I am looking to establish the relationship between Weyl groups and semisimple Lie algebras. So far I have found the root space decomposition (I am using $\mathfrak{sl}(3, \mathbb{C}) $as my example).
From here, I have constructed a Cartan matrix. I am now wondering where I go from here so that I can find the Weyl group from which the Lie algebra can be reconstructed. Basically I am trying to find the generators for a Lie algebra.....but I am getting very confused.
Any help would be greatly appreciated! Thanks!
Finding the Weyl group from the Cartan matrix $M$ is quite simple. The actual generators are the simple reflections $s_i$, where $$s_{i}(\alpha_j) = \alpha_j - M_{ij}\alpha_i$$ For the isomorphism class, you need to look at $$M_{ij}M_{ji}=4\cos^2\frac{\pi}{k_{ij}}$$ Then the generating relation is $$(s_is_j)^{k_{ij}}=1$$ The isomorphism class is then determined by the Dynkin diagram. Knowing the isomorphism class gives you a better idea of what the roots are without having to work them out yourself. For $\mathfrak{sl}(3,\mathbb{C})$, one choice of positive roots is $$t_i-t_j$$ for $1\leq i\lt j\leq 4$. Then the simple roots are $t_1-t_2$, $t_2-t_3$, and $t_3-t_4$. The reflections act as transpositions on the indices.