I. Heckenberger
The method of subquotients is developed and used to determine all finite dimensional rank 2 Nichols algebras of diagonal type over an arbitrary field of characteristic zero. Key Words: Hopf algebra, Nichols algebra
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The paper introduces a new method to determine all rank two Nichols algebras of diagonal type over fields of positive characteristic.
We determine the multiplicities of a class of roots for Nichols algebras of diagonal type of rank two, and identify the corresponding root vectors. Our analysis is based on a precise description of the relations of the Nichols algebra in the corresponding degrees.
Over fields of arbitrary characteristic we classify all rank three Nichols algebras of diagonal type with a finite root system. Our proof uses the classification of the finite Weyl groupoids of rank three.
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.
In this paper, our main aim is to determine the multiplicities of a class of roots for Nichols algebra of diagonal type over fields of arbitrary characteristic, which is a generalization of the results on the multiplicities of these roots over fields of characteristic zero obtained by I. Heckenberber and the author.
Yetter--Drinfel'd modules of diagonal type admit an equivalence relation which conjecturally preserves dimension and Gel'fand--Kirillov dimension of the corresponding Nichols algebras. This relation is determined explicitly for all rank 2 Yetter--Drinfeld modules where the Gel'fand--Kirillov dimension is known to be finite. Key Words: Brandt groupoid, Hopf algebra, pseudo-reflections, Weyl group
This paper is devoted to explore the freeness of Nichols algebras of diagonal type and to determine the dimension of the kernel of the shuffle map considered as an operator acting on the free algebra. Our proof is based on an inequality for the number of Lyndon words and on an identity for the shuffle map. For a particular family of examples, the freeness of the Nichols algebra is characterized in terms of solutions of a quadratic diophantine equation.
Nichols algebras naturally appear in the classification of finite dimensional pointed Hopf algebras. Assuming only that the base field has characteristic zero several new finite dimensional rank 2 Nichols algebras of diagonal type are listed. Each of them is described in terms of generators and relations. A Poincaré--Birkhoff--Witt basis of all of these Nichols algebras is given and their dimension is computed. Together with the continuation of this paper this gives an answer to a question of Andruskiewitsch. Key Words: Hopf algebra, Lyndon words, full binary tree
We obtain a presentation by generators and relations of any Nichols algebra of diagonal type with finite root system. We prove that the defining ideal is finitely generated. The proof is based in Kharchenko's theory of PBW basis of Lyndon words. We prove that the lexicographic order on Lyndon words is convex for such PBW generators and so the PBW basis is orthogonal with respect to the canonical non-degenerate form associated to the Nichols algebra.
The theory of Nichols algebras of diagonal type is known to be closely related to that of semisimple Lie algebras. In this paper the connection between both theories is made closer. For any Nichols algebra of diagonal type invertible transformations are introduced, which remind one of the action of the Weyl group on the root system associated to a semisimple Lie algebra. They give rise to the definition of a Brandt groupoid. As an application an alternative proof of classification results of Rosso, Andruskiewitsch, and Schneider is obtained without using any technical assumptions on the braidi…
Higher rank Nichols algebras of diagonal type with finite arithmetic root system in positive characteristic
本文分类了正特征域上秩r≥5且具有有限不可约根系的对角型Nichols代数，通过Weyl群胚、算术根系和Cartan图证明其对应于表1中的广义Dynkin图。
Between the braided vector spaces of diagonal type there exist some families whose associated Nichols algebras are infinite dimensional but with finite Gelfand-Kirillov dimension, called one-parameter families. We show that every pre-Nichols algebra of finite Gelfand-Kirillov dimension of this kind with connected diagram is necessarily the corresponding Nichols algebra, up to few exceptions. For each one of these exceptions we present a proper pre-Nichols algebra by generators and relations, and prove that it is the corresponding eminent pre-Nichols algebra. This paper essentially ends the det…
Let $(V,c)$ be a finite-dimensional braided vector space of diagonal type. We show that the Gelfand Kirillov dimension of the Nichols algebra $\mathfrak{B}(V)$ is finite if and only if the corresponding root system is finite, that is $\mathfrak{B}(V)$ admits a PBW basis with a finite number of generators. This had been conjectured in arXiv:1606.02521 and proved for $\dim V=2,3$ in arXiv:1803.08804, arXiv:2106.10143 respectively.
