Intuition behind the relation of commutative Hopf algebra and Groups 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
 A: The answer to your first question is that (for finite groups), those two hopf algebras are dual to each other. Here is this fact quite explicitly:
Suppose that $G$ is a finite group. We can build the usual hopf algebra associated to the vector space $\mathbb{C}[G]$, with the usual group algebra structure
$$ \eta: 1 \to e, \quad m: g \otimes h\mapsto gh$$
the coalgebra structure
$$ \varepsilon: g \mapsto 1, \quad \Delta: g \mapsto g \otimes g $$
where all of the maps above are defined on the basis of group elements.
Now, we can take the dual Hopf algebra $\mathbb{C}[G]^\vee$, which as a vector space are the functions on $G$: $\mathbb{C}[G]^\vee = \{f: G \to \mathbb{C}\}$, and has new operations being the transpose of all the old operations. Now check what happens to all of the Hopf algebra structure. The unit $\eta$ dualises to the new counit $\eta^\vee: \mathbb{C}[G]^\vee \to \mathbb{C}^\vee \cong \mathbb{C}$, defined by $\eta^\vee(f) = (f \circ \eta)(1)$, which on a function $f$ will be $\eta^\vee(f) = f(\eta(1)) = f(e)$. So the new counit maps a function to its value over the identity.
Similarly, you can dualise the other maps, and see what happens. Should should find that the new coproduct $m^\vee$ is a kind of convolution, taking an indicator function $f_g$ to the element $\sum_{xy = g} f_x \otimes f_y$, the new unit $\varepsilon^\vee$ takes $1$ to the constant function on $G$ with value $1$, and the new multiplication $\Delta^\vee$ is pointwise multiplication of functions.
As for your point (2): There is a correspondence between affine group schemes (which are kind of the algebraic analogue of Lie groups) and commutative Hopf algebras. 
