mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a Carnot group is a

simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...

nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the class ...

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...

, together with a derivation of its

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

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 In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called ''horizontal ...

. Carnot–Carathéodory metrics have metric dilations; they are asymptotic cones (see

Ultralimit In mathematics, an ultralimit is a geometric construction that assigns to a sequence of metric spaces ''Xn'' a limiting metric space. The notion of an ultralimit captures the limiting behavior of finite configurations in the spaces ''Xn'' and uses ...

) 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 In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ::\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ' ...

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 In mathematics, an element ''x'' of a Lie group or a Lie algebra is called an ''n''-Engel element, named after Friedrich Engel, if it satisfies the ''n''-Engel condition that the repeated commutator ..''x'',''y'y''">''x'',''y''.html" ;"title=". ...

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.

References

* * * * * Lie groups {{abstract-algebra-stub

Formal definition and basic properties

Examples

History

See also

References