Unité, Généralité Et Méthode Axiomatique Dans L’Article De Weyl De 1925-1926 Sur Les Groupes De Lie

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.

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.

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. 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. - second part: study of SO(n), O(n) and Sp(2n,C). - third part: the general theory of semi-simple Lie groups and Lie algebras. - fourth part: proof of the complete reducibility theorem in general, connection with invariant theory. The questions of generalization and generality, i.e. the relationship between particular cases and the general case are of the first importance in order to understand the structure of this article.

introduction ﬁrst part second part third part / conclusion General remarks : generality

How does Weyl succeed in proving his main theorem for all complex semisimple Lie groups ? - third part: the general theory of semi-simple Lie groups and Lie algebras. - fourth part: proof of the complete reducibility theorem in general, connection with invariant theory. The questions of generalization and generality, i.e. the relationship between particular cases and the general case are of the ﬁrst importance in order to understand the structure of this article.

introduction ﬁrst part second part third part / conclusion General remarks : generality

How does Weyl succeed in proving his main theorem for all complex semisimple Lie groups ? 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. - second part: study of SO(n), O(n) and Sp(2n,C). - third part: the general theory of semi-simple Lie groups and Lie algebras. - fourth part: proof of the complete reducibility theorem in general, connection with invariant theory. The questions of generalization and generality, i.e. the relationship between particular cases and the general case are of the ﬁrst importance in order to understand the structure of this article. In particular, Weyl describes with great details what we now call "Weyl’s unitarian Trick": - if the Lie algebra su(n) of SU(n) has the complete reducibility property, then so does the Lie algebra sl(n,C) of SL(n,C).

introduction ﬁrst part second part third part / conclusion General remarks : generality

The main steps of the general proof of the complete reducibility theorem are already sketched in the ﬁrst part of the article, i.e. when Weyl studies the representations of the special linear group. At the end of the third part, Weyl ﬁnds an Analogon to this unitarian trick in the general case, that is for all semisimple complex Lie groups. The group SL(n,C) plays the role of a paradigm or a paradigmatic example in Weyl’s paper. In fact, he initiates his general theory of semisimple groups by studying this particular case.

introduction ﬁrst part second part third part / conclusion General remarks : generality

The main steps of the general proof of the complete reducibility theorem are already sketched in the ﬁrst part of the article, i.e. when Weyl studies the representations of the special linear group. In particular, Weyl describes with great details what we now call "Weyl’s unitarian Trick": - if the Lie algebra su(n) of SU(n) has the complete reducibility property, then so does the Lie algebra sl(n,C) of SL(n,C). The group SL(n,C) plays the role of a paradigm or a paradigmatic example in Weyl’s paper. In fact, he initiates his general theory of semisimple groups by studying this particular case.

introduction ﬁrst part second part third part / conclusion General remarks : generality

At the end of the third part, Weyl ﬁnds an Analogon to this unitarian trick in the general case, that is for all semisimple complex Lie groups. introduction ﬁrst part second part third part / conclusion General remarks : generality

At the end of the third part, Weyl finds an Analogon to this unitarian trick in the general case, that is for all semisimple complex Lie groups. The group SL(n,C) plays the role of a paradigm or a paradigmatic example in Weyl’s paper. In fact, he initiates his general theory of semisimple groups by studying this particular case. Such a definition, based on the concept of a vector space, is due to Weyl and Noether. In Frobenius’ and Schur’s writings, a representation is defined as a matrix representation of a group.

- A representation (ρ,V ) is irreducible if there is no proper nonzero invariant subspace W of V .

introduction ﬁrst part second part third part / conclusion Representations of ﬁnite groups / Lie groups

A linear representation of a finite group G on a finite-dimensional complex vector space V is a group homomorphism ρ : G GL(V ), that is −→ - ρ(e) IdV where e denotes the identity of G, = 1 1 - ρ(g − ) ρ(g)− , for all g in G, = - ρ(gg 0) ρ(g)ρ(g 0) for all g, g 0 in G. = introduction first part second part third part / conclusion Representations of finite groups / Lie groups

A linear representation of a finite group G on a finite-dimensional complex vector space V is a group homomorphism ρ : G GL(V ), that is −→ - ρ(e) IdV where e denotes the identity of G, = 1 1 - ρ(g − ) ρ(g)− , for all g in G, = - ρ(gg 0) ρ(g)ρ(g 0) for all g, g 0 in G. = Such a definition, based on the concept of a vector space, is due to Weyl and Noether. In Frobenius’ and Schur’s writings, a representation is defined as a matrix representation of a group.

- A representation (ρ,V ) is irreducible if there is no proper nonzero invariant subspace W of V . Complete reducibility theorem for finite groups: every finite dimensional representation of a finite group satisfies the complete reducibility property. This theorem is discovered independently by Frobenius, Molien and Maschke in 1896-1897. Weyl’s paper could be simply interpreted as a generalization of this result for semisimple complex Lie groups, but - this assumption is based on an arbitrary comparison between Maschke’s Theorem and Weyl’s theorem. - it depends on a linear conception of the history of mathematics.

introduction ﬁrst part second part third part / conclusion Representations of ﬁnite groups / Lie groups

A group representation is said to have the complete reducibility property if it can be decomposed into a direct sum of irreducible subrepresentations. This theorem is discovered independently by Frobenius, Molien and Maschke in 1896-1897. Weyl’s paper could be simply interpreted as a generalization of this result for semisimple complex Lie groups, but - this assumption is based on an arbitrary comparison between Maschke’s Theorem and Weyl’s theorem. - it depends on a linear conception of the history of mathematics.

introduction ﬁrst part second part third part / conclusion Representations of ﬁnite groups / Lie groups

A group representation is said to have the complete reducibility property if it can be decomposed into a direct sum of irreducible subrepresentations. Complete reducibility theorem for finite groups: every finite dimensional representation of a finite group satisfies the complete reducibility property. This theorem is discovered independently by Frobenius, Molien and Maschke in 1896-1897. Weyl’s paper could be simply interpreted as a generalization of this result for semisimple complex Lie groups, but - this assumption is based on an arbitrary comparison between Maschke’s Theorem and Weyl’s theorem. - it depends on a linear conception of the history of mathematics. In his paper, Weyl mainly refers to - Hurwitz’s paper (1897) in which he introduces an integral method applied to closed continuous groups (for instance SU(n) and SO(n)) in the framework of invariant theory, - Cartan’s Thesis (1894) and his article (1913) which contains the theory of roots and weights for Lie algebras, - Schur’s articles on the representations of SO(n) and O(n) (1924).

introduction ﬁrst part second part third part / conclusion Representations of ﬁnite groups / Lie groups

In fact, the development of representation theory in the case of Lie groups doesn’t derive regularly and directly from representation theory of finite groups. introduction first part second part third part / conclusion Representations of finite groups / Lie groups

In fact, the development of representation theory in the case of Lie groups doesn’t derive regularly and directly from representation theory of ﬁnite groups. In his paper, Weyl mainly refers to - Hurwitz’s paper (1897) in which he introduces an integral method applied to closed continuous groups (for instance SU(n) and SO(n)) in the framework of invariant theory, - Cartan’s Thesis (1894) and his article (1913) which contains the theory of roots and weights for Lie algebras, - Schur’s articles on the representations of SO(n) and O(n) (1924). - In his paper (1897), Hurwitz refers exclusively to invariant theory. In 1924, Schur adapts Hurwitz’s integral method to representation theory of certain continuous groups. He doesn’t use Cartan’s method, although he mentions it in passing. - In 1925, Weyl continues these works and he uniﬁes Hurwitz’s, Cartan’s and Schur’s ideas. - connection between Cartan’s and Hurwitz’s methods, - generalization of Schur’s representation theory of continuous groups, - invariant theory interpreted in the framework of representation theory.

introduction ﬁrst part second part third part / conclusion Representations of ﬁnite groups / Lie groups

- In 1913, Cartan is not acquainted with representation theory of ﬁnite groups. He introduces concepts (roots, weights) which Weyl will explicitly interpret in the framework of representation theory of Lie algebras in 1925. - In 1925, Weyl continues these works and he uniﬁes Hurwitz’s, Cartan’s and Schur’s ideas. - connection between Cartan’s and Hurwitz’s methods, - generalization of Schur’s representation theory of continuous groups, - invariant theory interpreted in the framework of representation theory.

introduction ﬁrst part second part third part / conclusion Representations of ﬁnite groups / Lie groups

- In 1913, Cartan is not acquainted with representation theory of ﬁnite groups. He introduces concepts (roots, weights) which Weyl will explicitly interpret in the framework of representation theory of Lie algebras in 1925. - In his paper (1897), Hurwitz refers exclusively to invariant theory. In 1924, Schur adapts Hurwitz’s integral method to representation theory of certain continuous groups. He doesn’t use Cartan’s method, although he mentions it in passing. - connection between Cartan’s and Hurwitz’s methods, - generalization of Schur’s representation theory of continuous groups, - invariant theory interpreted in the framework of representation theory.

introduction ﬁrst part second part third part / conclusion Representations of ﬁnite groups / Lie groups

- In 1913, Cartan is not acquainted with representation theory of finite groups. He introduces concepts (roots, weights) which Weyl will explicitly interpret in the framework of representation theory of Lie algebras in 1925. - In his paper (1897), Hurwitz refers exclusively to invariant theory. In 1924, Schur adapts Hurwitz’s integral method to representation theory of certain continuous groups. He doesn’t use Cartan’s method, although he mentions it in passing. - In 1925, Weyl continues these works and he unifies Hurwitz’s, Cartan’s and Schur’s ideas. introduction first part second part third part / conclusion Representations of finite groups / Lie groups

- In 1913, Cartan is not acquainted with representation theory of finite groups. He introduces concepts (roots, weights) which Weyl will explicitly interpret in the framework of representation theory of Lie algebras in 1925. - In his paper (1897), Hurwitz refers exclusively to invariant theory. In 1924, Schur adapts Hurwitz’s integral method to representation theory of certain continuous groups. He doesn’t use Cartan’s method, although he mentions it in passing. - In 1925, Weyl continues these works and he unifies Hurwitz’s, Cartan’s and Schur’s ideas. - connection between Cartan’s and Hurwitz’s methods, - generalization of Schur’s representation theory of continuous groups, - invariant theory interpreted in the framework of representation theory. Indeed, a Lie group is characterized by a mixed structure. It can be defined as a set endowed simultaneously with the compatible structures of a group and an analytic manifold.

We do as if mathematical objects could be characterized independently from theoretical contexts and methods used by mathematicians in order to study them.

introduction ﬁrst part second part third part / conclusion Mathematical objects / mathematical domains

We may say that Weyl combines different mathematical domains because of the nature of continuous groups, that is Lie groups. - It seems rather obvious that these objects represent a confluence of algebra, topology and geometry. - But, this observation does not directly lead to effective methods in order to study these objects.

