We prove that a depth two Hopf subalgebra K of a semisimple Hopf algebra H is normal (where the ground field k is algebraically closed of characteristic zero). This means on the one hand that a Hopf subalgebra is normal when inducing (then restricting) modules several times as opposed to one time creates no new simple constituents. This point of view was taken in the paper [13] which establishe...

In the last lecture we have learned that the category of modules over a braided Hopf algebra H is a braided monoidal category. A braided Hopf algebra is a rather sophisticated algebraic object, it is not easy to give interesting nontrivial examples. In this text we develop a theory that will lead to a concrete recipe which produces a nontrivial braided Hopf algebra D(A) (called Drinfeld’s quant...

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...

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...

Let (H, σ) be a coquasitriangular Hopf algebra, not necessarily finite dimensional. Following methods of Doi and Takeuchi, which parallel the constructions of Radford in the case of finite dimensional quasitriangular Hopf algebras, we define Hσ , a sub-Hopf algebra of H, the finite dual of H. Using the generalized quantum double construction and the theory of Hopf algebras with a projection, we...

let $h$ be a hopf algebra and $a$ an $h$-bimodule algebra‎. ‎in this paper‎, ‎we investigate gorenstein global dimensions for hopf‎ ‎algebras and twisted smash product algebras $astar h$‎. ‎results from‎ ‎the literature are generalized‎.

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...

A C∞-Hopf algebra is a C∞-algebra which is also a convenient Hopf algebra with respect to the structure induced by the evaluations of smooth functions. We characterize those C∞-Hopf algebras which are given by the algebra C∞(G) of smooth functions on some compact Lie group G, thus obtaining an anti-isomorphism of the category of compact Lie groups with a subcategory of convenient Hopf algebras.

We define the notion of factorizable quasi-Hopf algebra by using a categorical point of view. We show that the Drinfeld double D(H) of any finite dimensional quasi-Hopf algebra H is factorizable, and we characterize D(H) when H itself is factorizable. Finally, we prove that any finite dimensional factorizable quasi-Hopf algebra is unimodular. In particular, we obtain that the Drinfeld double D(...