Equivalent definition of Weyl group?

Question

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".

Answer 1

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 $\{\lambda\in\mathfrak h^*\mid\langle\lambda,\alpha\rangle=0\}$.

Answer 2

Maybe the first obstacle in your understanding is to realize this: The roots which we are interested in are certain elements of $\mathfrak h^*$.

To expand: One can study root systems very abstractly, just from their definitions. That's how we viewed things e.g. in I don´t understand root systems. For that matter, roots are just certain vectors in some Euclidean space $E$ which are only allowed to have certain lengths, and certain angles between them. We really imagine "a root system" as a certain geometric object, kind of a crystal made out of arrows. Then one can very abstractly talk about the Weyl group as certain isometries (reflections, rotations) of this nice little crystal. That's your definition 1.

Well, it might be aesthetically pleasing to study such little crystals, and the many ways one can rotate and reflect them, but why do we do this? Because, as alluded to at the end of my answer to the above question, it turns out that in the study of Lie groups and Lie algebras, one can "distill" some information out of the group / algebra; this "distilled information" gives us kind of a "skeleton" of the group or algebra in question. This skeletal structure does not tell us everything, but surprisingly much about the group or algebra. This skeleton happens to be ... a root system!

But like every skeleton, it does not sit right at the surface. One has to look deep into the group or algebra to even see those roots.

That is what happens in your definition 2 for the Lie algebra, and it seems to be what also prompted the question Picture of Root System of $\mathfrak{sl}_{3}(\mathbb{C})$. In my answer to that I write that I understand a beginner's confusion here. What happens is that we start with a big complex vector space $\mathfrak h^*$ (which consists of linear maps from some subalgebra of our Lie algebra to the ground field ... a very abstract thing on first sight, not really "imaginable"). Then we single out something like six or ten or 24 special elements in that big vector space. It turns out that this little bunch of special elements, if one knows how (and this is the hard part you have to learn yourself through many examples and calculations), can be imagined as if they are just a bunch of six or ten or 24 vectors with certain lengths and angles between them: They are, indeed, a root system.

And that is what your definition 2 talks about. It blurs a little the distinction between the abstract ambient vector space $E$ of an abstract root system and the ambient vector space $\mathfrak h^*$ of the concrete root system $R$ of the Lie algebra constructed there. Technically, one would have to use something like $E := ( \mathbb R-$span of $R$ inside $\mathfrak h^*$) here. The Weyl group consists of automorphisms of this, but after a little practice, one sees that one can equivalently just talk of automorphisms of $\mathfrak h^*$; what is important, actually, is that the Weyl group induces isometric automorphisms (self-bijections) of our crystal $R$, the root system itself!

Finally, definition 3 lives in the even greater setting of Lie groups. Again, somewhere in the group there sits (as a hidden "skeleton") a certain root system. And with a lot of work one can see: The Weyl group of that root system (in the abstract sense of definition 1) can also be seen in a seemingly very different place: It is isomorphic to a certain quotient of a certain normalizer of a certain subgroup of the group we are studying. That these two ways of defining "the" Weyl group in this setting are equivalent is not at all trivial, but a very rewarding goal to understand through self-study. Do not expect it to be easily visible in any way. Maybe the first thing would be to go through all these definitions in certain easy examples with matrices, say the Lie group $G= SU(2)$, or the Lie algebra $\mathfrak g = \mathfrak{sl}_2(\mathbb C)$ with $\mathfrak h$ the diagonal matrices. Once you see what the roots here are (there's only two, and they are negatives of each other), and what the Weyl group is (in every definition, in the end it's just two automorphisms, one the idnetity and one the switching of those two roots); then you can go to $SU(3)$ and $\mathfrak{sl}_3$, and now things should become interesting. (Six roots, Weyl group consists of six isometries.)