About the intrinsic definition of the Weyl group of complex semisimple Lie algebras

Question

It may be a easy question for experts.

The definition of the Weyl group of a complex semisimple Lie algebra $\mathfrak{g}$ is well-known: We first $\textbf{choose}$ a Cartan subalgebra $\mathfrak{h}$ and we have the root space decomposition. The Weyl group now is the group generated by the reflections according to roots.

Naively this definition depends on the choices of the Cartan subalgebra $\mathfrak{h}$. Of course we can prove that for different choices the resulting Weyl groups are isomorphic.

My question is: can we define the Weyl group intrinsically such that we don't need the do check the unambiguity.

One thought is: we have the abstract Cartan subalgebra $\mathfrak{H}:=\mathfrak{b}/[\mathfrak{b},\mathfrak{b}]$ of $\mathfrak{g}$ (which is in fact not a subalgebra of $\mathfrak{g}$). Can we define the Weyl group along this way? Again is there any references for this?

Answer 1

Probably the earliest intrinsic definition of Weyl group occurs in section 1.2 of the groundbreaking paper "Representations of Reductive Groups Over Finite Fields" by Deligne and Lusztig (Ann. of Math. 103, 1976, available at JSTOR). This is done elegantly in the closely related but more general setting of a reductive algebraic group $G$ over an arbitrary algebraically closed field (though their interest is mainly in prime characteristic). Letting $X$ denote the set of all Borel subgroups of $G$, the set of $G$-orbits on $X \times X$ provides a natural model for a universal Weyl group of $G$ (or its Lie algebra).

[ADDED] In the algebraic group setting, this intrinsic definition depends just on knowing what a connected reductive (or semisimple) group is and what a Borel subgroup is (maximal closed connected solvable subgroup). But obviously one can't exploit the "Weyl group" without knowing more of the structure theory: conjugacy theorems, Bruhat decomposition. (Is it a group? finite?) In the easier characteristic 0 Lie algebra theory, where $X$ becomes the set of Borel subalgebras (whose definition requires some theory) with conjugation action by the adjoint group, this abstract notion of "Weyl group" similarly needs unpacking. But the Deligne-Lusztig definition is a good conceptual one for their purposes and sneaks in the underlying set $X$ of the flag variety of $G$. Any intrinsic definition of the Weyl group needs serious background in Lie theory.

In the treatment by Chriss and Ginzburg, even when one is primarily interested in the Lie algebra picture, the group in the background tends to play an important role. Indeed, in the early work of Borel and Chevalley on semisimple algebraic groups, the Weyl group appears most naturally in the guise of the finite quotient $W_G(T) :=N_G(T)/T$ for a fixed maximal torus $T$. Then one sees $W$ as generated by reflections relative to roots, etc. As in the parallel Lie algebra setting in characteristic 0, the maximal tori (or Cartan subalgebras) are all conjugate under the adjoint group action, but this falls short of giving an intrinsic definition of the sort provided by Deligne-Lusztig.

[Weyl himself gave the group an awkward name, but was mainly concerned with its use in the context of a compact Lie group. The notion basically originates earlier in the work of Cartan, but it took a while to see the root system and Weyl group as combinatorial objects including the Coxeter presentation of the group as a reflection group (carried over by Witt to Lie algebras).]

Answer 2

Let $g_r$ be the set of regular semi-simple elements of the Lie algebra, and $\tilde g_r$ be the set of these elements with a choice of Borel containing it. The Weyl group is the group of deck transformations of the cover $\tilde {g}_r\to {g}_r$.

Answer 3

Sometimes when you define a group using an arbitrary choice of object and then show the choice of object doesn't matter, you could have defined a groupoid without making an arbitrary choice.

For example, to define the fundamental group $\pi_1(X,x)$ of a path-connected space $X$ we need to choose a basepoint $x \in X$, but then we can show we get isomorphic groups no matter what basepoint we choose, with an isomorphism given by a homotopy class of paths between the basepoints. To avoid this maneuver we can work with the fundamental groupoid of $X$, whose objects are points of $X$ and whose morphisms are homotopy classes of paths. If $X$ is path-connected all objects in this groupoid are isomorphic, and thus the automorphism groups of all objects are isomorphic. The automorphism group of $x$ is just $\pi_1(X,x)$. The fundamental groupoid is thus equivalent, as a category, to the one-object groupoid corresponding to the group $\pi_1(X,x)$. But the advantage of the fundamental groupoid is that we can define it without choosing a basepoint, and it makes sense and works well even when $X$ is not path-connected.

Similarly, I think we can define the Weyl groupoid of a compact semisimple Lie group $G$ in a way that gives a groupoid equivalent to the usual Weyl group, but doesn't require a choice of maximal torus. The idea should go like this. The objects of the Weyl groupoid are maximal tori. A morphism $f : T \to T'$ in the Weyl groupoid is a Lie group isomorphism of the form

$$ t \mapsto g t g^{-1} \textrm{ for all } t \in T $$

for some $g \in G$. If I did this right, the automorphism group of any object $T$ in the Weyl groupoid is the usual Weyl group

$$ W_G(T) = N_G(T) / T ,$$

that is, the normalizer of $T \subset G$ modulo the centralizer of $T \subset G$, which is $T$ itself. If this is true, the Weyl groupoid will be equivalent, as a groupoid, to the usual Weyl group $W_G(T)$ for any maximal torus $T$.

Answer 4

Yes: this is the approach to defining the 'abstract Weyl group' introduced in "Representation Theory and Complex Geometry" by Chriss/Ginzburg on p. 135 (2nd Edition, Birkhauser).

Answer 5

I have heard that, originally, the Weyl group was designed (and worked out e.g. by Chevalley) as some Galois group which are then intrinsic.