In a vertex algebra setting, we consider non-local screening operators associated to the basis of any non-integral lattice. We have previously shown that, under certain restrictions, these screening operators satisfy the relations of a quantum shuffle algebra or Nichols algebra associated to a diagonal braiding, which encodes the non-locality and non-integrality. In the present article, we take all finite-dimensional diagonal Nichols algebras, as classified by Heckenberger, and find all lattice realizations of the braiding that are compatible with reflections. Usually, the realizations are uni…
Let $V$ be a braided vector space of diagonal type. Let $\mathfrak B(V)$, $\mathfrak L^-(V)$ and $\mathfrak L(V)$ be the Nichols algebra, Nichols Lie algebra and Nichols braided Lie algebra over $V$, respectively. We show that a monomial belongs to $\mathfrak L(V)$ if and only if that this monomial is connected. We obtain the basis for $\mathfrak L(V)$ of arithmetic root systems and the dimension for $\mathfrak L(V)$ of finite Cartan type. We give the sufficient and necessary conditions for $\mathfrak B(V) = F\oplus \mathfrak L^-(V)$ and $\mathfrak L^-(V)= \mathfrak L(V)$. We obtain an explici…
We give an explicit and essentially minimal list of defining relations of a Nichols algebra of diagonal type with finite root system. This list contains the well-known quantum Serre relations but also many new variations. A conjecture by Andruskiewitsch and Schneider states that any finite-dimensional pointed Hopf algebra over an algebraically closed field of characteristic zero is generated as an algebra by its group-like and skew-primitive elements. As an application of our main result, we prove the conjecture when the group of group-like elements is abelian.
The Nichols algebras of diagonal type with finite root system are either of standard, super or (yet) unidentified type. A concrete description of the defining relations of all those Nichols algebras was given in \cite{A-exp presentation}. In the present paper we use this result to give an explicit presentation of all Nichols algebras of unidentified type.
Let $\mathcal{B}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix $\mathfrak{q} \in \mathbf{k}^{θ\times θ}$, where $\mathbf{k}$ is an algebraically closed field of characteristic 0. Let $\mathcal{L}_{\mathfrak{q}}$ be the Lusztig algebra associated to $\mathcal{B}_{\mathfrak{q}}$, see http://arxiv.org/abs/1501.04518. We present $\mathcal{L}_{\mathfrak{q}}$ as an extension (as braided Hopf algebras) of $\mathcal{B}_{\mathfrak{q}}$ by $\mathfrak Z_{\mathfrak{q}}$ where $\mathfrak Z_{\mathfrak{q}}$ is isomorphic to the universal enveloping algebra…
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.
The concept of arithmetic root systems is introduced. It is shown that there is a one-to-one correspondence between arithmetic root systems and Nichols algebras of diagonal type having a finite set of (restricted) Poincare'-Birkhoff-Witt generators. This has strong consequences for both objects. As an application all rank 2 Nichols algebras of diagonal type having a finite set of (restricted) Poincare'-Birkhoff-Witt generators are determined. Key Words: Brandt groupoid, Hopf algebra, pseudo-reflections, Weyl group
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.
In this paper, the author gives a complete set of simple Yetter-Drinfeld modules over Suzuki algebra $A_{N\,2n+1}^{μλ}$ and investigates the Nichols algebras over those irreducible Yetter-Drinfeld modules. The finite dimensional Nichols algebras of diagonal type are of Cartan type $A_1$, $A_1\times A_1$, $A_2$, Super type ${A}_{2}(q;I_2)$ and the Nichols algebra $\mathfrak{ufo}(8)$. And the involved finite dimensional Nichols algebras of non-diagonal type are $12$, $4m$ and $m^2$ dimensional. The left three unsolved cases are set as open problems.
We show that every finite GK-dimensional pre-Nichols algebra for braidings of diagonal type with connected diagram of modular, supermodular or unidentified type is a quotient of the distinguished pre-Nichols algebra introduced by the first-named author, up to two exceptions. For both of these exceptional cases, we provide a pre-Nichols algebra that substitutes the distinguished one in the sense that it projects onto all finite GK-dimensional pre-Nichols algebras. We build these two substitutes as non-trivial central extensions with finite GK-dimension of the corresponding distinguished pre-Nic…
Let $H$ be the quantum double of a Nichols algebra of diagonal type. We compute the $R$-matrix of 3-uples of modules for general finite-dimensional highest weight modules over $H$. We calculate also a multiplicative formula for the universal $R$-matrix when $H$ is finite dimensional.
The simple finite-dimensional modules over the Drinfeld double of the bosonization of the Nichols algebra $\mathfrak{ufo}(7)$ are classified.
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 construct reflection functors for Yetter-Drinfeld modules over Nichols systems and discuss their fundamental properties. We will obtain properties about the geometry of the support of Nichols systems and their Yetter-Drinfeld modules, by looking at iterated reflections. We will also study the maximal subobject of Yetter-Drinfeld modules over Nichols systems and find a special morphism, that we name Shapovalov morphism, whose kernel coincides with this maximal subobject. Moreover, we will use this morphism to characterize properties about the reflections of the Yetter-Drinfeld modules. We ca…
Examples of Nichols algebras associated to upper triangular solutions of the Yang-Baxter equation in rank 3
本研究确定了若干带有非平凡二次关系的Nichols代数，这些代数与三维Yang-Baxter方程的多类上三角解相关，丰富了相关代数结构的研究实例。
Assume that $V$ is a braided vector space with diagonal type. It is shown that a monomial belongs to Nichols braided Lie algebra $\mathfrak L(V)$ if and only if this monomial is connected. A basis of Nichols braided Lie algebra and dimension of Nichols braided Lie algebra of finite Cartan type are obtained.
