Deformation Theory of Algebraic and Geometric Structures

S. Lie discovered in 1890 the "Lie pseudogroups", namely the groups of transformations solutions of systems of partial differential (PD) equations. During the next fifty years, these groups have only been studied by E. Cartan and E. Vessiot but the "Vessiot structure equations" are still unknown today. In the meantime, a "formal theory" of systems of PD equations has been pioneered by M. Janet in 1920. Then, the physicists E. Inonu and E.P. Wigner introduced in 1953 the concept of "deformation of a Lie algebra" by considering the speed of light as a parameter in the Lorentz composition of speeds. This idea led to the "deformation theory of algebraic structures" and the first applications of computer algebra. A few years later, a "deformation theory of geometric structures" has been introduced by D.C. Spencer and coworkers who used the formal theory of PD equations they had developped for studying "Lie pseudogroups". The existence of a link between these two deformation theories has been conjectured but never found. This book solves this conjecture for the first time by using new mathematical methods. It will be of interest for students and researchers in mathematics and physics.