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 $S^i$ 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 $(x,x') * (y, y') = (x%2By, j^mf_p(y) %2B y')$

With this group structure, $J^m(\mathbb R^n)$ is a Carnot group of class $m%2B1$ .

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.