Daniel Bulacu

Radford's biproduct for quasi-Hopf algebras and bosonization

Let B be a braided Hopf algebra (with bijective antipode) in the category of left Yetter– Drinfeld modules over a quasi-Hopf algebra H . As in the case of Hopf algebras (J. Algebra 92 (1985) 322), the smash product B#H de:ned in (Comm. Algebra 28(2) (2000) 631) and a kind of smash coproduct a=ord a quasi-Hopf algebra structure on B ⊗ H . Using this, we obtain the structure of quasi-Hopf algebras with a projection. Further we will use this biproduct to describe the Majid bosonization (J. Algebra 163 (1994) 165) for quasi-Hopf algebras. c � 2002 Elsevier

Integrals for (dual) quasi-Hopf algebras. Applications☆

Abstract A classical result in the theory of Hopf algebras concerns the uniqueness and existence of integrals: for an arbitrary Hopf algebra, the integral space has dimension ⩽1, and for a finite-dimensional Hopf algebra, this dimension is exactly 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 follows from the other axioms of a finite-dimensional quasi-Hopf algebra. We give a new version of the Fundamental Theorem for quasi-Hopf algebras. We show that a dual quasi-Hopf algebra is co-Frobenius if and only if it has a non-zero integral. In this case, the space of left or right integrals has dimension one.

Generalized Diagonal Crossed Products and Smash Products for Quasi-Hopf Algebras. Applications

In this paper we introduce generalizations of diagonal crossed products, two-sided crossed products and two-sided smash products, for a quasi-Hopf algebra H. The results we obtain may then be applied to H*-Hopf bimodules and generalized Yetter-Drinfeld modules. The generality of our situation entails that the “generating matrix” formalism cannot be used, forcing us to use a different approach. This pays off because as an application we obtain an easy conceptual proof of an important but very technical result of Hausser and Nill concerning iterated two-sided crossed products.

Yetter-Drinfeld Categories for Quasi-Hopf Algebras

ABSTRACT We show that all possible categories of Yetter-Drinfeld modules over a quasi-Hopf algebra H are isomorphic. We prove also that the category of finite dimensional left Yetter-Drinfeld modules is rigid, and then we compute explicitly the canonical isomorphisms in . Finally, we show that certain duals of H 0, the braided Hopf algebra (introduced in Bulacu and Nauwelaerts, 2002; Bulacu et al., 2000) are isomorphic as braided Hopf algebras if H is a finite dimensional triangular quasi-Hopf algebra. Communicated by M. Takeuchi.

Quasitriangular and Ribbon Quasi-Hopf Algebras

Abstract Following (Drinfeld, V. G. (1990a). Quasi-Hopf algebras.Leningrad Math. J. 1:1419–1457) a quasi-Hopf algebra has, by definition, its antipode bijective. In this note, we will prove that for a quasitriangular quasi-Hopf algebra with an R-matrix R, this condition is unnecessary and also the condition of invertibility of R. Finally, we will give a characterization for a ribbon quasi-Hopf algebra. This characterization has already been given in Altschuler and Coste (Altschuler, D., Coste, A. (1992). Quasi-quantum groups, knots, three-manifolds and topological field theory. Comm. Math. Phys. 150:83–107.), but with an additional condition. We will prove that this condition is unnecessary.

The Quantum Double for Quasitriangular Quasi-Hopf Algebras

Abstract Let D(H) be the quantum double associated to a finite dimensional quasi-Hopf algebra H, as in Hausser and Nill ((Hausser, F., Nill, F. (1999a). Diagonal crossed products by duals of quasi-quantum groups. Rev. Math. Phys. 11:553–629) and (Hausser, F., Nill, F. (1999b). Doubles of quasi-quantum groups. Comm. Math. Phys. 199:547–589)). In this note, we first generalize a result of Majid (Majid, S. (1991). Doubles of quasitriangular Hopf algebras. Comm. Algebra 19:3061–3073) for Hopf algebras, and then prove that the quantum double of a finite dimensional quasitriangular quasi-Hopf algebra is a biproduct in the sense of Bulacu and Nauwelaerts (Bulacu, D., Nauwelaerts, E. (2002). Radfords biproduct for quasi-Hopf algebras and bosonization. J. Pure Appl. Algebra 179:1–42.).

THE BRAIDED MONOIDAL STRUCTURES ON THE CATEGORY OF VECTOR SPACES GRADED BY THE KLEIN GROUP

Let

On the Antipode of a Co-Frobenius (Co)Quasitriangular Hopf Algebra

k

Doi–Hopf Modules and Yetter–Drinfeld Modules for Quasi-Hopf Algebras

be a field,

On Frobenius and separable algebra extensions in monoidal categories: applications to wreaths

k^*=k\setminus\{0\}