This paper deals with the theory of deformation of Lie algebras. A connection is established with the usual contraction theory, which leads to some ``more singular'' contractions. As a consequence it is shown that the only groups which can be contracted in the Poincaré group are SO(4, 1) and SO(3, 2).
Topics
Lie algebras
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Words being defined are given in italic.
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Recently, Michèle Vergne has found an example of an irreducible set of dimension greater than [Thèse de 3ème cycle, Paris, I.H.P. Mai 1966].
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W. T. Sharp, thesis, Princeton University (1960).
9.
Let us notice that the orbits are not connected; they are made of two connected pieces.
10.
11.
12.
This is not the convention of Saletan who chooses in order to have
13.
Which can also be written
14.
Indeed by writing we see that is sufficient to consider instead of u, and then the contracted algebra by is isomorphic to the contracted algebra by the isomorphism being w.
15.
We suppose also always nonsingular (even for ).
16.
E. Inonü, in Group Theoretical Concepts and Methods in Elementary Particle Physics, F. Gürsey, Ed. (Gordon and Breach Science Publishers, Inc., New York, 1965), p. 391.
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F. Gürsey, in Ref. 16, p. 365.
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L. Michel, in Lectures in Theoretical Physics (University of Colorado Press, Boulder, Colorado, 1964), Vol. VIIa, p. 117.
20.
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© 1967 The American Institute of Physics.
1967
The American Institute of Physics
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