I'm interested in the structures of categories like $Rep(GL\_n), Rep(SL\_n)$, etc. of algebraic representations of an algebraic group. I understand that there should be some relation between these and the categories of representations of the corresponding Lie algebras. 
However, it's not as intuitive to me what's going on here as with the case of, say, a Lie group, perhaps because the notion of a "tangent vector" is somewhat different.


So, how does one switch between the categories $Rep(G)$ and $Rep(\mathfrak{g})$ for $G$ an algebraic group and $\mathfrak{g}$ its Lie algebra---are there functors in each direction? Can this be used to prove that $Rep(G)$ is semisimple when $G$ is reductive?
In another direction, can the structure of $Rep(\mathfrak{g})$ as known from the representation theory of, say, semisimple Lie algebras give the structure of $Rep(G)$?



How do you switch between representations of an algebraic group and its Lie algebra?
If $G$ is semisimple simply connected in characteristic zero, the differential at $1$ gives an equivalence of (tensor) categories $Rep(G)\to Rep({\mathfrak g})$. If $G$ is not semisimple, this is not the case, but this functor is always fully faithful (i.e. an equivalence onto a full subcategory) if $G$ is connected. The essential image of this functor can be described explicitly. Namely, consider the Levi decomposition ${\mathfrak g}={\mathfrak l}\ltimes {\mathfrak u}$, where ${\mathfrak l}$ is reductive and ${\mathfrak u}=Lie(U)$, where $U$ is the unipotent radical of $G$. Then the image is those finite dimensional representations of ${\mathfrak g}$ for which ${\mathfrak u}$ acts nilpotently, and the weights for ${\mathfrak l}$ are integral (i.e. descend to chartacters of the maximal torus). 


