introduction first part second part third part / conclusion Unité, généralité et méthode axiomatique dans l’article de Weyl de 1925-1926 sur les groupes de Lie Christophe Eckes Université Paris VII 12 mars introduction first part second part third part / conclusion introduction - We should not reduce Weyl’s paper to this result. - how to describe the interplay between different mathematical domains (algebra, analysis situs) and different methods (Cartan’s, Hurwitz’s and Frobenius’s methods) ? - which relation can we find between the study of special cases (SL(n,C), O(n), Sp(2n,C)) and the general theory of semi-simple complex Lie groups ? introduction first part second part third part / conclusion General remarks Weyl’s article on the representation of semisimple groups is well known, because of one result in particular: The complete reducibility theorem for semisimple complex Lie groups. - how to describe the interplay between different mathematical domains (algebra, analysis situs) and different methods (Cartan’s, Hurwitz’s and Frobenius’s methods) ? - which relation can we find between the study of special cases (SL(n,C), O(n), Sp(2n,C)) and the general theory of semi-simple complex Lie groups ? introduction first part second part third part / conclusion General remarks Weyl’s article on the representation of semisimple groups is well known, because of one result in particular: The complete reducibility theorem for semisimple complex Lie groups. - We should not reduce Weyl’s paper to this result. introduction first part second part third part / conclusion General remarks Weyl’s article on the representation of semisimple groups is well known, because of one result in particular: The complete reducibility theorem for semisimple complex Lie groups. - We should not reduce Weyl’s paper to this result. - how to describe the interplay between different mathematical domains (algebra, analysis situs) and different methods (Cartan’s, Hurwitz’s and Frobenius’s methods) ? - which relation can we find between the study of special cases (SL(n,C), O(n), Sp(2n,C)) and the general theory of semi-simple complex Lie groups ? We shall determine the kind of things which have to be unified. (i) methods? (for instance Cartan’s local and algebraic method in the study of Lie algebras and Hurwitz’s integral and transcendant method applied to certain Lie groups) (ii) concepts?(Weyl defines axiomatically Lie algebras and he uses the language of linear algebra in the theory of Lie groups and in the representation theory of groups) (iii) mathematical domains?(algebra and analysis situs, in particular covering space theory which plays an essential role in Weyl’s proof of the complete reducibility theorem) (iv) mathematical theories?(Weyl has in mind that invariant theory can be reduced to representation theory) introduction first part second part third part / conclusion General remarks : unity of mathematics We will try to explain Weyl’s thesis concerning the unity of mathematics which derives from this article. (i) methods? (for instance Cartan’s local and algebraic method in the study of Lie algebras and Hurwitz’s integral and transcendant method applied to certain Lie groups) (ii) concepts?(Weyl defines axiomatically Lie algebras and he uses the language of linear algebra in the theory of Lie groups and in the representation theory of groups) (iii) mathematical domains?(algebra and analysis situs, in particular covering space theory which plays an essential role in Weyl’s proof of the complete reducibility theorem) (iv) mathematical theories?(Weyl has in mind that invariant theory can be reduced to representation theory) introduction first part second part third part / conclusion General remarks : unity of mathematics We will try to explain Weyl’s thesis concerning the unity of mathematics which derives from this article. We shall determine the kind of things which have to be unified. (ii) concepts?(Weyl defines axiomatically Lie algebras and he uses the language of linear algebra in the theory of Lie groups and in the representation theory of groups) (iii) mathematical domains?(algebra and analysis situs, in particular covering space theory which plays an essential role in Weyl’s proof of the complete reducibility theorem) (iv) mathematical theories?(Weyl has in mind that invariant theory can be reduced to representation theory) introduction first part second part third part / conclusion General remarks : unity of mathematics We will try to explain Weyl’s thesis concerning the unity of mathematics which derives from this article. We shall determine the kind of things which have to be unified. (i) methods? (for instance Cartan’s local and algebraic method in the study of Lie algebras and Hurwitz’s integral and transcendant method applied to certain Lie groups) (iii) mathematical domains?(algebra and analysis situs, in particular covering space theory which plays an essential role in Weyl’s proof of the complete reducibility theorem) (iv) mathematical theories?