发表机构
Univ. Littoral Côte d’Opale; UR 2597 LMPA, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville（滨海大学; 纯数学与应用数学约瑟夫·刘维尔实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为组合Hopf代数建立扭曲双代数新框架，定义CTB范畴并构造泛对象Exch，统一图、树等例子，并扩展至CTDB，给出多项式不变量与反极公式。
AI 中文摘要
本文为一系列组合Hopf代数提供了一个新框架。为此，我们采用物种与扭曲双代数（亦称Hopf幺半群）的语言。我们定义了组合扭曲双代数（简称CTB）范畴，这是一类满足乘积与余乘积同底层组合结构强兼容性条件的扭曲双代数。所考虑的态射也须在某种意义下尊重组合结构。许多经典例子符合此框架，包括图、根树、偏序集、超图、对称函数、布尔映射等的Hopf代数。大多数已知的组合Hopf代数可通过应用Fock函子或对偶从CTB获得。一个核心结果是引入一个由满足特定交换条件的组合构成的集合构建的泛CTB，记为$\mathbf{Exch}$。它具有特殊特征$\epsilon$，使得对$\mathbf{Exch},\epsilon$的作用类似于拟对称函数对分次连通Hopf代数的作用：它是带组合特征的CTB范畴中的终对象。这导致编码余乘积组合结构重要信息的泛态射。固定集合上组合的自然序用于定义$\mathbf{Exch}$的三个重要子对象$\mathbf{Exch}\\_-$, $\mathbf{Exch}\\_+$和$\mathbf{Exch}\\+$。我们还研究了与CTB相关的多项式不变量。这建立了特征与多项式不变量之间的对应关系。著名不变量如图的色多项式和偏序集的Ehrhart多项式在此框架内自然出现。我们证明，对于给定次数，仅存在有限多个多项式不变量。我们证明这些泛多项式不变量可用于在CTB条件下给出反极或Eulerian幂等元的无消去公式，并用于基于根树的例子，该例子来自语言学中使用的Hopf代数。对于满足交替符号条件或满足互反原理的多项式不变量族也获得了结果。我们还将理论扩展到组合扭曲双重双代数（简称CTDB），其涉及两个相互作用的余乘积结构。这些推广了由图、偏序集、超图和根树产生的先前已知例子。扭曲方法在提供统一代数框架的同时保留了这些结构的组合丰富性。我们证明并非任何CTB都能成为CTDB，这导致拟余交换性概念的引入。特别地，我们证明$\mathbf{Exch}\\_+$不是CTDB，并构造一个子对象，该子对象以唯一方式且对包含关系极大。在余交换情形中，自然发生简化，我们证明在此情形下，超图的CTB起核心作用，尤其是基于连通系统和反链覆盖的两个子对象。我们还证明，在余交换情形中，无论考虑哪个CTB，泛多项式不变量总是超图的色多项式。最后，忽略组合底层结构，我们借助Loday和Ronco关于无穷小双代数的刚性定理的扭曲版本研究$\mathrm{Exch}$的代数结构。
英文摘要
We provide in this text a new framework for a family of combinatorial Hopf algebras. We adopt for this the language of species and twisted bialgebras (also known as Hopf monoids). We define the category of combinatorial twisted bialgebras (briefly, CTBs), a class of twisted bialgebras satisfying strong combinatorial conditions of compatibilities of the product and the coproduct with the underlying combinatorial structure. The considered morphisms also have to respect the combinatorial structure, in some sense. Many classical examples fit into this framework, including Hopf algebras of graphs, rooted trees, posets, hypergraphs, symmetric functions, boolean maps, etc. Most of known combinatorial Hopf algebras can be obtained from a CTB by application of a Fock functor, or by duality. A central result is the introduction of a universal CTB denoted by $\mathbf{Exch}$, built from collections of compositions satisfying a certain exchange condition. It has a special character $ε$, such that the pair $(\mathbf{Exch},ε)$ plays a role analogous to that of quasi-symmetric functions for graded and connected Hopf algebras: it is the terminal object in the category of CTBs equipped with combinatorial characters. This leads to universal morphisms that encode important information about the combinatorial structure of coproducts. The natural order on the set of compositions on a fixed set is used to define three important sub-objects $\mathbf{Exch}\_-$, $\mathbf{Exch}\_+$ and $\mathbf{Exch}\_{+\hspace{-2mm}+}$ of $\mathbf{Exch}$. We also studies polynomial invariants associated with CTBs. It establishes a correspondence between characters and polynomial invariants. Well-known invariants such as the chromatic polynomial of graphs and the Ehrhart polynomials of posets arise naturally within this framework. We show that, for a given degree, only finitely many polynomial invariants can occur. We show that these universal polynomial invariants can be used to give cancellation-free formulas for the antipode or the Eulerian idempotent, under conditions on the CTB, and use this for an example based on rooted trees, coming from a Hopf algebra used in linguistics. Results are also obtained for families of polynomial invariants with the alternating sign condition, or satisfying a reciprocity principle. We also extend the theory to combinatorial twisted double bialgebras (briefly, CTDBs), which involve two interacting coproduct structures. These generalize previously known examples arising from graphs, posets, hypergraphs, and rooted trees. The twisted approach preserves the combinatorial richness of these structures while providing a unified algebraic framework. We show that not any CTB can be made a CTDB, leading to the introduction of the notion of quasi-cocommutativity. In particular, we show that $\mathbf{Exch}\_+$ is not a CTDB, and construct a sub-object which is, in a unique way, and maximal for the inclusion. In the cocommutative case, simplifications naturally occur, and we show that in this case, a CTB of hypergraphs plays a central role, especially two sub-objects based on connectedness systems and antichain covers. We also show that, in the cocommutative case, the universal polynomial invariants are always chromatic polynomials of hypergraphs, no matter which CTB is considered. Finally, forgetting the combinatorial underlying structure, we study the algebraic structure of $\mathrm{Exch}$, with the help of a twisted version of Loday and Ronco's rigidity theorem for infinitesimal bialgebras.
