The coordinate Hopf algebra of an affine group scheme

Let $k$ be a field and let $G$ be a group scheme of finite type over $k$ (Group schemes of finite type over a field) whose underlying scheme is affine (Affine schemes and their coordinate rings), say $G\cong \mathrm{Spec}A$ with $A=\mathcal{O}(G)=\mathrm{\Gamma }(G,{\mathcal{O}}_G)$ (Global functions on Spec A recover A); write $m\text{\hspace{0.17em}⁣}:G{\times }_{k}G\to G$, $e\text{\hspace{0.17em}⁣}:\mathrm{Spec}k\to G$ and $i\text{\hspace{0.17em}⁣}:G\to G$ for its multiplication, identity and inverse. Under the anti-equivalence between affine $k$-schemes and commutative $k$-algebras (Affine schemes are contravariantly equivalent to commutative rings) and the identification $\mathcal{O}(G{\times }_{k}G)\cong A{\otimes }_{k}A$ (Affine fibre products are spectra of tensor products), these morphisms correspond to $k$-algebra homomorphisms $\mathrm{\Delta }=\mathcal{O}(m)\text{\hspace{0.17em}⁣}:A\to A{\otimes }_{k}A,\epsilon =\mathcal{O}(e)\text{\hspace{0.17em}⁣}:A\to k,S=\mathcal{O}(i)\text{\hspace{0.17em}⁣}:A\to A,$ called the comultiplication, the counit and the antipode of $G$ (The map of affine spectra induced by a ring homomorphism). The three maps make $A$ a commutative Hopf algebra over $k$ in the sense of Commutative Hopf algebras over a field; the verification is The coordinate ring of an affine group scheme is a commutative Hopf algebra ↗, and this definition fixes the construction and the notation.

For a commutative unital $k$-algebra $R$, points $g,g_1,g_2\in G(R)$ and $f\in A$ one has, under the evaluation pairing $G(R)\times A\to R$, $(\mathrm{\Delta }f)(g_1,g_2)=f(g_1{g}_2),\epsilon f=f(e),(Sf)(g)=f(g^{-1})\mathrm{.}$

A morphism $f\text{\hspace{0.17em}⁣}:G\to H$ of affine group schemes (Morphisms and closed subgroup schemes of group schemes) induces the $k$-algebra homomorphism $\mathcal{O}(f)\text{\hspace{0.17em}⁣}:\mathcal{O}(H)\to \mathcal{O}(G)$ given by pullback of functions; it is a morphism of commutative Hopf algebras.