On pointed Hopf algebras associated with the Mathieu simple groups
Fernando Fantino
备注：
41 pages. Abstract and Theorem 1 (Table 1) modified; Subsection 1.3 added; Sections 2-6 (in version 1) put together in a Section 2 (in new version); Section 3 added. References: [AS1] removed, [AF3], [AFGV], [AHS] added. Cite in Theorem 1.2 corrected; Lemmata 1.4 and 1.5 (of v1) put together in Lemma 1.4 (of nv); Lemma 1.5 (of nv) briefly modified
发表信息：
J. Algebra Appl., Vol 8, No. 5 (2009), Pages 633-672
Let G be a Mathieu simple group, s in G, O_s the conjugacy class of s and ρan irreducible representation of the centralizer of s. We prove that either the Nichols algebra B(O_s,ρ) is infinite-dimensional or the braiding of the Yetter-Drinfeld module M(O_s, ρ) is negative. We also show that if G=M22 or M24, then the group algebra of G is the only (up to isomorphisms) finite-dimensional complex pointed Hopf algebra with group-likes isomorphic to G.
We study the Calabi-Yau property of pointed Hopf algebra $U(\mc{D},\lmd)$ of finite Cartan type. It turns out that this class of pointed Hopf algebras constructed by N. Andruskiewitsch and H.-J. Schneider contains many Calabi-Yau Hopf algebras. To give concrete examples of new Calabi-Yau Hopf algebras, we classify the Calabi-Yau pointed Hopf algebras $U(\mc{D},\lmd)$ of dimension less than 5.
We present examples of color Hopf algebras, i.e. Hopf algebras in color categories (braided tensor categories with braiding induced by a bicharacter on an abelian group), related with quantum doubles of pointed Hopf algebras. We also discuss semisimple color Hopf algebras.
The quiver Hopf algebras are classified by means of ramification systems with irreducible representations. This leads to the classification of Nichols algebras over group algebras and pointed Hopf algebras of type one.
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 discuss the relationship between Hopf superalgebras and Hopf algebras. We list the braided vector spaces of diagonal type with generalized root system of super type and give the defining relations of the corresponding Nichols algebras.
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.
We give a presentation in terms of generators and relations of Hopf algebras generated by skew-primitive elements and abelian group of group-like elements with action given via characters. This class of pointed Hopf algebras has shown great importance in the classification theory and can be seen as generalized quantum groups. As a consequence we get an analog presentation of Nichols algebras of diagonal type.
We show that all finite dimensional pointed Hopf algebras with the same diagram in the classification scheme of Andruskiewitsch and Schneider are cocycle deformations of each other. This is done by giving first a suitable characterization of such Hopf algebras, which allows for the application of results by Masuoka about Morita-Takeuchi equivalence and by Schauenburg about Hopf Galois extensions. We also outline a method to describe the deforming cocycles involved using the exponential map and its q-analogue.
We give a necessary and sufficient PBW basis criterion for Hopf algebras generated by skew-primitive elements and abelian group of group-like elements with action given via characters. This class of pointed Hopf algebras has shown great importance in the classification theory and can be seen as generalized quantum groups. We apply the criterion to classical examples and liftings of Nichols algebras which were determined in arxiv.org:1003.5882.
Any finite-dimensional complex pointed Hopf algebra with group of group-likes isomorphic to a sporadic group, with the possible exception of the Fischer groups Fi22, the Baby Monster B and the Monster M, is a group algebra.
We show that every finite-dimensional complex pointed Hopf algebra with group of group-likes isomorphic to a sporadic group is a group algebra, except for the Fischer group Fi22, the Baby Monster and the Monster. For these three groups, we give a short list of irreducible Yetter-Drinfeld modules whose Nichols algebra is not known to be finite-dimensional.
Let k be an algebraically closed field of characteristic 0 and let D_m be the dihedral group of order 2m with m= 4t, with t bigger than 2. We classify all finite-dimensional Nichols algebras over D_m and all finite-dimensional pointed Hopf algebras whose group of group-likes is D_m, by means of the lifting method. Our main result gives an infinite family of non-abelian groups where the classification of finite-dimensional pointed Hopf algebras is completed. Moreover, it provides for each dihedral group infinitely many non-trivial new examples.
