TITLE: From Weyl groups to Weyl groupoids?
QUESTION [11 upvotes]: 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:


Is this the correct categorification? 
How do I get the weyl group out of this? 
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?
Where can I find more about this kind of construction?

REPLY [6 votes]: 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.