A Centennial: Wilhelm Killing and the Exceptional Groups

In his article [3] in the Mathematical Intelligencer (vol. 11, no. 3) A. John Coleman gives a colorful biography of the mathematician Wilhelm Killing and offers an admiring appraisal of his paper [8]. The subject of this paper, the classification of the simple Lie algebras over C, has indeed turned out to be a milestone in the history of mathematics. At the conclusion of his article Coleman lists six reasons why he considers [8] to be such an epoch-making paper, the first one being that it furnished the impetus towards the problem of classifying finite simple groups. Here one could add that the answer to that problem was also motivated by and was partly provided by the classification of simple Lie algebras through Claude Chevalley's paper [2b]. In two sections of his article, entitled "Killing Intervenes" and "The Still Point of the Turning World," Coleman discusses the work of Killing and Elie Cartan on the classification of simple Lie algebras over C. I would like to add a few comments to his discussion (see also [7b]). While Sophus Lie and some of his associates in Leipzig at tempted the problem of classifying all local transformation groups of R n, Killing set himself the problem of finding all possible Zusammensetzungen of r-parameter groups. In other words, he was interested in all possible ways in which a vector space could be turned into a Lie algebra. While Lie was motivated by applications to differential equations, Killing was led to his problem from his work in geometry. Let ~ be a simple Lie algebra over C. Then g is isomorphic to the Lie algebra of linear transformations adX of g given by adX(Y) = IX, Y], with X running through g and [,] denoting the bracket product in the Lie algebra. To study this family it is natural to try to diagonalize the operators ad(X) as effectively as possible. This is the motivation for the definition of a Cartan subalgebra as a subalgebra ~ which is