Jet group

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In mathematics, a jet group is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point. Essentially a jet group describes how a Taylor polynomial transforms under changes of coordinate systems (or, equivalently, diffeomorphisms).

The k-th order jet group Gⁿ_k consists of jets of smooth diffeomorphisms

φ:Rⁿ→Rⁿ

such that φ(0)=0. Because of the properties of jets under function composition, Gⁿ_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.