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 ${\displaystyle k}$ is a connected, simply connected, finite-dimensional Lie group whose Lie algebra ${\displaystyle {\mathfrak {g}}}$ admits a step-${\displaystyle k}$ stratification. Namely, there exist nontrivial linear subspaces ${\displaystyle V_{1},\cdots ,V_{k}}$ such that

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

Note that this definition implies the first stratum ${\displaystyle V_{1}}$ generates the whole Lie algebra ${\displaystyle {\mathfrak {g}}}$.

The exponential map is a diffeomorphism from ${\displaystyle {\mathfrak {g}}}$ onto ${\displaystyle G}$. Using these exponential coordinates, we can identify ${\displaystyle G}$ with ${\displaystyle (\mathbb {R} ^{n},\star )}$, where ${\displaystyle n=\dim V_{1}+\cdots +\dim V_{k}}$ and the operation ${\displaystyle \star }$ is given by the Baker–Campbell–Hausdorff formula.

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

${\displaystyle z=(z_{1},\cdots ,z_{k})}$ with ${\displaystyle z_{i}\in \mathbb {R} ^{\dim V_{i}}}$ for ${\displaystyle i=1,\cdots ,k}$.

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

${\displaystyle \delta _{\lambda }(z_{1},\cdots ,z_{k}):=(\lambda z_{1},\cdots ,\lambda ^{k}z_{k})}$.

Examples

The real Heisenberg group is a Carnot group.

History

Carnot groups were introduced, under that name, by Pierre Pansu (1982, 1989) and John Mitchell (1985). 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 Pansu (1989)

References

• Folland, Gerald (1975), "Subelliptic estimates and function spaces on nilpotent Lie groups", Arkiv for Mat. 13 (2): 161-207.
• Mitchell, John (1985), "On Carnot-Carathéodory metrics", Journal of Differential Geometry, 21 (1): 35–45, ISSN 0022-040X, MR 0806700
• Pansu, Pierre (1982), Géometrie du groupe d'Heisenberg, Thesis, Université Paris VII
• Pansu, Pierre (1989), "Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un", Annals of Mathematics, Second Series, 129 (1): 1–60, doi:10.2307/1971484, ISSN 0003-486X, MR 0979599
• Bellaïche, André; Risler, Jean-Jacques, eds. (1996). Sub-Riemannian geometry. Progress in Mathematics. 144. Basel: Birkhäuser Verlag. doi:10.1007/978-3-0348-9210-0. MR 1421821.