On left-invariant Hörmander operators in $\mathbb R^N$ applications to Kolmogorov–Fokker–Planck equations

If $\mathcal L=\sum_{j=1}^mX_j^2+X_0$ is a Hörmander partial differential operator in $\mathbb R^N$, we give sufficient conditions on the $X_{j^\mathrm S}$ for the existence of a Lie group structure $\mathbb G=(\mathbb R^N,*)$, not necessarily nilpotent, such that $\mathcal L$ is left invariant on $\mathbb G$. We also investigate the existence of a global fundamental solution $\Gamma$ for $\mathcal L$, providing results that ensure a suitable left-invariance property of $\Gamma$. Examples are given for operators $\mathcal L$ to which our results apply: some are new; some have appeared in recent literature, usually quoted as Kolmogorov–Fokker–Planck-type operators. Nontrivial examples of homogeneous groups are also given.