On complements and the factorization problem for Hopf algebras
Sebastian Burciu
证明了关于半单霍普夫代数补数的两个新结果。它们推广了群论的一些著名结果。证明了半单完全可约霍普夫代数的Krull Schmidt Remak型分解的唯一性.
Two new results concerning complements in a semisimple Hopf algebra are proved. They extend some well known results from group theory. The uniqueness of Krull Schmidt Remak type decomposition is proved for semisimple completely reducible Hopf algebras.
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In this paper we introduce the notion of factorization in the Hopf quasigroup setting and we prove that, if $A$ and $H$ are Hopf quasigroups such that their antipodes are isomorphisms, a Hopf quasigroup $X$ admits a factorization as $X=AH$ iff $X$ is isomorphic to a double cross product $A\bowtie H$ as Hopf quasigroups.
The Factorization of Braided Hopf Algebras
给出了辫双代数或辫霍普夫代数的双因子分解，并给出了辫霍普夫代数及其因子的积分与半单性之间的关系.
We prove a variety results on tensor product factorizations of finite dimensional Hopf algebras (more generally Hopf algebras satisfying chain conditions in suitable braided categories). The results are analogs of well-known results on direct product factorizations of finite groups (or groups with chain conditions) such as Fitting's Lemma and the uniqueness of the Krull-Remak-Schmidt factorization. We analyze the notion of normal (and conormal) Hopf algebra endomorphisms, and the structure of endomorphisms and automorphisms of tensor products. The results are then applied to compute the automo…
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.
Factorization of simple modules for certain pointed Hopf algebras
本文研究了两类点霍普夫代数的表示：限制双参数量子群和幂零型秩一点霍普夫代数的Drinfel'd double.我们研究，特别是，在什么条件下，一个简单的模块可以被分解为张量积的一维模块与模块，自然是一个模块的商由中心组样的元素。
Classifying complements for Hopf algebras and Lie algebras
设$A \subseteq E$是Hopf（或李）代数的给定扩张.我们回答了\emph{classifying complements problem}（CCP）问题，它包括在$E$中描述和分类$A$的所有补语.如果$H$是一个给定的补数，那么所有其他的补数都是通过某种类型的变形从$H$获得的。我们建立了$E$中$A$的所有补集的同构类与上同调类型对象${\mathcal H}{\mathcal A}^{2} (H, A \, | \, (\triangleright, \triangleleft) )$之间的双射对应，其中$(\triangleright, \triangleleft)$是与$H$相关联的匹配对.因子分解指数$[E: A]^f$被引入作为（CCP）的数值度量。对于两个$n$ -次单位根，我们构造了一个$4n^2$ -维Hopf代数，其在群代数上的分解指数是任意大的.
Factorization in algebra is an important problem. In this paper, we first obtain a unique factorization in free Nijenhuis algebras. By using of this unique factorization, we then define a coproduct and a left counital bialgebraic structure on a free Nijenhuis algebra. Finally, we prove that this left counital bialgebra is connected and hence obtain a left counital Hopf algebra on a free Nijenhuis algebra.
On Hopf algebras with positive bases
证明了若${\bf C}$上的有限维霍普夫代数有一个基使得所有结构常数都非负，则该霍普夫代数必由有限群$G$和分解为两个子群的$G=G_+G_-$给出.我们还表明，霍普夫代数在有限集范畴的对应作为态射的分类在类似的方式。我们的结果可以用来从集合论的观点解释霍普夫代数中的一些结果。
Factorization of Laurent series over commutative rings
将洛朗级数的Wiener-Hopf分解推广到更一般的交换系数环，并给出了分解的显式公式.我们强调这种因式分解的代数性质。
Braided Hopf Algebras
在辫子张量范畴中，我们证明了Maschke定理，并给出了双交叉双积和交叉双积以及双积是双代数的充要条件.在对称张量范畴中，证明了对偶定理，构造了量子偶;在具有普通扭曲的普通向量空间范畴中，得到了代数$R$及其交叉积$R #_σH$的整体维数、弱维数、Jacobson根、Baer根之间的关系.我们还给出了余模分解与余代数分解之间的关系。我们对霍普夫代数进行了分类。在具有dim $L \le 3$的李代数$L$上得到了常值经典杨-巴克斯特方程（CYBE）的所有解.给出了$(L, \hbox {[ ]}, Δ_r, r)$是上边界（或三角）李双代数的充必要条件.
