Cosemisimple pointed Hopf algebras

Question

Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. Every cosemisimple pointed Hopf $\mathbb{K}$-algebra $A$ is easily seen to be cocommutative. Does this imply that $A$ is the group Hopf algebra $\mathbb{K}G$ of some group $G$.

Answer 1

Assuming that we are speaking about finite dimensional hopf algebras, the answer is yes:

Since $H$ is cosemisimple if and only if $H\cong Corad(H)$ (as coalgebras) and $H$ is pointed if and only if $Corad(H)\cong \bigoplus_{g\in G(H)} kg \equiv kG(H)$ (as coalgebras), then "cosemisimple + pointed" means that the hopf algebra $H$ is isomorphic (as a coalgebra) to the group hopf algebra $kG(H)$ of the group $G(H)$ of its grouplike elements: $$ H\cong kG(H) $$ But then $H$ is cocommutative (as mentioned in the OP) and by the Cartier-Kostant-Milnor-Moore theorem (applied in finite dimensions), the above isomorphism is also an isomorphism of Hopf algebras.
(If you want to bypass CKMM in the last step, you can take a look at: About the classification of commutative and of cocommutative, fin. dim. Hopf algebras as an alternative route; there it is shown that a cocommutative HA over an alg closed field of char zero is the group HA of its grouplikes).