TITLE: Hopf algebras and bijective antipodes
QUESTION [11 upvotes]: By a theorem of Larson and Sweedler, the antipode of every finite-dimensional Hopf algebra is bijective.
My question is the following: 
Is it true that in every noetherian Hopf algebra the antipode is bijective?

REPLY [12 votes]: 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:

S. Skryabin, New results on the bijectivity of antipode of a Hopf algebra, J. Algebra 306
  (2006), 622–633