TITLE: Classifying Hopf algebras that admit a single irreducible comodule
QUESTION [5 upvotes]: Is it possible to classify Hopf algebras $H$, over a field $k$, which admit a unique (up to isomorphism)  irreducible comodule, namely the trivial $1$-dim comodule
$$
k \to k \otimes H, ~~ k \mapsto k \otimes 1_H.
$$

REPLY [3 votes]: There have been several papers published on such Hopf algebras (which are referred to as "connected" Hopf algebras) over the past decade. In particular, over an algebraically closed field of characteristic 0, the discovery of non-commutative + non-cocommutative examples has helped spur on the interest in classifying these algebras. See, for example, the following:
Properties of pointed and connected Hopf algebras of finite Gelfand-Kirillov dimension

Quantum homogeneous spaces of connected Hopf algebras

Connected (graded) Hopf algebras
In general there is no known concrete classification of such connected Hopf algebras. In particular, the last of the papers listed above presents a non-cocommutative, non-commutative connected Hopf algebra which is not isomorphic as an algebra to the universal enveloping algebra of any lie algebra.