By a theorem of Larson and Sweedler, the antipode of every finitedimensional Hopf algebra is bijective. My question is the following:
Is it true that in every noetherian Hopf algebra the antipode is bijective?
By a theorem of Larson and Sweedler, the antipode of every finitedimensional Hopf algebra is bijective. My question is the following: Is it true that in every noetherian Hopf algebra the antipode is bijective? 


It is conjectured that the antipode is bijective for all noetherian Hopf algebras (Skryabin), but no proof is known. Take a look at this recent short survey, "Noetherian Hopf Algebras", by K.R. Goodearl, where this is listed as conjecture 1.9. Skryabin's original paper is:


