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