We study positively-graded Hopf algebras and obtain (dual) Gabriel-type results on graded Hopf algebras. Using it, we get certain (non-degenerate) graded Hopf pairings between quantum symmetric algebras.

Integrals in Hopf algebras are an essential tool in studying finite dimensional Hopf algebras and their action on rings. Over fields it has been shown by Sweedler that the existence of integrals in a Hopf algebra is equivalent to the Hopf algebra being finite dimensional. In this paper we examine how much of this is true for Hopf algebras over rings. We show that over any commutative ring R tha...

This paper offers an introduction to Hopf algebras. It is not a paper describing basic properties and applications of Hopf algebras. Rather, it is a paper providing the necessary structures in order to construct a Hopf algebra. It examines tensor products, algebras, coalgebras, bialgebras, and homology groups. After reading this paper, the reader will not be sufficiently experienced in Hopf alg...

The purpose of this paper is to introduce a cohomology theory for abelian matched pairs of Hopf algebras and to explore its relationship to Sweedler cohomology, to Singer cohomology and to extension theory. An exact sequence connecting these cohomology theories is obtained for a general abelian matched pair of Hopf algebras, generalizing those of Kac and Masuoka for matched pairs of finite grou...

The concept of biperfect (noncocommutative) weak Hopf algebras is introduced and their properties are discussed. A new type of quasi-bicrossed products are constructed by means of weak Hopf skew-pairs of the weak Hopf algebras which are generalizations of the Hopf pairs introduced by Takeuchi. As a special case, the quantum double of a finite dimensional biperfect (noncocommutative) weak Hopf a...

Group actions are ubiquitous in mathematics. To understand a mathematical object, it is often helpful to understand its symmetries as expressed by a group. For example, a group acts on a ring by automorphisms (preserving its structure). Analogously, a Lie algebra acts on a ring by derivations. Unifying these two types of actions are Hopf algebras acting on rings. A Hopf algebra is not only an a...

We give a Decomposition Theorem for a family of Hopf algebras containing the well-know family of Taft Hopf algebras. Therefore, those indecomposable codes over this family of algebras (cf. [4]) is an indecomposable code over the studied case. We use properties of Hopf algebras to show that dual (in the module sense) of an ideal code is again an ideal code.

Inspired by [3] we introduce the concept of extended Hopf algebra and consider their cyclic cohomology in the spirit of Connes-Moscovici [3, 4, 5]. Extended Hopf algebras are closely related, but different from, Hopf algebroids. Their definition is motivated by attempting to define cyclic cohomology of Hopf algebroids in general. Many of Hopf algebra like structures, including the Connes-Moscov...

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 general pointed Hopf algebras.

A classical result in the theory of Hopf algebras concerns the uniqueness and existence of inte-grals: for an arbitrary Hopf algebra, the integral space has dimension ≤ 1, and for a finite dimensional Hopf algebra, this dimension is exaclty one. We generalize these results to quasi-Hopf algebras and dual quasi-Hopf algebras. In particular, it will follow that the bijectivity of the antipode fol...