Group contraction

In theoretical physics, Eugene Wigner and Erdal Inönü 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 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 of null 4-vectors.) 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 Lorentz group to the Galilei group, as c → ∞; or the Moyal bracket Lie algebra (equivalent to quantum commutators) to the Poisson bracket Lie algebra, in the classical limit, ħ→0 .

References