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| {{unreferenced|date=November 2013}}
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| In [[mathematics]], a '''jet group''' is a generalization of the [[general linear group]] which applies to [[Taylor polynomial]]s instead of [[vector (mathematics)|vector]]s at a point. Essentially a jet group describes how a Taylor polynomial transforms under changes of [[coordinate system]]s (or, equivalently, [[diffeomorphism]]s).
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| The ''k''-th order '''jet group''' ''G''<sup>''n''</sup><sub>''k''</sub> consists of [[jet (mathematics)|jet]]s of smooth diffeomorphisms φ: '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup> such that φ(0)=0.
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| The following is a more precise definition of the jet group.
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| Let ''k'' ≥ 2. The gradient 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 ''J<sup>m</sup>''('''R'''<sup>''k''</sup>) = '''R'''<sup>''k''</sup> × ''W'', where
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| :<math>W = \mathbf R \times (\mathbf R^*)^k \times S^2( (\mathbf R^*)^k) \times \cdots \times S^{m} ( (\mathbf R^*)^k)</math>
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| and ''S<sup>i</sup>'' denotes the ''i''-th symmetric power. A function ''f:'' '''R'''<sup>''k''</sup> → '''R''' has a prolongation ''j<sup>m</sup>f'': '''R'''<sup>''n''</sup> → ''J<sup>m</sup>''('''R'''<sup>''n''</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''.
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| 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''.
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| Provide ''J<sup>m</sup>''('''R'''<sup>''n''</sup>) with a group structure by taking
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| :<math>(x,x') * (y, y') = (x+y, j^mf_p(y) + y')</math>
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| With this group structure, ''J<sup>m</sup>''('''R'''<sup>''n''</sup>) is a [[Carnot group]] of class ''m'' + 1.
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| Because of the properties of jets under [[function composition]], ''G''<sup>''n''</sup><sub>''k''</sub> 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.
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| [[Category:Lie groups]]
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| {{algebra-stub}}
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