Iván Angiono, Mikhail Kochetov, Mitja Mastnak
We study deformations of graded braided bialgebras using cohomological methods. In particular, we show that many examples of Nichols algebras, including the finite-dimensional ones arising in the Andruskiewitsch-Schneider program of classification of pointed Hopf algebras, are rigid. This result can be regarded as nonexistence of "braided Lie algebras" with nontrivial bracket.
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Nichols algebras of unidentified diagonal type
具有有限根系的对角型Nichols代数是标准型、超型或（尚未）确定型的。\cite{A-exp presentation}中给出了所有这些Nichols代数的定义关系的具体描述。在本文中，我们使用这个结果给出了一个明确的介绍，所有的Nichols代数的不明类型。
We compute all Nichols algebras of rigid vector spaces of dimension 2 that admit a non-trivial quadratic relation.
We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GK-dimension for short, through the study of Nichols algebras over $\mathbb{D}_{\infty}$, the infinite dihedral group. We find all the irreducible Yetter-Drinfeld modules $V$ over $\mathbb{D}_{\infty}$, and determine which Nichols algebras $\mathcal{B}(V)$ of $V$ are finite GK-dimensional.
On defining ideals and differential algebras of Nichols algebras
本文从辫群的群代数中特定元素的分解出发，研究Nichols代数的定义理想。找到了一族本原元，并提出了相应的算法.为了证明主要结果，构造了Nichols代数的微分代数.此外，还对Serre关系提出了另一种观点。
It was conjectured in \texttt{\small arXiv:1606.02521} that a Nichols algebra of diagonal type with finite Gelfand-Kirillov dimension has finite (generalized) root system. We prove the conjecture assuming that the rank is 2. We also show that a Nichols algebra of affine Cartan type has infinite Gelfand-Kirillov dimension.
Pointed Hopf algebras of odd dimension and Nichols algebras over solvable groups
本研究采用约化到正特征等方法，分类特征0下有限可解群上带不可分解支集的有限维Nichols代数，进而得出奇数阶群上Nichols代数均为对角型，完成所有奇数维点Hopf代数的刻画。
We prove that finite GK-dimensional pre-Nichols algebras of super and standard type are quotients of the corresponding distinguished pre-Nichols algebras, except when the braiding matrix is of type super A and the dimension of the braided vector space is three. For these two exceptions we explicitly construct substitutes as braided central extensions of the corresponding pre-Nichols algebras by a polynomial ring in one variable. Via bosonization this gives new examples of finite GK-dimensional Hopf algebras.
Calabi-Yau Nichols algebras of Hecke type
设$R$是Hecke型Nichols代数.本文证明了：如果$R$是Noether的且整体维数有限，则$R$有刚性对偶复形.然后给出了$R$是Calabi-Yau代数的一个充要条件。
Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid
研究了半单Yetter-Drinfeld模的Nichols代数的分次右余理想子代数。假设Yetter-Drinfeld模允许所有反射，Nichols代数是可分解的，我们构造了Weyl广群的态射与Nichols代数的分次右上理想子代数之间的一个保序和反序的内射映射.这里态射是关于右Duflo序和右共理想子代数是关于包含序。若Weyl广群是有限的，则证明了Nichols代数是可分解的，且上述映射是双射的。在量子化包络代数的Borel部分的特殊情况下，我们的结果隐含了Kharchenko的一个猜想。关键词：霍普夫代数，量子群，根系，外尔群
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.
Distinguished Pre-Nichols algebras
利用有限维Nichols代数$\mathcal{B}(V)$，形式化地定义并研究了对角型辫子向量空间$V$的特殊预Nichols代数$\widetilde{\mathcal{B}}(V)$.代数$\widetilde{\mathcal{B}}(V)$由比$\mathcal{B}(V)$更少的关系表示，因此它介于张量代数$T(V)$和$\mathcal{B}(V)$之间。杰出的前尼科尔斯代数的突出例子是量子化包络（超）代数及其多参数版本的正部分。证明了这些代数给出了有限Gelfand-Kirillov维数的Noether点霍普夫代数的新例子.研究了从$\widetilde{\mathcal{B}}(V)$到$\mathcal{B}(V)$的投影的核（在霍普夫代数的意义下），推广了De Concini和Procesi关于单位根上的量子群的结果.
