Continuous Groups of Transformations:´Elie Cartan's
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Continuous groups of transformations: Elie´ Cartan's structural approach Alberto Cogliati 2 Contents 1 Elie´ Joseph Cartan 13 2 Cartan's doctoral dissertation 23 2.1 Finite continuous groups . 23 2.1.1 Reduced form of a given group . 27 2.1.2 Integrability and Semisimplicity Criteria . 32 2.1.3 Radical and decomposition theorems . 35 2.2 Lie's theory of complete systems . 40 2.3 Complete systems and canonical reduction . 46 3 Infinite continuous groups 1883-1902 53 3.1 Lie's first contributions . 55 3.2 Differential invariants . 58 3.3 Engel's Habilitationsschrift . 65 3.4 Foundations of infinite continuous groups . 69 3.5 On a theorem by Engel . 80 3.6 Medolaghi's contributions . 83 3.7 Vessiot and his M´emoirecouronn´ee. 91 4 Exterior differential systems 105 4.1 Some technical preliminaries . 106 4.2 The state of the art in the early 1890's . 109 4.3 Engel's invariants theory of Pfaffian systems . 110 4.3.1 Invariant correspondences . 110 4.4 von Weber's contributions: 1898-1900 . 114 4.4.1 Character and characteristic transformations . 115 4.4.2 Pfaffian systems of character one, I . 118 4.4.3 Reducibility of a Pfaffian system to its normal form . 120 4.5 The foundations of the exterior differential calculus . 125 4.6 Cartan's theory of general Pfaffian systems . 127 4.6.1 Geometrical representation . 127 4.6.2 Cauchy's first theorem . 131 4.6.3 Genre and characters . 135 3 4 CONTENTS 4.6.4 Characteristic elements . 138 4.6.5 Pfaffian systems of character one, II . 140 5 Cartan's theory (1902-1909) 145 5.1 On the genesis of the theory . 145 5.2 Cartan's test for involutivity . 154 5.3 Cartan's theory of infinite continuous groups . 160 5.3.1 First fundamental theorem . 161 5.3.2 Second and third fundamental theorems . 165 5.4 Subgroups of a given continuous group . 167 5.5 Simple infinite continuous groups . 172 6 Cartan's method of moving frames 179 A Finite continuous groups 189 A.0.1 The three fundamental theorems . 190 A.0.2 The adjoint group . 192 B Picard-Vessiot theory 193 C Jules Drach, the Galois of his generation 203 CONTENTS 5 Figure 1: Elie´ Joseph Cartan (1869-1951). Archives of the Academy of Sciences of Paris, Elie´ Cartan's dossier biographique. 6 CONTENTS Introduction The problems dealt with by Cartan are among the most important, abstract and general problems of mathematics. As we have already said, group theory is, in a certain sense, mathematics itself, deprived of its matter and reduced to its pure form.1 With these praising words Henri Poincar´edescribed Elie´ Cartan's math- ematical works in the report which he wrote for the Faculty of Sciences of Paris in 1912. In the course of his survey, Poincar´esingled out two funda- mental characteristics of Cartan's mathematical production until that time: a high degree of unity due to his relentless commitment to group theory and a constant concern for issues of structural nature. In effect, since the beginning of his scientific career, Cartan had, almost exclusively, dealt with the theory of groups by considering different variants of the notion such as finite discontinuous groups, finite continuous groups and infinite continuous groups. Furthermore, in all his researches on the subject, Cartan had emphasized the importance of pursuing an abstract approach which was based on the fundamental notions of structure and isomorphism. Although it is doubtful that Cartan was willing to completely share Poincar´e'sview according to whom every mathematical theory was ulti- mately a branch of group theory, he admitted that the notion of group pro- vided a most precious tool by means of which apparently distinct theories could be reunited under common principles. Already in 1909, he clarified his ideas over this point by making recourse to two examples taken from pure geometry and theoretical physics. Lobachevskian geometry in 3-dimensional space, he asserted, is equivalent to projective geometry of real or imaginary figure upon a straight line. Similarly, the new cinematics of special relativity which is governed by the Lorentz group is equivalent to Laguerre geometry. The reason for such equivalences, Cartan explained, lays in the fact that their corresponding groups are isomorphic, i.e. they exhibit the same structure. 1\On voit que les probl`emestrait´espar M. Cartan sont parmi les plus importants, les plus abstraits et les plus g´en´eraux dont s'occupent les Math´ematiques;ainsi que nous l'avons dit, la th´eoriedes groupes est, pour ansi dire, la Math´ematiqueenti`ere, d´epouille de sa mati`ere et r´eduite`aune forme pure". [Poincar´e1914, p. 145]. 