We introduce the notion of factorizable quasi-Hopf algebra by using a categorical point of view. We show that the quantum 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 quantum double D(H) is a unimodular quasi-Hopf algebra.

We compute the quantum dimension of a finite-dimensional quasi-Hopf algebra H and of its quantum double D(H), within the rigid braided category of finite-dimensional left D(H)-modules. This involves the semisimplicity of D(H) and leads to the notion of involutory quasi-Hopf algebra, studied at the end of the chapter.

We introduce and study several kinds of crossed products corresponding to quasi-bialgebras and quasi-Hopf algebras (smash and quasi-smash products, diagonal crossed products, L–R-smash products). The chapter ends with a duality theorem for finite-dimensional quasi-Hopf algebras.

We define and characterize ribbon quasi-Hopf algebras by using properties of a ribbon category. Consequently, we have a one-to-one correspondence between ribbon elements for a quasi-Hopf algebra and a sort of grouplike elements of it. We also construct ribbon categories from left or right rigid monoidal categories and use this construction to introduce a special class of ribbon quasi-Hopf algebras.

We introduce the concepts of quasi-bialgebra and quasi-Hopf algebra by using a categorical point of view.We present the basic properties of these objects and study their invariance under a twist. We also introduce the dual notions, called dual quasi-bialgebra and dual quasi-Hopf algebra.

We introduce the categories of Yetter–Drinfeld modules by computing the left and right centers of a category of modules over a quasi-bialgebra H. We then show that all four categories of Yetter–Drinfeld modules are braided isomorphic. We also introduce the quasi-Hopf algebra structure of the quantum double of a finite-dimensional quasi-Hopf algebra.

We show that for a finite-dimensional quasi-Hopf algebra H the space of integrals in H, and the space of cointegrals on H, have dimension 1. We characterize semisimple and symmetric quasi-Hopf algebras with the help of integrals, and prove a formula for the fourth power of the antipode in terms of the modular elements by using the machinery provided by Frobenius algebras. The chapter ends with a freeness theorem stating that any finite-dimenisonal quasi-Hopf algebra is free over any quasi-Hopf subalgebra.

We define the notions of module (co)algebra and (bi)comodule algebra, respectively, over a quasi-bialgebra by using certain categorical points of view and by generalizing the axioms of a quasi-bialgebra, respectively. Then we give concrete classes of examples and the connections that exist between these structures.

We define the notions of algebra and coalgebra, and bialgebra and Hopf algebra, respectively, within a monoidal and a (pre-)braided monoidal category, respectively.

By using categorical tools, we introduce the concept of quasitriangular (QT) quasi-bialgebras. For QT quasi-Hopf algebras we show that the square of the antipode is an inner automorphism, and therefore bijective. We uncover the QT structure of the quantum double D(H) of a finite-dimensional quasi-Hopf algebra H, and characterize D(H) as a biproduct quasi-Hopf algebra in the case when H itself is QT.

We introduce the category of two-sided two-cosided Hopf modules over a quasi-bialgebra H and show that it is braided monoidally equivalent to the category of Yetter–Drinfeld modules over H, provided that H is a quasi-Hopf algebra. We use this equivalence to obtain structure theorems for bicomodule algebras and bimodule coalgebras over H. Finally, we show that a Hopf algebra within this braided monoidal category identifies with a quasi-Hopf algebra with projection.

We introduce the basic categorical language that will be used throughout this book. We define the concepts of monoidal and braided monoidal category and prove that any monoidal category is monoidally equivalent to a strict one.

We present some structure theorems for quasi-Hopf bimodules. We also show that for a quasi-Hopf algebra H the category of quasi-Hopf H-bimodules is monoidally equivalent to the category of left H-representations. As an application, we prove a structure theorem for quasi-Hopf comodule algebras.