We do as if mathematical objects could be characterized independently from theoretical contexts and methods used by mathematicians in order to study them.

introduction ﬁrst part second part third part / conclusion Mathematical objects / mathematical domains

We may say that Weyl combines diﬀerent mathematical domains because of the nature of continuous groups, that is Lie groups. Indeed, a Lie group is characterized by a mixed structure. It can be deﬁned as a set endowed simultaneously with the compatible structures of a group and an analytic manifold. We do as if mathematical objects could be characterized independently from theoretical contexts and methods used by mathematicians in order to study them.

introduction ﬁrst part second part third part / conclusion Mathematical objects / mathematical domains

- It seems rather obvious that these objects represent a confluence of algebra, topology and geometry. - But, this observation does not directly lead to effective methods in order to study these objects. introduction first part second part third part / conclusion Mathematical objects / mathematical domains

We do as if mathematical objects could be characterized independently from theoretical contexts and methods used by mathematicians in order to study them. In fact, he applies these terms to methods. - Lie groups can be studied algebraically, by describing the Lie algebras which are associated to them, - They can also be studied topologically, by using Hurwitz’s integral method. The uniﬁcation of "algebra" and "topology" which appears in Weyl’s paper is not only due to the nature of Lie groups, but also to a synthesis between diﬀerent viewpoints on Lie groups.

introduction ﬁrst part second part third part / conclusion Mathematical objects / mathematical domains

In his paper (1925), Weyl doesn’t use the terms "algebra" or analysis situs to classify mathematical objects according to their structural properties. The uniﬁcation of "algebra" and "topology" which appears in Weyl’s paper is not only due to the nature of Lie groups, but also to a synthesis between diﬀerent viewpoints on Lie groups.

introduction ﬁrst part second part third part / conclusion Mathematical objects / mathematical domains

In his paper (1925), Weyl doesn’t use the terms "algebra" or analysis situs to classify mathematical objects according to their structural properties. In fact, he applies these terms to methods. - Lie groups can be studied algebraically, by describing the Lie algebras which are associated to them, - They can also be studied topologically, by using Hurwitz’s integral method. The unification of "algebra" and "topology" which appears in Weyl’s paper is not only due to the nature of Lie groups, but also to a synthesis between different viewpoints on Lie groups. (ii) he underlines the specificity of topological methods and he criticizes a so-called hegemony of abstract algebra, (iii) for him, algebra can’t be considered as a self-sufficient domain. He takes the example of the complete reducibility theorem in the case of semi-simple groups.

introduction ﬁrst part second part third part / conclusion Mathematical objects / mathematical domains

In 1931, (i) Weyl draws a parallel between his book on Riemann Surfaces (1913) and his paper on representations of semisimple groups: in these two cases he unifies two distinct methods - in complex analysis, he refers to Weierstrass’s algebraic and local method, based on analytic continuation, and simultaneously to Riemann’s geometrical and topological method, based on the concept of a Riemann Surface. - in representation theory of Lie groups, we have already seen that he uses and he compares systematically Cartan’s and Hurwitz’s method. (ii) he underlines the specificity of topological methods and he criticizes a so-called hegemony of abstract algebra, (iii) for him, algebra can’t be considered as a self-sufficient domain. He takes the example of the complete reducibility theorem in the case of semi-simple groups. (i) How does Weyl realize the relevance of these works? (ii) How does Weyl use these different sources? Weyl is not passively influenced by Cartan, Hurwitz or Schur. - In fact, he combines these references, - he changes the shape of these works for his own goals, - he redefines some mathematical concepts already used by these three protagonists.

introduction ﬁrst part second part third part / conclusion The sources used by Weyl

We shall not merely notice that in 1924 Weyl is acquainted with Cartan’s thesis, Hurwitz’s paper on invariant theory and Schur’s paper on the representations of the orthogonal group. introduction ﬁrst part second part third part / conclusion The sources used by Weyl

We shall not merely notice that in 1924 Weyl is acquainted with Cartan’s thesis, Hurwitz’s paper on invariant theory and Schur’s paper on the representations of the orthogonal group.

(i) How does Weyl realize the relevance of these works? (ii) How does Weyl use these different sources? Weyl is not passively influenced by Cartan, Hurwitz or Schur. - In fact, he combines these references, - he changes the shape of these works for his own goals, - he redefines some mathematical concepts already used by these three protagonists. - By studying Schur’s paper on the representations of SO(n) and O(n), Weyl realizes the importance of Hurwitz’s paper (1897). - There exists a correspondence between Schur and Weyl in 1924. This correspondence will have a great influence on Weyl’s paper.

introduction ﬁrst part second part third part / conclusion The sources used by Weyl

- In 1923, Weyl is impressed by Cartan’s solution of the Raumproblem. This solution is based on the classification of simple Lie algebras (due to Killing and Cartan) and on the theory of semisimple Lie algebras. - In fact, Weyl begins to study precisely Cartan’s works on Lie algebras by developing his own research on the so-called group-theoretic foundation of tensor calculus. - By studying Schur’s paper on the representations of SO(n) and O(n), Weyl realizes the importance of Hurwitz’s paper (1897). - There exists a correspondence between Schur and Weyl in 1924. This correspondence will have a great influence on Weyl’s paper. (i) Burnside’s book on finite groups is a source of inspiration for Speiser and for Weyl. (ii) Weyl and Wigner will refer to Speiser’s book to justify the importance of group theory in physics. (iii) Weyl will mention the third edition of Speiser’s book) in his popular conferences on symmetries (developments on ornemental art).

introduction ﬁrst part second part third part / conclusion The sources used by Weyl

- In 1923, Speiser publishes his famous book Die Theorie der Gruppen von endlicher Ordnung, in which we can find three chapters devoted to representations of finite groups. - Speiser sums up the main results due to Frobenius, Schur, Maschke and Burnside in representation theory, - he underlines the effectiveness of group theory in cristallography. introduction first part second part third part / conclusion The sources used by Weyl

(i) Burnside’s book on ﬁnite groups is a source of inspiration for Speiser and for Weyl. (ii) Weyl and Wigner will refer to Speiser’s book to justify the importance of group theory in physics. (iii) Weyl will mention the third edition of Speiser’s book) in his popular conferences on symmetries (developments on ornemental art). Weyl admits with Brouwer that mathematics is a creation of the mind. He rejects simultaneously - essentialism — mathematical objects don’t have any existence by themselves; - formalism in a literal sense — mathematics is not an arbitrary play with symbols.

introduction ﬁrst part second part third part / conclusion Axiomatic method in Weyl’s article

In 1924, Weyl becomes aware that intuitionism puts too great restrictions on mathematics. According to Brouwer, - The principle of the excluded middle is no longer valid in the case of inﬁnite sets, - a mathematical object exists if and only if it can be constructed (intuitionism is a form of mathematical constructivism), - consequently, existence and noncontradiction are not equivalent. introduction ﬁrst part second part third part / conclusion Axiomatic method in Weyl’s article

Weyl admits with Brouwer that mathematics is a creation of the mind. He rejects simultaneously - essentialism — mathematical objects don’t have any existence by themselves; - formalism in a literal sense — mathematics is not an arbitrary play with symbols. Two diﬃculties: - Weyl’s own views on the foundations of mathematics have changed between 1918 and 1924, - during this period, his mathematical practice is not always in accordance with his conception of mathematics.

introduction ﬁrst part second part third part / conclusion Axiomatic method in Weyl’s article

Weyl rarely practises intuitionistic mathematics. However, the ﬁrst constructive proof of the fundamental theorem of algebra is due to Weyl in "Randbemerkungen zu Hauptproblem der mathematik" (1924). For instance, in Raum, Zeit, Materie (1rst edition 1918, 4th edition 1921) Weyl uses systematically implicit definitions. Weyl doesn’t apply dogmatically the intuitionist credo in his mathematical practice.

introduction ﬁrst part second part third part / conclusion Axiomatic method in Weyl’s article

Weyl rarely practises intuitionistic mathematics. However, the first constructive proof of the fundamental theorem of algebra is due to Weyl in "Randbemerkungen zu Hauptproblem der mathematik" (1924). Two difficulties: - Weyl’s own views on the foundations of mathematics have changed between 1918 and 1924, - during this period, his mathematical practice is not always in accordance with his conception of mathematics. introduction first part second part third part / conclusion Axiomatic method in Weyl’s article

Weyl rarely practises intuitionistic mathematics. However, the ﬁrst constructive proof of the fundamental theorem of algebra is due to Weyl in "Randbemerkungen zu Hauptproblem der mathematik" (1924). Two diﬃculties: - Weyl’s own views on the foundations of mathematics have changed between 1918 and 1924, - during this period, his mathematical practice is not always in accordance with his conception of mathematics.

For instance, in Raum, Zeit, Materie (1rst edition 1918, 4th edition 1921) Weyl uses systematically implicit definitions. Weyl doesn’t apply dogmatically the intuitionist credo in his mathematical practice. (i) In an implicit deﬁnition, we do not specify the nature of mathematical objects which correspond to a concept. (ii) A concept is implicitly deﬁned when it remains uninterpreted.

introduction ﬁrst part second part third part / conclusion Axiomatic method in Weyl’s article

implicit definition or definition by axioms: - a mathematical concept is to be defined by the fact that it satisfies axioms which must not lead to contradictions, - an implicit definition is independent from any intuitive content. "[Die Methode der impliziten Definition] hat den Vorteil, daß sie die wichtigsten Eigenschaften der zu definierenden Begriffe sogleich an die Spitze stellen kann, während sich diese Eigenschaften bei Zugrundelegung einer eigentlichen Definition vielleicht erst als sehr entfernte Konsequenzen der Definition ergeben würden". [Weyl Über die Definitionen der mathematischen Grundbegriffe, 1910]

introduction ﬁrst part second part third part / conclusion Axiomatic method in Weyl’s article

(i) In an implicit definition, we do not specify the nature of mathematical objects which correspond to a concept. (ii) A concept is implicitly defined when it remains uninterpreted. introduction first part second part third part / conclusion Axiomatic method in Weyl’s article

(i) In an implicit deﬁnition, we do not specify the nature of mathematical objects which correspond to a concept. (ii) A concept is implicitly deﬁned when it remains uninterpreted.

"[Die Methode der impliziten Definition] hat den Vorteil, daß sie die wichtigsten Eigenschaften der zu definierenden Begriffe sogleich an die Spitze stellen kann, während sich diese Eigenschaften bei Zugrundelegung einer eigentlichen Definition vielleicht erst als sehr entfernte Konsequenzen der Definition ergeben würden". [Weyl Über die Definitionen der mathematischen Grundbegriffe, 1910] Some implicit definitions in Weyl’s works: - Die Idee der Riemannschen Fläche (1913), (i) definition of a topological bi-dimensional manifold, (ii) definition of a Riemann surface as a complex analytic manifold of dimension 1, (iii) definition of a covering surface. - Raum, Zeit, Materie (1918 - 1923), (i) definition of a vector space, (ii) definition of an affine space, (iii) definition of an affine connection on a smooth manifold.

introduction ﬁrst part second part third part / conclusion Axiomatic method in Weyl’s article

Already in 1910, Weyl advocates for implicit definitions in very different domains of mathematics. This remains true during the 20’s in his mathematical practice and in his philosophical reflections on mathematics. - Raum, Zeit, Materie (1918 - 1923), (i) definition of a vector space, (ii) definition of an affine space, (iii) definition of an affine connection on a smooth manifold.

introduction ﬁrst part second part third part / conclusion Axiomatic method in Weyl’s article

Already in 1910, Weyl advocates for implicit definitions in very different domains of mathematics. This remains true during the 20’s in his mathematical practice and in his philosophical reflections on mathematics. Some implicit definitions in Weyl’s works: - Die Idee der Riemannschen Fläche (1913), (i) definition of a topological bi-dimensional manifold, (ii) definition of a Riemann surface as a complex analytic manifold of dimension 1, (iii) definition of a covering surface. introduction first part second part third part / conclusion Axiomatic method in Weyl’s article

Already in 1910, Weyl advocates for implicit definitions in very different domains of mathematics. This remains true during the 20’s in his mathematical practice and in his philosophical reflections on mathematics. Some implicit definitions in Weyl’s works: - Die Idee der Riemannschen Fläche (1913), (i) definition of a topological bi-dimensional manifold, (ii) definition of a Riemann surface as a complex analytic manifold of dimension 1, (iii) definition of a covering surface. - Raum, Zeit, Materie (1918 - 1923), (i) definition of a vector space, (ii) definition of an affine space, (iii) definition of an affine connection on a smooth manifold. More generally, in his article on semisimple groups, Weyl uses the axiomatic method in order to define his main concepts (Lie algebras, semisimple groups, etc.).

