相关论文: Hopf q-braces structures on rank one pointed Hopf …

In this paper we investigate pointed Hopf algebras via quiver methods. We classify all possible Hopf structures arising from minimal Hopf quivers, namely basic cycles and the linear chain. This provides full local structure information for…

In this paper, we study pointed rank one Hopf algebras and Hopf-Ore extensions of group algebras, over an arbitrary field $k$. It is proved that the rank of a Hopf-Ore extension of a group algebra is one or two or infinite. It is also shown…

Given a finite group $ G $, we study certain regular subgroups of the group of permutations of $ G $, which occur in the classification theories of two types of algebraic objects: skew left braces with multiplicative group isomorphic to $ G…

In this paper we classify the finite-dimensional pointed rank one Hopf algebras which are generated as algebras by the first element of the coradical filtration over a field of prime characteristic.

We give a systematic construction of Hopf algebra structures on braided cofree coalgebras. The relevant underlying structures are braided algebras and braided coalgebras. We provide some interesting examples of these algebras and coalgebras…

We investigate the structures of Hopf $\ast$-algebra on the Radford algebras over $\mathbb {C}$. All the $*$-structures on $H$ are explicitly given. Moreover, these Hopf $*$-algebra structures are classified up to equivalence.

We classify pointed Hopf algebras of discrete corepresentation type over an algebraically closed field K with characteristic zero. For such algebras $H$, we explicitly determine the algebra structure up to isomorphism for the link…

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.

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…

Using the fact that Hopf-Galois structures on separable extensions and skew bracoids are both intrinsically connected to transitive subgroups of the holomorph of a finite group, we present algorithms to classify and enumerate these objects…

Skew braces have recently attracted attention as a method to study set-theoretical solutions of the Yang-Baxter equation. Here, we present a new approach to these solutions by studying Hopf algebras in the category, $\mathrm{SupLat}$, of…

Let $n\geq 1$ be an integer, $p$, $q$ be distinct odd primes. Let ${G}$, $N$ be two groups of order $p^nq$ with their Sylow-$p$-subgroups being cyclic. We enumerate the Hopf-Galois structures on a Galois ${G}$-extension, with type $N$. This…

We classify all skew braces of Heisenberg type for a prime number $ p>3 $. Furthermore, we determine the automorphism group of each one of these skew braces (as well as their socle and annihilator). Hence, by utilising a link between skew…

We give some general results on the ring structure of Hochschild cohomology of smash products of algebras with Hopf algebras. We compute this ring structure explicitly for a large class of finite dimensional Hopf algebras of rank one.

Let $\mathbb{K}$ be the set of hybrid numbers. This paper is to look for all the weak Hopf structures on $\mathbb{K}$. Once $\mathbb{K}$ is endowed with a structure of a weak Hopf algebra, we shall compute the source algebra and target…

We study the realizations of certain braided vector spaces of rack type as Yetter-Drinfeld modules over a cosemisimple Hopf algebra $H$. We apply the strategy developed in arXiv:1212.5279 to compute their liftings and use these results to…

This paper introduces Hopf braces, a new algebraic structure related to the Yang-Baxter equation which include Rump's braces and their non-commutative generalizations as particular cases. Several results of classical braces are still valid…

The variety of skew braces contains several interesting subcategories as subvarieties, as for instance the varieties of radical rings, of groups and of abelian groups. In this article the methods of non-abelian homological algebra are…

This article studies the construction of Hopf algebras $H$ acting on a given algebra $K$ in terms of algebra morphisms $ \sigma \colon K \rightarrow \mathrm{M}_n(K)$. The approach is particularly suited for controlling whether these actions…

This is a survey on pointed Hopf algebras over algebraically closed fields of characteristic 0. We propose to classify pointed Hopf algebras $A$ by first determining the graded Hopf algebra $\gr A$ associated to the coradical filtration of…