Homogeneous Carnot groups related to sets of vector fields

Bonfiglioli, Andrea

Bollettino della Unione matematica italiana, Série 8, 7B (2004) no. 1, pp. 79-107

Résumé

In this paper, we are concerned with the following problem: given a set of smooth vector fields $X_{1}, \ldots , X_{m}$ on $\mathbb{R}^{N}$, we ask whether there exists a homogeneous Carnot group $\mathbb{G}=(\mathbb{R}^{N}, \circ, \delta_{\lambda} )$ such that $\sum_{i} X_{i}^{2}$ is a sub-Laplacian on $\mathbb{G}$. We find necessary and sufficient conditions on the given vector fields in order to give a positive answer to the question. Moreover, we explicitly construct the group law i as above, providing direct proofs. Our main tool is a suitable version of the Campbell-Hausdorff formula. Finally, we exhibit several non-trivial examples of our construction.