# 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** *G*^{n}_{k} consists of jets of smooth diffeomorphisms φ: **R**^{n} → **R**^{n} such that φ(0)=0.^{[1]}

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

Let *k* ≥ 2. The gradient 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 *J ^{m}*(

**R**

^{k}) =

**R**

^{k}×

*W*, where

Here **R*** is the dual vector space to **R**, and *S ^{i}* denotes the

*i*-th symmetric power. A function

*f:*

**R**

^{k}→

**R**has a prolongation

*j*:

^{m}f**R**

^{n}→

*J*(

^{m}**R**

^{n}) defined at each point

*p*∈

**R**

^{k}by placing the

*i*-th partials of

*f*at

*p*in the

*S*((

^{i}**R***)

^{k}) component of

*W*.

Consider a point . There is a unique polynomial *f _{p}* in

*k*variables and of order

*m*such that

*p*is in the image of

*j*. That is, . The differential data

^{m}f_{p}*x′*may be transferred to lie over another point

*y*∈

**R**

^{n}as

*j*, the partials of

^{m}f_{p}(y)*f*over

_{p}*y*.

Provide *J ^{m}*(

**R**

^{n}) with a group structure by taking

With this group structure, *J ^{m}*(

**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.

## Notes

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## References

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