For a quasi-Hopf algebra H, we study two types of 1-cycle deformations for a coalgebra C within the category of Yetter-Drinfeld modules over H, _H^H𝒴D . The two deformations produce C-comodule structures in _H^H𝒴D and new coalgebra structures on C in _H^H𝒴D , respectively. We show that the isomorphism types of these structures are described by a 1-homology ℋ^1_H(C, H_0) that we will introduce. Then we apply our results to the so called symplectic fermion quasi-Hopf algebras, algebras recently introduced by Farsad, Gainutdinov and Runkel.

Inspired by the work of Radford, for H an arbitrary quasi-Hopf algebra we describe all the Hopf algebras of dimension 2 within the braided category of left Yetter-Drinfeld modules over H and determine the biproduct quasi-Hopf algebras defined by them. Classes of such biproduct quasi-Hopf algebras are obtained by taking H as the Hopf algebra of functions on a group G, endowed with the quasi-Hopf algebra structure provided by a non-trivial 3-cocycle on G (especially when G is a finite cyclic group or the double dihedral group), or as being a quasi-Hopf algebra with radical of codimension two. In this way we uncover new classes of basic quasi-Hopf algebras of even dimension, as well as new classes of tensor categories.

We construct an explicit isomorphism between the quasitriangular quasi-Hopf algebra $D^\omega(H)$ defined in \cite{bp} and a certain quantum double quasi-Hopf algebra. We give also new characterizations for a quasitriangular quasi-Hopf algebra to be ribbon and use them to construct some ribbon elements for $D^\omega(H)$.

We introduce the notions of sovereign, spherical and balanced quasi-Hopf algebra. We investigate the connections between these, as well as their connections with the class of pivotal, involutory and ribbon quasi-Hopf algebras, respectively. Examples of balanced and ribbon quasi-Hopf algebras are obtained from a sort of double construction which associates to a braided category (resp. rigid braided) a balanced (resp. ribbon) one.

We survey results on Frobenius algebras and illustrate their importance to the structure of some generalizations of the notion of Hopf algebra, as well as their connections to topics like monoidal categories, 2-categories, functors, topological quantum field theories, etc.

We prove a uniqueness type theorem for (weak, total) integrals on a Frobenius cowreath in a monoidal category. When the cowreath is, moreover, pre-Galois, we construct a Morita context relating the subalgebra of coinvariants and a certain wreath algebra. Then we see that the strictness of the Morita context is related to the Galois property of the cowreath and the existence of a weak total integral on it. We apply our results to quasi-Hopf algebras.

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.

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 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.

If A is a finite dimensional Hopf algebra and ∫ ⊆ A is the space of integrals in A, it is well known that dim( ∫ ) = 1. The proof given in [12] actually shows the existence and uniqueness of integrals in A∗ and it relies on the structure of Hopf modules over A, namely one has to prove that A∗ is a right A-Hopf module and then the result follows from the fundamental theorem for Hopf modules (see [12] for details). It is very natural to ask if the result remains true if A is not a Hopf algebra, but a quasi-Hopf algebra (this question arose in [9], where the following version of Maschke’s theorem for quasi-Hopf algebras was proved: A is semisimple if and only if e( ∫ ) 6= 0). The answer is positive for some particular quasi-Hopf algebras, for instance for Dijkgraaf-Pasquier-Roche’s quasi-Hopf algebras D(G) (where G is a finite group and ω is a normalized 3-cocycle on G) and for their generalizations D(H) introduced in [1] (where H is a finite dimensional cocommutative Hopf algebra and ω : H⊗H⊗H → k is a normalized 3-cocycle in Sweedler’s cohomology). But if one tries to generalize the proof given in [12] to quasi-Hopf algebras some problems occur, for example it is not clear which could be the appropriate definition for a Hopf module over a quasi-Hopf algebra. The existence and uniqueness of integrals for finite dimensional Hopf algebras have been reproved in [11], [8] by avoiding the use of Hopf modules. In this note we shall prove the existence of integrals for finite dimensional quasi-Hopf algebras, by generalizing the short and direct proof given by A. Van Daele in [11] for the Hopf algebra case. It seems that the method in [11] does not yield a proof for the uniqueness property.

This is the first book to be dedicated entirely to Drinfeld's quasi-Hopf algebras. Ideal for graduate students and researchers in mathematics and mathematical physics, this treatment is largely self-contained, taking the reader from the basics, with complete proofs, to much more advanced topics, with almost complete proofs. Many of the proofs are based on general categorical results; the same approach can then be used in the study of other Hopf-type algebras, for example Turaev or Zunino Hopf algebras, Hom-Hopf algebras, Hopfish algebras, and in general any algebra for which the category of representations is monoidal. Newcomers to the subject will appreciate the detailed introduction to (braided) monoidal categories, (co)algebras and the other tools they will need in this area. More advanced readers will benefit from having recent research gathered in one place, with open questions to inspire their own research.