7 8 CONTENTS In a certain sense, he concluded, the logical content of many geometrical theories coincides with the structure of their corresponding groups. In actual fact, geometry will assume a dominant role in Cartan's research priorities only later in his career, namely starting from the 1910's. It is cer- tainly true that such a favourable attitude towards geometrical applications of group theory may have contributed to guide Cartan's interests even in his early researches. Nonetheless, as it will be shown in great detail, appli- cations to integration theory of differential equations seem to have played a by far prevalent role in driving Cartan to conceive a structural theory of continuous groups of transformations. Until at least 1910, the notion of group structure constituted the main, if not unique, object of Cartan's researches. He defined it to be the law of composition of the transformations of a group when these are considered independently of the nature of the objects upon which they act. However, as he hastened to remark, depending upon the type of groups under consideration, finite or infinite continuous groups, the study of the structure of groups took on different forms and required quite different tech- niques. As for the case of finite continuous groups, forefathers of modern Lie groups, Cartan's theory was indeed based upon consideration of infinites- imal transformations. First introduced by Lie, they had been profitably exploited by W. Killing in his monumental classification work in which the grounds for the modern theory of the structure of (complex) Lie algebras were laid. Infinitesimal transformations had proved to be a valuable techni- cal tool essentially in consequence of the fact that they led to the existence of constants which fully characterize the structure of a given group. On the contrary, in the case of infinite continuous groups (which nowa- days we would call Lie pseudogroups), since still at the beginning of the last century no structural approach was available, it was up to Cartan to build up a brand new theory which introduced, for the first time, structural considerations in the infinite domain. In this respect, Cartan's innovative theory of exterior differential systems turned out to be an essential tool. As a consequence of this, infinitesimal transformations were replaced by in- variant exterior forms whose exterior derivatives provided generalization of classical structure constants. In view of Cartan's constant concern for structural issues, it is not sur- prising that his figure and work became a kind of benchmark for the Bour- baki group. Dieudonn´e,for example, saw him as a tutelary deity of incoming generations of mathematicians. By directly addressing Cartan, on the occa- sion of his seventieth birthday, he said for example: \vous ^etesun jeune, et vous comprenez les jeunes2". Nonetheless, we should not disregard the fact that Cartan's commitment to the structural theory of continuous groups 2See [Jubil´e,p. 49]. CONTENTS 9 must be situated in the appropriate historical context in which his work saw the light. Indeed, an attentive analysis of the motivations at the basis of his researches in this field reveals, at the same time, a marked inclination towards concrete applications3. This emerges quite clearly both from his early works on finite continuous groups and from his subsequent studies on infinite continuous ones. Indeed, one of the main driving forces guiding Cartan's first contributions on the structure of finite continuous groups was represented by the wide variety of applications to the theory of differential equations and, in particular, to Lie's integration theory of complete systems of first order linear PDE's. Similarly, the applications to the theory of general systems of PDE's played a major role in orienting Cartan's research priorities in the realm of infinite continuous groups, too. Furthermore, his peculiar approach to infinite continuous groups pro- vided him with essential technical tools later on to be profitably employed in differential geometry, namely in his method of moving frames. Indeed, the systematic use of exterior differential forms not only turned out to be indispensable for treating infinite groups, but it also provided a reformu- lation of Lie's theory of finite continuous groups in terms of the so-called Maurer-Cartan forms which proved to be the most suitable one for geomet- rical applications. This constant search for a balance between abstraction and application represented a crucial characteristic of Cartan's entire work, all the more so, since this peculiarity of his mathematical activity frequently reflected his natural tendency to develop general (algorithmic) methods which found application in a large variety of specific problems. His general approach to PDE systems by means of exterior forms only, the theory of equivalence of differential structures and the method of moving frames are the most significant and well known examples of such a tendency. From a certain point of view, one may even say that the real greatness of Cartan's entire work coincides precisely with the generality and the power of his methods and technical tools rather than with specific achievements in a particular branch of mathematics.