- Weyl mainly refers to axiomatization in order to formulate implicit deﬁnitions.

introduction ﬁrst part second part third part / conclusion Axiomatic method in Weyl’s article

In his Mathematische Analyse des Raumproblems (1923) [Anhang 8], Weyl defines axiomatically an "abstract" infinitesimal group (i.e. a Lie algebra). - He explicitly refers to the axiomatization of a vector space in RZM as a source of inspiration. - This implicit definition will occur in the first part of his paper on semisimple groups. - Weyl mainly refers to axiomatization in order to formulate implicit definitions.

introduction ﬁrst part second part third part / conclusion Axiomatic method in Weyl’s article

More generally, in his article on semisimple groups, Weyl uses the axiomatic method in order to deﬁne his main concepts (Lie algebras, semisimple groups, etc.). introduction ﬁrst part second part third part / conclusion Axiomatic method in Weyl’s article

More generally, in his article on semisimple groups, Weyl uses the axiomatic method in order to deﬁne his main concepts (Lie algebras, semisimple groups, etc.).

- Weyl mainly refers to axiomatization in order to formulate implicit definitions. According to Weyl, axiomatization can’t be considered by itself as an instrument of discovery. In 1923-1924, Weyl tries to find a compromise between intuitionism and formalism, that is between to kinds of procedure, an axiomatic one and a constructive one. His article on semisimple groups reflects this compromise.

introduction ﬁrst part second part third part / conclusion Axiomatic method in Weyl’s article

Weyl ascribes three functions to axiomatic method: - generalization of mathematical concepts because they are considered abstractly, - clarification of these concepts: we know exactly which axioms they do satisfy, - rigor in the reasoning: we can determine in which level a theorem remains true / is no longer verified. In 1923-1924, Weyl tries to find a compromise between intuitionism and formalism, that is between to kinds of procedure, an axiomatic one and a constructive one. His article on semisimple groups reflects this compromise.

introduction ﬁrst part second part third part / conclusion Axiomatic method in Weyl’s article

According to Weyl, axiomatization can’t be considered by itself as an instrument of discovery. His article on semisimple groups reﬂects this compromise.

introduction ﬁrst part second part third part / conclusion Axiomatic method in Weyl’s article

According to Weyl, axiomatization can’t be considered by itself as an instrument of discovery. In 1923-1924, Weyl tries to ﬁnd a compromise between intuitionism and formalism, that is between to kinds of procedure, an axiomatic one and a constructive one. His article on semisimple groups reﬂects this compromise. Second part: SL(n,C) as a paradigmatic example. - Meaning of a paradigmatic example. - The complete reducibility theorem established in a particular case (SL(n,C)) and then proved in the general case (all semisimple groups). - Weyl’s paper as a compromise between constructive and axiomatic procedures.

introduction ﬁrst part second part third part / conclusion Plan of our presentation

First part: situation of Weyl’s article in the history of Lie groups / representation theory. - three main sources: Cartan, Schur, Hurwitz - the question of unifying two methods, an algebraic one and an integral one, - A "reform" of representation theory in Weyl’s article? introduction ﬁrst part second part third part / conclusion Plan of our presentation

Second part: SL(n,C) as a paradigmatic example. - Meaning of a paradigmatic example. - The complete reducibility theorem established in a particular case (SL(n,C)) and then proved in the general case (all semisimple groups). - Weyl’s paper as a compromise between constructive and axiomatic procedures. - Cartan on the use of analysis situs in the theory of Lie groups, - Wigner, von Neumann and the use of representation theory in quantum mechanics (1926-1928), - Integral methods in the case of topological groups (Haar, von Neumann, Weil, 1933-1940), - The need of an algebraic proof of the complete reducibility theorem (van der Waerden, Casimir, 1935).

In 1931 Weyl refers to this paper in order to criticize a so-called hegemony of abstract algebra at the beginning of the 1930’s. According to Weyl, the representation theory of semisimple groups is fruitful because it is located at the intersection of topology and algebra.

introduction ﬁrst part second part third part / conclusion Plan of our presentation

Third part: Reception of Weyl’s article In 1931 Weyl refers to this paper in order to criticize a so-called hegemony of abstract algebra at the beginning of the 1930’s. According to Weyl, the representation theory of semisimple groups is fruitful because it is located at the intersection of topology and algebra.

introduction ﬁrst part second part third part / conclusion Plan of our presentation

Third part: Reception of Weyl’s article - Cartan on the use of analysis situs in the theory of Lie groups, - Wigner, von Neumann and the use of representation theory in quantum mechanics (1926-1928), - Integral methods in the case of topological groups (Haar, von Neumann, Weil, 1933-1940), - The need of an algebraic proof of the complete reducibility theorem (van der Waerden, Casimir, 1935). introduction ﬁrst part second part third part / conclusion Plan of our presentation

ﬁrst part In a paragraph entitled "Gruppentheoretische Grundlagen der Lieschen Theorie", Weyl applies Lie’s theory of continuous groups to the group of fractional linear transformations acting on the Riemann Sphere, that is transformations f of Cb into itself which verify: az b f (z) + , ad bc 0. = cz d − 6= + - In this course and in his articles on the Raumproblem, Weyl mainly refers to Lie’s and Scheffers’s Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen (1893).

introduction ﬁrst part second part third part / conclusion Lie groups in Weyl’s work before 1925

Weyl doesn’t discover suddenly the theory of Lie groups in 1921 by resolving the Raumproblem. He already mentions continuous groups in his course on functions of a complex variable at Göttingen (1910-1911): - In this course and in his articles on the Raumproblem, Weyl mainly refers to Lie’s and Scheffers’s Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen (1893).

introduction ﬁrst part second part third part / conclusion Lie groups in Weyl’s work before 1925

Weyl doesn’t discover suddenly the theory of Lie groups in 1921 by resolving the Raumproblem. He already mentions continuous groups in his course on functions of a complex variable at Göttingen (1910-1911): In a paragraph entitled "Gruppentheoretische Grundlagen der Lieschen Theorie", Weyl applies Lie’s theory of continuous groups to the group of fractional linear transformations acting on the Riemann Sphere, that is transformations f of Cb into itself which verify: az b f (z) + , ad bc 0. = cz d − 6= + - In this course and in his articles on the Raumproblem, Weyl mainly refers to Lie’s and Scheffers’s Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen (1893). (iii) There exists a unique aﬃne connection compatible with their metric structure. (iv) To prove this assumption, Weyl rewrites his problem in the theoretical framework of (linear) Lie groups and Lie algebras. In other words, he solves this problem by using group-theoretical methods.

introduction ﬁrst part second part third part / conclusion Lie groups in Weyl’s work before 1925

Weyl’s problem of space:

introduction ﬁrst part second part third part / conclusion Lie groups in Weyl’s work before 1925

Weyl’s problem of space:

(i) Weyl builds up a first version of the problem of space in his commentary to Riemann’s Habilitationsvortrag, (ii) Weyl’s problem of space (second version 1921-1923) consists in the characterization of the so-called "infinitesimally pythagorean manifolds" (differentiable manifolds with a metric structure defined by a non-degenerate quadratic form). (iii) There exists a unique affine connection compatible with their metric structure. (iv) To prove this assumption, Weyl rewrites his problem in the theoretical framework of (linear) Lie groups and Lie algebras. In other words, he solves this problem by using group-theoretical methods. (i) Cartan discovers Weyl’s Raumproblem in the french translation of Raum, Zeit, Materie (4th edition, 1921). (ii) Cartan solves directly in general this problem with a great simplicity: "Sur un théorème fondamental de M. H. Weyl dans la théorie de l’espace métrique", CRAS (1922) / Journal de math. pures et appliquées (1923) (iii) Weyl mentions Cartan’s proof in his "mathematische Analyse des Raumproblems" (1923).

We may think that Weyl, impressed by Cartan’s solution of the Raumproblem", starts studying immediately in detail Cartan’s thesis and his papers on continuous groups published in 1913-1914.

introduction ﬁrst part second part third part / conclusion Lie groups in Weyl’s work before 1925

He becomes aware of the eﬀectiveness of Cartan’s works on Lie groups in 1922-1923. We may think that Weyl, impressed by Cartan’s solution of the Raumproblem", starts studying immediately in detail Cartan’s thesis and his papers on continuous groups published in 1913-1914.

introduction ﬁrst part second part third part / conclusion Lie groups in Weyl’s work before 1925

He becomes aware of the eﬀectiveness of Cartan’s works on Lie groups in 1922-1923.

(i) Cartan discovers Weyl’s Raumproblem in the french translation of Raum, Zeit, Materie (4th edition, 1921). (ii) Cartan solves directly in general this problem with a great simplicity: "Sur un théorème fondamental de M. H. Weyl dans la théorie de l’espace métrique", CRAS (1922) / Journal de math. pures et appliquées (1923) (iii) Weyl mentions Cartan’s proof in his "mathematische Analyse des Raumproblems" (1923). introduction ﬁrst part second part third part / conclusion Lie groups in Weyl’s work before 1925

He becomes aware of the eﬀectiveness of Cartan’s works on Lie groups in 1922-1923.

We may think that Weyl, impressed by Cartan’s solution of the Raumproblem", starts studying immediately in detail Cartan’s thesis and his papers on continuous groups published in 1913-1914. "But as far as the space problem was concerned, the extensive detour required by Cartan’s approach was not deemed appropriate by Weyl, who was still fascinated by his own approach. I would suggest that Weyl had not yet found suﬃcient reason to take on the nontrivial task of mastering the details of Cartan’s papers so as to put them to his own use. Eventually he did — and he was perhaps the ﬁrst mathematician to do so — but the motivation to do so seems to have come not from the space problem but from the calculus of tensors".

