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 Gⁿ_k consists of jets of smooth diffeomorphisms

φ:Rⁿ→Rⁿ

such that φ(0)=0.

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

Let $k \geq 2$. The gradient of a function $f: \R^k \rightarrow \R$ can be interpreted as a section of the cotangent bundle of $\mathbb R^K$ given by $df: \mathbb R^k \rightarrow T^*\mathbb R^k$. Similarly, derivatives of order up to m are sections of the jet bundle $J^{m}(\mathbb R^k)=\mathbb R^k \times W$, where

$W = \mathbb R \times (\mathbb R^*)^k \times S^2( (\mathbb R^*)^k) \times \cdots \times S^{m} ( (\mathbb R^*)^k)$

and Sⁱ denotes the i^th symmetric power. A function $f: \mathbb R^k \rightarrow \mathbb R$ has a prolongation $j^{m}f: \mathbb R^n \rightarrow J^{m}(\mathbb R^n)$ defined at each point $p \in \mathbb R^k$ by placing the i^th partials of f at p in the $S^{i} ( (\mathbb R^*)^k)$ component of W.

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

Provide $J^m(\mathbb R^n)$ with a group structure by taking (x,x') * (y,y') = (x + y,j^mf_p(y) + y')

With this group structure, $J^m(\mathbb R^n)$ is a Carnot group of class m + 1.

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