Noetherian pointed Hopf algebras are affine
该研究引入词上的约化序与约化因子分解概念，证明域上右或左诺特有点Hopf代数是仿射的，部分回应了Wu和Zhang提出的诺特Hopf代数仿射性经典问题。
In these notes we review and further explore the Lie enveloping algebra of a post-Lie algebra. From a Hopf algebra point of view, one of the central results, which will be recalled in detail, is the existence of a second Hopf algebra structure. By comparing group-like elements in suitable completions of these two Hopf algebras, we derive a particular map which we dub post-Lie Magnus expansion. These results are then considered in the case of Semenov-Tian-Shansky's double Lie algebra, where a post-Lie algebra is defined in terms of solutions of modified classical Yang-Baxter equation. In this c…
We calculate Frobenius-Schur indicator values for some fusion categories obtained from inclusions of finite groups $H\subset G$, where more concretely $G$ is symmetric or alternating, and $H$ is a symmetric, alternating or cyclic group. Our work is strongly related to earlier results by Kashina-Mason-Montgomery, Jedwab-Montgomery, and Timmer for bismash product Hopf algebras obtained from exact factorizations of groups. We can generalize some of their results, settle some open questions and offer shorter proofs; this already pertains to the Hopf algebra case, while our results also cover fusio…
A Noetherian Hopf algebra is affine iff its Hopf coradical is affine
本文通过引入非交换约化序、因子分解理论和广义提升方法，建立了诺特Hopf代数仿射性的充要条件，即左或右诺特Hopf代数仿射当且仅当其Hopf余根仿射，从而推进了Wu-Zhang问题的解决。
Protomodularity of cocommutative Hopf monoids in duoidal categories and quasitriangular Hopf algebras
本文将余交换Hopf代数范畴的原模范性扩展到拟三角情形，通过修改对极概念等方法，研究双幺半范畴中Hopf幺半群相关性质，证明了一些分解和引理，得到多个范畴的原模范性及相关结果，并扩展了双交叉积。
This is a study on pattern Hopf algebras in combinatorial structures. We introduce the notion of combinatorial presheaf, by adapting the algebraic framework of species to the study of substructures in combinatorics. Afterwards, we consider functions that count the number of patterns of objects and endow the linear span of these functions with a product and a coproduct. In this way, any well behaved family of combinatorial objects that admits a notion of substructure generates a Hopf algebra, and this association is functorial. For example, the Hopf algebra on permutations studied by Vargas in …
A noncommutative and noncocommutative Hopf algebra on finite topologies H_T is introduced and studied (freeness, cofreeness, self-duality...). Generalizing Stanley's definition of P-partitions associated to a special poset, we define the notion of T-partitions associated to a finite topology, and deduce a Hopf algebra morphism from H_T to the Hopf algebra of packed words WQSym. Generalizing Stanley's decomposition by linear extensions, we deduce a factorization of this morphism, which induces a combinatorial isomorphism from the shuffle product to the quasi-shuffle product of WQSym. It is stro…
A quantum analog of the Poincare-Birkhoff-Witt theorem
我们减少的基础作图题的霍普夫代数产生的斜本原半不变量的特殊元素的研究，称为"超级字母“，这是定义的Shirshov标准字。通过这种方式，我们证明了上述霍普夫代数总是有PBW生成元集（“硬”超级字母）。它还表明，这些霍普夫代数不超过1000个“硬”超字母共享的泛包络代数的有限维李代数的一些性质。证明的背景是一个过滤的建设，使相关的分次代数是通过迭代的斜多项式建设，可能其次是因式分解。
In this note we further explore the properties of universal enveloping algebras associated to a post-Lie algebra. Emphasizing the role of the Magnus expansion, we analyze the properties of group like-elements belonging to (suitable completions) of those Hopf algebras. Of particular interest is the case of post-Lie algebras defined in terms of solutions of modified classical Yang-Baxter equations. In this setting we will study factorization properties of the aforementioned group-like elements.
