埃里.嘉当和他的数学工作

ÉLIE CARTAN AND HIS MATHEMATICAL WORK SHIING-SHEN CHERN AND CLAUDE CHEVALLEY After a long illness Élie Cartan died on May 6, 1951, in Paris. His death came at a time when his reputation and the influence of his ideas were in full ascent. Undoubtedly one of the greatest mathe- maticians of this century, his career was characterized by a rare harmony of genius and modesty. Êlie Cartan was born on April 9, 1869 in Dolomieu (Isère), a village in the south of France. His father was a blacksmith. Cartan's elementary education was made possible by one of the state stipends for gifted children. In 1888 he entered the "École Normale Su- périeure," where he learned higher mathematics from such masters as Tannery, Picard, Darboux, and Hermite. His research work started with his famous thesis on continuous groups, a subject sug- gested to him by his fellow student Tresse, recently returned from studying with Sophus Lie in Leipzig. Cartan's first teaching position was at Montpellier, where he was "maître de conférences" ; he then went successively to Lyon, to Nancy, and finally in 1909 to Paris. He was made a professor at the Sorbonne in 1912. The report on his work which was the basis for this promotion was written by Poincaré;1 this was one of the circumstances in his career of which he seemed to have been genuinely proud. He remained at the Sorbonne until his retirement in 1940. Cartan was an excellent teacher; his lectures were gratifying intel- lectual experiences, which left the student with a generally mistaken idea that he had grasped all there was on the subject. It is therefore the more surprising that for a long time his ideas did not exert the influence they so richly deserved to have on young mathematicians. This was perhaps partly due to Cartan's extreme modesty. Unlike Poincaré, he did not try to avoid having students work under his direction. However, he had too much of a sense of humor to organize around himself the kind of enthusiastic fanaticism which helps to form a mathematical school. On the other hand, the bulk of the mathematical research which was accomplished at the beginning of this century in France centered around the theory of analytic func- tions; this subject, made glamorous by the achievement represented 1 This report was in part published in Acta Math. vol. 38 (1921) pp. 137-145. It should be of considerable historic interest to have now a complete version of this report. 217 218 ÉLIE CARTAN AND HIS MATHEMATICAL WORK [March by Picard's theorem, offered many not too difficult problems for a young mathematician to tackle. In the minds of inexperienced begin- ners in mathematics, Cartan's teaching, mostly on geometry, was sometimes very wrongly mistaken for a remnant of the earlier Darboux tradition of rather hollow geometric elegance. When, largely under the influence of A. Weil, a breeze of fresh air from the outside came to blow on French mathematics, it was a great temptation to concentrate entirely on the then ultra-modern fields of topology or modern algebra, and the ideas of Cartan once more, though for other reasons, partially failed to attract the amount of attention which was their due. This regrettable situation was partly corrected when Cartan's work was taken (at the suggestion of A. Weil) in 1936 to be the central theme of the seminar of mathematics organized by Julia. In 1939, at the celebration of Cartan's scientific jubilee, J. Dieudonné could rightly say to him: " . . . vous êtes un "jeune," et vous com- prenez les jeunes"—it was then beginning to be true that the young understood Cartan. In foreign countries, particularly in Germany, his recognition as a great mathematician came earlier. It was perhaps H. Weyl's funda- mental papers on group representations published around 1925 that established Cartan's reputation among mathematicians not in his own field. Meanwhile, the development of abstract algebra naturally helped to attract attention to his work on Lie algebra. However, the reception of his contributions to differential geometry was varied. This was partly due to his approach which, though lead- ing more to the heart of the problem, was unconventional, and partly due to inadequate exposition. Thus Weyl, in reviewing one of Cartan's books [41 ],2 wrote in 1938:8 "Cartan is undoubtedly the greatest living master in differential geometry. . . . I must admit that I found the book, like most of Cartan's papers, hard reading. . . ." This sentiment was shared by many geometers. Cartan was elected to the French Academy in 1931. In his later years he received several other honors. Thus he was a foreign mem- ber of the National Academy of Sciences, U.S.A., and a foreign Fel- low of the Royal Society. In 1936 he was awarded an honorary degree by Harvard University. Closely interwoven with Cartan's life as a scientist and teacher has been his family life, which was filled with an atmosphere of happiness and serenity. He had four children, three sons, Henri, Jean, 2 Numbers in brackets refer to the bibliography at the end of the paper. 