Английская Википедия:Group contraction

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Шаблон:Short description

In theoretical physics, Eugene Wigner and Erdal İnönü have discussed[1] the possibility to obtain from a given Lie group a different (non-isomorphic) Lie group by a group contraction with respect to a continuous subgroup of it. That amounts to a limiting operation on a parameter of the Lie algebra, altering the structure constants of this Lie algebra in a nontrivial singular manner, under suitable circumstances.[2][3]

For example, the Lie algebra of the 3D rotation group Шаблон:Math, Шаблон:Math, etc., may be rewritten by a change of variables Шаблон:Math, Шаблон:Math, Шаблон:Math, as

Шаблон:Math.

The contraction limit Шаблон:Math trivializes the first commutator and thus yields the non-isomorphic algebra of the plane Euclidean group, Шаблон:Math. (This is isomorphic to the cylindrical group, describing motions of a point on the surface of a cylinder. It is the little group, or stabilizer subgroup, of null four-vectors in Minkowski space.) Specifically, the translation generators Шаблон:Math, now generate the Abelian normal subgroup of Шаблон:Math (cf. Group extension), the parabolic Lorentz transformations.

Similar limits, of considerable application in physics (cf. correspondence principles), contract

Notes

Шаблон:Reflist

References