# 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 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* → 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*→ ∞; 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.

## Remarks

## 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 779059. - 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. (subscription required (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.