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- Group_contraction abstract "In theoretical physics, Eugene Wigner and Erdal İnönü have discussed the possibility to obtain from a given Lie group a different (non-isomorphic) Lie groupby 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 SO(3), [X1, X2] = X3, etc, may be rewrittenby 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 → ∞; 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.↑ ↑ ↑ ↑".
- Group_contraction wikiPageID "34976899".
- Group_contraction wikiPageLength "4704".
- Group_contraction wikiPageOutDegree "28".
- Group_contraction wikiPageRevisionID "700165110".
- Group_contraction wikiPageWikiLink American_Mathematical_Society.
- Group_contraction wikiPageWikiLink Category:Lie_algebras.
- Group_contraction wikiPageWikiLink Category:Lie_groups.
- Group_contraction wikiPageWikiLink Category:Mathematical_physics.
- Group_contraction wikiPageWikiLink Classical_limit.
- Group_contraction wikiPageWikiLink Correspondence_principle.
- Group_contraction wikiPageWikiLink De_Sitter_space.
- Group_contraction wikiPageWikiLink Dover_Publications.
- Group_contraction wikiPageWikiLink Erdal_İnönü.
- Group_contraction wikiPageWikiLink Euclidean_group.
- Group_contraction wikiPageWikiLink Eugene_Wigner.
- Group_contraction wikiPageWikiLink Four-vector.
- Group_contraction wikiPageWikiLink Galilean_transformation.
- Group_contraction wikiPageWikiLink Group_action.
- Group_contraction wikiPageWikiLink Group_extension.
- Group_contraction wikiPageWikiLink Lie_algebra.
- Group_contraction wikiPageWikiLink Lie_group.
- Group_contraction wikiPageWikiLink Lorentz_group.
- Group_contraction wikiPageWikiLink Minkowski_space.
- Group_contraction wikiPageWikiLink Moyal_bracket.
- Group_contraction wikiPageWikiLink Normal_subgroup.
- Group_contraction wikiPageWikiLink Planck_constant.
- Group_contraction wikiPageWikiLink Poincaré_group.
- Group_contraction wikiPageWikiLink Poisson_bracket.
- Group_contraction wikiPageWikiLink Speed_of_light.
- Group_contraction wikiPageWikiLink Structure_constants.
- Group_contraction wikiPageWikiLinkText "Group contraction".
- Group_contraction wikiPageWikiLinkText "contracts".
- Group_contraction wikiPageWikiLinkText "group contraction".
- Group_contraction wikiPageUsesTemplate Template:Abstract-algebra-stub.
- Group_contraction wikiPageUsesTemplate Template:Cite_book.
- Group_contraction wikiPageUsesTemplate Template:Cite_journal.
- Group_contraction wikiPageUsesTemplate Template:Math.
- Group_contraction wikiPageUsesTemplate Template:Reflist.
- Group_contraction subject Category:Lie_algebras.
- Group_contraction subject Category:Lie_groups.
- Group_contraction subject Category:Mathematical_physics.
- Group_contraction type Algebra.
- Group_contraction comment "In theoretical physics, Eugene Wigner and Erdal İnönü have discussed the possibility to obtain from a given Lie group a different (non-isomorphic) Lie groupby a group contraction with respect to a continuous subgroup of it.".
- Group_contraction label "Group contraction".
- Group_contraction sameAs Q5611219.
- Group_contraction sameAs m.0j679m8.
- Group_contraction sameAs Q5611219.
- Group_contraction wasDerivedFrom Group_contraction?oldid=700165110.
- Group_contraction isPrimaryTopicOf Group_contraction.