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 <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.
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 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 {{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, 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 | 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 | 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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