I am new to representation theory and only know an informal definition of Weyl group – it is a group of isometries generated by some transformations (I think reflections) of hyperplanes associated to the roots of the given space.
Then, in Dixmier´s book Enveloping algebras, I have seen this definition: Weyl group is a group of automorphisms of $\mathfrak{h}^*$. If we have a Lie algebra $\mathfrak{g}$, its Cartan subalgebra is denoted $\mathfrak{h}$ in the book, so $\mathfrak{h}^*$ should be functions from $\mathfrak{h}$. My question is whether this is true and how this definition belows connects with the other definition about isometries.
Thank you for advice, if there is any problem, I can edit.
Edit: found another different definition of Weyl group. Let´s call it Definition 3, the isometry definition can be refered to as "Definition 1 and the $\mathfrak{h}^*$ definition can be "Definition 2".
Best Answer
Note that $\mathfrak h^*$ is the dual of $\mathfrak h$, that is, it's the set of all linear maps from $\mathfrak h$ into the field that you are working with.
And those $s_\alpha$'s are reflections: if $\langle\lambda,\alpha\rangle=0$, then $s_\alpha(\lambda)=\lambda$. And if $\lambda$ is a multiple of $\alpha$, then $s_\alpha(\lambda)=-\lambda$. So, $s_\alpha$ is a reflection on the hyperplane $\{\lambda\in\mathfrak h^*\mid\langle\lambda,\alpha\rangle=0\}$.