  The aim of this paper is to prove the existence and several selected
properties of a global fundamental Heat kernel $\Gamma$ for the parabolic
operators $\mathcal{H}=\sum_{j=1}^m X_j^2-\partial_t$, where $X_1,\ldots,X_m$
are smooth vector fields on $\mathbb{R}^n$ satisfying H\"ormander'snrank
condition, and enjoying a suitable homogeneity assumption with respect to a
family of non-isotropic dilations. The proof of the existence of $\Gamma$ is
based on a (algebraic) global lifting technique, together with a representation
of $\Gamma$ in terms of the integral (performed over the lifting variables) of
the Heat kernel for the Heat operator associated with a suitable sub-Laplacian
on a homogeneous Carnot group. Among the features of $\Gamma$ we prove:
homogeneity and symmetry properties; summability properties; its vanishing at
infinity; the uniqueness of the bounded solutions of the related Cauchy
problem; reproduction and density properties; an integral representation for
the higher-order derivatives.
