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 $SO(3)$ , $[X 1, X 2] = X 3$ , 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 $ε \to 0$ 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: $R \to \infty$ ; 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.

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. MR 0779059. {{cite journal}}: Invalid |ref=harv (help)
Gilmore, Robert (2006). Lie Groups, Lie Algebras, and Some of Their Applications. Dover Books on Mathematics. Dover Publications. ISBN 0486445291. MR 1275599. {{cite book}}: Invalid |ref=harv (help)
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. {{cite journal}}: Invalid |ref=harv (help)
Saletan, E. J. (1961). "Contraction of Lie Groups". Journal of Mathematical Physics. 2 (1): 1. Bibcode:1961JMP.....2....1S. doi:10.1063/1.1724208. {{cite journal}}: Invalid |ref=harv (help)
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. {{cite journal}}: Invalid |ref=harv (help)

[1] Inönü & Wigner 1953

[2] Segal 1951, p. 221

[3] Saletan 1961, p. 1

[4] Gilmore 2006

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