- Cartan’s papers on continuous groups becomes Weyl’s main reference in the theory of Lie groups only in 1924. - Weyl interprets them in the language of representation theory.

introduction ﬁrst part second part third part / conclusion Lie groups in Weyl’s work before 1925

This hypothesis is quite too simple and it has been rightly criticized by Hawkins: - Cartan’s papers on continuous groups becomes Weyl’s main reference in the theory of Lie groups only in 1924. - Weyl interprets them in the language of representation theory.

introduction ﬁrst part second part third part / conclusion Lie groups in Weyl’s work before 1925

This hypothesis is quite too simple and it has been rightly criticized by Hawkins: "But as far as the space problem was concerned, the extensive detour required by Cartan’s approach was not deemed appropriate by Weyl, who was still fascinated by his own approach. I would suggest that Weyl had not yet found sufficient reason to take on the nontrivial task of mastering the details of Cartan’s papers so as to put them to his own use. Eventually he did — and he was perhaps the first mathematician to do so — but the motivation to do so seems to have come not from the space problem but from the calculus of tensors". introduction first part second part third part / conclusion Lie groups in Weyl’s work before 1925

- Cartan’s papers on continuous groups becomes Weyl’s main reference in the theory of Lie groups only in 1924. - Weyl interprets them in the language of representation theory. A Lie algebra g is a vector space over R or C together with a bracket, i.e. a bilinear map

[ , ]: g g g × → which is skew-symmetric :

[X,Y ] [Y ,X] = − and which satisﬁes the Jacobi Identity

[X,[Y ,Z]] [Y ,[Z,X]] [Z,[X,Y ]] 0. + + =

introduction ﬁrst part second part third part / conclusion Cartan and Weyl

Cartan’s main results: - Classiﬁcation of simple complex Lie algebras (Killing, Cartan), - Cartan’s criterion for semi-simplicity. introduction ﬁrst part second part third part / conclusion Cartan and Weyl

Cartan’s main results: - Classiﬁcation of simple complex Lie algebras (Killing, Cartan), - Cartan’s criterion for semi-simplicity.

A Lie algebra g is a vector space over R or C together with a bracket, i.e. a bilinear map

[ , ]: g g g × → which is skew-symmetric :

[X,Y ] [Y ,X] = − and which satisﬁes the Jacobi Identity

[X,[Y ,Z]] [Y ,[Z,X]] [Z,[X,Y ]] 0. + + = - in RZM, he underlines that the tangent space at a point of a smooth variety admits the structure of a vector space. - in 1923, he axiomatizes the concept of an inﬁnitesimal group by refering to his deﬁnition of a vector space which is formulated at the beginning of RZM.

A representation ρ of a Lie algebra g on a vector space V is a map of Lie algebras ρ : g gl(V ) End(V ) → = i.e. a linear map that preserves the brackets.

introduction ﬁrst part second part third part / conclusion Cartan and Weyl

This deﬁnition of an "inﬁnitesimal group" is due to Weyl. Geometrically, the Lie algebra of a Lie group is the tangent space at the identity to the Lie group.Weyl is aware of this geometrical interpretation: A representation ρ of a Lie algebra g on a vector space V is a map of Lie algebras ρ : g gl(V ) End(V ) → = i.e. a linear map that preserves the brackets.

introduction ﬁrst part second part third part / conclusion Cartan and Weyl

- in RZM, he underlines that the tangent space at a point of a smooth variety admits the structure of a vector space. - in 1923, he axiomatizes the concept of an infinitesimal group by refering to his definition of a vector space which is formulated at the beginning of RZM. introduction first part second part third part / conclusion Cartan and Weyl

A representation ρ of a Lie algebra g on a vector space V is a map of Lie algebras ρ : g gl(V ) End(V ) → = i.e. a linear map that preserves the brackets. (ii) In 1913, he doesn’t use the language of representation theory.

In other words, Weyl interprets Cartan’s papers in his own theoretical framework.

introduction ﬁrst part second part third part / conclusion Cartan and Weyl

(i) In 1894, Cartan doesn’t define axiomatically a Lie algebra. In fact, he constructs explicitly an "infinitesimal group" (Lie algebra) by determining the infinitesimal generators of a continuous group (Lie group). - Cartan focuses his theory on the study of Lie algebras. According to Weyl, this is the reason why Cartan’s point of view is algebraic. - Retrospectively, a Lie algebra is a purely algebraic structure which is very useful to describe the geometry of a Lie group. - In the third part of his own article (1925), Weyl continues Cartan’s "algebraic" approach. But he also ascribes great importance to the topological properties of a Lie group.

introduction ﬁrst part second part third part / conclusion Cartan and Weyl

In other words, Weyl interprets Cartan’s papers in his own theoretical framework. introduction ﬁrst part second part third part / conclusion Cartan and Weyl

In other words, Weyl interprets Cartan’s papers in his own theoretical framework.

- Cartan focuses his theory on the study of Lie algebras. According to Weyl, this is the reason why Cartan’s point of view is algebraic. - Retrospectively, a Lie algebra is a purely algebraic structure which is very useful to describe the geometry of a Lie group. - In the third part of his own article (1925), Weyl continues Cartan’s "algebraic" approach. But he also ascribes great importance to the topological properties of a Lie group. Cartan’s dissertation becomes a key-reference in the theory of inﬁnitesimal groups at the end of the nineteenth century. - cf. the Enzyklopädie der mathematischen Wissenschaften, II. 1. article by Maurer and Burckhardt on continuous groups of transformations.

introduction ﬁrst part second part third part / conclusion Cartan and Weyl

In his doctoral thesis (1894), Cartan clarifies Killing’s reasoning and he gives a synthetic presentation of Killing’s classification of complex simple Lie algebras. Cartan, notice 1931 : "Les résultats les plus saillants [de Killing] sont relatifs à la détermination de toutes les structures possibles de groupes simples : en dehors des quatre grandes classes de groupes simples trouvées par S. Lie, il n’y a que cinq structures possibles de groupes simples, qui ont respectivement 14, 52, 78, 133 et 248 paramètres". Cartan’s dissertation becomes a key-reference in the theory of infinitesimal groups at the end of the nineteenth century. - cf. the Enzyklopädie der mathematischen Wissenschaften, II. 1. article by Maurer and Burckhardt on continuous groups of transformations. In his doctoral dissertation, Cartan reformulates Killing’s classification of complex simple Lie algebras. - Lie was already acquainted with the first four types — the so-called "classical" complex simple Lie algebras. - between 1886 and 1890, Killing discovered progressively the five other types — the so-called "exceptional" simple Lie algebras — abstractly, i.e. without having in mind any concrete realization of these algebras.

introduction ﬁrst part second part third part / conclusion Cartan and Weyl

(i)A Lie subalgebra h g of a Lie algebra g is an ideal if ⊂ [X,Y ] h for all X h, Y g, ∈ ∈ ∈ (ii) A Lie algebra g is simple if dim g 2 and g contains no ≥ nontrivial ideals, (iii) A Lie algebra is semisimple if it has no nonzero solvable ideals. A semisimple Lie algebra can be written as the direct sum of simple Lie algebras. introduction ﬁrst part second part third part / conclusion Cartan and Weyl

In his doctoral dissertation, Cartan reformulates Killing’s classification of complex simple Lie algebras. - Lie was already acquainted with the first four types — the so-called "classical" complex simple Lie algebras. - between 1886 and 1890, Killing discovered progressively the five other types — the so-called "exceptional" simple Lie algebras — abstractly, i.e. without having in mind any concrete realization of these algebras. Weyl organizes the first and the second part of his paper according to this classification: "Begonnen wurde die Untersuchung der Struktur der halb-einfachen kontinuierlichen Gruppen endlicher Parameterzahl von Killing ; aber erst Cartan gelang in seiner Thèse (Paris, 1894) ein einwandfreier Aufbau. Das resultat war eine vollständige Tabelle aller abstrakten einfachen Gruppen (deren direkte Produkte die halb-einfachen sind) : zu den drei großen Klassen der Gruppen g [sl(n,C)], c [sp(2n,C)], d [so(n)] deren Darstellungstheorie in den beiden vorigen Kapiteln entwickelt wurde, traten nur noch weitere fünf einzelne Gruppen hinzu".

introduction ﬁrst part second part third part / conclusion the four classes of classical simple Lie algebras

- For l 1, Al sl(l 1) ≥ = + - For l 2, Bl so(2l 1) ≥ = + - For l 3, Cl sp(2l) ≥ = - For l 4, Dl so(2l) ≥ = "Begonnen wurde die Untersuchung der Struktur der halb-einfachen kontinuierlichen Gruppen endlicher Parameterzahl von Killing ; aber erst Cartan gelang in seiner Thèse (Paris, 1894) ein einwandfreier Aufbau. Das resultat war eine vollständige Tabelle aller abstrakten einfachen Gruppen (deren direkte Produkte die halb-einfachen sind) : zu den drei großen Klassen der Gruppen g [sl(n,C)], c [sp(2n,C)], d [so(n)] deren Darstellungstheorie in den beiden vorigen Kapiteln entwickelt wurde, traten nur noch weitere fünf einzelne Gruppen hinzu".

introduction ﬁrst part second part third part / conclusion the four classes of classical simple Lie algebras

- For l 1, Al sl(l 1) ≥ = + - For l 2, Bl so(2l 1) ≥ = + - For l 3, Cl sp(2l) ≥ = - For l 4, Dl so(2l) ≥ = Weyl organizes the first and the second part of his paper according to this classification: introduction first part second part third part / conclusion the four classes of classical simple Lie algebras

- For l 1, Al sl(l 1) ≥ = + - For l 2, Bl so(2l 1) ≥ = + - For l 3, Cl sp(2l) ≥ = - For l 4, Dl so(2l) ≥ = Weyl organizes the first and the second part of his paper according to this classification: "Begonnen wurde die Untersuchung der Struktur der halb-einfachen kontinuierlichen Gruppen endlicher Parameterzahl von Killing ; aber erst Cartan gelang in seiner Thèse (Paris, 1894) ein einwandfreier Aufbau. Das resultat war eine vollständige Tabelle aller abstrakten einfachen Gruppen (deren direkte Produkte die halb-einfachen sind) : zu den drei großen Klassen der Gruppen g [sl(n,C)], c [sp(2n,C)], d [so(n)] deren Darstellungstheorie in den beiden vorigen Kapiteln entwickelt wurde, traten nur noch weitere fünf einzelne Gruppen hinzu". Cartan’s criterion is based on the analysis of the characteristic equation ∆(ω) det[adX ωIdg] 0. = − = Pr Let X1, X2, ..., Xr be a basis of g, and X ei Xi a fixed = i 1 element of g, the caracteristic equation ∆(ω) can= be written as follows : r r r 1 r 2 r ∆(ω) ( 1) [ω ψ1(e)ω − ψ2(e)ω − ... ( 1) ψr (e)] = − − + − + − where the coefficients ψi (e) are homogeneous polynomials in e1, e2, ..., er of degree i. Cartan’s criterion for semisimplicity: g is semisimple if and only if ψ2 is non degenerate.

introduction ﬁrst part second part third part / conclusion Cartan’s criterion for semisimplicity

