I've heard that a commutative Hopf algebra can be thought an algebra construction over the space of functions on a group to the ground field. The product should be the pointwise multiplication, Coproduct maps $f$ to the functions on 2 variables $(x,y)↦f(xy)$. Counit is evaluation $f↦f(1)$. Antipode maps f to $x↦f(x^−1)$. Everything is fine so far. Here's my question:
1) How it is related this construction with the other canonical group algebra $\mathbb CG$ where the coproduct is given by $$\triangle\left(g\right)=\left(g\otimes g\right),$$ with $g\in G$. Is it the same construction with different notation? How can I show it's the same?
2) I also don't get if there's an equivalence between commutative Hopf algebras and Groups and to which extent (does the group have to be finite or it can be a Lie Group)?
3) how this is related if it is with the definition of Quantum Groups as non-commutative nor co-commutative Hopf Algebras?
Please if someone can do a systematic and intuitive argument for 1 and for 2 which I suppose to be strictly related. Thank you