Classifying Hopf algebras that admit a single irreducible comodule 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
$$
k \to k \otimes H, ~~ k \mapsto k \otimes 1_H.
$$
 A: The HAs you are describing are again the connected (=irreducible) ones. I am using the terminology here as in my answer to your previous question: Name for a Hopf algebra whose only grouplike element is the identity?.
Their classification, is in general an open project:

*

*In the cocommutative case, over a field of $chark=0$, they are all universal enveloping algebras of lie algebras. If $k=\mathbb{C}$, this is by an old result of Milnor and Moore. You can find the statement for any field of char zero at Montgomery's book.

*In the finite dimensional case they appear only over fields of positive characteristic. I think this is an old result of Masuoka (but i do not have the exact reference right now). See arXiv:1309.0286 [math.RA], arXiv:1310.7073 [math.RA] (and the references therein) for recent results on their classification.

A: 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.
