了多元法阿·迪·布鲁诺公式。
Semi-pointed partition posets and Species
我们定义了半点划分偏序集，这是一个推广的划分偏序集，并表明他们是科恩-麦考利。然后利用多重链计算了这些对称群作用于它们的同调的维数和特征标。最后研究了关联霍普夫代数，它类似于Fa{à} di Bruno霍普夫代数.
Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras
贝尔多项式出现在数学中的几个组合结构中。也许最自然的是在组合学的集合划分，但也当研究的组成，向量空间和流形上的同胚，并在研究累积量和时刻在概率论。我们构造交换和非交换贝尔多项式，并解释他们如何产生Faà迪布鲁诺霍普夫代数。我们使用的语言的关联霍普夫代数，并沿着这条道路提供了一个新的描述的对极在非交换的关联霍普夫代数，涉及拟行列式。我们还讨论了由Bell多项式构造的某些霍普夫代数中的Möbius反演。
Partial multivariate Bell polynomials have been defined by E.T. Bell in 1934. These polynomials have numerous applications in Combinatorics, Analysis, Algebra, Probabilities, etc. Many of the formulae on Bell polynomials involve combinatorial objects (set partitions, set partitions in lists, permutations, etc.). So it seems natural to investigate analogous formulae in some combinatorial Hopf algebras with bases indexed by these objects. The algebra of symmetric functions is the most famous example of a combinatorial Hopf algebra. In a first time, we show that most of the results on Bell polyno…
We introduce the Hopf monoid of sets of cycles and paths, which contains the Faà di Bruno Hopf monoid as a submonoid. We give cancellation-free and grouping-free formulas for its antipode, one in terms of tubings and one in terms of \emph{pointed} noncrossing partitions. We provide an explicit description of the group of characters of this Hopf monoid in terms of pairs of power series. Using graph associahedra, we relate paths and cycles to associahedra and cyclohedra, respectively. We give formulas for inversion in the group of characters in terms of the faces of these polytopes.
Polynomial realizations of natural Hopf algebras of nonsymmetric operads
本文为自然Hopf代数引入新基，通过变量字母表构造多项式实现，并揭示其与多个已知Hopf代数（如词拟对称函数、Connes-Kreimer装饰版本等）的联系。
Formulas for the Connes-Moscovici Hopf algebra
给出了Connes-Moscovici霍普夫代数$\mathcal{H}_{\tmop{CM}}$的余积和对极的显式表达式.要做到这一点，我们首先限制自己的一个子霍普夫代数$\mathcal{H}^1_{\tmop{CM}}$包含非平凡的元素，即那些的余积和对极是非平凡的。有两种方法可以获得显式公式。一方面，代数$\mathcal{H}^1_{\tmop{CM}}$同构于单位切同态群上坐标的Faà di Bruno霍普夫代数，并且使用替换自同构而不是自同构使计算变得容易。另一方面，在向量场的形式共轭和解析共轭的框架下，代数$\mathcal{H}^1_{\tmop{CM}}$同构于在重构理论中自然出现的经典shuffle霍普夫代数的子霍普夫代数.利用洗牌霍普夫代数的简单结构，我们再次导出了$\mathcal{H}^1_{\tmop{CM}}$中余积和对极的显式公式.
Polynomial realizations of Hopf algebras built from nonsymmetric operads
本文利用带一元与二元关系的非交换变量字母表，为非对称 operad 的自然 Hopf 代数构造多项式实现，并将其与多种经典 Hopf 代数联系起来。
In this paper, we discuss centralizers in the Riordan group. We will see that Faà di Bruno's formula is an application of the Fundamental Theorem of Riordan arrays. Then the composition group of formal power series in ${\cal F}_1$ is studied to construct the centralizers of Bell type and Lagrange type Riordan arrays. Our tools are the $A$-sequences of Riordan arrays and Faà di Bruno's formula. Some combinatorial explanation and discussion about related algebraic topics are also given.
