Jacobi and the Birth of Lie’s Theory of Groups

Abstract

It is a well known fact, documented in Lie’s correspondence with Adolph Mayer, that it was during the winter of 1873–74 that Lie began to systematically develop what became his theory of continuous transformation groups. Until recently, Lie’s decision to devote himself to this enterprise had struck me as rather mysterious. He had begun his career as a geometer and under the influence of Felix Klein had come to appreciate the potential importance that a theory of continuous transformation groups could have in applications to geometry and to the theory of differential equations.¹ By 1871, however, very little of that potential had been realized. In [Klein, Lie 1870] Klein and Lie had called attention to a class of curves and surfaces defined by one and two parameter continuous groups of commuting projective transformations. These W-curves and W-surfaces (as they named them) included examples previously studied by other geometers. Klein and Lie suggested that the entire class was of special geometrical interest because of their invariance with respect to the group that generated them, but they apparently did not impress other geometers with their novel ideas, most of which were only sketched. They only published the details regarding W-curves in the plane [Klein, Lie 1871] because, in general, in order to determine the “W-configurations„ in n-dimensional projective space, they first had to classify the groups of commuting projective transformations that defined them, and already for n = 3 the classification problem became so involved that they abandoned their original plans to present a detailed discussion of W-curves and surfaces in space.

On this aspect of Lie’s early geometrical work (1869–71), see [Hawkins 1989].