Affine algebra of an algebraic group From what I understand there are two approaches to defining an algebraic group. One can start talking about varieties and the Zariski topology and such and get to a definition of an algebraic group. One can also define an algebraic group as a (representable) functor from the category of $k$-algebras to the category of sets.
If one chooses the functorial definition of an (affine) algebraic group $G$, what is the definition of the affine algebra of $G$?

By affine algebra I assume you mean the coordinate ring, as in $\mathbb A^n$ has coordinate ring $k[x_1, \ldots, x_n]$.
Given an affine group scheme $G$ over $k$ there is a natural homomorphism $G \simeq \hom_k(A, -)$ where $A$ is some $k$-algebra.  By the Yoneda lemma the algebra $A$ is unique up to isomorphism and the coordinate ring of $G$ is defined to be $k[G] = A$.
