Related papers: Iterated Hopf Ore Extensions over Group Rings

A class of finite-dimensional Hopf algebras which generalise the notion of Taft algebras is studied. We give necessary and sufficient conditions for these Hopf algebras to omit a pair in involution, that is, to not have a group-like and a…

It is known that there is a Hopf algebra structure on the vector space with basis all heap-ordered trees. We give a new bialgebra structure on the space with basis all permutations and show that there is a direct bialgebra isomorphism…

Classifying all Hopf algebras of a given finite dimension over the complex numbers is a challenging problem which remains open even for many small dimensions, not least because few general approaches to the problem are known. Some useful…

It is proved in this paper that for any finite-dimensional nonsemisimple Hopf algebra $A$ there exists a Hopf algebra $H$ containing $A$ as a Hopf subalgebra such that $H$ is not flat over $A$. On the other hand, there is a class of…

A Hopf Galois structure on a finite field extension $L/K$ is a pair $(H,\mu)$, where $H$ is a finite cocommutative $K$-Hopf algebra and $\mu$ a Hopf action. In this paper we present a program written in the computational algebra system…

Let $\mathcal{G}=\mathrm{Spec}(A)$ be a finite and flat group scheme over the ring of algebraic integers $R$ of a number field $K$ and suppose that the generic fiber of $\mathcal{G}$ is the constant group scheme over $K$ for a finite group…

We show that certain twisting deformations of a family of supersolvable groups are simple as Hopf algebras. These groups are direct products of two generalized dihedral groups. Examples of this construction arise in dimensions 60 and…

Let $H$ be a semisimple Hopf algebras over an algebraically closed field $k$ of characteristic $0.$ We define Hopf algebraic analogues of commutators and their generalizations and show how they are related to $H',$ the Hopf algebraic…

In this paper we revisit and extend the constructions of Glauberman and Doro on groups with triality and Moufang loops to Hopf algebras. We prove that the universal enveloping algebra of any Lie algebra with triality is a Hopf algebra with…

The goal of this paper is to give a new method of constructing finite-dimensional semisimple triangular Hopf algebras, including minimal ones which are non-trivial (i.e. not group algebras). The paper shows that such Hopf algebras are quite…

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…

The notion of Hopf center and Hopf cocenter of a Hopf algebra is investigated by the extension theory of Hopf algebras. We prove that each of them yields an exact sequence of Hopf algebras. Moreover the exact sequences are shown to satisfy…

In this paper we contribute to the classification of Hopf algebras of dimension pq, where p,q are distinct prime numbers. More precisely, we prove that if p and q are odd primes with p<q<2p+3, then any complex Hopf algebra of dimension pq…

We prove that the structure algebra of a Bruhat moment graph of a finite real root system is a Hopf algebroid with respect to the Hecke and the Weyl actions. We introduce new techniques (reconstruction and push-forward formula of a product,…

This paper describes the $K$-theory structure for three algebra classes. For cyclic $p$-group rings and truncated polynomial rings over $\mathbb{Z}/p^s\mathbb{Z}$, we determine reduced $K_2$-structures via a common algebraic framework. For…

We discuss algebraic and representation theoretic structures in braided tensor categories C which obey certain finiteness conditions. Much interesting structure of such a category is encoded in a Hopf algebra H in C. In particular, the Hopf…

The notion of inner linear Hopf algebra is a generalization of the notion of discrete linear group. In this paper, we prove two general results that enable us to enlarge the class of Hopf algebras that are known to be inner linear: the…

The character ring \CGL of covariant irreducible tensor representations of the general linear group admits a Hopf algebra structure isomorphic to the Hopf algebra \Sym$ of symmetric functions. Here we study the character rings \CO and \CSp…

We construct and study a family of finitely generated Hopf algebra domains $H$ of Gelfand-Kirillov dimension two such that $\Ext^1_H(k,k)=0$. Consequently, we answer a question of Goodearl and the second-named author.

Let $p,q$ be prime numbers with $p^4<q$, and $k$ an algebraically closed field of characteristic 0. We show that semisimple Hopf algebras of dimension $p^2q^2$ can be constructed either from group algebras and their duals by means of…