TITLE: How do you switch between representations of an algebraic group and its Lie algebra?
QUESTION [11 upvotes]: 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 $Rep(G)$?

REPLY [2 votes]: One point which seems not to have been mentioned (but is implicit in Pavel's answer) is that the small question posed at the end of the main question, whether $\mathrm{Rep}(\mathfrak{g})$ semisimple implies $\mathrm{Rep}(G)$ semisimple, has a simple answer regardless of most technical conditions.  In fact, in characteristic zero it is true.
The relevant theorem is:

Theorem: Suppose $k$ has characteristic zero, let $G$ be a connected affine algebraic $k$-group, and let $\mathfrak{g}$ be its Lie algebra.  If $V$ is a $G$-representation and $W \subset V$ is a subspace, then $W$ is $G$-stable if and only if it is $\mathfrak{g}$-stable.

The proof for faithful representations is in Humphreys' "Linear Algebraic Groups", Theorem 13.2, and to extend it to any representation one need only show that the Lie algebra is compatible with fibered products, which is tautological.  (Possibly this result must be stated over an algebraically closed field; this is so in the book, and I am badly acquainted with rationality properties.)
As a consequence, $V$ is irreducible or completely reducible for $G$ if and only if it is for $\mathfrak{g}$.