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), [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 → ∞; 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.

References

• Dooley, A. H.; Rice, J. W. (1985). "On contractions of semisimple Lie groups" (PDF). Transactions of the American Mathematical Society. 289 (1): 185–202. doi:10.2307/1999695. ISSN 0002-9947. JSTOR 1999695. MR 0779059.
• Gilmore, Robert (2006). Lie Groups, Lie Algebras, and Some of Their Applications. Dover Books on Mathematics. Dover Publications. ISBN 0486445291. MR 1275599.
• Inönü, E.; Wigner, E. P. (1953). "On the Contraction of Groups and Their Representations". Proc. Natl. Acad. Sci. 39 (6): 510–24. Bibcode:1953PNAS...39..510I. doi:10.1073/pnas.39.6.510. PMC 1063815. PMID 16589298.
• Saletan, E. J. (1961). "Contraction of Lie Groups". Journal of Mathematical Physics. 2 (1): 1–21. Bibcode:1961JMP.....2....1S. doi:10.1063/1.1724208.
• Segal, I. E. (1951). "A class of operator algebras which are determined by groups". Duke Mathematical Journal. 18: 221. doi:10.1215/S0012-7094-51-01817-0.