I have probably a stupid question about representations of algebraic groups:


Let $G$ be an algebraic group and $L$ be a Lie algebra of $G$. What is the connection between
categories of representations of $G$ and $L$ (are they equivalent)?


Now, let $V$ be an irreducible representation of $G$, how to prove that $V$ is also 
an irreducible representation of $L$ (in case it is true)? It should be true and I believe,
proof is easy, but still, I do not see that and I am looking for a nice proof of that. 


And vice versa, let $V$ be an irreducible representation of $L$, is it also irreducible representation of $G$ ? 



algebraic groups and their Lie algebras
I suggest the following lecture notes of Bruhat: 


www.math.tifr.res.in/~publ/ln/tifr14.pdf


Chapter 3 & 4 should answer most of your questions.


For example, there are statements like this :


Proposition 1(pg.19). To every analytic representation h : G −→ G′ there
corresponds a map dh : U(G) → U(G′) which is a representation of
algebras such that ( f ◦ h) = (dh() f ) ◦ h.


Corollary (pg.36). Let G and H be two Lie groups having g and J as their Lie
algebras. If G is connected and simply connected, to every representation
π of g in J, there corresponds one and only one representation f
of G → H such that d f = π.


If you are interested in semisimple, connected, simply connected groups only, both give you an isomorphism between the categories of complex representations. Equivalence of categories is weaker than isomorphism. Moreover, the categories of representations are both semisimple in this case, i.e., reps decompose into irreducible ones. Thus, also your second and third question can be answered affirmative in this case.


