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For a Hopf algebra A of arbitrary dimension over a field K, it is well-known that if A has nonzero integrals, or, in other words, if the coalgebra A is co-Frobenius, then the space of integrals is one-dimensional and the antipode of A is bijective. Bulacu and Caenepeel recently showed that if H is a dual quasi-Hopf algebra with nonzero integrals, then the space of integrals is one-dimensional, and the antipode is injective. In this short note we show that the antipode is bijective.

The Antipode of a Dual Quasi-Hopf Algebra with Nonzero Integrals is Bijective