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

Question 2: Let $G$ be a Lie group. Suppose that $Q$ is a quantum deformation of the universal enveloping algebra $U(\mathfrak{g})$. How are you supposed to think about the category of finite dimensional $Q$-modules?

Thanks!

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!