Group contraction

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 SO(3), [X₁, X₂] = X₃, etc., may be rewritten by a change of variables Y₁ = εX₁, Y₂ = εX₂, Y₃ = X₃, as

[Y₁, Y₂] = ε² Y₃,     [Y₂, Y₃] = Y₁,     [Y₃, Y₁] = Y₂.

The contraction limit ε → 0 trivializes the first commutator and thus yields the non-isomorphic algebra of the plane Euclidean group, E₂ ~ 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 Y₁, Y₂, now generate the Abelian normal subgroup of E₂ (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 → ∞;^[4] 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.