Jet group

In mathematics, a jet group is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point. A jet group is a group of jets that describes how a Taylor polynomial transforms under changes of coordinate systems (or, equivalently, diffeomorphisms).
Overview.
The "k"-th order jet group "G""n""k" consists of jets of smooth diffeomorphisms φ: R"n" → R"n" such that φ(0)=0.
The following is a more precise definition of the jet group.
Let "k" ≥ 2. The differential of a function "f:" R"k" → R can be interpreted as a section of the cotangent bundle of R"K" given by "df:" R"k" → "T*"R"k". Similarly, derivatives of order up to "m" are sections of the jet bundle "Jm"(R"k") = R"k" × "W", where
Here R* is the dual vector space to R, and "Si" denotes the "i"-th symmetric power. A smooth function "f:" R"k" → R has a prolongation "jmf": R"k" → "Jm"(R"k") defined at each point "p" ∈ R"k" by placing the "i"-th partials of "f" at "p" in the "Si"((R*)"k") component of "W".
Consider a point formula_2. There is a unique polynomial "fp" in "k" variables and of order "m" such that "p" is in the image of "jmfp". That is, formula_3. The differential data "x′" may be transferred to lie over another point "y" ∈ R"n" as "jmfp(y)" , the partials of "fp" over "y".
Provide "Jm"(R"n") with a group structure by taking
With this group structure, "Jm"(R"n") is a Carnot group of class "m" + 1.
Because of the properties of jets under function composition, "G""n""k" is a Lie group. The jet group is a semidirect product of the general linear group and a connected, simply connected nilpotent Lie group. It is also in fact an algebraic group, since the composition involves only polynomial operations.