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?