During a classification problem I came across a set of algebras given as the path algebra of a quiver with relations. As an example the local ones: $k\langle x,y\rangle/x^2,y^2, xy-qyx$, where $q\in k$, sometimes called truncated quantum plane.

My question is whether there are necessary/sufficient criteria for (these) algebras to be a block of a Hopf algebra (up to Morita equivalence). For example, $q$ obviously has to be a root of unity, because otherwise the Nakayama automorphism will not be of finite order. A sufficient criteria is $q=1$, but I don't know of other instances. (Maybe it also depends on the characteristic of the underlying field).

Weaker results would also be of interest to me, for example [Farnsteiner: Polyhedral groups, McKay quivers, and the finite algebraic groups with tame principal blocks, Proposition 7.4.3] shows that they will not be the principal block of a cocommutative Hopf algebra if $char k \geq 3$.