The R-matrix of quantum doubles of Nichols algebras of diagonal type
设$H$是对角型Nichols代数的量子双。我们计算了$H$上一般有限维最高权模块的3个模块的$R$矩阵。当$H$为有限维时，我们还计算了泛$R$ -矩阵的乘法公式.
This is a survey on Nichols algebras of diagonal type with finite dimension, or more generally with arithmetic root system. The knowledge of these algebras is the cornerstone of the classification program of pointed Hopf algebras with finite dimension, or finite Gelfand-Kirillov dimension; and their structure should be indispensable for the understanding of the representation theory, the computation of the various cohomologies, and many other aspects of finite dimensional pointed Hopf algebras. These Nichols algebras were classified in [H-classif RS] as a notable application of the notions of …
We compute the finite-dimensional Nichols algebras over the sum of two simple Yetter-Drinfeld modules V and W over non-abelian quotients of a certain central extension of the dihedral group of eight elements or SL(2,3), and such that the Weyl groupoid of the pair (V,W) is finite. These central extensions appear in the classification of non-elementary finite-dimensional Nichols algebras with finite Weyl groupoid of rank two. We deduce new information on the structure of primitive elements of finite-dimensional Nichols algebras over groups.
This paper contributes to the proof of the conjecture posed in arXiv:1606.02521, stating that a Nichols algebra of diagonal type with finite Gelfand-Kirillov dimension has finite (generalized) root system. We prove the conjecture assuming that the rank is 3 or that the braiding is of Cartan type.
We classify infinite-dimensional decomposable braided vector spaces arising from abelian groups whose components are either points or blocks such that the corresponding Nichols algebras have finite Gelfand-Kirillov dimension. In particular we exhibit examples with $\operatorname{GKdim} = n$ for any natural number $n$.
Rank 4 Nichols Algebras of Pale Braidings
本文分类了秩为4、有限GK维数且由交换群上非点块直和的Yetter-Drinfeld模生成的Nichols代数。
We show that except in several cases conjugacy classes of classical Weyl groups $W(B_n)$ and $W(D_n)$ are of type {\rm D}. We prove that except in three cases Nichols algebras of irreducible Yetter-Drinfeld ({\rm YD} in short )modules over the classical Weyl groups are infinite dimensional.
The classification of Nichols algebras over groups with finite root system of rank two
对群G和G上的绝对单Yetter-Drinfeld模对（V，W）进行分类，使得V与W的直和的支集生成G，V与W的辫的平方不是单位元，V与W的直和的Nichols代数有有限根系.作为一个副产品，我们确定了这类Nichols代数的维数，并得到了几个新的有限维Nichols代数族。我们的主要工具是Weyl广群的绝对简单的Yetter-Drinfeld模对在群上。
In this article, we explicitly construct new finite-dimensional, link-indecomposable Nichols algebras with Dynkin diagrams of type An,Cn,Dn,E6,E7,E8,F4 over any group G with commutator subgroup isomorphic to Z_2.The construction is generic in the sense that the type just depends on the rank and center of G, and thus positively answers for all groups of this class a question raised by Susan Montgomory in 1995 [Mont95][AS02]. Our construction uses the new notion of a covering Nichols algebra as a special case of a covering Hopf algebra [Len12] and produces non-faithful Nichols algebras. However,…
Let $\mathcal{B}_\mathfrak{q}$ be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix $\mathfrak{q}$. We consider the graded dual $\mathcal{L}_{\mathfrak{q}}$ of the distinguished pre-Nichols algebra $\widetilde{\mathcal{B}}_{\mathfrak{q}}$ from [A3] and the divided powers algebra $\mathcal{U}_{\mathfrak{q}}$, a suitable Drinfeld double of $\mathcal{L}_{\mathfrak{q}} # \mathbf{k} \mathbb{Z}^θ$. We provide basis and presentations by generators and relations of $\mathcal{L}_{\mathfrak{q}}$ and $\mathcal{U}_{\mathfrak{q}}$, and prove that they are noetherian and have f…
