mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a jet group is a generalization of the

general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...

which applies to

Taylor polynomial In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...

s instead of

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

s at a point. A jet group is a group of jets that describes how a Taylor polynomial transforms under changes of

coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...

s (or, equivalently,

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...

s).

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 In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...

to R, and ''Sⁱ'' denotes the ''i''-th

symmetric power In mathematics, the ''n''-th symmetric power of an object ''X'' is the quotient of the ''n''-fold product X^n:=X \times \cdots \times X by the permutation action of the symmetric group \mathfrak_n. More precisely, the notion exists at least in the ...

. 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 In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...

, ''G''^''n''_''k'' is a

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the add ...

. The jet group is a

semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in wh ...

of the general linear group and a connected, simply connected nilpotent Lie group. It is also in fact an

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...

, since the composition involves only polynomial operations.

Notes

References

* * * Lie groups {{algebra-stub