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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?

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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

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