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Published online by Cambridge University Press: 21 February 2019
We show that for a finite-dimensional quasi-Hopf algebra H the space of integrals in H, and the space of cointegrals on H, have dimension 1. We characterize semisimple and symmetric quasi-Hopf algebras with the help of integrals, and prove a formula for the fourth power of the antipode in terms of the modular elements by using the machinery provided by Frobenius algebras. The chapter ends with a freeness theorem stating that any finite-dimenisonal quasi-Hopf algebra is free over any quasi-Hopf subalgebra.
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