group contraction

{{Short description|Construct in theoretical physics}}

In theoretical physics, Eugene Wigner and Erdal İnönü have discussed{{harvnb|Inönü|Wigner|1953}} 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.{{harvnb|Segal|1951|p=221}}{{harvnb|Saletan|1961|p=1}}

For example, the Lie algebra of the 3D rotation group {{math|SO(3)}}, {{math|1=[X₁, X₂] = X₃}}, etc., may be rewritten by a change of variables {{math|1=Y₁ = εX₁}}, {{math|1=Y₂ = εX₂}}, {{math|1=Y₃ = X₃}}, as

: {{math|1=[Y₁, Y₂] = ε² Y₃,     [Y₂, Y₃] = Y₁,     [Y₃, Y₁] = Y₂}}.

The contraction limit {{math|ε → 0}} trivializes the first commutator and thus yields the non-isomorphic algebra of the plane Euclidean group, {{math|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 {{math|Y₁, Y₂}}, now generate the Abelian normal subgroup of {{math|E₂}} (cf. Group extension), the parabolic Lorentz transformations.

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

• the de Sitter group {{math|SO(4, 1) ~ Sp(2, 2)}} to the Poincaré group {{math|ISO(3, 1)}}, as the de Sitter radius diverges: {{math|R → ∞}}; or
• the super-anti-de Sitter algebra to the super-Poincaré algebra as the AdS radius diverges {{math|R → ∞}}; or
• the Poincaré group to the Galilei group, as the speed of light diverges: {{nowrap|c → ∞}};{{harvnb|Gilmore|2006}} 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: {{nowrap|ħ → 0}}.