Deformation cohomology of algebraic and geometric structures

Abstract

In 1953, the physicists E. Inonü 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 (Poincaré group) should produce the composition law $(u,v) \rightarrow u + v $ used in classical mechanics (Galilée 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 $ϕ/c^2$ where $ϕ=GM/r$ 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 $0 \rightarrow Θ\rightarrow T \rightarrow F_0 \rightarrow F_1 \rightarrow ... \rightarrow F_n \rightarrow 0$ as a resolution of the sheaf $Θ\subset T$ of infinitesimal transformations and to induce a purely algebraic " deformation sequence " with finite dimensional vector spaces and linear maps, even if $Θ$ 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.