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?



Hopf algebras and bijective antipodes
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"](http://arxiv.org/abs/1201.4854), 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](http://www.sciencedirect.com/science/article/pii/S0021869306002328), J. Algebra 306
>  (2006), 622–633
> 
> 
> 


