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I recently came across Kac algebra. They are roughly Hopf algebras and $C^*$-algebras with compatible structures. It follows from Artin–Wedderburn theorem that every semisimple complex Hopf algebra ...

Give an algebra $A$, a bialgebra $B$, and an action, that is, a bilinear map $\triangleright: B \times A \to A$ such that $$ (b_1b_2) \triangleright a = b_1\triangleright(b_2 \triangleright a). $$ ...

I have been recently studying different methods to construct Hopf algebras. In Theorem IX.2.3 of "Quantum groups" by Kassel the bicrossed product of a pair of matched bialgebras (or Hopf ...

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 ...

It's all in the question! What is an example of a sub-bialgebra of a Hopf algebra that is not a Hopf subalgebra?