8 Bull. Amer. Math. Soc. vol. 44 (1938) p. 601. i95*] ÉLIE CARTAN AND HIS MATHEMATICAL WORK 219 and Louis, and a daughter, Hélène. Jean Cartan oriented himself towards music, and already appeared to be one of the most gifted composers of his generation when he was cruelly taken by death. Louis Cartan was a physicist; arrested by the Germans at the be- ginning of the Résistance, he was murdered by them after a long period of detention. There is no need to say here that Henri Cartan followed in the footsteps of his father to become a mathematician. Cartan's mathematical work can be roughly classified under three main headings: group theory, systems of differential equations, and geometry. These themes are, however, constantly interwoven with each other in his work. Almost everything Cartan did is more or less connected with the theory of Lie groups. S. Lie introduced the groups which were named after him as groups of transformations, i.e., as systems of analytic transforma- tions on n variables such that the product of any two transforma- tions of the system still belongs to the system and each trans- formation of the system has an inverse in the system. The idea of considering the abstract group which underlies a given group of transformations came only later; it is more or less implicit in Killing's work and appears quite explicitly already in the first paper by Cartan. Whereas, for Lie, the problem of classification consisted in finding all possible transformation groups on a given number of variables—a far too difficult problem in the present stage of mathe- matics as soon as the number of variables is not very small—for Killing and Cartan, the problem was to find all possible abstract structures of continuous groups; and their combined efforts solved the problem completely for simple groups. Once the structures of all simple groups were known, it became possible to look for all possible realizations of any one of these structures by transformations of a specified nature, and, in particular, for their realizations as groups of linear transformations. This is the problem of the determination of the representations of a given group; it was solved completely by Cartan for simple groups. The solution led in particular to the dis- covery, as early as 1913, of the spinors, which were to be re-dis- covered later in a special case by the physicists. Cartan also investigated the infinite Lie groups, i.e., the groups of transformations whose operations depend not on a finite number of continuous parameters, but on arbitrary functions. In that case, one does not have the notion of the abstract underlying group. Cartan and Vessiot found, at about the same time and independently of each other, a substitute for this notion of the abstract group which con- sists in defining when two infinite Lie groups are to be considered as 220 ÉLIE CARTAN AND HIS MATHEMATICAL WORK [March isomorphic. Cartan then proceeded to classify all possible types of non-isomorphic infinite Lie groups. Cartan paid also much attention to the study of topological prop- erties of groups considered in the large. He showed how many of these topological problems may be reduced to purely algebraic questions; by so doing, he discovered the very remarkable fact that many properties of the group in the large may be read from the in- finitesimal structure of the group, i.e., are already determined when some arbitrarily small piece of the group is given. His work along these lines resembles that of the paleontologist reconstructing the shape of a prehistoric animal from the peculiarities of some small bone. The idea of studying the abstract structure of mathematical ob- jects which hides itself beneath the analytical clothing under which they appear at first was also the mainspring of Cartan's theory of differential systems. He insisted on having a theory of differential equations which is invariant under arbitrary changes of variables. Only in this way can the theory uncover the specific properties of the objects one studies by means of the differential equations they satisfy, in contradistinction to what depends only on the particular representation of these objects by numbers or sets of numbers. In order to achieve such an invariant theory, Cartan made a sys- tematic use of the notion of the exterior differential of a differential form, a notion which he helped to create and which has just the re- quired property of being invariant with respect to any change of variables. Raised in the French geometrical tradition, Cartan had a constant interest in differential geometry. He had the unusual combination of a vast knowledge of Lie groups, a theory of differential systems whose invariant character was particularly suited for geometrical