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).

1.
M.
Gerstenhaber
,
Ann. Math.
79
, No.
1
(
1964
).
2.
Words being defined are given in italic.
3.
W. V. D. Hodge and D. Pedoe, Methods of Algebraic Geometry (Cambridge University Press, New York, 1964), Vol. II.
4.
M.
Raynaud
,
Compt. Rend.
258
,
2457
(
1964
);
M.
Raynaud
,
260
,
4391
(
1965
).,
Compt. Rend.
5.
Recently, Michèle Vergne has found an example of an irreducible set of dimension greater than 2n3/27. [Thèse de 3ème cycle, Paris, I.H.P. Mai 1966].
6.
A. Nijenhuis and R. W. Richardson, Bull. Am. Math. Soc. January (1966), p. 1.
7.
N. Jacobson, Lie Algebras (Interscience Publishers Inc., New York, 1962), p. 11.
8.
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.
E.
Inonü
and
E. P.
Wigner
,
Proc. Natl. Acad. Sci. U.S.
39
,
510
(
1953
).
11.
E. J.
Saletan
,
J. Math. Phys.
2
,
1
(
1961
).
12.
This is not the convention of Saletan who chooses υ = 1−u, in order to have Φ1 = 1.
13.
Which can also be written [a,b](1) = u−1[ua,ub]R+δu(a, b)N.
14.
Indeed by writing Φt = (uψt−1+1)ψt, we see that is sufficient to consider w = uv−1 instead of u, and then the contracted algebra by Φt is isomorphic to the contracted algebra by w+t, the isomorphism being w.
15.
We suppose also ψt always nonsingular (even for t = 0).
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.
17.
F. Gürsey, in Ref. 16, p. 365.
18.
A. O.
Barut
and
A.
Böhm
,
Phys. Rev.
139
,
B1107
(
1965
).
19.
L. Michel, in Lectures in Theoretical Physics (University of Colorado Press, Boulder, Colorado, 1964), Vol. VIIa, p. 117.
20.
G.
Hochschild
and
J. P.
Serre
,
Ann. Math.
57
,
603
(
1953
).
This content is only available via PDF.
You do not currently have access to this content.