We describe how to find quantum determinants and antipode formulas from braided vector spaces using the FRT-construction and finite-dimensional Nichols algebras. It generalizes the construction of quantum function algebras using quantum grassmanian algebras. Given a finite-dimensional Nichols algebra B, our method provides a Hopf algebra H such that B is a braided Hopf algebra in the category of H-comodules. It also serves as source to produce Hopf algebras generated by cosemisimple subcoalgebras, which are of interest for the generalized lifting method. We give several examples, among them qu…
Classifying partial (co)actions of Taft and Nichols Hopf algebras on their base fields
该研究完成了Taft和Nichols Hopf代数在其基域上所有部分作用、部分余作用的分类，且证明这类部分（余）作用均为对称的，丰富了Hopf代数部分作用理论。
Point modules of Nichols algebras over non-abelian groups
该研究通过点模系统研究非阿贝尔群上的Nichols代数，确定了相关有限维Nichols代数的截断点空间，发现3-截断点模空间为空，并找到了其他重要族的对应截断点模。
Let ${\mathcal B}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra of diagonal type with braiding matrix $\mathfrak{q}$, let $\mathcal{L}_{\mathfrak{q}}$ be the corresponding Lusztig algebra as in arXiv:1501.04518 and let $\operatorname{Fr}_{\mathfrak{q}}: \mathcal{L}_{\mathfrak{q}} \to U(\mathfrak{n}^{\mathfrak{q}})$ be the corresponding quantum Frobenius map as in arXiv:1603.09387. We prove that the finite-dimensional Lie algebra $\mathfrak{n}^{\mathfrak{q}}$ is either 0 or else the positive part of a semisimple Lie algebra $\mathfrak{g}^{\mathfrak{q}}$ which is determined for each $\…
Left coideal subalgebras of Nichols algebras
本研究确定了满足一次生成条件的特定Yetter-Drinfeld模的Nichols代数，提出低次生成元扩展左余理想子代数的策略，并探讨更一般Hopf代数情形下的构造方法。
On the classification of finite GK-dimensional pre-Nichols algebras and quasi-quantum groups
本文证明了在扭曲Yetter-Drinfeld范畴中非对角对象的预尼科尔代数具有无限GK维，从而完全刻画了该范畴中有限GK维的尼科尔代数，通过bosonization给出了有限GK维的指针余Hopf代数分类。
We classify finite-dimensional pointed Hopf algebras with abelian coradical, up to isomorphism, and show that they are cocycle deformations of the associated graded Hopf algebra. More generally, for any braided vector space of diagonal type $V$ with a principal realization in the category of Yetter-Drinfeld modules of a cosemisimple Hopf algebra $H$ and such that the Nichols algebra $\mathfrak{B}(V)$ is finite-dimensional, thus presented by a finite set $\mathcal G$ of relations, we define a family of Hopf algebras $\mathfrak{u}(\boldsymbolλ)$, $\boldsymbolλ\in \Bbbk^{\mathcal G}$, which are c…
Nichols algebras over groups with finite root system of rank two II
我们对所有非阿贝尔群G进行分类，使得存在G上的一对（V，W）绝对单Yetter-Drinfeld模，使得V与W的直和的Nichols代数在两个假设下是有限维的：V与W的辫的平方不是单位元，G由V与W的支集生成.作为推论，我们证明了这样的V和W的维数至多为6。作为一个工具，我们使用（V，W）的Weyl广群。
Hopf ideals of the generalized quantum double associated to skew-paired Nichols algebras
量化包络代数$U_q$被构造为与自然斜对$ τ: U^{\leq 0}_q \otimes U^{\geq 0}_q \to k $相关的广义量子双$ U^{\leq 0}_q \cmdbicross_τ U^{\geq 0}_q $的商。这个对偶被$\cmdcalD = (\cmdfrakB (V) \cmddotrtimes F) \cmdbicross_τ (\cmdfrakB (W) \cmddotrtimes G)$推广，其中$F$，$G$是交换群，$ V \in {}^F_F \cmdcalYD $，$ W \in {}^G_G \cmdcalYD $是Yetter-Drinfeld模，$ \cmdfrakB (V) $，$ \cmdfrakB (W) $是它们的Nichols代数.我们证明了$ \cmdcalD $的Hopf理想的一些结果，包括我们所谓的薄的Hopf理想的表征。作为应用，我们给出了由斜基元生成的特征为零的极小拟三角点Hopf代数的一个显式描述。
