TITLE: Relation between root systems and representations of complex semisimple Lie algebras
QUESTION [15 upvotes]: I'm trying to understand the machinery of root systems for the purpose of classifying complex semisimple Lie algebras. During this process i lost the overview, espacially when it came to highest weight representations and all that, so i'm looking for a short summary:

Do isomorphic root systems imply isomorphic Lie algebras (to good to be true :-))?
If two Lie algebras have isomorphic root systems, which relation does this imply at the level of (irreducible) representations?
One can often read that it is possible to reduce the representation theory of those Lie algebras to the problem of classifying their root systems. What does this mean?

I know that it is possible to construct a Lie algebra from a given root system with isomorphic resulting root system, but if there can be ambiguities at the level of representations, even for isomorphic root systems, this property doesn't help at all, does it?

REPLY [8 votes]: As the previous answer nicely sums up the relationship between Lie algebras and root systems, I will say a little bit about representation theory. 
As explained above, there is a 1-1 correspondence between complex semi-simple Lie algebras and irreducible root systems, which in turn are in 1-1 correspondence with Dynkin diagrams and Cartan matrices. Thus isomorphic Lie algebras imply isomorphic root systems, and vice versa. Hence complex semisimple Lie algebras with isomorphic root systems (i.e. isomorphic Lie algebras) have the same representation theory.
Warning: if we also consider Lie algebras which are not complex, then we may have several Lie algebras corresponding to the same root system. For example, the simple real Lie algebra su(n) also has the root system A(n). This is reflected by the fact that sl(n) is the complexification of su(n). 
Classification by root systems is of extreme importance to representation theory. Weights interact in very interesting ways with roots, in fact roots determine the possible weight spaces. The Weyl group, i.e., the group of permutations of the roots, plays a crucial role in the classification of finite-dimensional representations and character theory. 
To grasp the connection between root systems and representation theory you should read about Verma modules - they are highest weight modules induced from the trivial representation of the Borel subalgebra. Since they have a single generator, a lot of the structure of the Lie algebra (or its enveloping algebra) is reflected in the representation. This is why the roots have such a big impact on weights. 
This said, one can hardly "reduce" the representation theory of Lie algebras to the classification of root systems. The classification is rather a first step, but the representation theory is whole new chapter!