investiga- tions, and, most important of all, a remarkable geometrical intuition. As a result, he was able to see the geometrical content of very compli- cated calculations, and even to substitute geometrical arguments for some of the computations. The latter practice has often been baffling to his readers. But it is an art whose presence is usually identical with the vigor of a geometrical thinker. In the 1920's the general theory of relativity gave a new impulse to differential geometry. This gave rise to a feverish search of spaces with a suitable local structure. The most notable example of such a local structure is a Riemann metric. It can be generalized in various ways, by modifying the form of the integral which defines the arc length in Riemannian geometry (Finsler geometry), by studying only those properties pertaining to the geodesies or paths (geometry 1952I ÉLIE CARTAN AND HIS MATHEMATICAL WORK 221 of paths of Eisenhart, Veblen, and T. Y. Thomas), by studying the properties of a family of Riemann metrics whose fundamental forms differ from each other by a common factor (conformai geometry), etc. While in all these directions the definition of a parallel displacement is considered to be the major concern, the approach of Car tan to these problems is most original and satisfactory. Again the notion of group plays the central rôle. Roughly speaking, a generalized space (espace généralisé) in the sense of Cartan is a space of tangent spaces such that two infinitely near tangent spaces are related by an in- finitesimal transformation of a given Lie group. Such a structure is known as a connection. The tangent spaces may not be the spaces of tangent vectors. This generality, which is absolutely necessary, gave rise to misinterpretation among differential geometers. As we shall show below, it is now possible to express these concepts in a more satisfactory way, by making use of the modern notion of fiber bundles. We can perhaps conclude from the above brief description that Cartan's mathematical work, unlike that of Poincaré or Hadamard, centers around a few major concepts. This is partly due to the richness of the field, in which his pioneering work has opened avenues where much further development is undoubtedly possible. While many of Cartan's ideas have received clarification in recent years, the difficulties of conceiving the proper concepts at the early stage of development can hardly be overestimated. Thus in writing on the psychology of mathematical thinking, Hadamard had to ad- mit "the insuperable difficulty in mastering more than a rather ele- mentary and superficial knowledge of Lie groups."4 Thanks to the development of modern mathematics, such difficulties are now eased. Besides several books Cartan published about 200 mathematical papers. It is earnestly to be hoped that the publication of his col- lected works may be initiated in the near future. Not only do they fully deserve to find their place on the bookshelves of our libraries at the side of those of other great mathematicians of the past, but they will be, for a long time to come, a most indispensable tool for all those who will attempt to proceed further in the same directions. We now proceed to give a more detailed review of some of the most important of Cartan's mathematical contributions. I. GROUP THEORY Cartan's papers on group theory fall into two categories, dis- tinguished from each other both by the nature of the questions 4 J. Hadamard, The psychology of invention in the mathematical field, Princeton, 1945, p. 115. 222 ÉLIE CARTAN AND HIS MATHEMATICAL WORK [March treated and by the time at which they were written. The papers of the first cycle are purely algebraic in character; they are more con- cerned with what are now called Lie algebras than with group theory proper. In his thesis [3], Car tan gives the complete classification of all simple Lie algebras over the field of complex numbers. They fall into four general classes (which are the Lie algebras of the uni- modular groups, of the orthogonal groups in even or odd numbers of variables, and of the symplectic groups) and a system of five "excep- tional" algebras, of dimensions 14, 52, 78, 133, and 248. Killing had already discovered the fact that, outside the four general classes, there can exist only these five exceptional Lie algebras; but his proofs were incorrect a t several important points, and, as to the exceptional algebras, it is not clear from his paper whether he ever proved that they actually existed. Moreover, in his work, the algebra of dimen- sion 52 appears under two different forms, whose equivalence he did not recognize. Cartan gave rigorous proofs that the classification into four general classes and five exceptional algebras is complete, and constructed explicitly the exceptional algebras. Let g be any Lie algebra; to every element X of g there is asso- ciated a linear