We apply a combinatorial formula of the first author and Rosso, for products in Hopf quiver algebras, to determine the structure of Nichols algebras. We illustrate this technique by explicitly constructing new examples of Nichols algebras in positive characteristic. We further describe the corresponding Radford biproducts and some liftings of these biproducts, which are new finite dimensional pointed Hopf algebras.
We formulate the generation of finite dimensional pointed Hopf algebras by group-like elements and skew-primitives in geometric terms. This is done through a more general study of connected and coconnected Hopf algebras inside a braided fusion category $\mathcal{C}$. We describe such Hopf algebras as orbits for the action of a reductive group on an affine variety. We then show that the closed orbits are precisely the orbits of Nichols algebras, and that all other algebras are therefore deformations of Nichols algebras. For the case where the category $\mathcal{C}$ is the category $^G_G\mathcal…
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.
Let $H_8$ be the neither commutative nor cocommutative semisimple eight dimensional Hopf algebra, which is also called Kac-Paljutkin algebra \cite{MR0208401}. All simple Yetter-Drinfel'd modules over $H_8$ are given. As for simple objects and direct sums of two simple objects in ${}_{H_8}^{H_8}\mathcal{YD}$, we calculated dimensions for the corresponding Nichols algebras, except four semisimple cases which are generally difficult. Under the assumption that the four undetermined Nichols algebras are all infinite dimensional, we determine all the finite dimensional Nichols algebras over $H_8$. I…
We show that the definition of unrolled Hopf algebras can be naturally extended to the Nichols algebra $\mathcal{B}$ of a Yetter-Drinfeld module $V$ on which a Lie algebra $\mathfrak g$ acts by biderivations. Specializing to Nichols algebras of diagonal type, we find unrolled versions of the small, the De Concini-Procesi and the Lusztig divided power quantum group, respectively.
Classifying partial (co)actions of Taft and Nichols Hopf algebras on their base fields
Instituto Federal Sul-rio-grandense（南里奥格兰德联邦学院）；Universidade Federal do Rio Grande（里奥格兰德联邦大学）；IME, Universidade Federal do Rio Grande do Sul（里约格兰德州立大学数学与统计学院）
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 complete the classification of Hopf algebras whose infinitesimal braiding is a principal Yetter-Drinfeld realization of a braided vector space of Cartan type $G_2$ over a cosemisimple Hopf algebra. We develop a general formula for a class of liftings in which the quantum Serre relations hold. We give a detailed explanation of the procedure for finding the relations, based on the recent work of Andruskiewitsch, Angiono and Rossi Bertone.
Let $H$ be the dual of $16$-dimensional nontrivial semisimple Hopf algebra $H_{b:1}$ in the classification work of Kashina \cite{K00}. We completely determine all finite-dimensional Nichols algebras satisfying $\mathcal{B}(N)\cong \bigotimes_{i\in I}\mathcal{B}(N_i)$, where $N=\bigoplus_{i\in I}N_i$, each $N_i$ is a simple object in $_H^H\mathcal{YD}$. Under this assumption, we classify all those Hopf algebras of finite-dimensional growth from the semisimple Hopf algebra $H$ via the relevant Nichols algebras $\mathcal B(N)$.
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。
We construct finite-dimensional Hopf algebras whose coradical is the group algebra of a central extension of an abelian group. They fall into families associated to a semisimple Lie algebra together with a Dynkin diagram automorphism. We show conversely that every finite-dimensional pointed Hopf algebra over a non-abelian group with non-simple infinitesimal braiding of rank at least 4 is of this form. We follow the steps of the Lifting Method by Andruskiewitsch--Schneider. Our starting point is the classification of finite-dimensional Nichols algebras over non-abelian groups by Heckenberger--V…
We translate the concept of restriction of an arrangement in terms of Hopf algebras. In consequence, every Nichols algebra gives rise to a simplicial complex decorated by Nichols algebras with restricted root systems. As applications, some of these Nichols algebras provide Weyl groupoids which do not arise for Nichols algebras over finite groups and in fact we realize all root systems of finite Weyl groupoids of rank greater than three. Further, our result explains the root systems of the folded Nichols algebras over nonabelian groups and of generalized Satake diagrams.
Nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classification of pointed Hopf algebras by lifting method of Andruskiewitsch and Schneider. Arithmetic root systems are invariants of Nichols algebras of diagonal type with a certain finiteness property. In the present paper, all rank 4 Nichols algebras of diagonal type with a finite arithmetic root system over fields of arbitrary characteristic are classified. Our proof uses the classification of the finite arithmetic root systems of rank 4.
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…
Using the standard filtration associated with a generalized lifting method, we determine all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose coradical generates a Hopf subalgebra isomorphic to the smallest non-pointed non-cosemisimple Hopf algebra ${\mathcal{K}}$ and the corresponding infinitesimal module is an indecomposable object in ${}^{\mathcal{K}}_{\mathcal{K}}\mathcal{YD}$ (we assume that the diagrams are Nichols algebras). As a byproduct, we obtain new Nichols algebras of dimension 8 and new Hopf algebras of dimension 64.
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…
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 present a framework for the computation of the Hopf 2-cocycles involved in the deformations of Nichols algebras over semisimple Hopf algebras. We write down a recurrence formula and investigate the extent of the connection with invariant Hochschild cohomology in terms of exponentials. As an example, we present detailed computations leading to the explicit description of the Hopf 2-cocycles involved in the deformations of a Nichols algebra of Cartan type $A_2$ with $q=-1$, a.k.a. the positive part of the small quantum group $\mathfrak{u}^+_{\sqrt{\text{-1}}}(\mathfrak{sl}_3)$. We show that t…
We present new Hopf algebras with the dual Chevalley property by determining all semisimple Hopf algebras Morita-equivalent to a group algebra over a finite group, for a list of groups supporting a non-trivial finite-dimensional Nichols algebra.
A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver
V. Ginzburg，并独立地R. Bocklandt和L. Le Bruyn，定义了一个无限维的“项链”李代数，它与任何一个李代数正则相关。根据V.Turaev，P.Etingof和Ginzburg的建议，我们定义了一个上括号，并证明了它定义了一个李双代数结构。给出了一个量子化该李双代数的霍普夫代数，并证明了它是一个满足PBW性质的霍普夫代数.我们提出的表示到空间上的微分算子的表示，它的轨迹表示的李代数的Ginzburg。
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.
In this paper, we continue to investigate finite-dimensional Nichols algebras over simple Yetter-Drinfeld modules of the Suzuki algebras $A_{N\, n}^{μλ}$. It is finished for the case $A_{N\, 2n}^{μλ}$. As for the case $A_{N\, 2n+1}^{μλ}$, it boils down to the long-standing open problem: calculate dimensions of Nichols algebras of dihedral rack type $\Bbb D_{2n+1}$. It is interesting to see that the Suzuki algebras are of set-theoretical type. We pose some question or problems for our future research. In particular, we are curious about how to generalize the correspondence between braidings of …
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.