This note is an application of classification results for finite-dimensional Nichols algebras over groups. We apply these results to generalizations of Fomin--Kirillov algebras to complex reflection groups. First, we focus on the case of cyclic groups where the corresponding Nichols algebras are only finite-dimensional up to order four, and we include results about the existence of Weyl groupoids and finite-dimensional Nichols subalgebras for this class. Second, recent results by Heckenberger--Vendramin [ArXiv e-prints, 1412.0857 (December 2014)] on the classification of Nichols algebras of se…
Finite-dimensional Nichols algebras of simple Yetter-Drinfeld modules (over groups) of prime dimension
该研究在特征零域下确定了所有生成有限维Nichols代数的素维数群上绝对不可约Yetter-Drinfeld模，通过引入辫子向量空间的阶并研究其退化与特化实现了这一目标。
Posets of finite GK-dimensional graded pre-Nichols algebras of diagonal type
本文分类了具有有限 Gelfand-Kirillov 维数的对角型分次 pre-Nichols 代数，通过正根子集的偏序集同构进行刻画，推广了小量子群的中心扩张，并分类了相关幂么代数群函数代数的分次商。
Over fields of arbitrary characteristic we classify all braid-indecomposable tuples of at least two absolutely simple Yetter-Drinfeld modules over non-abelian groups such that the group is generated by the support of the tuple and the Nichols algebra of the tuple is finite-dimensional. Such tuples are classified in terms of analogs of Dynkin diagrams which encode much information about the Yetter-Drinfeld modules. We also compute the dimensions of these finite-dimensional Nichols algebras. Our proof uses the Weyl groupoid of a tuple of simple Yetter-Drinfeld modules.
In this paper we give the relationship between the connected components of pure generalized Dynkin graphs and Nichols braided Lie algebras.
Lecture notes on Nichols algebras
本文从范畴论视角介绍了Nichols代数的定义、性质及其在量子群表示范畴构建中的作用，同时探讨其在共形场论中的应用。
Twist equivalence for Nichols algebras over Coxeter groups
本文证明Coxeter群上与反射类及2-上循环q^+相关的Nichols代数扭等价于常数上循环情形，从而得到相同Hilbert级数及二次性等价，并补全二面体群上有限维Nichols代数的分类。
After the classification of the finite-dimensional Nichols algebras of diagonal type arXiv:math/0411477, arXiv:math/0605795, the determination of its defining relations arXiv:1008.4144, arXiv:1104.0268, and the verification of the generation in degree one conjecture arXiv:1104.0268, there is still one step missing in the classification of complex finite-dimensional Hopf algebras with abelian group, without restrictions on the order of the latter: the computation of all deformations or liftings. A technique towards solving this question was developed in arXiv:1212.5279, built on cocycle deforma…
We introduce the concept of a covering of a graded pointed Hopf algebra. The theory developed shows that coverings of a bosonized Nichols algebra can be concretely expressed by biproducts using a quotient of the universal coalgebra covering group of the Nichols algebra. If there are enough quadratic relations, the universal coalgebra covering is given by the bosonization by the enveloping group of the underlying rack.
Partial (co)actions of Taft and Nichols Hopf algebras on algebras
该研究刻画了Taft与Nichols Hopf代数在代数上的适宜部分（余）作用，证明其具有对称性，且特定代数的全部此类作用均被涵盖，推广了作者此前的相关研究成果。
Quadratic Lie Algebras
本文给出了[A. Ardizzoni，\emph{A First Sight Towards Primitively Generated Connected Braided Bialgebras}，已提交。（arXiv：0805.3391v3）]专门用于其尼科尔斯代数是二次代数的编织向量空间的情况。在这种设置的分类泛包络代数的编织向量空间的尺寸不大于2处理。作为应用，我们研究了连续生成的连通辫双代数的结构，它的本原元的辫向量空间构成一个Nichols代数，而Nichols代数是二次代数。
Classification of PM Quiver Hopf Algebras
我们用参数描述了一类非霍普夫代数.这导致了多重Taft代数的分类以及指向的Yetter-Drinfeld模和它们对应的Nichols代数。特别地，当基场$k$是复域，$G$是有限阿贝尔群时，我们对$G$上的多重Taft霍普夫代数，$G$上的多重Taft代数和$^{kG}_{kG} {\cal YD}$上的Nichols代数进行了分类.证明了复半单李代数的量子包络代数是半路霍普夫代数的商.
