Group contraction


In theoretical physics, Eugene Wigner and Erdal İnönü have discussed the possibility to obtain from a given Lie group a different 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,. Specifically, the translation generators, now generate the Abelian normal subgroup of, the parabolic Lorentz transformations.
Similar limits, of considerable application in physics, contract