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 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.
Notes
Inönü & Wigner 1953
Segal 1951, p. 221
Saletan 1961, p. 1
Gilmore 2006
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.
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. 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.
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