# 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. Essentially a jet group describes how a Taylor polynomial transforms under changes of coordinate systems (or, equivalently, diffeomorphisms).

The k-th order jet group Gnk consists of jets of smooth diffeomorphisms φ: RnRn such that φ(0)=0.

The following is a more precise definition of the jet group.

Let k ≥ 2. The gradient of a function f: RkR can be interpreted as a section of the cotangent bundle of RK given by df: RkT*Rk. Similarly, derivatives of order up to m are sections of the jet bundle Jm(Rk) = Rk × W, where

$W=\mathbf {R} \times (\mathbf {R} ^{*})^{k}\times S^{2}((\mathbf {R} ^{*})^{k})\times \cdots \times S^{m}((\mathbf {R} ^{*})^{k}).$ Here R* is the dual vector space to R, and Si denotes the i-th symmetric power. A function f: RkR has a prolongation jmf: RnJm(Rn) defined at each point pRk by placing the i-th partials of f at p in the Si((R*)k) component of W.

Consider a point $p=(x,x')\in J^{m}(\mathbf {R} ^{n})$ . There is a unique polynomial fp in k variables and of order m such that p is in the image of jmfp. That is, $j^{k}(f_{p})(x)=x'$ . The differential data x′ may be transferred to lie over another point yRn as jmfp(y) , the partials of fp over y.

Provide Jm(Rn) with a group structure by taking

$(x,x')*(y,y')=(x+y,j^{m}f_{p}(y)+y')$ With this group structure, Jm(Rn) is a Carnot group of class m + 1.

Because of the properties of jets under function composition, Gnk 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.