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.<ref>{{harvtxt|Kolář|Michor|Slovák|1993|pp=128-131}}</ref>

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

:<math>W = \mathbf R \times (\mathbf R^*)^k \times S^2( (\mathbf R^*)^k) \times \cdots \times S^{m} ( (\mathbf R^*)^k).</math>

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 <math>p=(x,x')\in J^m(\mathbf R^n)</math>. 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, <math>j^k(f_p)(x)=x'</math>. 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

:<math>(x,x') * (y, y') = (x+y, j^mf_p(y) + y')</math>

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== {{Reflist}}

==References== * {{citation|last1=Kolář|first1=Ivan|last2=Michor|first2=Peter|last3=Slovák|first3=Jan|url=http://www.emis.de/monographs/KSM/kmsbookh.pdf|format=PDF|title=Natural operations in differential geometry|year=1993|publisher=Springer-Verlag|access-date=2014-05-02|archive-url=https://web.archive.org/web/20170330154524/http://www.emis.de/monographs/KSM/kmsbookh.pdf|archive-date=2017-03-30|url-status=dead}} * {{citation|last1 = Krupka|first1=Demeter|last2=Janyška|first2=Josef|title=Lectures on differential invariants|year = 1990|publisher = Univerzita J. E. Purkyně V Brně|isbn=80-210-0165-8}} * {{citation|last1 = Saunders|first1 = D.J.|title = The geometry of jet bundles|year = 1989|publisher = Cambridge University Press|isbn = 0-521-36948-7|url-access = registration|url = https://archive.org/details/geometryofjetbun0000saun}}

Category:Lie groups

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