Let g be a Lie algebra of finite dimension r and X a fixed element in g, the adjoint mapping adX : g gl(g) can be defined as follows: → adX (Y ) [X,Y ] for all Y g. = ∈ Pr Let X1, X2, ..., Xr be a basis of g, and X ei Xi a fixed = i 1 element of g, the caracteristic equation ∆(ω) can= be written as follows : r r r 1 r 2 r ∆(ω) ( 1) [ω ψ1(e)ω − ψ2(e)ω − ... ( 1) ψr (e)] = − − + − + − where the coefficients ψi (e) are homogeneous polynomials in e1, e2, ..., er of degree i. Cartan’s criterion for semisimplicity: g is semisimple if and only if ψ2 is non degenerate.

introduction ﬁrst part second part third part / conclusion Cartan’s criterion for semisimplicity

Let g be a Lie algebra of finite dimension r and X a fixed element in g, the adjoint mapping adX : g gl(g) can be defined as follows: → adX (Y ) [X,Y ] for all Y g. = ∈ Cartan’s criterion is based on the analysis of the characteristic equation ∆(ω) det[adX ωIdg] 0. = − = Cartan’s criterion for semisimplicity: g is semisimple if and only if ψ2 is non degenerate.

introduction ﬁrst part second part third part / conclusion Cartan’s criterion for semisimplicity

Let g be a Lie algebra of finite dimension r and X a fixed element in g, the adjoint mapping adX : g gl(g) can be defined as follows: → adX (Y ) [X,Y ] for all Y g. = ∈ Cartan’s criterion is based on the analysis of the characteristic equation ∆(ω) det[adX ωIdg] 0. = − = Pr Let X1, X2, ..., Xr be a basis of g, and X ei Xi a fixed = i 1 element of g, the caracteristic equation ∆(ω) can= be written as follows : r r r 1 r 2 r ∆(ω) ( 1) [ω ψ1(e)ω − ψ2(e)ω − ... ( 1) ψr (e)] = − − + − + − where the coefficients ψi (e) are homogeneous polynomials in e1, e2, ..., er of degree i. Cartan’s criterion for semisimplicity: g is semisimple if and only if ψ2 is non degenerate. Cartan, thèse: "on dit qu’un groupe est intégrable [résoluble] lorsque l’ordre de ses groupes dérivés successifs va constamment en diminuant jusqu’à ce que l’un d’eux se réduise à la transformation identique". n 1 In other words, a Lie algebra g is solvable if D + (g) 0 for some n. = Cartan’s criterion for solvability: g is solvable if and only if ψ2 vanishes on D(g).

introduction ﬁrst part second part third part / conclusion Cartan’s criterion for solvability

- The derived Lie algebra D(g) [g,g] of g is generated by the = set of values [X,Y ], where X,Y g. ∈ - the derived series of g can be formulated as follows: D0(g) g, D1(g) D(g) and, for all i 1, i 1 = i =i ≥ D + (g) [D (g),D (g)]. = Cartan, thèse: "on dit qu’un groupe est intégrable [résoluble] lorsque l’ordre de ses groupes dérivés successifs va constamment en diminuant jusqu’à ce que l’un d’eux se réduise à la transformation identique". n 1 In other words, a Lie algebra g is solvable if D + (g) 0 for some n. = introduction ﬁrst part second part third part / conclusion Cartan’s criterion for solvability

- The derived Lie algebra D(g) [g,g] of g is generated by the = set of values [X,Y ], where X,Y g. ∈ - the derived series of g can be formulated as follows: D0(g) g, D1(g) D(g) and, for all i 1, i 1 = i =i ≥ D + (g) [D (g),D (g)]. = Cartan, thèse: "on dit qu’un groupe est intégrable [résoluble] lorsque l’ordre de ses groupes dérivés successifs va constamment en diminuant jusqu’à ce que l’un d’eux se réduise à la transformation identique". n 1 In other words, a Lie algebra g is solvable if D + (g) 0 for some n. = Cartan’s criterion for solvability: g is solvable if and only if ψ2 vanishes on D(g). Weyl sums up these two main results in the third part, § 3 of his paper which is entitled: "Cartans Kriterium für die auﬂösbaren und die halb-einfachen Gruppen".

introduction ﬁrst part second part third part / conclusion Cartan’s criterion for solvability

Cartan: "Il suffit (...) que le groupe dérivé d’un groupe G d’ordre r annule identiquement le coefficient ψ2(e) de l’équation caractéristique de G pour que G soit intégrable [résoluble]". (this condition is also necessary: see Cartan, Thesis, chapter III, theorem V.) Weyl doesn’t base his reasoning on the coefficient ψ2(e) but on the Killing form : K(X,X) Tr(adX adX ), for all X g. The = ◦ 2 ∈ Killing form is related to ψ2(e) by K [ψ1(e)] 2ψ2(e): = − "Von besonderer Wichtigkeit wird für uns die quadratische Form 2 ψ2(t) oder die Spur ϕ(t) der Transformation T ; es ist 2 ϕ(t) ψ 2ψ2". = 1 −

introduction ﬁrst part second part third part / conclusion Cartan’s criterion for solvability

Cartan: "Il suffit (...) que le groupe dérivé d’un groupe G d’ordre r annule identiquement le coefficient ψ2(e) de l’équation caractéristique de G pour que G soit intégrable [résoluble]". (this condition is also necessary: see Cartan, Thesis, chapter III, theorem V.) Weyl sums up these two main results in the third part, § 3 of his paper which is entitled: "Cartans Kriterium für die auflösbaren und die halb-einfachen Gruppen". introduction first part second part third part / conclusion Cartan’s criterion for solvability

Cartan: "Il suffit (...) que le groupe dérivé d’un groupe G d’ordre r annule identiquement le coefficient ψ2(e) de l’équation caractéristique de G pour que G soit intégrable [résoluble]". (this condition is also necessary: see Cartan, Thesis, chapter III, theorem V.) Weyl sums up these two main results in the third part, § 3 of his paper which is entitled: "Cartans Kriterium für die auflösbaren und die halb-einfachen Gruppen".

Weyl doesn’t base his reasoning on the coeﬃcient ψ2(e) but on the Killing form : K(X,X) Tr(adX adX ), for all X g. The = ◦ 2 ∈ Killing form is related to ψ2(e) by K [ψ1(e)] 2ψ2(e): = − "Von besonderer Wichtigkeit wird für uns die quadratische Form 2 ψ2(t) oder die Spur ϕ(t) der Transformation T ; es ist 2 ϕ(t) ψ 2ψ2". = 1 − - Cartan describes semisimple Lie algebras by analyzing their simple elements (a semisimple Lie algebra consists of simple Lie algebras). - Weyl develops the theory of semisimple complex Lie algebras for itself, without decomposing them into their simple parts.

introduction ﬁrst part second part third part / conclusion Diﬀerences between Cartan and Weyl

- For Weyl, the representation theory of Lie groups must be located at the intersection of algebra and topology. - The "algebraic tradition" embodied by Cartan must be completed by an integral method based on the study of the topological properties of a Lie group. "Cartans methode besteht also in einer direkten algebraischen Konstruktion; sie baut die Darstellungen von unten her durch Komposition auf. Die halb-eifachen Gruppen werden aus den einfachen zusammengesetzt".

introduction ﬁrst part second part third part / conclusion Diﬀerences between Cartan and Weyl

- For Weyl, the representation theory of Lie groups must be located at the intersection of algebra and topology. - The "algebraic tradition" embodied by Cartan must be completed by an integral method based on the study of the topological properties of a Lie group. - Cartan describes semisimple Lie algebras by analyzing their simple elements (a semisimple Lie algebra consists of simple Lie algebras). - Weyl develops the theory of semisimple complex Lie algebras for itself, without decomposing them into their simple parts. introduction ﬁrst part second part third part / conclusion Diﬀerences between Cartan and Weyl

- For Weyl, the representation theory of Lie groups must be located at the intersection of algebra and topology. - The "algebraic tradition" embodied by Cartan must be completed by an integral method based on the study of the topological properties of a Lie group. - Cartan describes semisimple Lie algebras by analyzing their simple elements (a semisimple Lie algebra consists of simple Lie algebras). - Weyl develops the theory of semisimple complex Lie algebras for itself, without decomposing them into their simple parts.

"Cartans methode besteht also in einer direkten algebraischen Konstruktion; sie baut die Darstellungen von unten her durch Komposition auf. Die halb-eifachen Gruppen werden aus den einfachen zusammengesetzt". Let us suppose that P is a polynomial on Cn, P is said to be an absolute invariant if it remains unchanged under the action of 1 n SL(n,C), i.e. P(g − x) P(x), for all g SL(n,C) and x C . · = ∈ ∈ In 1897, Hurwitz publishes "Über die Erzeugung der Invarianten durch Integration". His motivation is to propose a new method, based on integration over continuous groups, in order to construct all the invariants under SO(n,C) and SL(n,C).

introduction ﬁrst part second part third part / conclusion Hurwitz’s "unitarian trick"

Roughly speaking, invariant theory consists in studying homogeneous polynomials of some degree in various indeterminates that remain invariant under linear changes of the indeterminates. In 1897, Hurwitz publishes "Über die Erzeugung der Invarianten durch Integration". His motivation is to propose a new method, based on integration over continuous groups, in order to construct all the invariants under SO(n,C) and SL(n,C).

introduction ﬁrst part second part third part / conclusion Hurwitz’s "unitarian trick"

Roughly speaking, invariant theory consists in studying homogeneous polynomials of some degree in various indeterminates that remain invariant under linear changes of the indeterminates. Let us suppose that P is a polynomial on Cn, P is said to be an absolute invariant if it remains unchanged under the action of 1 n SL(n,C), i.e. P(g − x) P(x), for all g SL(n,C) and x C . · = ∈ ∈ - To this end, Hurwitz reasons by analogy: he sums up a well-known construction of invariants for a ﬁnite linear group. He then sketches an analogue of this construction in the case of certain continuous groups.

introduction ﬁrst part second part third part / conclusion Hurwitz’s "unitarian trick"

Roughly speaking, invariant theory consists in studying homogeneous polynomials of some degree in various indeterminates that remain invariant under linear changes of the indeterminates. Let us suppose that P is a polynomial on Cn, P is said to be an absolute invariant if it remains unchanged under the action of 1 n SL(n,C), i.e. P(g − x) P(x), for all g SL(n,C) and x C . · = ∈ ∈ In 1897, Hurwitz publishes "Über die Erzeugung der Invarianten durch Integration". His motivation is to propose a new method, based on integration over continuous groups, in order to construct all the invariants under SO(n,C) and SL(n,C). introduction ﬁrst part second part third part / conclusion Hurwitz’s "unitarian trick"

Roughly speaking, invariant theory consists in studying homogeneous polynomials of some degree in various indeterminates that remain invariant under linear changes of the indeterminates. Let us suppose that P is a polynomial on Cn, P is said to be an absolute invariant if it remains unchanged under the action of 1 n SL(n,C), i.e. P(g − x) P(x), for all g SL(n,C) and x C . · = ∈ ∈ In 1897, Hurwitz publishes "Über die Erzeugung der Invarianten durch Integration". His motivation is to propose a new method, based on integration over continuous groups, in order to construct all the invariants under SO(n,C) and SL(n,C).

