Line Geometry, Differential Equations And the Birth of Lie's Theory of Groups

Abstract

This chapter presents some of the conclusions and implications of a research project. It discusses Lie's early work and its significance for the birth of Lie's theory. By Lie's account, the birth of his theory of groups occurred during the winter of 1873–74. At that time, he posed the problem of determining all continuous groups of transformations acting on n-dimensional space, that is, all groups of transformations generated by a finite number of infinitesimal transformations of n-dimensional space. Lie acquired along the way the mathematical tools that convinced him of the feasibility of the above group classification problem. Lie's prodigious mathematical research was not dominated by group-related considerations, but involved a diverse spectrum of mathematical ideas. Considerations of closed systems of commutative transformations have an intimate relation to investigations, which occur in the theory of substitutions and the attendant theory of algebraic equations.