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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 $$ k \to k \otimes H, ~~ v \mapsto v \otimes 1_H. $$

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 $$ k \to k \otimes H, ~~ k \mapsto k \otimes 1_H. $$

Is it possible to classify Hopf algebras $H$ which admit a unique (up to isomorphism) irreducible comodule $V$, namely the trivial $1$-dim comodule $$ V \to V \otimes H, ~~ v \mapsto v \otimes 1_H. $$

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 $$ k \to k \otimes H, ~~ v \mapsto v \otimes 1_H. $$