The antipode of a dual quasi-Hopf algebra with nonzero integrals is bijective

Beattie, Margaret; Iovanov, Miodrag Cristian; Raianu, Serban

The antipode of a dual quasi-Hopf algebra with nonzero integrals is bijective

Authors

Beattie, Margaret

Iovanov, Miodrag Cristian

Raianu, Serban

Description

For $A$ a Hopf algebra 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.
4 pages