New results on the bijectivity of antipode of a Hopf algebra. (Q858734)

The two main results of this paper are the following: If the Hopf algebra \(H\) is weakly finite, then its antipode is injective, and it is bijective whenever \(H\) can be embedded into a left perfect ring \(Q\) which is an essential extension of \(H\) as right \(H\)-modules. For any Hopf algebra \(H\) over the field \(k\), the finite dual \(H^\circ\) is weakly finite, so its antipode \(s^\circ\) is injective by the first result above. The second result says that \(s^\circ\) is bijective if \(k\) is either finite or an algebraic closure of a finite field. If this is not the case, and if \(H\) is the free Hopf algebra on the matrix coalgebra \(C=\text{Mat}_2(k)\), then \(s^\circ\) is not surjective. It is conjectured that Noetherian Hopf algebras over a field have bijective antipodes.