Yinhuo Zhang

ACTIONS OF MULTIPLIER HOPF ALGEBRAS

For an action a of a group G on an algebra R (over C), the crossed product Rxα G is the vector space of R-valued functions with finite support in G, together with the twisted convolution product given by where p∈G. This construction has been extended to the theory of Hopf algebras. Given an action of a Hopf algebra A on an algebra R, it is possible to make the tensor productR⊗A into an algebra by using a twisted product, involving the action. In this case, the algebra is called the smash product and denoted by R#A. In the group case, the action a of G on R yields an action of the group algebra CG as a Hopf algebra on R and the crossed Rxα G coincides with the smash product R#CG. In this paper we extend the theory of actions of Hopf algebras to actions of multiplier Hopf algebras. We also construct the smash product and we obtain results very similar as in the original situation for Hopf algebras. The main result in the paper is a duality theorem for such actions. We consider dual pairs of multiplier Hopf ...

Galois Theory for Multiplier Hopf Algebras with Integrals

In this paper, we introduce a generalized Hopf Galois theory for regular multiplier Hopf algebras with integrals, which might be viewed as a generalization of the Hopf Galois theory of finite-dimensional Hopf algebras. We introduce the notion of a coaction of a multiplier Hopf algebra on an algebra. We show that there is a duality for actions and coactions of multiplier Hopf algebras with integrals. In order to study the Galois (co)action of a multiplier Hopf algebra with an integral, we construct a Morita context connecting the smash product and the coinvariants. A Galois (co)action can be characterized by certain surjectivity of a canonical map in the Morita context. Finally, we apply the Morita theory to obtain the duality theorems for actions and coactions of a co-Frobenius Hopf algebra.

The Green rings of Taft algebras

We compute the Green ring of the Taft algebra

The quantum double of a cofrobenius hopf algebra

H_n(q)

The Brauer group of Sweedler’s Hopf algebra ₄

, where

Embedding the Hopf automorphism group into the Brauer group

n

H-module endomorphism rings

is a positive integer greater than 1, and

Derived H-module endomorphism rings

q

The crossed coproduct theorem and Galois cohomology

is an

Hopf frobenius extensions of algebras

n