26 research outputs found
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 of any
finite dimensional quasi-Hopf algebra is factorizable, and we characterize
when itself is factorizable. Finally, we prove that any finite
dimensional factorizable quasi-Hopf algebra is unimodular. In particular, we
obtain that the Drinfeld double 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 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 , a sub-Hopf algebra of , the finite dual of
. Using the generalized quantum double construction and the theory of Hopf
algebras with a projection, we associate to a braided Hopf algebra
structure in the category of Yetter-Drinfeld modules over .
Specializing to , we obtain explicit formulas which endow
with a braided Hopf algebra structure within the category of
left Yetter-Drinfeld modules over .Comment: 43 pages, 1 figur