I have an algebraic group (a representable functor) $(G,\mu,\eta,\sigma)$ and it's coordinate algebra ${\cal{O}}(G)$.
In my course it is stated without proof that the category of algebraic groups ${\bf{AlgGrp}}_k$ is anti-equivalent to the category of Hopf algebras ${\bf{Hpf}}_k$.
I would like two know which are the two functors that work.
One is easy
$$F:{\bf{Hpf}}_k^{op}\rightarrow {\bf{AlgGrp}}_k \\H\mapsto (\text{Hom}_{Alg}(H,-),\mu,\eta,\sigma)\\f\mapsto\bar{f}\text{ where }\bar{f}(\phi) = \phi\circ f$$
with $\mu,\eta,\sigma$ given by the rules of $H$.
However defining the opposite functor is proving more difficult.
Any help?
The precise equivalence is between affine group schemes over $k$ and the opposite of commutative Hopf algebras over $k$. ("Algebraic group" is a very ambiguous term - for most people it means the underlying scheme has finite type, and for some people it has to be a variety - and noncommutative Hopf algebras don't enter the story.) It goes like this:
The category of affine schemes over $k$ is equivalent to the opposite of the category of commutative $k$-algebras. This is either a definition or a theorem depending on your definition of affine schemes.
An affine group scheme over $k$ is a group object in the category of affine schemes over $k$.
A commutative Hopf algebra over $k$ is a cogroup object in the category of commutative $k$-algebras. (This is not true if "commutative" is dropped!)
The functor from affine group schemes to Hopf algebras is given by taking the coordinate algebra, which for any scheme $X$ is given by $\text{Hom}_{\text{Sch}}(X, \mathbb{A}^1)$.