The classification of finite-dimensional pointed Hopf algebras with group S_3 was finished in "The Nichols algebra of a semisimple Yetter-Drinfeld module", arXiv:0803.2430v1 [math.QA], by Andruskiewitsch, Heckenberger and Schneider: there are exactly two of them, the bosonization of a Nichols algebra of dimension 12 and a non-trivial lifting. Here we determine all simple modules over any of these Hopf algebras. We also find the Gabriel quivers, the projective covers of the simple modules, and prove that they are not of finite representation type. To this end, we first investigate the modules o…
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 introduce a method to construct explicitly multiplicative 2-cocycles for bosonizations of Nichols algebras B(V) over Hopf algebras H. These cocycles arise as liftings of H-invariant linear functionals on V tensor V and give a close formula to deform braided commutator-type relations. Using this construction, we show that all known finite dimensional pointed Hopf algebras over the dihedral groups D_m with m=4t > 11, over the symmetric group S_3 and some families over S_4 are cocycle deformations of bosonizations of Nichols algebras.
This is a sequel to arXiv:0807.2406. It is shown that the Nichols algebras over the symmetric groups S_m, m > 4, are all infinite-dimensional, except (maybe) those related to the transpositions considered by Fomin and Kirillov, resp. Milinski and Schneider, and the class of type (2,3) in S_5.
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 prove that Nichols algebras of irreducible Yetter-Drinfeld modules over classical Weyl groups $A \rtimes \mathbb S_n$ supported by $\mathbb S_n$ are infinite dimensional, except in three cases. We give necessary and sufficient conditions for Nichols algebras of Yetter-Drinfeld modules over classical Weyl groups $A \rtimes \mathbb S_n$ supported by $A$ to be finite dimensional.
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.
In this paper, we study the representations of the new finite-dimensional pointed Hopf algebras in positive characteristic given in \cite{Cib09}. We find that these Hopf algebras are symmetric algebras. We determine the simple modules and their projective covers over these Hopf algebras. We show that these Hopf algebras are of wild representation type.
Let k be an algebraically closed field of characteristic 0. We conclude the classification of finite dimensional pointed Hopf algebras whose group of group-likes is S_4. We also describe all pointed Hopf algebras over S_5 whose infinitesimal braiding is associated to the rack of transpositions.
We describe all finite-dimensional pointed Hopf algebras whose infinitesimal braiding is a fixed Yetter-Drinfeld module decomposed as the sum of two simple objects: a point and the one of transpositions of the symmetric group in three letters. We give a presentation by generators and relations of the corresponding Nichols algebra and show that Andruskiewitsch-Schneider Conjecture holds for this kind of pointed Hopf algebras.
This article serves a two-fold purpose. On the one hand, it is a survey about the classification of finite-dimensional pointed Hopf algebras with abelian coradical, whose final step is the computation of the liftings or deformations of graded Hopf algebras. On the other, we present a step-by-step guide to carry out the strategy developed to construct the liftings. As an example, we conclude the work with the classification of pointed Hopf algebras of type $B_2$.
We show that any finite-dimensional pointed Hopf algebra over an abelian group $Γ$ such that its infinitesimal braiding is of standard type is generated by group-like and skew-primitive elements. This fact agrees with the long-standing conjecture by Andruskiewitsch and Schneider. We also show that the quantum Serre relations hold in any coradically graded pointed Hopf algebra over $Γ$ of finite dimension and determine how these relations are lifted in the standard case.
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.
We study the pointed or copointed liftings of Nichols algebras associated to affine racks and constant cocycles for any finite group admitting a principal YD-realization of these racks. In the copointed case we complete the classification for the six affine racks whose Nichols algebra is known to be of finite dimension. In the pointed case our method allows us to finish four of them. In all of the cases the Hopf algebras obtained turn out to be cocycle deformations of their associated graded Hopf algebras. All of them are new examples of finite-dimensional copointed or pointed Hopf algebras ov…
We examine actions of finite-dimensional pointed Hopf algebras on central simple division algebras in characteristic 0. (By a Hopf action we mean a Hopf module algebra structure.) In all examples considered, we show that the given Hopf algebra does admit a faithful action on a central simple division algebra, and we construct such a division algebra. This is in contrast to earlier work of Etingof and Walton, in which it was shown that most pointed Hopf algebras do not admit faithful actions on fields. We consider all bosonizations of Nichols algebras of finite Cartan type, small quantum groups…
We develop some techniques to the study of exact module categories over some families of pointed finite-dimensional Hopf algebras. As an application we classify exact module categories over the tensor category of representations of the small quantum groups $u_q(\mathfrak{sl}_2)$.
Every non quasi- -1-type Nichols algebra is infinite dimensional. All quasi- -1-type Nichols algebra over sporadic simple groups ${\rm HS}$ and ${\rm Co3}$ are found.
