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 (see Ultralimit) 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 (or stratified) group of step $k$ is a connected, simply connected, finite-dimensional Lie group whose Lie algebra $𝔤$ admits a step-$k$ stratification. Namely, there exist nontrivial linear subspaces $V_1,\cdots ,V_k$ such that

$𝔤=V_1\oplus \cdots \oplus V_k$, $[V_1,V_i]=V_{i+1}$ for $i=1,\cdots ,k-1$, and $[V_1,V_k]=\{0\}$.

Note that this definition implies the first stratum $V_1$ generates the whole Lie algebra $𝔤$.

The exponential map is a diffeomorphism from $𝔤$ onto $G$. Using these exponential coordinates, we can identify $G$ with $({ℝ}^n,\star )$, where $n=\dim V_1+\cdots +\dim V_k$ and the operation $\star$ is given by the Baker–Campbell–Hausdorff formula.

Sometimes it is more convenient to write an element $z\in G$ as

$z=(z_1,\cdots ,z_k)$ with $z_i\in {ℝ}^{\dim V_i}$ for $i=1,\cdots ,k$.

The reason is that $G$ has an intrinsic dilation operation ${\delta }_{\lambda }:G\to G$ given by

${\delta }_{\lambda }(z_1,\cdots ,z_k):=(\lambda z_1,\cdots ,{\lambda }^k{z}_k)$.

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 Template:Harvs and Template:Harvs. However, the concept was introduced earlier by Gerald Folland (1975), under the name stratified group.

See also

• Pansu derivative, a derivative on a Carnot group introduced by Template:Harvtxt

References

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