- To this end, Hurwitz reasons by analogy: he sums up a well-known construction of invariants for a ﬁnite linear group. He then sketches an analogue of this construction in the case of certain continuous groups. If G is continuous, Hurwitz’s idea is to replace summation by integration over G, after having constructed a parametrization of G. ½µ a b¶ ¯ ¾ Example : SU(2) ¯ a,b C, a 2 b 2 1 , with = b a ¯ ∈ | | + | | = − a x1 ix2, b x3 ix4. The group SU(2) can be parametrized by = + = + the Euler angles ϕ, ψ, θ ( ϕ , ψ π/2, θ [0,2π]) | | | | ≤ ∈ x1 cosψ cosϕ cosθ x3 cosψ sinϕ = · · = · x2 cosψ cosϕ sinθ x4 sinψ = · · =

introduction ﬁrst part second part third part / conclusion Hurwitz’s "unitarian trick"

If G is a ﬁnite linear group of order N acting on Cn and P a polynomial on Cn, then

1 X 1 n P∗(x) P(g − x), x C = N · g G · ∈ ∈ is invariant under G. Hurwitz recalls this construction of invariants at the beginning of his paper. introduction ﬁrst part second part third part / conclusion Hurwitz’s "unitarian trick"

If G is a ﬁnite linear group of order N acting on Cn and P a polynomial on Cn, then

1 X 1 n P∗(x) P(g − x), x C = N · g G · ∈ ∈ is invariant under G. Hurwitz recalls this construction of invariants at the beginning of his paper. If G is continuous, Hurwitz’s idea is to replace summation by integration over G, after having constructed a parametrization of G. ½µ a b¶ ¯ ¾ Example : SU(2) ¯ a,b C, a 2 b 2 1 , with = b a ¯ ∈ | | + | | = − a x1 ix2, b x3 ix4. The group SU(2) can be parametrized by = + = + the Euler angles ϕ, ψ, θ ( ϕ , ψ π/2, θ [0,2π]) | | | | ≤ ∈ x1 cosψ cosϕ cosθ x3 cosψ sinϕ = · · = · x2 cosψ cosϕ sinθ x4 sinψ = · · = The polynomial P∗ is invariant under SU(2). In fact, Hurwitz tries to construct absolute invariants which remain unchanged under the action of SL(n,C). - It is not possible to apply directly his method to SL(n,C). - This is due to the fact that SL(n,C) is not compact: the integrals to be considered will not converge.

He uses the "unitarian trick" i.e. the restriction to SU(n) in order to avoid this diﬃculty. We can construct the absolute invariants by restricting ourselves to this compact subgroup of SL(n,C).

introduction ﬁrst part second part third part / conclusion Hurwitz’s "unitarian trick"

Let P be a polynomial on Cn. By using Hurwitz’s integral method, we can construct the polynomial Z 1 1 P∗ P(g − x)dv = 8π SU2 · with x Cn and dv cosψ cosϕ dψ dϕ dθ. ∈ = · · · · In fact, Hurwitz tries to construct absolute invariants which remain unchanged under the action of SL(n,C). - It is not possible to apply directly his method to SL(n,C). - This is due to the fact that SL(n,C) is not compact: the integrals to be considered will not converge.

He uses the "unitarian trick" i.e. the restriction to SU(n) in order to avoid this diﬃculty. We can construct the absolute invariants by restricting ourselves to this compact subgroup of SL(n,C).

introduction ﬁrst part second part third part / conclusion Hurwitz’s "unitarian trick"

Let P be a polynomial on Cn. By using Hurwitz’s integral method, we can construct the polynomial Z 1 1 P∗ P(g − x)dv = 8π SU2 · with x Cn and dv cosψ cosϕ dψ dϕ dθ. ∈ = · · · · The polynomial P∗ is invariant under SU(2). He uses the "unitarian trick" i.e. the restriction to SU(n) in order to avoid this diﬃculty. We can construct the absolute invariants by restricting ourselves to this compact subgroup of SL(n,C).

introduction ﬁrst part second part third part / conclusion Hurwitz’s "unitarian trick"

Let P be a polynomial on Cn. By using Hurwitz’s integral method, we can construct the polynomial Z 1 1 P∗ P(g − x)dv = 8π SU2 · with x Cn and dv cosψ cosϕ dψ dϕ dθ. ∈ = · · · · The polynomial P∗ is invariant under SU(2). In fact, Hurwitz tries to construct absolute invariants which remain unchanged under the action of SL(n,C). - It is not possible to apply directly his method to SL(n,C). - This is due to the fact that SL(n,C) is not compact: the integrals to be considered will not converge. introduction ﬁrst part second part third part / conclusion Hurwitz’s "unitarian trick"

Let P be a polynomial on Cn. By using Hurwitz’s integral method, we can construct the polynomial Z 1 1 P∗ P(g − x)dv = 8π SU2 · with x Cn and dv cosψ cosϕ dψ dϕ dθ. ∈ = · · · · The polynomial P∗ is invariant under SU(2). In fact, Hurwitz tries to construct absolute invariants which remain unchanged under the action of SL(n,C). - It is not possible to apply directly his method to SL(n,C). - This is due to the fact that SL(n,C) is not compact: the integrals to be considered will not converge.

He uses the "unitarian trick" i.e. the restriction to SU(n) in order to avoid this diﬃculty. We can construct the absolute invariants by restricting ourselves to this compact subgroup of SL(n,C). - in 1923-1924, Schur interprets Hurwitz’s method in the framework of representation theory of continuous groups. - Schur studies mainly the representations of SO(n) and O(n). - Weyl realizes the fruitfulness of this method by reading Schur’s papers on the representations of the orthogonal group. - Weyl uses Hurwitz’s "unitarian trick" with full generality in his proof of the complete reducibility theorem for semisimple groups.

introduction ﬁrst part second part third part / conclusion Hurwitz’s "unitarian trick"

At the end of his article, Hurwitz intends to generalize his method to all continuous groups with a ﬁnite number of parameters. Although Hurwitz is Weyl’s colleague at the ETH, Weyl becomes aware of the importance of Hurwitz’s methods (in representation theory) thanks to Schur. introduction ﬁrst part second part third part / conclusion Hurwitz’s "unitarian trick"

- in 1923-1924, Schur interprets Hurwitz’s method in the framework of representation theory of continuous groups. - Schur studies mainly the representations of SO(n) and O(n). - Weyl realizes the fruitfulness of this method by reading Schur’s papers on the representations of the orthogonal group. - Weyl uses Hurwitz’s "unitarian trick" with full generality in his proof of the complete reducibility theorem for semisimple groups. This is the reason why Weyl relates this method to topology or analysis situs.

- Hurwitz applies this method in the framework of a purely algebraic theory, namely invariant theory. - Schur uses this method in order to construct a theory of characters in the case of continuous groups. Weyl generalizes this project. - The "unitarian trick", which is the key in Weyl’s proof of the complete reducibility theorem, is based on topological considerations.

In 1931, Weyl will mention all these facts in order to show that "algebra" is not self-suﬃcient.

introduction ﬁrst part second part third part / conclusion Hurwitz’s "unitarian trick"

Hurwitz’s integral method is only possible if a continuous group satisﬁes a topological condition : it must be compact. - Hurwitz applies this method in the framework of a purely algebraic theory, namely invariant theory. - Schur uses this method in order to construct a theory of characters in the case of continuous groups. Weyl generalizes this project. - The "unitarian trick", which is the key in Weyl’s proof of the complete reducibility theorem, is based on topological considerations.

In 1931, Weyl will mention all these facts in order to show that "algebra" is not self-suﬃcient.

introduction ﬁrst part second part third part / conclusion Hurwitz’s "unitarian trick"

Hurwitz’s integral method is only possible if a continuous group satisﬁes a topological condition : it must be compact. This is the reason why Weyl relates this method to topology or analysis situs. In 1931, Weyl will mention all these facts in order to show that "algebra" is not self-suﬃcient.

introduction ﬁrst part second part third part / conclusion Hurwitz’s "unitarian trick"

In 1931, Weyl will mention all these facts in order to show that "algebra" is not self-suﬃcient. In 1925-1926, Weyl creates "for semisimple groups a continuous analogue of Frobenius’s theory" (Hawkins). In other words, Frobenius’s theory is a central reference in Weyl’s paper.

introduction ﬁrst part second part third part / conclusion Frobenius and Schur

Frobenius made great achievements in the representation theory of finite groups at the turn of the twentieth century: - definition of a character in the case of abelian and non-abelian finite groups, - definition of a matrix representation, - proof of the complete reducibility theorem (independently from Maschke and Molien) - proof of the regular representation theorem, - representation theory of the symmetric group. introduction first part second part third part / conclusion Frobenius and Schur

In 1925-1926, Weyl creates "for semisimple groups a continuous analogue of Frobenius’s theory" (Hawkins). In other words, Frobenius’s theory is a central reference in Weyl’s paper. (a) Representation theory of ﬁnite groups is well-known at the ETH in the 1920s. - cf. Speiser’s monography on ﬁnite groups, which contains the main results of Frobenius’s theory, is edited in 1923, - nevertheless, Weyl has undoubtly also studied Frobenius’s papers on representations of the symmetric group.

introduction ﬁrst part second part third part / conclusion Frobenius and Schur

(a) How does Weyl become aware of Frobenius’ main results? (b) Why does he consider Frobenius’ theory as a "model" for his own research? (b) Weyl’s ﬁrst motivation consists in proving the complete reducibility theorem for semisimple groups. For this task, he just needs Hurwitz’s "unitarian trick". The problem of developing a continuous analogue of Frobenius’s theory has been suggested to Weyl by Schur.

introduction ﬁrst part second part third part / conclusion Frobenius and Schur

(a) How does Weyl become aware of Frobenius’ main results? (b) Why does he consider Frobenius’ theory as a "model" for his own research?

(a) Representation theory of finite groups is well-known at the ETH in the 1920s. - cf. Speiser’s monography on finite groups, which contains the main results of Frobenius’s theory, is edited in 1923, - nevertheless, Weyl has undoubtly also studied Frobenius’s papers on representations of the symmetric group. introduction first part second part third part / conclusion Frobenius and Schur

(a) How does Weyl become aware of Frobenius’ main results? (b) Why does he consider Frobenius’ theory as a "model" for his own research?

(b) Weyl’s ﬁrst motivation consists in proving the complete reducibility theorem for semisimple groups. For this task, he just needs Hurwitz’s "unitarian trick". The problem of developing a continuous analogue of Frobenius’s theory has been suggested to Weyl by Schur. Weyl underlines the importance of Cartan’s theory of weights which Schur hardly knows. On the other hand, Schur advises Weyl to develop a theory of characters for semisimple groups. The exchange of letters between Schur and Weyl has a great impact on the ﬁnal structure of Weyl’s paper which contains - a synthesis on Cartan’s theory of weights - a proof of the complete reducibility theorem - a continuous analogue of Frobenius’s theory

introduction ﬁrst part second part third part / conclusion Frobenius and Schur

- In 1924, Schur formulates the continuous analogue of Frobenius’s theory in one case, the orthogonal group. - Simultaneously we can ﬁnd an important correspondence between Weyl and Schur. introduction ﬁrst part second part third part / conclusion Frobenius and Schur

- In 1924, Schur formulates the continuous analogue of Frobenius’s theory in one case, the orthogonal group. - Simultaneously we can ﬁnd an important correspondence between Weyl and Schur.