(Weyl has in mind that invariant theory can be reduced to representation theory) introduction first part second part third part / conclusion General remarks : unity of mathematics We will try to explain Weyl’s thesis concerning the unity of mathematics which derives from this article. We shall determine the kind of things which have to be unified. (i) methods? (for instance Cartan’s local and algebraic method in the study of Lie algebras and Hurwitz’s integral and transcendant method applied to certain Lie groups) (ii) concepts?(Weyl defines axiomatically Lie algebras and he uses the language of linear algebra in the theory of Lie groups and in the representation theory of groups) (iv) mathematical theories?(Weyl has in mind that invariant theory can be reduced to representation theory) introduction first part second part third part / conclusion General remarks : unity of mathematics We will try to explain Weyl’s thesis concerning the unity of mathematics which derives from this article. We shall determine the kind of things which have to be unified. (i) methods? (for instance Cartan’s local and algebraic method in the study of Lie algebras and Hurwitz’s integral and transcendant method applied to certain Lie groups) (ii) concepts?(Weyl defines axiomatically Lie algebras and he uses the language of linear algebra in the theory of Lie groups and in the representation theory of groups) (iii) mathematical domains?(algebra and analysis situs, in particular covering space theory which plays an essential role in Weyl’s proof of the complete reducibility theorem) introduction first part second part third part / conclusion General remarks : unity of mathematics We will try to explain Weyl’s thesis concerning the unity of mathematics which derives from this article. We shall determine the kind of things which have to be unified. (i) methods? (for instance Cartan’s local and algebraic method in the study of Lie algebras and Hurwitz’s integral and transcendant method applied to certain Lie groups) (ii) concepts?(Weyl defines axiomatically Lie algebras and he uses the language of linear algebra in the theory of Lie groups and in the representation theory of groups) (iii) mathematical domains?(algebra and analysis situs, in particular covering space theory which plays an essential role in Weyl’s proof of the complete reducibility theorem) (iv) mathematical theories?(Weyl has in mind that invariant theory can be reduced to representation theory) - The complete reducibility theorem can be considered as a purely algebraic result. But Weyl doesn’t reason as an algebraist by proving this theorem. - At the beginning of the 30’s, an algebraic proof of this theorem is still needed. It will be obtained by Casimir and Van der Waerden in 1935, then by Brauer in 1936. Weyl refuses to consider algebra and algebraic methods as a solution to unify mathematics. Weyl’s paper on Lie groups differs from what we call "mathematical structuralism", although he defines axiomatically the concepts of a Lie algebra or of a semi-simple Lie group. introduction first part second part third part / conclusion General remarks : unity of mathematics We will show that Weyl’s conception of the unity of mathematics is very specific at that time: Weyl’s paper on Lie groups differs from what we call "mathematical structuralism", although he defines axiomatically the concepts of a Lie algebra or of a semi-simple Lie group. introduction first part second part third part / conclusion General remarks : unity of mathematics We will show that Weyl’s conception of the unity of mathematics is very specific at that time: - The complete reducibility theorem can be considered as a purely algebraic result. But Weyl doesn’t reason as an algebraist by proving this theorem. - At the beginning of the 30’s, an algebraic proof of this theorem is still needed. It will be obtained by Casimir and Van der Waerden in 1935, then by Brauer in 1936. Weyl refuses to consider algebra and algebraic methods as a solution to unify mathematics. introduction first part second part third part / conclusion General remarks : unity of mathematics We will show that Weyl’s conception of the unity of mathematics is very specific at that time: - The complete reducibility theorem can be considered as a purely algebraic result. But Weyl doesn’t reason as an algebraist by proving this theorem. - At the beginning of the 30’s, an algebraic proof of this theorem is still needed. It will be obtained by Casimir and Van der Waerden in 1935, then by Brauer in 1936. Weyl refuses to consider algebra and algebraic methods as a solution to unify mathematics. Weyl’s paper on Lie groups differs from what we call "mathematical structuralism", although he defines axiomatically the concepts of a Lie algebra or of a semi-simple Lie group. Structure of his article. - first part: study of SL(n,C); proof of the complete reducibility theorem in this case; theory of characters; connection between the irreducible representations of SL(n,C) and the irreducible representations of the symmetric group.