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.Overview
The k-th order jet group Gnk consists of jets of smooth diffeomorphisms φ: Rn → Rn such that φ=0.
The following is a more precise definition of the jet group.
Let k ≥ 2. The differential of a function f: Rk → R can be interpreted as a section of the cotangent bundle of RK given by df: Rk → T*Rk. Similarly, derivatives of order up to m are sections of the jet bundle Jm = Rk × W, where
Here R* is the dual vector space to R, and Si denotes the i-th symmetric power. A smooth function f: Rk → R has a prolongation jmf: Rk → Jm defined at each point p ∈ Rk by placing the i-th partials of f at p in the Si component of W.
Consider a point. There is a unique polynomial fp in k variables and of order m such that p is in the image of jmfp. That is,. The differential data x′ may be transferred to lie over another point y ∈ Rn as jmfp, the partials of fp over y.
Provide Jm with a group structure by taking
With this group structure, Jm 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.
