On Semisimple Hopf Algebras of Dimension pq r

Let p and q be distinct prime numbers. We prove a result on the existence of nontrivial group-like elements in a certain class of semisimple Hopf algebras of dimension pq ^r . We conclude the classification of semisimple Hopf algebras A of dimension pq ² over an algebraically closed field k of characteristic zero, such that both A and A ^* are of Frobenius type. We also complete the classification of semisimple Hopf algebras of dimension pq ²<100.     

Classification of Pointed Hopf Algebras of Dimension p 2 over any Algebraically Closed Field

Let p be a prime. We complete the classification of pointed Hopf algebras of dimension p ² over an algebraically closed field k. When char k?≠?p, our result is the same as the well-known result for char k?=?0. When char k?=?p, we obtain 14 types of pointed Hopf algebras of dimension p ², including a unique noncommutative and noncocommutative type.     

On Semisimple Hopf Algebras of Dimension pq2, II

We obtain further classification results for semisimple Hopf algebras of dimension pq ² over an algebraically closed field k of characteristic zero. We complete the classification of semisimple Hopf algebras of dimension 28.     

Forms of pointed Hopf algebras

Using descent theory, we study Hopf algebra forms of pointed Hopf algebras. It turns out that the set of isomorphism classes of such forms are in one-to-one correspondence to other known invariants, for example the set of isomorphism classes of Galois extensions with a certain group F, or the set of isometry classes of m-ary quadratic forms. Our theory leads to a classification of all Hopf algebras over a field of characteristic zero that become pointed after a base extension, in dimension p, p ² and p ³, with p odd. Received: 22 November 1998     

On Two Families of Hopf Algebras of Dimension 2 m

Abstract

In this paper we construct two families of semisimple Hopf algebras of dimension 2 ⁿ⁺¹, n ≥ 3. They are all constructed as Radford's biproducts. For these examples and their duals we compute their grouplike elements, centers, character algebras and Grothendieck rings. Comparing these facts we are able to show that depending on the dimension, representatives of one of the families are selfdual. We also prove that Hopf algebras from these families are neither triangular nor cotriangular and that their cocycle deformations are trivial.     

Structure Theorems for Semisimple Hopf Algebras of Dimension pq 3

Let p, q be prime numbers with p > q ³, and k an algebraically closed field of characteristic 0. In this article, we obtain the structure theorems for semisimple Hopf algebras of dimension pq ³.     

Finite generation of some cohomology rings via twisted tensor product and Anick resolutions

Over a field of prime characteristic $p>2$, we prove that the cohomology rings of some pointed Hopf algebras of dimension $p^3$ are finitely generated. These are Hopf algebras arising in the ongoing classification of finite dimensional pointed Hopf algebras in positive characteristic. They include bosonizations of Nichols algebras of Jordan type in a general setting. When $p=3$, we also consider their Hopf algebra liftings, that is Hopf algebras whose associated graded algebra with respect to the coradical filtration is given by such a bosonization. Our proofs are based on an algebra filtration and a lemma of Friedlander and Suslin, drawing on both twisted tensor product resolutions and Anick resolutions to locate the needed permanent cocycles in May spectral sequences.     

Weakly group-theoretical and solvable fusion categories

We introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups – weakly group-theoretical categories and solvable categories. These are fusion categories that are Morita equivalent to iterated extensions (in the world of fusion categories) of arbitrary, respectively solvable finite groups. Weakly group-theoretical categories have integer dimension, and all known fusion categories of integer dimension are weakly group-theoretical. Our main results are that a weakly group-theoretical category C has the strong Frobenius property (i.e., the dimension of any simple object in an indecomposable C-module category divides the dimension of C), and that any fusion category whose dimension has at most two prime divisors is solvable (a categorical analog of Burnside's theorem for finite groups). This has powerful applications to classification of fusion categories and semsisimple Hopf algebras of a given dimension. In particular, we show that any fusion category of integer dimension <84 is weakly group-theoretical (i.e. comes from finite group theory), and give a full classification of semisimple Hopf algebras of dimensions pqr and pq², where p,q,r are distinct primes.     

On Semisimple Hopf Algebras of Dimension 2<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>, II

In this paper we classify, up to equivalence, all semisimple nontrivial Hopf algebras of dimension 2²ⁿ⁺¹ for n ≥ 2 over an algebraically closed field of characteristic 0 with the group of group-like elements isomorphic to \(\mathbb {Z}_{2^{n}}\times \mathbb {Z}_{2^{n}}\). Moreover we classify all such nonisomorphic Hopf algebras of dimension 32 and show that they are not twist-equivalent to each other. More generally, given an abelian group of order 2 ^m?1 we give an upper bound for the number of nonisomorphic nontrivial Hopf algebras of dimension 2 ^m which have this group as their group of group-like elements.     

Finite dimensional Nichols algebras over certain p-groups

We investigate pointed Hopf algebras over finite nilpotent groups of odd order, with nilpotency class 2. For such a group G, we show that if its commutator subgroup coincides with its center, then there exists no non-trivial finite-dimensional pointed Hopf algebra with kG as its coradical. We apply these results to non-abelian groups of order p³, p⁴ and p⁵, and list all the pointed Hopf algebras of order p⁶, whose group of grouplikes is non-abelian.     

