What is an example of a Hopf algebra with a non-invertible antipode?
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Theorem of Takeuchi (in Free Hopf algebras generated by coalgebras, 1971) asserts that free Hopf algebra $H(C)$ over a coalgebra $C$ has injective antipode, and it is bijective precisely (at least over alg. closed field) when $C$ is pointed.
On the other hand, in a paper Faithful flatness over Hopf subalgebras - counterexamples, 2000 P. Schauenburg constructs a Hopf algebra with surjective, but non-injective antipode. Example is constructed as follows. There's a "free Hopf-with-bijective-antipode" functor from coalgebras to Hopf algebras; one can find a biideal in such free-with-bijective-antipode Hopf algebra over matrix coalgebra $M_4(k)$ that is stable under antipode, but not under its inverse, and quotient will be the example. Paper can be found on author's homepage http://schauenburg.perso.math.cnrs.fr/personnelle.html
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$\begingroup$ What exactly is the 'free Hopf algebra 𝐻(𝐶) over a coalgebra 𝐶"? $\endgroup$ Mar 6, 2022 at 10:56
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1$\begingroup$ The definition is in Takeuchi's paper: projecteuclid.org/journals/… $\endgroup$ Mar 6, 2022 at 13:36