On integrals and cointegrals for quasi-Hopf algebras

Using the theory of Frobenius algebras, we study how the antipode of a quasi-Hopf algebra H acts on the space of left and right integrals and cointegrals. We obtain formulas that allow us to find out the explicit form of the integrals and cointegrals for the Drinfeld double D(H), in terms of the integrals and cointegrals of H. This leads to an answer to a conjecture made by Hausser and Nill at the end of the nineties.