Weyl underlines the importance of Cartan’s theory of weights which Schur hardly knows. On the other hand, Schur advises Weyl to develop a theory of characters for semisimple groups. The exchange of letters between Schur and Weyl has a great impact on the ﬁnal structure of Weyl’s paper which contains - a synthesis on Cartan’s theory of weights - a proof of the complete reducibility theorem - a continuous analogue of Frobenius’s theory (c) Weyl doesn’t refer to Frobenius’ papers just in order to construct a theory of characters for continuous groups. - the representations of the symmetric group which have been studied in detail by Frobenius in 1900 and 1903 play a central role in the Schur-Weyl duality, - Weyl tries to ﬁnd the analogue of the regular representation theorem for certain continuous groups, cf. his paper with Peter in 1927.

introduction ﬁrst part second part third part / conclusion Frobenius and Schur

Hawkins : "The focus of Weyl’s attention seems to have been on the complete reducibility theorem and its extension to all semisimple groups (...). Judging by the Schur-Weyl correspondence (...), it would seem that Schur’s reply to Weyl’s communication [on the complete reducibility theorem] inspired — and to a certain extent guided — Weyl’s successful efforts to develop a theory of characters for semisimple groups that would provide formulas for the irreducible characters and the degrees of the corresponding representations". (c) Weyl doesn’t refer to Frobenius’ papers just in order to construct a theory of characters for continuous groups. - the representations of the symmetric group which have been studied in detail by Frobenius in 1900 and 1903 play a central role in the Schur-Weyl duality, - Weyl tries to find the analogue of the regular representation theorem for certain continuous groups, cf. his paper with Peter in 1927. Frobenius’ regular representation theorem consists in analyzing the complete reduction of the regular representation ρR of a finite group. - The regular representation contains all the inequivalent irreducible representations of a finite group,

introduction ﬁrst part second part third part / conclusion Frobenius and Schur

Let G be a finite group of order g and let V be a vector space of dim g with a basis (et )t G indexed by the elements t of G. For ∈ s G, let ρs be the linear map of V into V which sends et to est ; ∈ this defines a linear representation which is called the (left) regular representation of G. Frobenius’ regular representation theorem consists in analyzing the complete reduction of the regular representation ρR of a finite group. - The regular representation contains all the inequivalent irreducible representations of a finite group,

- more precisely, if ρi denotes an irreducible representation of G and µi its degree, then ρi occurs µi times in the complete reduction of the regular representation ρR . Weyl : "[Die reguläre Darstellung] liefert durch ihre Reduktion mit einem Schlage alle irreduziblen Darstellungen". - two different methods: Cartan’s algebraic and infinitesimal method; Hurwitz’s integral and topological method; - two different theories: Cartan’s theory of weights; Frobenius’s theory of characters; - comparison and interplay between the representation theory of finite groups and the representation theory of continuous groups; - reference to different mathematical domains: mainly algebra and topology.

Weyl conceives the unity of mathematics as a connection between diﬀerent kinds of methods, theories and domains. This connection is obtained by resolving a speciﬁc problem (for instance the proof of the complete reducibility theorem).

introduction ﬁrst part second part third part / conclusion A wide synthesis

Weyl’s paper is characterized by its polyphony: Weyl conceives the unity of mathematics as a connection between diﬀerent kinds of methods, theories and domains. This connection is obtained by resolving a speciﬁc problem (for instance the proof of the complete reducibility theorem).

introduction ﬁrst part second part third part / conclusion A wide synthesis

Weyl’s paper is characterized by its polyphony:

- two different methods: Cartan’s algebraic and infinitesimal method; Hurwitz’s integral and topological method; - two different theories: Cartan’s theory of weights; Frobenius’s theory of characters; - comparison and interplay between the representation theory of finite groups and the representation theory of continuous groups; - reference to different mathematical domains: mainly algebra and topology. introduction first part second part third part / conclusion A wide synthesis

Weyl’s paper is characterized by its polyphony:

Weyl (1931) : "In topology we begin with the notion of continuous connection, and in the course of specialization we add, step by step, relevant structural features. In algebra this order is, in a sense, reversed. Algebra views the operations as the beginning of all mathematical thinking and admits continuity, or some algebraic surrogate of continuity, at the last step of the specialization. The two methods follow opposite directions. (...) In the last few years, in the theory of representation of continuous groups by means of linear substitutions, I have experienced most poignantly how diﬃcult it is to serve these two masters at the same time". introduction ﬁrst part second part third part / conclusion

second part (i) In 1924-1925 he focuses his research on a so-called "group-theoretic foundation of tensor calculus". Precisely, tensor calculus plays a central role in the mathematization of general theory of relativity. There is a continuity between - Weyl’s works on general theory of relativity (for instance Raum, Zeit, Materie) - his research on Lie groups. (ii) In Die Idee der Riemannschen Fläche (1913), Weyl constructs systematically the theory of covering spaces in order to clarify Koebe’s proof of the uniformization theorem. In 1925, Weyl reformulates this theory in the context of Lie groups so as to prove the complete reducibility theorem.

introduction ﬁrst part second part third part / conclusion Riemann’s geometric ideas and group theory

- We have already shown that Weyl doesn’t discover suddenly the theory of Lie groups in 1921. - Conversely, his paper on semisimple groups is not isolated in his whole work. (ii) In Die Idee der Riemannschen Fläche (1913), Weyl constructs systematically the theory of covering spaces in order to clarify Koebe’s proof of the uniformization theorem. In 1925, Weyl reformulates this theory in the context of Lie groups so as to prove the complete reducibility theorem.

introduction ﬁrst part second part third part / conclusion Riemann’s geometric ideas and group theory

- We have already shown that Weyl doesn’t discover suddenly the theory of Lie groups in 1921. - Conversely, his paper on semisimple groups is not isolated in his whole work.

(i) In 1924-1925 he focuses his research on a so-called "group-theoretic foundation of tensor calculus". Precisely, tensor calculus plays a central role in the mathematization of general theory of relativity. There is a continuity between - Weyl’s works on general theory of relativity (for instance Raum, Zeit, Materie) - his research on Lie groups. (ii) In Die Idee der Riemannschen Fläche (1913), Weyl constructs systematically the theory of covering spaces in order to clarify Koebe’s proof of the uniformization theorem. In 1925, Weyl reformulates this theory in the context of Lie groups so as to prove the complete reducibility theorem. Klein describes is own work [on automorphic functions] as a "Verschmelzung von Galois und Riemann". [quoted by Weyl in "Felix Kleins Stellung in der mathematischen Gegenwart". (1930)]

(i) In his article "Strenge Begründung der Charakteristikentheorie auf zweiseitigen Flächen" (1916), Weyl points out the great analogy between the theory of covering spaces and the Galois theory.

introduction ﬁrst part second part third part / conclusion Riemann’s geometric ideas and group theory

Weyl continues in a certain way Klein’s project which consists in unifying group theory and Riemann’s geometric ideas. (i) In his article "Strenge Begründung der Charakteristikentheorie auf zweiseitigen Flächen" (1916), Weyl points out the great analogy between the theory of covering spaces and the Galois theory.

introduction ﬁrst part second part third part / conclusion Riemann’s geometric ideas and group theory

Weyl continues in a certain way Klein’s project which consists in unifying group theory and Riemann’s geometric ideas. Klein describes is own work [on automorphic functions] as a "Verschmelzung von Galois und Riemann". [quoted by Weyl in "Felix Kleins Stellung in der mathematischen Gegenwart". (1930)] "Eine reguläre (unverzweigte, unbegrenzte) Überlagerungsﬂäche F über F kann man gemäß der in der Zahlenkörpertheorie üblichen Terminologie als eine Galoissche Überlagerungsﬂäche (relativ zu F) bezeichnen und die Gruppe der Decktransformationen als die Galoische Gruppe dieser Fläche (relativ zu F)".

introduction ﬁrst part second part third part / conclusion Riemann’s geometric ideas and group theory

(i) In his article "Strenge Begründung der Charakteristikentheorie auf zweiseitigen Flächen" (1916), Weyl points out the great analogy between the theory of covering spaces and the Galois theory. introduction ﬁrst part second part third part / conclusion Riemann’s geometric ideas and group theory

(i) In his article "Strenge Begründung der Charakteristikentheorie auf zweiseitigen Flächen" (1916), Weyl points out the great analogy between the theory of covering spaces and the Galois theory.

"Eine reguläre (unverzweigte, unbegrenzte) Überlagerungsﬂäche F über F kann man gemäß der in der Zahlenkörpertheorie üblichen Terminologie als eine Galoissche Überlagerungsﬂäche (relativ zu F) bezeichnen und die Gruppe der Decktransformationen als die Galoische Gruppe dieser Fläche (relativ zu F)". (iii) Weyl sums up - his main results in the theory of covering spaces (1913-1916) - his solution to the Raumproblem (1921-1923) in a writing entitled Riemanns geometrische Ideen, ihre Auswirkung und ihre Verknüpfung mit der Gruppentheorie (1925).

introduction ﬁrst part second part third part / conclusion Riemann’s geometric ideas and group theory

(ii) Weyl begins to construct the "Raumproblem" in his comment on Riemann’s Habilitationsvortrag (1919). The theory of linear Lie groups plays a central role in the formulation and the resolution of this problem (1921-1923). In his paper on semisimple groups, he ﬁnds new connections between Riemann’s geometric ideas and group theory: the theory of covering spaces is required to fullﬁl the proof of the complete reducibility theorem.

introduction ﬁrst part second part third part / conclusion Riemann’s geometric ideas and group theory

(iii) Weyl sums up - his main results in the theory of covering spaces (1913-1916) - his solution to the Raumproblem (1921-1923) in a writing entitled Riemanns geometrische Ideen, ihre Auswirkung und ihre Verknüpfung mit der Gruppentheorie (1925). introduction ﬁrst part second part third part / conclusion Riemann’s geometric ideas and group theory

In his paper on semisimple groups, he ﬁnds new connections between Riemann’s geometric ideas and group theory: the theory of covering spaces is required to fullﬁl the proof of the complete reducibility theorem. Which function do they have in the construction of Weyl’s theory? - We may think that they just illustrate implicitly the general theory of semisimple groups. - We could also think that they are isolated cases.

introduction ﬁrst part second part third part / conclusion A paradigmatic example

The status of the examples studied by Weyl in his paper requires a ﬁne analysis. - First part : SL(n,C), - second part : SO(n) and Sp(n,C). introduction ﬁrst part second part third part / conclusion A paradigmatic example

The status of the examples studied by Weyl in his paper requires a ﬁne analysis. - First part : SL(n,C), - second part : SO(n) and Sp(n,C).

Which function do they have in the construction of Weyl’s theory? - We may think that they just illustrate implicitly the general theory of semisimple groups. - We could also think that they are isolated cases.

