Group contraction
In theoretical physics, Eugene Wigner and Erdal İnönü have discussed 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. For example, the Lie algebra of the 3D rotation group , , etc., may be rewritten by a change of variables , , , as

 . The contraction limit  trivializes the first commutator and thus yields the non-isomorphic algebra of the plane Euclidean group, . (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 , now generate the Abelian normal subgroup of   (cf. Group extension), the parabolic Lorentz transformations. Similar limits, of considerable application in physics (cf. correspondence principles),  contract
 the de Sitter group  to the Poincaré group , as the de Sitter radius diverges: ; or
 the super-anti-de Sitter algebra to the super-Poincaré algebra as the AdS radius diverges ; or
 the Poincaré group to the Galilei group, as the speed of light diverges: ; or
 the Moyal bracket Lie algebra (equivalent to quantum commutators) to the Poisson bracket Lie algebra, in the classical limit as the Planck constant vanishes: . Notes

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Lie algebras
Lie groups
Mathematical physics