transformation, the adjoint ad X of X, operating on the space g, which transforms any element F of g into [-X", F ] . Be- cause of the relation [X, X]=0, this linear transformation always admits 0 as a characteristic root; those elements X of g for which 0 is a characteristic root of least possible multiplicity of ad X are called regular elements. Let H be a regular element; then those ele- ments of g which are mapped into 0 by powers of ad H are seen to form a certain subalgebra Ï) of g, and this subalgebra is always nil- potent (which means that, in the adjoint representation of such an algebra, every element has 0 as its only characteristic roots). A sub- algebra such as Ï) has been called a Cartan subalgebra of g. I t is a kind of inner core of the algebra g, and many properties of the big algebra g are reflected in properties of this subalgebra fy. In the case where g is semi-simple, Ï) is always abelian (which means, for a Lie algebra, that [X, F ] is always 0 for any X and F in the algebra). Moreover, g has a base which is composed of elements which are eigenvectors simultaneously for all adjoint operations of elements of Ï). The factors by which these elements are multiplied when bracketed with elements of t) are called the roots of the Lie algebra; it is the study of the properties of these roots which leads to the classification of simple Lie algebras. In establishing these properties, Cartan made a systematic use of the "fundamental quadratic form" of g, whose value at an element X is the trace of the square of ad X (if g is 1952] ÉLIE CARTAN AND HIS MATHEMATICAL WORK 223 semi-simple, or more generally if it coincides with its derived algebra, then the trace of ad X itself is always zero). One of the most im- portant results of Cartan's thesis is that a necessary and sufficient condition for g to be semi-simple is that its fundamental quadratic form be nondegenerate (i.e. that its rank be equal to the dimension of g). Incidentally, Cartan also applied similar methods to the study of systems of hypercomplex numbers (cf. [4]) and obtained in this manner the main structure theorems for associative algebras over the fields of real and of complex numbers; however, these results were superseded by the work of Wedderburn, which applies to algebras over arbitrary basic fields. By studying those algebras which have only one integrable (or, as we say now, solvable) ideal, Cartan also laid the foundations in his thesis for his subsequent study of linear representations of simple Lie algebras; in particular, he determined, for each class of simple groups, the linear representation of smallest possible degree. The general theory of linear representations is the object of the paper [5]. As above let g be any semi-simple Lie algebra over the field of complex numbers (any algebraically closed field of characteristic 0 would do just as well); a linear representation of g is a law which assigns to every X in g a linear transformation p(X) on some finite- dimensional space; p(X) depends linearly on X, and is such that p([X, Y])=p{X)p(Y)-p(Y)p(X) for any X and Y in g. Let $ be a Cartan subalgebra of g. Then it turns out that the matrices which represent the elements of § may all be reduced simultaneously to the diagonal form ; the diagonal coefficients which occur in these matrices, considered as linear functions of the element which is represented, are called the weights of the representation. The roots of g are the weights of a particular linear representation, viz. the adjoint representation. Cartan proved that all relations between weights of one or several representations are consequences of certain linear relations with rational coefficients between these weights, a fact which can now be explained in two different manners: it reflects the properties of characters of compact abelian groups, and also the properties of alge- braic groups of linear transformations. Cartan then introduces an order relation in the system of all weights and roots, and proves that any irreducible representation is uniquely determined by its highest weight for this order relation. The problem of finding all irreducible linear representations of g is thereby reduced to th^t of finding all possible highest weights of representations. The sum of the highest weights of two irreducible representations is again the highest weight of an irreducible representation, which is contained in the tensor 224 EUE CARTAN AND HIS MATHEMATICAL WORK [March product (or rather, sum, if we speak of representations of Lie algebras and not of groups) of the two given representations. If r is the rank of the Lie algebra g (i.e. the dimension of any Cartan subalgebra of ô), Cartan established that all possible highest weights of irreducible representations may be written as linear combinations with non- negative integral coefficients of r particular linear functions which depend only on