Group contraction
# 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 SO(3), $[X_1,$ $X_2]$ = $X_3,$ etc., may be rewritten by a change of variables $Y_1$ = $εX_1,$ $Y_2$ = $εX_2,$ $Y_3$ = $X_3,$ as

    $[Y_1,$ $Y_2]$ = $ε^2$ $Y_3,$  $[Y_2,$ $Y_3]$ = $Y_1,$  $[Y_3,$ $Y_1]$ = $Y_2.$

The contraction limit ε → 0 trivializes the first commutator and thus yields the non-isomorphic algebra of the plane Euclidean group, $E_2$ ~ ISO(2). (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 $Y_1,$ $Y_2,$ now generate the Abelian normal subgroup of $E_2$ (cf. Group extension), the parabolic Lorentz transformations.

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

* the de Sitter group SO(4, 1) ~ Sp(2, 2) to the Poincaré group ISO(3, 1), as the de Sitter radius diverges: R → ∞; or
* the super-anti-de Sitter algebra to the super-Poincaré algebra as the AdS radius diverges R → ∞; or
* the Poincaré group to the Galilei group, as the speed of light diverges: c → ∞; 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: ħ → 0.
