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). Radford's biproduct for quasi-Hopf algebras and bosonization. J. Pure Appl. Algebra 179:1–42.).

Key Words:

• Quasitriangular quasi-Hopf algebras
• Quantum double
• Braided Hopf algebras
• Biproduct

Acknowledgments

Research supported by the bilateral project “Hopf Algebras in Algebra, Topology, Geometry and Physics” of the Flemish and Romanian governments. This paper was written while the first author was visiting the Free University of Brussels, VUB (Belgium); he would like to thank VUB for its warm hospitality.