# Jet group

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In [mathematics](/source/mathematics), a '''jet group''' is a generalization of the [general linear group](/source/general_linear_group) which applies to [Taylor polynomial](/source/Taylor_polynomial)s instead of [vector](/source/vector_(mathematics))s at a point.  A jet group is a [group](/source/group_(mathematics)) of [jet](/source/Jet_(mathematics))s that describes how a Taylor polynomial transforms under changes of [coordinate system](/source/coordinate_system)s (or, equivalently, [diffeomorphism](/source/diffeomorphism)s).

==Overview==
The ''k''-th order '''jet group''' ''G''<sup>''n''</sup><sub>''k''</sub> consists of [jet](/source/jet_(mathematics))s of smooth diffeomorphisms φ: '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup> 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'''<sup>''k''</sup> → '''R''' can be interpreted as a section of the cotangent bundle of '''R'''<sup>''K''</sup> given by ''df:'' '''R'''<sup>''k''</sup> → ''T*'''''R'''<sup>''k''</sup>. Similarly, derivatives of order up to ''m'' are sections of the [jet bundle](/source/jet_bundle) ''J<sup>m</sup>''('''R'''<sup>''k''</sup>) = '''R'''<sup>''k''</sup> × ''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](/source/dual_vector_space) to '''R''', and ''S<sup>i</sup>'' denotes the ''i''-th [symmetric power](/source/symmetric_power). A smooth function ''f:'' '''R'''<sup>''k''</sup> → '''R''' has a prolongation ''j<sup>m</sup>f'': '''R'''<sup>''k''</sup> → ''J<sup>m</sup>''('''R'''<sup>''k''</sup>) defined at each point ''p'' ∈ '''R'''<sup>''k''</sup> by placing the ''i''-th partials of ''f'' at ''p'' in the ''S<sup>i</sup>''(('''R'''*)<sup>''k''</sup>) component of ''W''.

Consider a point <math>p=(x,x')\in J^m(\mathbf R^n)</math>. There is a unique polynomial ''f<sub>p</sub>'' in ''k'' variables and of order ''m'' such that ''p'' is in the image of ''j<sup>m</sup>f<sub>p</sub>''. That is, <math>j^k(f_p)(x)=x'</math>. The differential data ''x′'' may be transferred to lie over another point ''y'' ∈ '''R'''<sup>''n''</sup> as  ''j<sup>m</sup>f<sub>p</sub>(y)'' , the partials of ''f<sub>p</sub>''  over ''y''.

Provide ''J<sup>m</sup>''('''R'''<sup>''n''</sup>) with a group structure by taking

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

With this group structure, ''J<sup>m</sup>''('''R'''<sup>''n''</sup>) is a [Carnot group](/source/Carnot_group) of class ''m'' + 1.

Because of the properties of jets under [function composition](/source/function_composition), ''G''<sup>''n''</sup><sub>''k''</sub> is a [Lie group](/source/Lie_group). The jet group is a [semidirect product](/source/semidirect_product) of the general linear group and a connected, simply connected [nilpotent Lie group](/source/nilpotent_Lie_group). It is also in fact an [algebraic group](/source/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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