Élie Cartan

Élie Cartan (9 April 1869 - 6 May 1951) was a French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. He was born in Dolomieu in Savoie, and became a student at the École Normale in Paris in 1888. After his doctorate in 1894, he took lecturing positions in Montpellier and Lyons, becoming a professor in Nancy in 1903. He took a lecturing position in Paris in 1909, becoming professor in 1912, and retiring in 1942. He died in Paris. He was the father of the mathematician Henri Cartan.

By his own account, in his Notice sur les travaux scientifiques, the main theme of his works (numbering 186 and published throughout the period 1893-1947) was the theory of Lie groups. He began by working over the foundational material on the complex simple Lie algebras, tidying up the previous work by Engel and Killing. This proved definitive, as far as the classification went, with the identification of the four main families and the five exceptional cases. He also introduced the algebraic group concept, which was not to be developed seriously before 1950.

He defined the general notion of anti-symmetric differential form, in the style now used; his approach to Lie groups through the Maurer-Cartan equations required 2-forms for their statement. At that time what were called Pfaffian systems (i.e. first-order differential equations given as 1-forms) were in general use; by the introduction of fresh variables for derivatives, and extra forms, they allowed for the formulation of quite general PDE systems. Cartan added the exterior derivative, as an entirely geometric and coordinate-independent operation. It naturally leads to the need to discuss p-forms, of general degree p. Cartan writes of the influence on him of Riquier’s general PDE theory.

With these basics – Lie groups and differential forms – he went on to produce a very large body of work, and also some general techniques such as moving frames, that were gradually incorporated into the mathematical mainstream.

In the Travaux, he breaks down his work into 15 areas. Using modern terminology, they are these:

1. Lie groups
2. Representations of Lie groups
3. Hypercomplex numbers, division algebras
4. Systems of PDEs, Cartan-Kähler theorem
5. Theory of equivalence
6. Integrable systems, theory of prolongation and systems in involution
7. Infinite-dimensional groups and pseudogroups
8. Differential geometry and moving frames
9. Generalised spaces with structure groups and connections, Cartan connection, holonomy, Weyl tensor
10. Geometry and topology of Lie groups
11. Riemannian geometry
12. Symmetric spaces
13. Topology of compact groups and their homogeneous spaces
14. Integral invariants and classical mechanics
15. Relativity, spinors

Most of these topics have been worked over thoroughly by later mathematicians. That cannot be said of all of them: while Cartan’s own methods were remarkably unified, in the majority of cases the subsequent work can be said to have removed his characteristic touch. That is, it became more algebraic.

To look at some of those less mainstream areas:

• the PDE theory has to take into account singular solutions (i.e. envelopes), such as are seen in Clairaut’s equation;
• the prolongation method is supposed to terminate in a system in involution (this is an analytic theory, rather than smooth, and leads to the theory of formal integrability and Spencer cohomology);
• the equivalence problem, as he put it, is to construct differential isomorphisms of structures (and discover thereby the invariants) by forcing their graphs to be integral manifolds of a differential system;
• the moving frames method, as well as being connected to principal bundles and their connections, should also use frames adapted to geometry;
• these days, the jet bundle method of Ehresmann is applied to use contact as a systematic equivalence relation.

There is a sense, therefore, in which the distinctive side of Cartan’s work is still being digested by mathematicians. This is constantly seen in areas such as calculus of variations, Bäcklund transformations and the general theory of differential systems; roughly speaking those parts of differential algebra which feel that the existing, Galois theory-led model of symmetry is too narrow and requires something more analogous to a category of relations.