Cocycle deformations and Galois objects for semisimple Hopf algebras of dimension p3 and pq2

Let p and q be distinct prime numbers. We study the Galois objects and cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf algebras of odd dimension $p^3$ and of dimension $p{q}^2$. We obtain that the $p+1$ non-isomorphic self-dual semisimple Hopf algebras of dimension $p^3$ classified by Masuoka have no non-trivial cocycle deformations, extending his previous results for the 8-dimensional Kac–Paljutkin Hopf algebra. This is done as a consequence of the classification of categorical Morita equivalence classes among semisimple Hopf algebras of odd dimension $p^3$, established by the third-named author in an appendix.     

On Semisimple Hopf Algebras of Dimension 2 m

In this paper we classify all nontrivial semisimple Hopf algebras of dimension 2 ^{n +1} with the group of grouplikes isomorphic to _{2 n–1}×₂. Moreover, we extend some results on irreducible representations from groups to semisimple Hopf algebras and prove that certain semisimple Hopf algebras, including the ones classified in this paper, satisfy the generalized power map property.     

Breuil–Kisin Modules and Hopf Orders in Cyclic Group Rings

For K, a finite extension of ? _p with ring of integers R, we show how Breuil–Kisin modules can be used to determine Hopf orders in K-Hopf algebras of p-power dimension. We find all cyclic Breuil–Kisin modules and use them to compute all of the Hopf orders in the group ring KΓ where Γ is cyclic of order p or p ². We also give a Laurent series interpretation of the Breuil–Kisin modules that give these Hopf orders.     

Finite quantum groups over abelian groups of prime exponent

We classify pointed finite-dimensional complex Hopf algebras whose group of group-like elements is abelian of prime exponent p, p>17. The Hopf algebras we find are members of a general family of pointed Hopf algebras we construct from Dynkin diagrams. As special cases of our construction we obtain all the Frobenius-Lusztig kernels of semisimple Lie algebras and their parabolic subalgebras. An important step in the classification result is to show that all these Hopf algebras are generated by group-like and skew-primitive elements.     

Automorphismes des algèbres cubiques

Manin associated to a quadratic algebra (quantum space) the quantum matrix group of its automorphisms. This Note aims to demonstrate that Manin's construction can be extended for quantum spaces which are non-quadratic homogeneous algebras. The Artin–Schelter classification of regular algebras of global dimension three contains two types of algebra: quadratic and cubic. Ewen and Ogievetsky classified the quantum matrix groups which are deformations of GL(3) corresponding to the quadratic algebras in the Artin–Schelter classification. In this Note we consider the cubic Artin–Schelter algebras as quantum spaces and construct Hopf algebras of their automorphisms. To cite this article: T. Popov, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Hopf algebra deformations of binary polyhedral groups

We show that semisimple Hopf algebras having a self-dual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z2. We prove that nontrivial Hopf algebras arising in this way can be regarded as deformations of binary polyhedral groups and describe its category of representations. We also prove a strengthening of a result of Nichols and Richmond on cosemisimple Hopf algebras with a two-dimensional irreducible comodule in the finite-dimensional context. Finally, we give some applications to the classification of certain classes of semisimple Hopf algebras.     

Quiver Hopf algebras

In this paper we study subHopfalgebras of graded Hopf algebra structures on a path coalgebra kQ^c. We prove that a Hopf structure on some subHopfquivers can be lifted to a Hopf structure on the whole Hopf quiver. If Q is a Schurian Hopf quiver, then we classified all simple-pointed subHopfalgebras of a graded Hopf structure on kQ^c. We also prove a dual Gabriel theorem for pointed Hopf algebras.     

Graded Associative Conformal Algebras of Finite Type

In this paper, we consider graded associative conformal algebras. The class of these objects includes pseudo-algebras over non-cocommutative Hopf algebras of regular functions on some linear algebraic groups. In particular, an associative conformal algebra which is graded by a finite group Γ is a pseudo-algebra over the coordinate Hopf algebra of a linear algebraic group G such that the identity component G ⁰ is the affine line and G/G ⁰???Γ. A classification of simple and semisimple graded associative conformal algebras of finite type is obtained.     

On formal quantum group laws

There exist natural generalizations of the concept of formal groups laws for noncommutative power series. This is a note on formal quantum group laws and quantum group law chunks. Formal quantum group laws correspond to noncommutative (topological) Hopf algebra structures on free associative power series algebras ká áx₁,...,x_m ? ?k\langle\! \langle x_1,\dots,x_m \rangle\! \rangle , k a field. Some formal quantum group laws occur as completions of noncommutative Hopf algebras (quantum groups). By truncating formal power series, one gets quantum group law chunks. ¶If the characteristic of k is 0, the category of (classical) formal group laws of given dimension m is equivalent to the category of m-dimensional Lie algebras. Given a formal group law or quantum group law (chunk), the corresponding Lie structure constants are determined by the coefficients of its chunk of degree 2. Among other results, a classification of all quantum group law chunks of degree 3 is given. There are many more classes of strictly isomorphic chunks of degree 3 than in the classical case.