Linear simplicity constraint in Loop Quantum Gravity

[Georges Simenon]

Homework Statement

Hi,
I am studying covariant LQG from the recent book by Rovelli & Vidotto, and i am struggling with the linear simplicity constraint. My problems are not with its proof, that i understand, but rather with the physical interpretation in terms of boost generators.
I will try to make my question as much self-consistent as possible. I refer to ch. 3 of the Book.
They start from the Holst action
$$S=\int B\wedge F\quad\text{where}\quad B=\star(e\wedge e)+\frac{1}{\gamma}e\wedge e$$
Then, in sec. 3.3.1, they define the two 2-forms
$$K^I=n_J B^{IJ}|_{\Sigma}\quad\text{and}\quad L^I=n_J(\star B^{IJ})|_{\Sigma}$$
restricted on a spatial hypersurface $\Sigma$ with unit timelike normal $n_I$.
Using these definitions, they prove that the following equation
$$\vec{K}=\gamma\vec{L}\qquad\text{linear simplicity constraint}$$
holds in the "time gauge" $n_Ie^I|_{\Sigma}$.
In sec. 3.4.3, they interpret the components of the vector $\vec{K}$ as the canonical generators of Lorentz boosts. It is this interpretation that i don't understand, and I'm trying to give sense to it.
The same claim you can find in the Zakopane lectures by C. Rovelli. In the notation used there, he identifies
$$(\star(e\wedge e)+\frac{1}{\gamma}e\wedge e)|_{\Sigma}$$
with the $SL(2,\mathbb{C})$ generator.

Homework Equations

The Attempt at a Solution

If i am asked to write the Lorentz generators in tetrad notation, i write
$$G^{IJ}=e^I\wedge e^J$$
up to an overall proportionality factor.
This makes sense. Consider, for definiteness, the 2-form
$$e^0\wedge e^1 = d\xi^0\otimes d\xi^1-d\xi^1\otimes d\xi^0$$
It generates Lorentz boosts along the local inertial axis 1. Note: the $\xi$'s are local inertial coordinates.[/B]

 

[sgd37]

Have you checked if

$$(\star(e\wedge e)+\frac{1}{\gamma}e\wedge e)|_{\Sigma}$$

obeys the so(1,3) algebra commutation relations