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A Hopf algebra is called pointed if all its simple left (or right) comodules are one-dimensional. See for example this question for a discussion. Now every Hopf algebra $H$ admits a one-dimensional ...

For a $k$-Hopf algebra $H$ and element $h \in H$ is called grouplike is $\Delta(h) = h \otimes h$ and $\epsilon(h)=1_k$ ($\epsilon$ is the counit). The identity $1_H$ is clearly grouplike, but in ...