A fundamental step in the classification of finite-dimensional complex pointed Hopf algebras is the determination of all finite-dimensional Nichols algebras of braided vector spaces arising from groups. The most important class of braided vector spaces arising from groups is the class of braided vector spaces (CX, c^q), where C is the field of complex numbers, X is a rack and q is a 2-cocycle on X with values in C^*. Racks and cohomology of racks appeared also in the work of topologists. This leads us to the study of the structure of racks, their cohomology groups and the corresponding Nichols algebras. We will show advances in these three directions. We classify simple racks in group-theoretical terms; we describe projections of racks in terms of general cocycles; we introduce a general cohomology theory of racks contaninig properly the existing ones. We introduce a "Fourier transform" on racks of certain type; finally, we compute some new examples of finite-dimensional Nichols algebras.
In this paper we focus on a certain self-distributive multiplication on coalgebras, which leads to so-called rack bialgebra. Inspired by semi-group theory (adapting the Suschkewitsch theorem), we do some structure theory for rack bialgebras and cocommutative Hopf dialgebras. We also construct canonical rack bialgebras (some kind of enveloping algebras) for any Leibniz algebra and compare to the existing constructions. We are motivated by a differential geometric procedure which we call the Serre functor: To a pointed differentible manifold with multiplication is associated its distribution spa…
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.
Let H be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed field of characteristic zero. In this paper we describe all annihilator ideals of indecomposable H-modules by generators. In particular, we give the classification of all ideals of finite-dimensional pointed Hopf algebra of rank one of nilpotent type over Klein 4-group.
Let $\mathbb{k}$ be an algebraically closed field. We give a complete classification of non-connected pointed Hopf algebras of dimension $16$ with char$\,\mathbb{k}=2$ that are generated by group-like elements and skew-primitive elements. It turns out that there are infinitely many classes (up to isomorphism) of pointed Hopf algebras of dimension 16. In particular, we obtain infinitely many new examples of non-commutative non-cocommutative finite-dimensional pointed Hopf algebras.
We study the realizations of certain braided vector spaces of rack type as Yetter-Drinfeld modules over a cosemisimple Hopf algebra $H$. We apply the strategy developed in arXiv:1212.5279 to compute their liftings and use these results to obtain the classification of finite-dimensional copointed Hopf algebras over $\mathbb{S}_4$.
We compute higher Frobenius-Schur indicators of pq-dimensional pointed Hopf algebras in characteristic p through their associated graded Hopf algebras. These indicators are gauge invariants for the monoidal categories of representations of these algebras.
群为S_3的有限维尖霍普夫代数的分类在Andruskiewitsch，Heckenberger和Schneider的“The Nichols algebra of a semisimple Yetter-Drinfeld module”，arXiv：0803.2430v1 [math.QA]中完成：它们中正好有两个，12维Nichols代数的玻色化和非平凡提升。在这里，我们确定所有的单模在任何这些霍普夫代数。我们还找到了单模的投射覆盖Gabriel箭图，并证明了它们不是有限表示型的.为此，我们首先研究了Andruskiewitsch和Graña的论文“Examples of liftings of Nichols algebras over racks”和G. Garcia和作者，他们对类群的群的限制是一维模的直和。
本文综述了特征为0的代数闭域上的点霍普夫代数。我们提出对点霍普夫代数$A$进行分类，首先确定与$A$的余根滤子相关的分次霍普夫代数$\gr A$. $A_{0}$ -共不变元形成了一个辫子Hopf代数$R$，它是上基$A_{0} = \ku Γ$，$Γ$，$A$和$\gr A \simeq R # A_{0}$上的类群元群上的Yetter-Drinfeld模范畴。我们称$R$的原始元素的编织为$A$的无穷小编织。如果这种编织是Cartan型\cite{AS2}，那么通常可以确定$R$，证明$R$是由其本原元素生成的代数，最后计算所有的变形或提升，即指向霍普夫代数，使得$\gr A \simeq R # \ku Γ$。在最后一章中，作为该方法的一个具体例子，我们明确地描述了所有有限维点霍普夫代数$A$，它具有类群$G(A)$的交换群和$A_{n}$型的无穷小辫子（直到某些例外情况）。换句话说，我们计算了$A_n$型的所有提升，这一结果是本文的主要新贡献。
We present new examples of finite-dimensional Nichols algebra over fields of characteristic 2 starting from braided vector spaces that are not of diagonal type, admit realizations as Yetter-Drinfeld modules over finite abelian groups and are analogous to braidings over fields of odd characteristic with finite-dimensional Nichols algebras presented in arXiv:1905.03074. As these last ones, they are related to the Nichols algebras of finite Gelfand-Kirillov dimension in characteristic 0 described in arXiv:1606.02521. New finite-dimensional pointed Hopf algebras over fields of characteristic 2 are…
We classify finite-dimensional Nichols algebras over finite nilpotent groups of odd order in group-theoretical terms. The main step is to show that the conjugacy classes of such finite groups are either abelian or of type C; this property also holds for finite conjugacy classes of finitely generated nilpotent groups whose torsion has odd order. To extend our approach to the setting of finite GK-dimension, we propose a new Conjecture on racks of type C. We also prove that the bosonization a of Nichols algebra of a Yetter-Drinfeld module over a group whose support is an infinite conjugacy class …
We survey a vast array of known results and techniques in the area of polynomial identities in pointed Hopf algebras. Some new results are proven in the setting of Hopf algebras that appeared in papers of D. Radford and N. Andruskiewitsch - H.-J. Schneider.
Finite Dimensional Pointed Hopf Algebras with Abelian Coradical and Cartan matrices
在以前的工作\cite{AS2}中，我们展示了如何附加到一个点的Hopf代数A与corradical $\kΓ$，一个严格分次的Hopf代数R的范畴$_Γ^Γ\Cal{YD}$的Yetter-Drinfeld模块$Γ$。本文考虑A的另一个不变量，即R的由本原元空间V生成的子代数R'。这类代数是已知的，因为开创性的工作尼科尔斯。结果表明，R'完全由编织物c决定：V \otimes V \to V \otimes V。我们表示R' = B（V）。我们进一步假设$Γ$是有限阿贝尔的。则c由矩阵（b_{ij}）给出，其元素是单位根;我们还假设它们具有奇数阶。我们介绍这些编织的概念，“编织的Cartan型”，我们附加一个广义Cartan矩阵的Cartan型编织。证明了B（V）是有限维的，如果它对应的矩阵是有限Cartan型的，并给出了逆命题的充分条件.作为结果，我们得到了许多新的家庭的点霍普夫代数。当$Γ$是素数阶群的副本的直和时，条件成立且任何矩阵是Cartan型的。作为一个例子，我们对有限维余根分次的、一次生成的且余根为奇素维p的点霍普夫代数进行了分类，并刻画了一次生成的p^4阶余根分次点霍普夫代数.
