Factorizable quasi-Hopf algebras. Applications

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 $D(H)$ is a unimodular quasi-Hopf algebra.Comment: 35 page

Quasi-bialgebra Structures and Torsion-free Abelian Groups

We describe all the quasi-bialgebra structures of a group algebra over a torsion-free abelian group. They all come out to be triangular in a unique way. Moreover, up to an isomorphism, these quasi-bialgebra structures produce only one (braided) monoidal structure on the category of their representations. Applying these results to the algebra of Laurent polynomials, we recover two braided monoidal categories introduced in \cite{CG} by S. Caenepeel and I. Goyvaerts in connection with Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras)

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 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.Comment: 41 pages, no figure

Braided Hopf algebras obtained from coquasitriangular Hopf algebras

Let $(H, \sigma)$ 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 $H_\sigma$, a sub-Hopf algebra of $H^0$, the finite dual of $H$. Using the generalized quantum double construction and the theory of Hopf algebras with a projection, we associate to $H$ a braided Hopf algebra structure in the category of Yetter-Drinfeld modules over $H_\sigma^{\rm cop}$. Specializing to $H={\rm SL}_q(N)$, we obtain explicit formulas which endow ${\rm SL}_q(N)$ with a braided Hopf algebra structure within the category of left Yetter-Drinfeld modules over $U_q^{\rm ext}({\rm sl}_N)^{\rm cop}$.Comment: 43 pages, 1 figur