Abstract. In the last years a new theory of Hopf algebras has begun to be developed: that of Hopf algebras in braided categories, or, briefly, braided Hopf algebras. This is a survey of general aspects of the theory with emphasis in H HYD, the Yetter–Drinfeld category over H, where H is the group algebra of a non abelian finite group Γ. We discuss a special class of braided graded Hopf algebras from different points of view following Lusztig, Nichols and Schauenburg. We present some finite dimensional examples arising in an unpublished work by Milinski and Schneider. Sinopsis. En los últimos años comenzó a ser desarrollada una nueva teoría de álgebras de Hopf en categorías trenzadas, o brevemente, álgebras de Hopf trenzadas. Presentamos aquí aspectos generales de la teoría con énfasis en H HYD, la categoría de Yetter–Drinfeld sobre H, donde H es el álgebra de grupo de un grupo finito no abeliano Γ. Discutimos una clase especial de álgebras de Hopf trenzadas graduadas desde diferentes puntos de vista, siguiendo a Lusztig, Nichols y Schauenburg. Presentamos algunos ejemplos de dimensión finita que aparecen en un trabajo inédito de Milinski y Schneider. 0. Introduction and notations 0.1. Introduction.

Algebras and coalgebras are fundamental notions for large parts of mathematics. The basic constructions from universal algebra are now expressed in the language of categories and thus are accessible to classical algebraists and topologists as well as to logicians and computer scientists. Some of them have developed specialised parts of the theory and often reinvented constructions already known in a neighbouring area. One purpose of this survey is to show the connection between results from different fields and to trace a number of them back to some fundamental papers in category theory from the early 70’s. Another intention is to look at the interplay between algebraic and coalgebraic notions. Hopf algebras are one of the most interesting objects in this setting. While knowledge of algebras and coalgebras are folklore in general category theory, the notion of Hopf algebras is usually only considered for monoidal categories. In the course of the text we do suggest how to overcome this defect by defining a Hopf monad on an arbitrary category as a monad and comonad satisfying some compatibility conditions and inducing an equivalence between

Braided non-commutative differential geometry is studied. We investigate the theory of (bicovariant) differential calculi in braided abelian categories. Previous results on crossed modules and Hopf bimodules in braided categories are used to construct higher order bicovariant differential calculi over braided Hopf algebras out of first order ones. These graded objects are shown to be braided differential Hopf algebras with universal bialgebra properties. The article extends Woronowicz’s results on (bicovariant) differential calculi to the braided non-commutative case.

Abstract. We show that, under some mild conditions, a bialgebra in an abelian and coabelian braided monoidal category has a weak projection onto a formally smooth (as a coalgebra) subbialgebra with antipode; see Theorem 1.12. In the second part of the paper we prove that bialgebras with weak projections are cross product bialgebras; see Theorem 2.12. In the particular case when the bialgebra A is cocommutative and a certain cocycle associated to the weak projection is trivial we prove that A is a double cross product, or biproduct in Madjid’s terminology. The last result is based on a universal property of double cross products which, by Theorem 2.15, works in braided monoidal categories. We also investigate the situation when the right action of the associated matched pair is trivial.

We construct new (finite dimensional) Hopf algebras using (finite dimensional) braided Hopf algebras in Yetter-Drinfeld categories by means of bosinization and Drinfeld double. We give the relation between crossed product and H-Galois extension in braided tensor

The relation between crossed product and H-Galois extension in braided tensor categories is established. It is shown that A = B#σH is a crossed product algebra if and only if the extension A/B is Galois, the inverse can −1 of the canonical morphism can factors through object A ⊗B A and A is isomorphic as left B-modules and right H-comodules to B⊗H in braided tensor categories. For the Yetter-Drinfeld modules category Q Q YD, the condition that can−1 factors can be thrown off. Keywords: braided Hopf algebra, crossed product algebra, H-Galois extension. 0

We construct new (finite dimensional) Hopf algebras using (finite dimensional) braided Hopf algebras in Yetter-Drinfeld categories by means of bosinization and Drinfeld double. We give the relation between crossed product and H-Galois extension in braided tensor

The term “quantum group ” was popularized by Drinfeld in his address to the International Congress of Mathematicians in Berkeley. However, the concepts of quantum groups and quasitriangular Hopf algebras are the same. Therefore, Hopf algebras have close connections with various areas of mathematics and physics. The development of Hopf algebras can be divided into five stages. The first stage is that integral and Maschke’s theorem are found. Maschke’s theorem gives a nice criterion of semisimplicity, which is due to M.E. Sweedler. The second stage is that the Lagrange’s theorem is proved, which is due to W.D. Nichols and M.B. Zoeller. The third stage is the research of the actions of Hopf algebras, which unifies actions of groups and Lie algebras. S. Montgomery’s “Hopf algebras and their actions on rings ” contains the main results in this area. S. Montgomery and R.J. Blattner show the duality theorem. S. Montegomery, M. Cohen, H.J. Schneider, W. Chin, J.R. Fisher and the author study the relation between algebra R and the crossed product R#σH of the semisimplicity, semiprimeness, Morita equivalence and radical. S. Montgomery and M. Cohen ask whether R#σH is semiprime when R is semiprime for a finite-dimensional semiprime Hopf algebra H. This is a famous semiprime problem. The

In this chapter, we study the structures of Hopf algebras living in a symmetric braided tensor category (C, C). We obtain that (H, R) is a quasitriangular bialgebra living in C iff (HM, C R) is a braided tensor category. We show that the antipode of (co)quasitriangular Hopf algebra living in C is invertible. Next we also prove that S 2 is inner when almost cocommutative Hopf algebra (H, R) living in C has an invertible antipode S. In particular, we structure the Drinfeld (co)double D(H) in symmetric braided tensor category with left dual and we prove that it is (co)quasitriangular. We give some basic concepts as follows: Let H and A be two bialgebras in a symmetric braided tensor categories (C, C), and for any U, V, W ∈ obC, assume that R: I → H ⊗ H, r: H ⊗ H → I, τ: H ⊗ A → I, σ: H ⊗ H → I,

The term “quantum group ” was popularized by Drinfeld in his address to the International Congress of Mathematicians in Berkeley. However, the concepts of quantum groups and quasitriangular Hopf algebras are the same. Therefore, Hopf algebras have close connections with various areas of mathematics and physics. The development of Hopf algebras can be divided into five stages. The first stage is that integral and Maschke’s theorem are found. Maschke’s theorem gives a nice criterion of semisimplicity, which is due to M.E. Sweedler. The second stage is that the Lagrange’s theorem is proved, which is due to W.D. Nichols and M.B. Zoeller. The third stage is the research of the actions of Hopf algebras, which unifies actions of groups and Lie algebras. S. Montgomery’s “Hopf algebras and their actions on rings ” contains the main results in this area. S. Montgomery and R.J. Blattner show the duality theorem. S. Montegomery, M. Cohen, H.J. Schneider, W. Chin, J.R. Fisher and the author study the relation between algebra R and the crossed product R#σH of the semisimplicity, semiprimeness, Morita equivalence and radical. S. Montgomery and M. Cohen ask whether R#σH is semiprime when R is semiprime for a finite-dimensional semiprime Hopf algebra H. This is a famous semiprime problem. The