I have been learning some representation theory and have some questions about the following pattern:


 **Instance 1:**  If we have a finite group $G$ and a field $k$, a representation of $G$ over $k$ consists of a finite dimensional $k$-vector space $V$ and a homomorphism $G \to GL(V)$. We have a Hopf algebra $kG$ and an equivalence between representations of $G$ over $k$ and finite dimensional $kG$-modules.


 **Instance 2:** If we have a Lie group $G$, a representation of $G$ consists of a finite dimensional $\mathbb{R}$-vector space $V$ and a smooth homomorphism $ G \to GL(V) $. Let $\mathfrak{g}$ be the lie algebra of $G$. Then we have a Hopf algebra $U(\mathfrak{g})$ and an equivalence between representations of $G$ and finite dimensional $U(\mathfrak{g})$-modules.



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> >  **Question:**  If $G$ is an algebraic group over some field $k$, is there a Hopf algebra floating around?
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Thanks!



Naive question about the representation theory of algebraic groups and hopf algebras
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 $G$. 


