On the stochastic Lie algebra

We study the structure of the Lie algebra $\mathfrak{s}(n,\mathbb{R})$ corresponding to the so-called stochastic Lie group $$\mathcal{S}$ (n,\mathbb R)$. We obtain the Levi decomposition of the Lie algebra, classify Levi factor and classify the representation of the factor in ${\mathbb{R}}^n$. We discuss isomorphism of $\mathcal{S}(n,\mathbb{R})$ with the group of invertible affine maps $Aff(n-1,\mathbb{R})$. We prove that $\mathfrak{s}(n,\mathbb{R})$ is generated by two generic elements.