From Weyl groups to Weyl groupoids?

Question

I'm trying to find a framework where the choices in the classical construction of a root system of a semi-simple lie algebra are not needed.

Let $\mathfrak{g}$ be a semisimple lie algebra.

Definition (Weyl groupoid): Let $\mathcal{W}$ be the following groupoid:

• $Obj(\mathcal{W})=\{\mathfrak{b}\subset \mathfrak{g} | \space \mathfrak{ b} \text{ is a maximal solvable algebra}\}$

• $Mor(\mathcal{W}) = \{Ad(e^x):\mathfrak{b}_1 \to \mathfrak{b}_2 \space |\space x \in \mathfrak{g}\}$

Here's what I like about this so far:

• Objects in $\mathcal{W}$ are in 1-1 correspondence with choosing: cartan subalgebra & Weyl chamber (root basis).
• Morally there's sort of an exponential map $exp: \mathcal{W} \to G/B$ (this is not meant to be a precise statement).

A couple of questions:

1. Is this the correct categorification?

2. How do I get the weyl group out of this?

3. I read somewhere that a choice of borel subalgebra is equivalent to a choice of complex structure on the complexified tangent bundle of $G/B$. Can this groupoid be interpreted in terms of a groupoid of complex structures?

4. Where can I find more about this kind of construction?

Answer 1

Warning: This answer went through a massive editing.

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This answer does not regard the specific questions asked, but the motivation behind: finding a framework where the choices in the classical construction of a root system of a semi-simple Lie algebra are not needed.

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Let $\mathfrak{g}$ be a Lie algebra and $G=\text{Aut}(\mathfrak{g})^0$. Let $C^+$ be the set of pairs (Cartan,Borel) where the Cartan subalgebra is contained in the Borel subalgebra ($C^+$ is a subset of a product of two Grassmannians of $\mathfrak{g}$). Note that $G$ acts transitively on $C^+$, stabilizers being centralizers of maximal tori.

It is not hard to see that the centralizer of the $G$-action on $C^+$ is isomorphic to the Weyl group (it acts trivially on the Cartan coordinate and transitively on all Borels including a given Cartan). Let's call it "the choice free Weyl group" $W$.

There is a natural vector bundle $H$ over $C^+$, the fiber over (Cartan,Borel) being the vector space underlying the Cartan. Note that (since the stabilizer of a point acts trivially on its fiber) the vector space of all $G$-invariant sections is isomorphic to each fiber. Let's call it "the choice free Cartan" $\mathfrak{h}$. We get a natural action of $W$ on $\mathfrak{h}$.

Note that the dual space, $\mathfrak{h}^*$, could be identified with the space of invariant sections to the dual vector bundle. We can single out those invariant sections which are in the root system at each fiber and obtain "the choice free root system" $\Phi$. Note that the second coordinate in $C^+$ is a Borel. This gives a choice of positivity at each fibers, thus we can also single out "the choice free positive roots" $\Phi^+$.

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For each parabolic type we can consider the space of parabolic subalgebras of this given type $P$. Note that between two such, $P,P'$, either there exists a unique $G$-equivariant map or none. If there exists such we will say $P\leq P'$. We may regard a $G$-equivariant maps $C^+\to P$ as a simplex whose faces are the simplices $C^+\to P'$ for each $P'$ such that $P\leq P'$. We will get a simplicial complex on which $W$ acts, "the choice free Coxeter complex".

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The following is unrelated, just a comment regarding the question's title. The name "Weyl Groupoid" should be reserved to the following category.

• Objects: Cartan subalgebras.
• Morphisms: A morphism between two Cartan algebras is a Lie-algebra isomorphism which could be extended to a global automorphism in $G$.

This is a connected groupoid and the isotropy group of every object is isomorphic to the Weyl group.

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Remark: I was tacitly assuming above that everything is over an algebraically closed field, or split. One can easily adjust the definitions to handle the general case.

Answer 2

It's a bit unclear to me precisely what you want, but the usual solution is take the abstract Cartan (discussed a bit here), which is defined to be $\mathbb{T}=B/[B,B]$ for any Borel $B$. This is canonical, since any two Borels are conjugate, and every Borel is self-normalizing so $\mathbb{T}$ is unique up to the map induced by conjugation by an element of $B$. The action of the Weyl group is given by looking at all the isomorphisms induced between $\mathbb{T}$ and any fixed torus $T$ by all the different Borels containing $T$ (which are all conjugate under $N(T)$). Composing one of these isomorphisms with the inverse of another gives all the elements of the Weyl group.