We find a natural compatible condition between the Rota-Baxter operator and Turaev's (Hopf) group-(co)algebras, which leads to the concept of Rota-Baxter Turaev's (Hopf) group-(co)algebra. Two characterizations of Rota-Baxter Turaev's group-algebras (abbr. T-algebras) are obtained: one by Atkinson factorization and the other by T-quasi-idempotent elements. The relations among some related Turaev's group algebraic structures (such as (tri)dendriform T-algebras, Zinbiel T-algebras, pre-Lie T-algebras, Lie T-algebras) are discussed, and some concrete examples from the algebras of dimensions 2,3 a…
On the Semi-Abelianness of Affine Group Schemes
本文证明了在域$k$上交换Hopf代数的范畴是co-semi-abelian的，并推导出仿射群$k$-方案的范畴是semi-abelian的。
In this paper we explore minimum odd and minimum even depth sub algebra pairs in the context of double cross products of finite dimensional Hopf algebras. We start by defining factorization algebras and outline how subring depth in this context relates with the module depth of the regular left module representation of the given subalgebra. Next we study minimum odd depth for double cross product Hopf subalgebras and determine their value in terms of their related module depth, we conclude that minimum odd depth of Drinfeld double Hopf subalgebras is 3. Finaly we produce a necessary and suffici…
Post-Hopf algebra in non-commutative probability theory
该研究探讨了李代数上相容的$\mathcal{O}$-算子与后李乘积的群乘积分解问题，并将其应用于非交换概率论，通过引入新变换简化了算子值非交换分布的条件自由和条件单调乘法卷积计算。
The factorization problem for Jordan algebras. Applications
该研究在约当代数框架下探讨因式分解与分类补问题，引入匹配对与双交叉积并证明相关同构结论，还提出新型约当代数形变作为解决分类补问题的关键步骤。
Factorizations, classifying complements problem and deformation maps for Lie-Yamaguti algebras
研究Lie-Yamaguti代数的分解问题、强补的分类以及形变映射的统一理论，证明形变映射推广了同态、导子、交叉同态和相对Rota-Baxter算子，并定义了其上同调与Maurer-Cartan刻画。
Spitzer's Identity and the Algebraic Birkhoff Decomposition in pQFT
在这篇文章中，我们继续探讨的概念Rota-Baxter代数的背景下的Hopf代数方法的重整化理论在微扰量子场论。我们表明，在非常简单的代数条件下，递归定义的公式的Birkhoff因式分解的正则化的Hopf代数字符，即费曼规则的解决方案，自然给出了一个非交换的推广著名的斯皮策的身份。基本的抽象代数结构进行了分析，在完全过滤Rota-Baxter代数。
These lecture notes aim to present the algebraic theory of regularity structures as developed in arXiv:1303.5113, arXiv:1610.08468, and arXiv:1711.10239. The main aim of this theory is to build a systematic approach to renormalisation of singular SPDEs; together with complementary analytic results, these works give a general solution theory for a wide class of semilinear parabolic singular SPDEs in the subcritical regime. We demonstrate how "positive" and "negative" renormalisation can be described using interacting Hopf algebras, and how the renormalised non-linearities in SPDEs can be comput…
Algebraic Birkhoff decomposition and its applications
在康纳斯和克雷默的微扰量子场论的重整化的霍普夫代数方法中，中心是他们的代数Birkhoff分解。在这篇教程文章中，我们介绍了它们的分解，并通过Rota-Baxter代数中的Atkinson分解来证明它。然后，我们给出了这种分解在发散积分和多重zeta值的研究中的一些应用。
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