TITLE: Naive question about the representation theory of algebraic groups and hopf algebras
QUESTION [6 upvotes]: 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.


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


Thanks!

REPLY [2 votes]: There is a direct way to define group algebras for different classes of groups (locally compact groups, Lie groups, algebraic groups, etc.). 
For an (affine) algebraic group $G$ one should take the algebra ${\mathcal R}(G)$ of polynomials (i.e. regular functions) on $G$ and then consider the dual algebra ${\mathcal R}^\star(G)$ of linear functionals on ${\mathcal R}(G)$ with the natural topology (in this case the topology of poitwise convergence on ${\mathcal R}(G)$). Then ${\mathcal R}^\star(G)$ becomes a (topological) group algebra for $G$: each  "polynomial representation" of $G$ corresponds bijectively to a continuous representation of ${\mathcal R}^\star(G)$. At the same time both ${\mathcal R}(G)$ and ${\mathcal R}^\star(G)$ are Hopf algebras in the monoidal category of stereotype spaces, see details in: S.S.Akbarov, Pontryagin duality in the theory of topological vector spaces and in topological algebra, Journal of Mathematical Sciences, 113(2): 179-349, 2003 (Theorem 10.12).