About Weyl group

Question

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Let $W$ be its Weyl group.

I would like to know whether $W$ is always finite? If so, why?

Answer 1

Yes, it is always finite. Let $\mathfrak h$ be a Cartan subalgebra of $\mathfrak g$, and let $\Phi$ be the root system of $(\mathfrak g,\mathfrak h)$. By definition the Weyl group of $(\mathfrak g,\mathfrak h)$ is generated by all reflection $s_\alpha$ ($\alpha\in\Phi$), where$$s_\alpha(v)=v-2\frac{\langle\alpha, v\rangle}{\langle\alpha,\alpha\rangle}\alpha$$and $\langle\cdot,\cdot\rangle$ is the inner product in $\mathfrak h^*$ induced by the Killing form. It turns out that each $s_\alpha$ preserves $\Phi$ and that therefore each element of the Weyl group preserves $\Phi$. But $\Phi$ generates $\mathfrak h^*$ and therefore the Weyl group can be seen as a subgroup of the group of permutations of $\Phi$, which is finite.