Group contraction

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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), [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

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