  In 1953, the physicists E. Inon\"u and E.P. Wigner introduced the concept of
deformation of a Lie algebra by claiming that the limit $1/c \rightarrow 0$,
when c is the speed of light, of the composition law $(u,v) \rightarrow
(u+v)/(1+(uv/c^2))$ of speeds in special relativity (Poincar\'e group) should
produce the composition law $(u,v) \rightarrow u + v $ used in classical
mechanics (Galil\'ee group). However, the dimensionless composition law
$(u'=u/c,v'=v/c) \rightarrow (u'+v')/(1+u'v')$ does not contain any longer a
perturbation parameter. Nevertheless, this idea brought the birth of the "
deformation theory of algebraic structures", culminating in the use of the
Chevalley-Eilenberg cohomology of Lie algebras and one of the first
applications of computer algebra in the seventies. One may also notice that the
main idea of general relativity is to deform the Minkowski metric of space-time
by means of the small dimensionless parameter $\phi/c^2$ where $\phi=GM/r$ is
the gravitational potential at a distance r of a central attractive mass M with
gravitational constant G. A few years later, a " deformation theory of
geometric structures " on manifolds of dimension n was introduced and one may
quote riemannian, symplectic or complex analytic structures. Though often
conjectured, the link between the two approaches has never been exhibited and
the aim of this paper is to provide the solution of this problem by new
methods. The key tool is made by the " Vessiot structure equations " (1903) for
Lie groups or Lie pseudogroups of transformations, which, contrary to the "
Cartan structure equations ", are still unknown today and contain " structure
constants " which, like in the case of constant riemannian curvature, have in
general nothing to do with any Lie algebra. The main idea is then to introduce
the purely differential Janet sequence $0 \rightarrow \Theta \rightarrow T
\rightarrow F_0 \rightarrow F_1 \rightarrow ... \rightarrow F_n \rightarrow 0$
as a resolution of the sheaf $\Theta \subset T$ of infinitesimal
transformations and to induce a purely algebraic " deformation sequence " with
finite dimensional vector spaces and linear maps, even if $\Theta$ is infinite
dimensional. The infinitesimal equivalence problem for geometric structures has
to do with the local exactness at $ F_0 $ of the Janet sequence while the
deformation problem for algebraic structures has to do with the exactness of
the deformation sequence at the invariant sections of $F_1 $, that is ONE STEP
FURTHER ON in the sequence and this unexpected result explains why the many
tentatives previously quoted have not been successful. Finally, we emphasize
through examples the part that could be played by computer algebra in any
explicit computation.
