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. A jet group is a group of jets that describes how a Taylor polynomial transforms under changes of coordinate systems (or, equivalently, diffeomorphisms).

Overview

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..

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

Let ''k'' ≥ 2. The differential 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

:$W = \mathbf R \times (\mathbf R^*)^k \times S^2( (\mathbf R^*)^k) \times \cdots \times S^ ( (\mathbf R^*)^k).$

Here R* is the dual vector space to R, and ''Sⁱ'' denotes the ''i''-th symmetric power. A smooth function ''f:'' R^''k'' → R has a prolongation ''j^mf'': R^''k'' → ''J^m''(R^''k'') defined at each point ''p'' ∈ R^''k'' 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 ''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'' ∈ R^''n'' as ''j^mf_p(y)'' , the partials of ''f_p'' over ''y''.

Provide ''J^m''(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''(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

References

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Lie groups

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