Liftings of Nichols algebras of type $B_3$
该研究在特征零的代数闭域上给出了有限阿贝尔群上对角Cartan型$B_3$的所有Nichols代数的提升对应的有限维尖点Hopf代数族的显式表示。
A technique is developed to reduce the investigation of Nichols algebras of integrable U_q(g)-modules to the investigation of Nichols algebras of braided vector spaces with diagonal braiding. The results are applied to obtain information on the Gelfan'd-Kirillov dimension of these Nichols algebras and their defining relations if the braiding is of a special type and q is not a root of unity. For simple g the cases of finite Gelfand-Kirillov dimension are determined completely.
We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, $\operatorname{GKdim}$ for short, through the study of Nichols algebras over abelian groups. We deal first with braided vector spaces over $\mathbb Z$ with the generator acting as a single Jordan block and show that the corresponding Nichols algebra has finite $\operatorname{GKdim}$ if and only if the size of the block is 2 and the eigenvalue is $\pm 1$; when this is 1, we recover the quantum Jordan plane. We consider next a class of braided vector spaces that are direct sums of blocks and points that …
We compute liftings of the Nichols algebra of a Yetter-Drinfeld module of Cartan type $B_2$ subject to the small restriction that the diagonal elements of the braiding matrix are primitive $n$th roots of 1 with odd $n\neq 5$. As well, we compute the liftings of a Nichols algebra of Cartan type $A_2$ if the diagonal elements of the braiding matrix are cube roots of 1; this case was not completely covered in previous work of Andruskiewitsch and Schneider. We study the problem of when the liftings of a given Nichols algebra are quasi-isomorphic. The Appendix (with I. Rutherford) contains a genera…
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 …
Nichols algebras are a fundamental building block of pointed Hopf algebras. Part of the classification program of finite-dimensional pointed Hopf algebras with the lifting method of Andruskiewitsch and Schneider is the determination of the liftings, i.e., all possible deformations of a given Nichols algebra. Based on recent work of Heckenberger about Nichols algebras of diagonal type we compute explicitly the liftings of all Nichols algebras with Cartan matrix of type A_2, some Nichols algebras with Cartan matrix of type B_2, and some Nichols algebras of two Weyl equivalence classes of non-sta…
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…
In the structure theory of quantized enveloping algebras, the algebra isomorphisms determined by Lusztig led to the first general construction of PBW bases of these algebras. Also, they have important applications to the representation theory of these and related algebras. In the present paper the Drinfel'd double for a class of graded Hopf algebras is investigated. Various quantum algebras, including multiparameter quantizations of semisimple Lie algebras and of Lie superalgebras, are covered by the given definition. For these Drinfel'd doubles Lusztig maps are defined. It is shown that these…
It is shown that if $\mathfrak B(V) $ is connected Nichols algebra of diagonal type with $\dim V>1$, then $\dim (\mathfrak L^-(V)) = \infty$ $($resp. $ \dim (\mathfrak L(V)) = \infty $$)$ $($ resp. $ \dim (\mathfrak B(V)) = \infty $$)$ if and only if $Δ(\mathfrak B(V)) $ is an arithmetic root system and the quantum numbers (i.e. the fixed parameters) of generalized Dynkin diagrams of $V$ are of finite order. Sufficient and necessary conditions for $m$-fold adjoint action in $\mathfrak B(V)$ equal to zero, viz. $\overline{l}_{x_{i}}^{m}[x_{j}]^ -=0$ for $x_i,~x_j\in \mathfrak B(V)$, are given.
Let $H$ be the Hopf algebra $H_{c: σ_{0}}$ of Kashina [J. Algebra, 232(2000),pp.617-663]. We give all simple Yetter-Drinfel'd modules $V$ over $H$, then classify all finite-dimensional Nichols algebras of $V$. The finite dimensional Nichols algebras of diagonal type are either $A_{1}, A_{2}$ or quantum planes, and non-diagonal type ones are $8$ or $16$ dimensional.
