Related papers: Triangular Hopf algebras with the Chevalley proper…

In this paper we study the properties of Drinfeld's twisting for finite-dimensional Hopf algebras. We determine how the integral of the dual to a unimodular Hopf algebra $H$ changes under twisting of $H$. We show that the classes of…

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…

We prove an analog of Deligne's theorem for finite symmetric tensor categories $\mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $2$. Namely, we prove that every such category $\mathcal{C}$…

We discuss some general results on finite-dimensional Hopf algebras over an algebraically closed field k of characteristic zero and then apply them to Hopf algebras H of dimension p^{3} over k. There are 10 cases according to the group-like…

In a previous paper we prove that any semisimple triangular Hopf algebra A over an algebraically closed field of characteristic 0 (say the field of complex numbers C) is obtained from a finite group after twisting the ordinary…

We prove a structure theorem for Yetter-Drinfel'd Hopf algebras over groups of prime order that are nontrivial, cocommutative, and cosemisimple: Under certain assumptions on the base field, these algebras can be decomposed into a tensor…

Let $H$ be the $16$-dimensional nontrivial (namely, noncommutative and noncocommutative) semisimple Hopf algebra $H_{b:x^2y}$ classified by Kashina. We figure out all simple Yetter-Drinfeld $H$-modules, and then determine all…

For any affine Hopf algebra $H$ which admits a large central Hopf subalgebra, $H$ can be endowed with a Cayley-Hamilton Hopf algebra structure in the sense of De Concini-Procesi-Reshetikhin-Rosso. The category of finite-dimensional modules…

In 1999, Y. Kashina introduced the exponent of a Hopf algebra. In this paper, we prove that the exponent of a finite dimensional non-cosemisimple Hopf algebra with Chevalley property in characteristic 0 is infinite, and the exponent of a…

A basis B of a finite dimensional Hopf algebra H is said to be positive if all the structure constants of H relative to B are non-negative. A quasi-triangular structure $R\in H\otimes H$ is said to be positive with respect to B if it has…

We establish a 3-manifold invariant for each finite-dimensional, involutory Hopf algebra. If the Hopf algebra is the group algebra of a group $G$, the invariant counts homomorphisms from the fundamental group of the manifold to $G$. The…

A modular tensor category is a non-degenerate ribbon finite tensor category. And a ribbon factorizable Hopf algebra is exactly the Hopf algebra whose finite-dimensional representations form a modular tensor category. The goal of this paper…

We survey variety theory for modules of finite dimensional Hopf algebras, recalling some definitions and basic properties of support and rank varieties where they are known. We focus specifically on properties known for classes of examples…

We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective special linear group over a finite field, corresponding to semisimple orbits, have infinite dimension. We introduce a new criterium to determine when a…

The aim of this paper is to study quasitriangular structures on a class of semisimple Hopf algebras constructed through abelian extensions of $Z_2$ for an abelian group $G$. We prove that there are only two forms of them. Using such…

We show that semisimple Hopf algebras having a self-dual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z_2. We prove that nontrivial Hopf algebras arising in this way can be regarded as…

This paper introduces methods for classifying actions of finite-dimensional Hopf algebras on path algebras of quivers, and more generally on tensor algebras $T_B(V)$ where $B$ is semisimple. We work within the broader framework of finite…

We say that a Hopf algebra H is semicocommutative if the right adjoint coaction factorizes through the tensor product of H with the center of H. For instance the commutative and the cocommutative Hopf algebras are semicocommutative. The…

We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field by using the braiding structures of A. We investigate basic properties of the polynomial invariants including stability…

We prove that every finite symmetric integral tensor category $\mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $p>2$ admits a symmetric fiber functor to $\text{sVec}$. This proves Ostrik's…