Carnot group

In mathematics, a Carnot group is a simply connected nilpotent Lie group, together with a derivation of its Lie algebra such that the subspace with eigenvalue 1 generates the Lie algebra. The subbundle of the tangent bundle associated to this eigenspace is called horizontal. On a Carnot group, any norm on the horizontal subbundle gives rise to a Carnot–Carathéodory metric. Carnot–Carathéodory metrics have metric dilations; they are asymptotic cones of finitely-generated nilpotent groups, and of nilpotent Lie groups, as well as tangent cones of sub-Riemannian manifolds.

Formal definition and basic properties

A Carnot group of step is a connected, simply connected, finite-dimensional Lie group whose Lie algebra admits a step- stratification. Namely, there exist nontrivial linear subspaces such that
Note that this definition implies the first stratum generates the whole Lie algebra.
The exponential map is a diffeomorphism from onto. Using these exponential coordinates, we can identify with, where and the operation is given by the Baker–Campbell–Hausdorff formula.
Sometimes it is more convenient to write an element as
The reason is that has an intrinsic dilation operation given by

Examples

The real Heisenberg group is a Carnot group which can be viewed as a flat model in Sub-Riemannian geometry as Euclidean space in Riemannian geometry. The Engel group is also a Carnot group.

History

Carnot groups were introduced, under that name, by and. However, the concept was introduced earlier by Gerald Folland, under the name stratified group.