I have a small problem, could you tell me what is wrong with my reasoning?
I have 2 gaussian variables $X$ and $Y$, with correlation $p$, clearly $X + Y$ is not gaussian because $X$ and $Y$ are not independant.
But if I use the Cholesky decomposition and set $Y = \sqrt(p)X + \displaystyle \sqrt(1-p)U$ with $X$ and $U$ independant and gaussian, we have that $X + Y = (1 + \sqrt(p)X + \sqrt(1-p)U$ which is gaussian.
What is my mistake here?
You have a couple of mistakes here: first of all, $X + Y$ may be Gaussian, even if they are not independent; as an example, let $W$ and $Z$ be independent and Gaussian, and set $X = W + Z$ and $Y = W$. Then $X$ and $Y$ aren't independent, but $X + Y = 2W + Z$, which is Gaussian since $2W$ and $Z$ are independent and Gaussian. More generally, if $(X,Y)$ is jointly Gaussian, then $X + Y$ is Gaussian (this need not go in the reverse manner).
The Cholesky decomposition that you're written down only works if $(X,Y)$ is jointly Gaussian. As an example, consider taking $X$ and $W$ to be independent standard normals and set $Y = \mathrm{sgn}(X) W$.