Kaplansky's second conjecture (on Hopf algebras) deals with "admissible" coalgebras: He calls a coalgebra admissible, if there is an algebra structure making it a Hopf algebra. The conjecture states that:

a coalgebra $C$ is admissible if and only if any finite subset of $C$ lies in a finite dimensional admissible subcoalgebra.

As far as I know there are counterexamples refuting the conjecture. (the first one was stated by Larson if I remember correctly).

Inspired by the above, let us lay the following definition of the notion of admissible algebra:

Definition: An algebra $A$ will be called admissible if there is a coalgebra structure on $A$ and a suitable linear map $S:A\to A$, such that all these data (the algebra, the coalgebra and the linear map $S$) constitute a Hopf algebra with antipode $S$.

Now, my question is whether there is some criterion (necessary, sufficient or both) regarding to when an algebra is admissible (or non-admissible) in the sense of the above definition. In other words:

• Given an algebra $A$, under which conditions can a coalgebra structure be found on $A$ such that it will be turned into a Hopf algebra?
• When would such a coalgebra structure be unique?
• What could be examples of a non-admissible algebras, in the sense of the above definition?