# Carnot group

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In [mathematics](/source/mathematics), a '''Carnot group''' is a [simply connected](/source/simply_connected_space) [nilpotent](/source/nilpotent_group) [Lie group](/source/Lie_group), together with a derivation of its [Lie algebra](/source/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](/source/Carnot%E2%80%93Carath%C3%A9odory_metric). Carnot–Carathéodory metrics have metric dilations; they are asymptotic cones (see [Ultralimit](/source/Ultralimit)) of finitely-generated nilpotent groups, and of nilpotent Lie groups, as well as tangent cones of [sub-Riemannian manifold](/source/sub-Riemannian_manifold)s.

==Formal definition and basic properties==

A Carnot (or stratified) group of step <math>k</math> is a connected, simply connected, finite-dimensional Lie group whose Lie algebra <math>\mathfrak{g}</math> admits a step-<math>k</math> stratification. Namely, there exist nontrivial linear subspaces <math>V_1, \cdots, V_k</math> such that 

:<math>\mathfrak{g} = V_1\oplus \cdots \oplus V_k</math>, <math>[V_1, V_i] = V_{i+1}</math> for <math>i = 1, \cdots, k-1</math>, and <math>[V_1,V_k] = \{0\}</math>.

Note that this definition implies the first stratum <math>V_1</math> generates the whole Lie algebra <math>\mathfrak{g}</math>.

The exponential map is a diffeomorphism from <math>\mathfrak{g}</math> onto <math>G</math>. Using these exponential coordinates, we can identify <math>G</math> with <math>(\mathbb{R}^n, \star)</math>, where <math>n = \dim V_1 + \cdots + \dim V_k </math> and the operation <math>\star</math> is given by the [Baker–Campbell–Hausdorff formula](/source/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula).

Sometimes it is more convenient to write an element <math>z \in G</math> as

:<math>z = (z_1, \cdots, z_k)</math> with <math>z_i \in \R^{\dim V_i}</math> for <math>i = 1, \cdots, k</math>.

The reason is that <math>G</math> has an intrinsic dilation operation <math>\delta_\lambda : G \to G</math> given by

:<math>\delta_\lambda(z_1, \cdots, z_k) := (\lambda z_1, \cdots, \lambda^k z_k)</math>.

==Examples==

The real [Heisenberg group](/source/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](/source/Engel_group) is also a Carnot group.

==History== 
Carnot groups were introduced, under that name, by {{harvs|txt|last=Pansu|first=Pierre|authorlink=Pierre Pansu|year1=1982|year2=1989}} and {{harvs|txt|first=John|last=Mitchell|year=1985}}.  However, the concept was introduced earlier by Gerald Folland (1975), under the name '''stratified group'''.

==See also==
*[Pansu derivative](/source/Pansu_derivative), a derivative on a Carnot group introduced by {{harvtxt|Pansu|1989}}

==References==

*{{Citation | last1=Folland |first1=Gerald |year=1975 |title=Subelliptic estimates and function spaces on nilpotent Lie groups |journal=Arkiv för Matematik |volume=13 |issue=2 |pages=161–207 |doi=10.1007/BF02386204|bibcode=1975ArM....13..161F |s2cid=121144337 |doi-access=free }}
*{{Citation | last1=Mitchell | first1=John | title=On Carnot-Carathéodory metrics | url=http://projecteuclid.org/getRecord?id=euclid.jdg/1214439462 | mr=806700 | year=1985 | journal=[Journal of Differential Geometry](/source/Journal_of_Differential_Geometry) | issn=0022-040X | volume=21 | issue=1 | pages=35–45| doi=10.4310/jdg/1214439462 | doi-access=free }}
*{{citation|last=Pansu | first=Pierre |authorlink=Pierre Pansu|  title=Géometrie du groupe d'Heisenberg|series=Thesis|place=Université Paris VII|year=1982|url=http://www.math.u-psud.fr/~pansu/pansu_These_1982.html}}
*{{Citation | last1=Pansu | first1=Pierre |authorlink=Pierre Pansu |  title=Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un | doi=10.2307/1971484 | mr=979599 | year=1989 | journal=[Annals of Mathematics](/source/Annals_of_Mathematics) | volume=129 | issue=1 | pages=1–60| jstor=1971484 }}
*{{cite book | editor1-first=André|editor1-last=Bellaïche | editor2-first=Jean-Jacques | editor2-last=Risler | title=Sub-Riemannian geometry | url=https://www.springer.com/gb/book/9783764354763 | publisher=Birkhäuser Verlag| location=Basel |series = Progress in Mathematics |volume=144| year = 1996|mr=1421821|doi=10.1007/978-3-0348-9210-0|isbn=978-3-0348-9946-8 }}

Category:Lie groups

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Adapted from the Wikipedia article [Carnot group](https://en.wikipedia.org/wiki/Carnot_group) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Carnot_group?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.