Relation between root systems and representations of complex semisimple Lie algebras

Solution 1:

The below is a short exposition I wrote a few months ago about the proof of the classification, attempting to give a "big-picture" view of it. I have copied it directly from the .tex file I wrote (and then attempted to fix all those things that fails due to that), so if anyone sees any weird things, let me know. If anyone wants the original .pdf instead, feel free to ping me on the chat.

In this paper, I will provide an outline of the proof of the classification of finite dimensional semisimple Lie algebras over an algebraically closed field of characteristic $0$. So from now on, the term Lie algebra will refer to one with the aforementioned properties. All references are to Humphreys ``Introduction to Lie algebras and representation theory''.

The overall strategy of the proof is to associate to each Lie algebra $L$ a root system $\Phi_L$ such that we have the following properties:

1. If $L\cong L'$ then $\Phi_L\cong \Phi_{L'}$
2. If $\Phi_L\cong \Phi_{L'}$ then $L\cong L'$
3. For each root system $\Phi$ there is a Lie algebra $L$ such that $\Phi \cong \Phi_L$
4. If $L = L_1 \oplus L_2$ then $\Phi_L = \Phi_{L_1} \sqcup \Phi_{L_2}$ (here $\sqcup$ denotes the orthogonal union)
5. $L$ is simple iff $\Phi_L$ is irreducible

Once the above properties have been established, we have a nice correspondence between Lie algebras and root systems, allowing us to classify the Lie algebras by classifying the root systems. Due to properties 4 and 5, it is clear that we really just need to classify the irreducible root systems, which then give us the simple Lie algebras (and we know that semisimple Lie algebras are completely reducible by Theorem 6.3). The classification of irreducible root systems can be found in 11.4.

So how does one obtain such a correspondence? First, one needs to find a way to get a root system from a Lie algebra. At first, this is done by choosing a Cartan subalgebra $H$ of $L$ and then associating a root system to the pair $(L,H)$ as described in Section 8 (here a Cartan subalgebra is called a maximal toral subalgebra, but these are equivalent terms in our case due to Corollary 15.3). Then one needs to make sure that this does not actually depend on the choice of $H$. To do this, one shows that for any two cartan subalgebras $H_1$ and $H_2$ of $L$ there is an automorphism $\varphi$ of $L$ such that $\varphi(H_1) = H_2$, which is done in Section 16. This automorphism then induces an isomorphism of the associated root systems. Once we have established this, it is clear that we also get property 1, as the image of a Cartan subalgebra under an isomorphism is again a Cartan subalgebra.

Properties 4 and 5 are established in 14.1.

Property 2 is established in Theorem 14.2.

There are two ways to establish property 3. One is to note that we only need to find a corresponding simple Lie algebra for each irreducible root system. One can then go through the classical simple Lie algebras in 1.2 and check that they have the correct root systems (as well as constructing suitable Lie algebras for the exceptional cases). Another approach is to define the Lie algebras via the root systems (more precisely via the Cartan matrix), as summed up in 18.4.

Solution 2:

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!