The Suzuki algebra $A_{Nn}^{μλ}$ was introduced by Suzuki Satoshi in 1998, which is a class of cosemisimple Hopf algebras. It is not categorically Morita-equivalent to a group algebra in general. In this paper, the author gives a complete set of simple Yetter-Drinfeld modules over the Suzuki algebra $A_{N\,2n}^{μλ}$ and investigates the Nichols algebras over those simple Yetter-Drinfeld modules. The involved finite dimensional Nichols algebras of diagonal type are of Cartan type $A_1$, $A_1\times A_1$, $A_2$, $A_2\times A_2$, Super type ${\bf A}_{2}(q;I_2)$ and the Nichols algebra ufo(8). Ther…
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。
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 $\…
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…
On the Cartan Graphs of Nichols Algebras over Coquasi-Hopf Algebras
研究余拟霍普夫代数上Nichols代数反射理论，证明特定条件下半Cartan图是Cartan图，给出Nichols代数有限维等价条件，表明Cartan图在特定辫幺半等价下不变，并应用于对角型Nichols代数。
We provide a tool to obtain local descriptions of quiddity cycles. As an application, we classify rank two affine Nichols algebras of diagonal type.
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 extend the main result of [N. Andruskiewitsch and H.-J. Schneider, A characterization of quantum groups], see math/0201095, to braided vector spaces of generic diagonal type using results of Heckenberger.
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 $n$.
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.
We construct symmetric pairs for Drinfeld doubles of pre-Nichols algebras of diagonal type and determine when they possess an Iwasawa decomposition. This extends G. Letzter's theory of quantum symmetric pairs. Our results can be uniformly applied to Kac-Moody quantum groups for a generic quantum parameter, for roots of unity in respect to both big and small quantum groups, to quantum supergroups and to exotic quantum groups of ufo type. We give a second construction of symmetric pairs for Heisenberg doubles in the above generality and prove that they always admit an Iwasawa decomposition. For …
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.
We study the factorization of the Hilbert series and more general graded traces of a Nichols algebra into cyclotomic polynomials. Nichols algebras play the role of Borel subalgebras of finite-dimensional quantum groups, so this observation can be viewed as an analog to the factorization of the order of finite Lie groups into cyclotomic polynomials. We prove results on this factorization and give a table of many examples of rank 1 over nonabelian groups.
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,…
We classify all groups G and all pairs (V,W) of absolutely simple Yetter-Drinfeld modules over G such that the support of the direct sum of V and W generates G, the square of the braiding between V and W is not the identity, and the Nichols algebra of the direct sum of V and W admits a finite root system. As a byproduct, we determine the dimensions of such Nichols algebras, and several new families of finite-dimensional Nichols algebras are obtained. Our main tool is the Weyl groupoid of pairs of absolutely simple Yetter-Drinfeld modules over groups.
This is a contribution to the classification program of pointed Hopf algebras. We give a generalization of the quantum Serre relations and propose a generalization of the Frobenius-Lusztig kernels in order to compute Nichols algebras of diagonal group type. With this, we classify Nichols algebras B(V) with dimension < 32 or with dimension p^3, p a prime number, when V lies in a Yetter-Drinfeld category over a finite group. With the so called Lifting Procedure, this allows to classify pointed Hopf algebras of index < 32 or p^3. In recent articles by Etingof-Schedler-Soloviev and Lu-Yan-Zhu, the…
In this paper we define two Lie operations, and with that we define the bicharacter algebras, Nichols bicharacter algebras, quantum Nichols bicharacter algebras, etc. We obtain explicit bases for $\mathfrak L(V)${\tiny $_{R}$} and $\mathfrak L(V)${\tiny $_{L}$} over (i) the quantum linear space $V$ with $\dim V=2$; (ii) a connected braided vector $V$ of diagonal type with $\dim V=2$ and $p_{1,1}=p_{2,2}= -1$. We give the sufficient and necessary conditions for $\mathfrak L(V)${\tiny $_{R}$}$= \mathfrak L(V)$, $\mathfrak L(V)${\tiny $_{L}$}$= \mathfrak L(V)$, $\mathfrak B(V) = F\oplus \mathfrak…
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。
The paper purposes to contribute to the classification of pointed Hopf algebras by the method of Andruskiewitsch and Schneider. The structure of arithmetic root systems is enlightened such that their relation to ordinary root systems associated to semi-simple Lie algebras becomes more astounding. As an application all arithmetic root systems of rank 3 are determined. The result gives in particular all finite dimensional rank 3 Nichols algebras of diagonal type over a field of characteristic zero. Key Words: Weyl groupoid, Hopf algebra, Nichols algebra
We classify all non-abelian groups G such that there exists a pair (V,W) of absolutely simple Yetter-Drinfeld modules over G such that the Nichols algebra of the direct sum of V and W is finite-dimensional under two assumptions: the square of the braiding between V and W is not the identity, and G is generated by the support of V and W. As a corollary, we prove that the dimensions of such V and W are at most six. As a tool we use the Weyl groupoid of (V,W).
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.
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