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 $\mathfrak$ admits a step-$k$ stratification. Namely, there exist nontrivial linear subspaces $V_1, \cdots, V_k$ such that

:$\mathfrak = V_1\oplus \cdots \oplus V_k$, $$_1, V_i= V_ for $i = 1, \cdots, k-1$, and _1,V_k= \.

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

The exponential map is a diffeomorphism from $\mathfrak$ onto $G$. Using these exponential coordinates, we can identify $G$ with $(\mathbb^n, \star)$, 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 $z_i \in \R^$ 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 and . 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

References

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Lie groups

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