If $G$ is an affine algebraic group, its coordinate ring $\mathcal O(G)$ is a Hopf algebra. The multiplication is the usual (commutative) pointwise multiplication of functions. The comultiplication is pullback under the map $m:G\times G \to G$ given by the group structure. The algebraic group $G$ is determined by the Hopf algebra $\mathcal O(G)$.

Representations of $G$ are the same as comodules for $\mathcal O(G)$. So this Hopf algebra is dual to the notion of group algebra for finite groups.

Note that this Hopf algebra is dual to the notion of group algebra of a finite group.

Similarly, the universal enveloping algebra $U(\mathfrak g)$ is (kind of) dual to the coordinate ring as Hopf algebras. More precisely (I think!), the dual of $U(\mathfrak g)$ is the coordinate ring of the formal neighbourhood of the identity in $G$.

If $G$ is an affine algebraic group, its coordinate ring $\mathcal O(G)$ is a Hopf algebra. The multiplication is the usual (commutative) pointwise multiplication of functions. The comultiplication is pullback under the map $m:G\times G \to G$ given by the group structure (this is only cocommutative if $G$ is abelian). 

The algebraic group $G$ is determined by the Hopf algebra $\mathcal O(G)$.

Representations of $G$ are the same as comodules for $\mathcal O(G)$. So this Hopf algebra is dual to the notion of group algebra for finite groups.

Similarly, the universal enveloping algebra $U(\mathfrak g)$ is (kind of) dual to the coordinate ring as Hopf algebras. More precisely (I think!), the dual of $U(\mathfrak g)$ is the coordinate ring of the formal neighbourhood of the identity in $G$.