Quasitriangular and ribbon quasi-Hopf algebras

Abstract

Following brinfeld (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.