Killing and the Structure of Lie Algebras

Abstract

As will be seen in Section 2 below, shortly after Killing published his essay The Concept of Space Extended [1884], he learned that Lie had earlier introduced similar ideas. Before long, Killing decided to temporarily postpone his research on his theory of space forms and to apply the underlying ideas he had developed to the problem of classifying Lie algebras, with simple algebras particularly in mind. Thanks to his extensive correspondence with Engel, we are able to glimpse the excitement, frustration, and uncertainty that attended what we can now see—thanks largely to the subsequent work of Élie Cartan (Chapter 6)— as the birth of the theory of the structure of semisimple Lie algebras.¹