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
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.