We present new examples of finite-dimensional Nichols algebras over fields of positive characteristic. The corresponding braided vector spaces are not of diagonal type, admit a realization as Yetter-Drinfeld modules over finite abelian groups and are analogous to braidings over fields of characteristic zero whose Nichols algebras have finite Gelfand-Kirillov dimension described in arXiv:1606.02521. We obtain nex examples of finite-dimensional pointed Hopf algebras by bosonization with group algebras of suitable finite abelian groups.
Finite-dimensional Pointed or Copointed Hopf algebras over affine racks
本文研究了任意有限群的仿射架和常余圈上的Nichols代数的点提升或余点提升，这些架允许一个主YD-实现。在copointed的情况下，我们完成了分类的六仿射架的尼科尔斯代数是已知的有限维。在特定的情况下，我们的方法允许我们完成其中的四个。在所有的情形下，所得到的霍普夫代数都是其相应的分次霍普夫代数的余圈变形.它们都是非阿贝尔群上有限维余点或点霍普夫代数的新例子。
In positive characteristic the Jordan plane covers a finite-dimensional Nichols algebra that was described by Cibils, Lauve and Witherspoon and we call the restricted Jordan plane. In this paper the characteristic is odd. The defining relations of the Drinfeld double of the restricted Jordan plane are presented and its simple modules are determined. A Hopf algebra that deserves the name of double of the Jordan plane is introduced and various quantum Frobenius maps are described. The finite-dimensional pre-Nichols algebras intermediate between the Jordan plane and its restricted version are cla…
On the finite generation of the cohomology of bosonizations
该研究拓展已有结果，利用（霍普夫）代数的形变序列证明合适的辫子霍普夫代数经有限维霍普夫代数玻色化后上同调有限生成，结果可推广至smash积，还复现了已知结论并给出新的例子。
Higher Fomin-Kirillov algebras are infinite-dimensional
本文证明Fomin-Kirillov代数$FK_n$在$n\geq6$时无限维，回答1997年猜想，通过发展部分半简化方法处理Nichols代数。
Fomin-Kirillov algebras
这是在Oberwolfach迷你研讨会“尼科尔斯代数和Weyl群胚”在2012年10月的谈话的扩展摘要。
PBW bases for a class of braided Hopf algebras
证明了P（R）的辫子子空间V生成的辫子霍普夫代数R的Poincare-Birkhoff-Witt型基的存在性，如果V上的辫子满足三角条件.我们将所得结果应用于具有交换余根的点霍普夫代数和低维单模U_q（sl_2）的Nichols代数。
Computing finite Weyl groupoids
该研究提出算法计算对角型Nichols代数与余伴随李超代数的广义根系，进而得到计算对应Kharchenko PBW基中Lyndon字及超字的算法，为获得有限根系对角型Nichols代数的极小表示提供关键数据。
In this paper we focus on algebraic aspects of contractions of Lie and Leibniz algebras. The rigidity of algebras plays an important role in the study of their varieties. The rigid algebras generate the irreducible components of this variety. We deal with Leibniz algebras which are generalizations of Lie algebras. In Lie algebras case, there are different kind of rigidities (rigidity, absolutely rigidity, geometric rigidity and e.c.t.). We explore the relations of these rigidities with Leibniz algebra rigidity. Necessary conditions for a Lie algebra to be Leibniz rigid are discussed.
A survey on stability and rigidity results for Lie algebras
对李代数、李子代数和李代数同态的稳定性和刚性的已知结果给出了简单而统一的证明。此外，我们还研究了当一个李代数同态在其上域的所有自同构（包括外自同构）下稳定时的情形。
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