There is a very natural and well-behaved Hopf algebra morphism from quasisymmetric functions to peak algebra, which we call it Theta map. This paper focuses on generalizing the peak algebra by constructing generalized Theta maps for an arbitrary combinatorial Hopf algebra. The image of Theta maps lies in the odd Hopf subalgebras, so we present a strategy to find odd Hopf subalgebra of any combinatorial Hopf algebra. We also give a combinatorial description of a family of Theta maps for Malvenuto-Reutenauer Hopf algebra of permutations $\operatorname{\mathsf{\mathfrak{S}Sym}}$ whose images are …
A family of permutations called 2-clumped permutations forms a basis for a sub-Hopf algebra of the Malvenuto-Reutenauer Hopf algebra of permutations. The 2-clumped permutations are in bijection with certain decompositions of a square into rectangles, called generic rectangulations. Thus, we can describe the Hopf algebra of 2-clumped permutations using generic rectangulations (we call this isomorphic Hopf algebra the Hopf algebra of generic rectangulations). In this paper, we describe the cover relations in a lattice of generic rectangulations that is a lattice quotient of the right weak order …
Word Hopf algebras
对称函数的霍普夫代数的两个重要推广是非交换对称函数的霍普夫代数及其分次对偶拟对称函数的霍普夫代数。后者的一个常见推广是置换的自对偶霍普夫代数（MPR霍普夫代数）。后一个霍普夫代数可以看作是一个霍普夫代数的自同态的霍普夫代数。事实证明，这是一个富有成效的方式看待事物，并引起广泛的进一步推广，如字霍普夫代数和双字霍普夫代数（和许多更多）。
Four explicit formulas for the prolongations of an infinitesimal Lie symmetry and multivariate Faa di Bruno formulas
1979年，在S. Lie的（部分）微分方程的对称性理论，P. J. Olver制定了归纳公式，适用于计算无穷小Lie对称性到喷流空间的转换，对于任意数量n \geq 1的自变量（x^1，.，x^n）和对于任意数量m \geq 1的因变量（y^1，.，y^m）。本文致力于阐述一种基于多个克罗内克符号的形式主义，它使人们能够处理这些“无法管理”的解释，并发现潜在的复杂组合学。逐步进行，我们写下四种情况下的封闭显式公式：n=m=1; n \geq 1，m=1; n=1，m \geq 1;一般情况n \geq 1，m \geq 1。作为所获得的公式的子部分，我们恢复四个可能的版本（多元）Faà di Bruno公式。我们不采用基于对称代数的经典形式（{ \it cf.}例如H. Federer's book，p. 222），因为它在符号紧化中隐藏了几个显式和，并且因为补充复杂性的存在（{ \it e.g.}指数的分裂、偏导数的组合学）阻碍了我们连贯地应用这种紧化。我们的阐述方法是归纳的：我们通过分析几个彻底组织的公式，通过将它们放在一起进行比较，并通过与L的古典风格同源的“漂移”走向一般性来进行推理。欧拉
Cocommutative Hopf algebras of permutations and trees
考虑Loday和Ronco的平面二叉树的霍普夫代数和Malvenuto和Alfretenauer的置换的余根滤子.我们给出了明确的同构，表明相关的分次霍普夫代数是对偶的上交换霍普夫代数在20世纪80年代末引进的Grossman和Larson。这些霍普夫代数分别由有序树和堆有序树构成。这些结果来自于这样一个事实，即只要从一个霍普夫代数出发，它是一个余自由分次余代数，则相应的分次霍普夫代数是一个shuffle霍普夫代数.
Hopf algebras of endomorphisms of Hopf algebras
在过去的十年中，对称函数的霍普夫代数的两个推广出现了，并显示出它们的重要性，非交换对称函数的霍普夫代数NSymm和拟对称函数的霍普夫代数QSymm。同样清楚的是，理解对称函数的霍普夫代数Symm等重要结构的非交换形式也很重要。尤其是因为正确的非交换形式通常比交换形式更漂亮（并非都被计数系数弄得乱七八糟）。NSymm和QSymm并不是真正的完全的非对易推广。一个是极大非对易但余对易的，另一个是极大非余对易但交换的。有一个常见的自对偶推广，即Malvenuto，Poirier，and Apertenauer置换的霍普夫代数（MPR）。我觉得，最好把它理解为自同态的霍普夫代数。在任何情况下，这一观点提出了大量的推广，导致霍普夫代数的自同态和字霍普夫代数，本文所关注的。这个观点也揭示了MPR的一些神秘的公式，以及所有额外的结构（如自对偶性）从何而来的问题。最后，本文对MPR的结构、NSymm的霍普夫代数自然包含到MPR的代数收缩问题以及MPR到QSymm的自然投影问题作了一些讨论。
The Hopf algebra of uniform block permutations. Extended abstract
引入了一致块置换的霍普夫代数，并证明了它是自对偶的、自由的和余自由的.这些结果与一致块置换构成可分解逆幺半群密切相关。这个霍普夫代数包含了Malvenuto和Littenauer的置换的霍普夫代数和Gebhard、Rosas和Sagan的非交换变量的对称函数的霍普夫代数。
Baxter Algebras and Hopf Algebras
利用自由Baxter代数的一个新构造，得到了一类新的霍普夫代数，它推广了经典的除幂霍普夫代数.我们还研究了这些霍普夫代数同构的条件。
Hopf Algebras of B-Diagrams and Boson Normal Ordering: Exploring the Dual Structures
本文研究B-图的Hopf代数及其对偶结构，探索其与海森堡代数及非交换对称多项式的联系，揭示玻色正常排列问题的组合学本质。
Three equivalent methods allow to compute the antipode of the Hopf algebras of Feynman diagrams in perturbative quantum field theory (QFT): the Dyson-Salam formula, the Bogoliubov formula, and the Zimmermann forest formula. Whereas the first two hold generally for arbitrary connected graded Hopf algebras, the third one requires extra structure properties of the underlying Hopf algebra but has the nice property to reduce drastically the number of terms in the expression of the antipode (it is optimal in that sense).The present article is concerned with the forest formula: we show that it genera…
This paper introduces the braidings of dendriform algebras and tridendriform algebras. By studying free braided dendriform algebras, we obtain braidings of the Hopf algebras of Loday and Ronco of planar binary rooted trees. We also give a variation of the braiding of Foissy for the noncommutative Connes-Kreimer (a.k.a the Foissy-Holtkamp) Hopf algebra of planar rooted forests so that the well-known isomorphism between this Hopf algebra and the Loday-Ronco Hopf algebra is extended to the braided context. As free braided tridendriform algebras, we also give braided extension of the Hopf algebra …
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