In fact, the proof of the complete reducibility theorem for SL(n,C) can be generalized to all semisimple Lie groups. In other words, Weyl bases the construction of the general theory of semisimple groups on the study of this particular case. Weyl already sketches his general theory by describing SL(n,C) and its representations. His reasoning has exactly the same structure in the ﬁrst part and in the last two parts of his paper. We may diﬀerentiate three kinds of examples - illustrative examples - paradigmatic examples - problematic examples

introduction ﬁrst part second part third part / conclusion A paradigmatic example

It seems inappropriate to divide Weyl’s article into two main parts: (i) particular cases (part I, part II) (ii) general theory of semisimple Lie groups (part III, part IV). We may diﬀerentiate three kinds of examples - illustrative examples - paradigmatic examples - problematic examples

introduction ﬁrst part second part third part / conclusion A paradigmatic example

It seems inappropriate to divide Weyl’s article into two main parts: (i) particular cases (part I, part II) (ii) general theory of semisimple Lie groups (part III, part IV).

Weyl already sketches his general theory by describing SL(n,C) and its representations. His reasoning has exactly the same structure in the first part and in the last two parts of his paper. We may differentiate three kinds of examples - illustrative examples - paradigmatic examples - problematic examples A paradigmatic example is not merely a particular case : it plays a decisive role in order to construct a general theory. - For instance, SL(n,C) is a paradigmatic example in Weyl’s paper, because it guides him in order to find the proof of the complete reducibility theorem for all semisimple groups.

introduction ﬁrst part second part third part / conclusion A paradigmatic example

An illustrative example is just a particular realization of a general concept. - For instance, when we say "the complex plane is a Riemann surface", the object "complex plane" is just an illustration of a Riemann surface. A problematic example (counterexample) plays a central role in order to determine the conditions for the validity of a theorem. - For instance, Weyl shows that GL(n,C) is not semisimple and it doesn’t satisfy the complete reducibility property.

introduction ﬁrst part second part third part / conclusion A paradigmatic example

A paradigmatic example is not merely a particular case : it plays a decisive role in order to construct a general theory. - For instance, SL(n,C) is a paradigmatic example in Weyl’s paper, because it guides him in order to ﬁnd the proof of the complete reducibility theorem for all semisimple groups. introduction ﬁrst part second part third part / conclusion A paradigmatic example

A problematic example (counterexample) plays a central role in order to determine the conditions for the validity of a theorem. - For instance, Weyl shows that GL(n,C) is not semisimple and it doesn’t satisfy the complete reducibility property. (i) The special linear group SL(n,C) is the group of n n × invertible matrices with complex entries having determinant one. (ii) Its Lie algebra sl(n,C) is the space of all n n complex × matrices with trace zero. (iii) The special unitary group SU(n) is the group of unitary matrices with determinant one. - A unitary matrix A is a n n complex matrix which satisﬁes × the condition A∗A AA∗ In, where In is the identity matrix = = and A∗ is the adjoint of A, i.e. (A∗ ) (Aj i ). i,j = , (iv) Its real Lie algebra su(n) is the space of all n n skew-hermitian matrices with trace zero. ×

- A matrix A is skew-hermitian, if (A∗ ) (Aj i ). i,j = − ,

introduction ﬁrst part second part third part / conclusion

The complete reducibility theorem for SL(n,C)

In the first part (§ 5) of his article, Weyl clarifies the connection between the representations of SL(n,C), sl(n,C), SU(n) and su(n). (iii) The special unitary group SU(n) is the group of unitary matrices with determinant one. - A unitary matrix A is a n n complex matrix which satisfies × the condition A∗A AA∗ In, where In is the identity matrix = = and A∗ is the adjoint of A, i.e. (A∗ ) (Aj i ). i,j = , (iv) Its real Lie algebra su(n) is the space of all n n skew-hermitian matrices with trace zero. ×

- A matrix A is skew-hermitian, if (A∗ ) (Aj i ). i,j = − ,

introduction ﬁrst part second part third part / conclusion

The complete reducibility theorem for SL(n,C)

In the ﬁrst part (§ 5) of his article, Weyl clariﬁes the connection between the representations of SL(n,C), sl(n,C), SU(n) and su(n).

The complete reducibility theorem for SL(n,C)

In the ﬁrst part (§ 5) of his article, Weyl clariﬁes the connection between the representations of SL(n,C), sl(n,C), SU(n) and su(n).

(i) The special linear group SL(n,C) is the group of n n × invertible matrices with complex entries having determinant one. (ii) Its Lie algebra sl(n,C) is the space of all n n complex × matrices with trace zero. (iii) The special unitary group SU(n) is the group of unitary matrices with determinant one. - A unitary matrix A is a n n complex matrix which satisﬁes × the condition A∗A AA∗ In, where In is the identity matrix = = and A∗ is the adjoint of A, i.e. (A∗ ) (Aj i ). i,j = , (iv) Its real Lie algebra su(n) is the space of all n n skew-hermitian matrices with trace zero. ×

- A matrix A is skew-hermitian, if (A∗ ) (Aj i ). i,j = − , A connected matrix Lie group is said to be simply connected if every loop in G can be shrunk continuously to a point in G. - Already in Die Idee der Riemannschen Fläche, Weyl was acquainted with the property of simple connectivity which is all the more important in the formulation of the uniformization theorem (Koebe, Poincaré, 1907). Hurwitz’s integral method can be applied to SU(n) because it is compact. in 1924, Schur proves that SU(n) has the complete reducibility property because of its compactness. - Neither Schur nor Weyl are aware that the complete reducibility theorem is valid for any topological compact group conceived in abstracto. Haar proves this theorem in 1933.

introduction ﬁrst part second part third part / conclusion

The complete reducibility theorem for SL(n,C)

The special unitary group has two topological properties which will play a decisive role in Weyl’s proof : SU(n) is compact and simply connected, for n 2. ≥ Hurwitz’s integral method can be applied to SU(n) because it is compact. in 1924, Schur proves that SU(n) has the complete reducibility property because of its compactness. - Neither Schur nor Weyl are aware that the complete reducibility theorem is valid for any topological compact group conceived in abstracto. Haar proves this theorem in 1933.

introduction ﬁrst part second part third part / conclusion

The complete reducibility theorem for SL(n,C)

The special unitary group has two topological properties which will play a decisive role in Weyl’s proof : SU(n) is compact and simply connected, for n 2. ≥ A connected matrix Lie group is said to be simply connected if every loop in G can be shrunk continuously to a point in G. - Already in Die Idee der Riemannschen Fläche, Weyl was acquainted with the property of simple connectivity which is all the more important in the formulation of the uniformization theorem (Koebe, Poincaré, 1907). Hurwitz’s integral method can be applied to SU(n) because it is compact. in 1924, Schur proves that SU(n) has the complete reducibility property because of its compactness. - Neither Schur nor Weyl are aware that the complete reducibility theorem is valid for any topological compact group conceived in abstracto. Haar proves this theorem in 1933. Weyl has to prove the following implications (i) If SU(n) has the complete reducibility property, then so does su(n), (ii) If su(n) has the complete reducibility property, then so does sl(n,C).

introduction ﬁrst part second part third part / conclusion

The complete reducibility theorem for SL(n,C)

How to deduce that SL(n,C) has the complete reducibility property from the fact that any finite dimensional representation of SU(n) is fully reducible? Such a problem can’t be solved directly, i.e. without referring to the Lie algebras of SL(n,C) and SU(n). He first proves the second implication, which can be called "Weyl’s unitarian trick". He then realizes that the first implication is not self-evident: although a Lie group has the complete reducibility property, its Lie algebra need not have it.

introduction ﬁrst part second part third part / conclusion

The complete reducibility theorem for SL(n,C)

How to deduce that SL(n,C) has the complete reducibility property from the fact that any ﬁnite dimensional representation of SU(n) is fully reducible? Such a problem can’t be solved directly, i.e. without referring to the Lie algebras of SL(n,C) and SU(n). Weyl has to prove the following implications (i) If SU(n) has the complete reducibility property, then so does su(n), (ii) If su(n) has the complete reducibility property, then so does sl(n,C). introduction ﬁrst part second part third part / conclusion

The complete reducibility theorem for SL(n,C)

How to deduce that SL(n,C) has the complete reducibility property from the fact that any ﬁnite dimensional representation of SU(n) is fully reducible? Such a problem can’t be solved directly, i.e. without referring to the Lie algebras of SL(n,C) and SU(n). Weyl has to prove the following implications (i) If SU(n) has the complete reducibility property, then so does su(n), (ii) If su(n) has the complete reducibility property, then so does sl(n,C).

He first proves the second implication, which can be called "Weyl’s unitarian trick". He then realizes that the first implication is not self-evident: although a Lie group has the complete reducibility property, its Lie algebra need not have it. In fact su(n)C and sl(n,C) are isomorphic. In other words, there is a one-to-one correspondence between the irreducible representations of sl(n,C) and the irreducible representations of su(n) (by complexification).

introduction ﬁrst part second part third part / conclusion su(n) / sl(n,C)

Let us recall that su(n) is a real Lie algebra. We shall consider its complexification su(n)C. We have to mention first two general results: - Let g be a finite-dimensional real Lie algebra. The bracket operation on g has a unique extension to gC which is defined by the following relations :

[X,iY ] i[X,Y ] and [iX,iY ] [X,Y ] = = − for all X, Y in g. - There is a one-to-one correspondence between the (complex) representations of g and the complex representations of gC. introduction ﬁrst part second part third part / conclusion su(n) / sl(n,C)

[X,iY ] i[X,Y ] and [iX,iY ] [X,Y ] = = − for all X, Y in g. - There is a one-to-one correspondence between the (complex) representations of g and the complex representations of gC. In fact su(n)C and sl(n,C) are isomorphic. In other words, there is a one-to-one correspondence between the irreducible representations of sl(n,C) and the irreducible representations of su(n) (by complexiﬁcation). two problems : (a) Hawkins : "the irreducible representations of a Lie algebra g associated to a given Lie group G do not necessarily correspond to representations of G but rather to its (...) universal covering group Ge ". (b) The implication "if a Lie group has the compactness property, then so does its universal covering group" is false: the unit circle on the complex plane is compact, its covering group is an inﬁnite spiral which is not compact.

introduction ﬁrst part second part third part / conclusion

SU(n) / su(n)

Let G be a connected Lie group. A universal covering group of G is a connected, simply connected Lie group Ge together with a Lie group homomorphism φ : Ge G with the following properties → - φ maps Ge onto G, - There is a neighborhood U of I in Ge which maps homeomorphically under φ onto a neighborhood V of I in G. two problems : (a) Hawkins : "the irreducible representations of a Lie algebra g associated to a given Lie group G do not necessarily correspond to representations of G but rather to its (...) universal covering group Ge ". (b) The implication "if a Lie group has the compactness property, then so does its universal covering group" is false: the unit circle on the complex plane is compact, its covering group is an inﬁnite spiral which is not compact. - The universal covering group of SU(n) can be identiﬁed with SU(n) itself, - in particular, the universal covering group of SU(n) is compact. For this reason, it has the complete reducibility property.

introduction ﬁrst part second part third part / conclusion

The complete reducibility theorem for SL(n,C)

Weyl proves the following lemma in order to overcome these diﬃculties: the special unitary group is simply connected. Thus, a topological property — i.e. the simple connectivity of SU(n) — plays a central role in Weyl’s proof of an algebraic theorem: the complete reducibility theorem in the case of SL(n,C).

introduction ﬁrst part second part third part / conclusion

The complete reducibility theorem for SL(n,C)

Weyl proves the following lemma in order to overcome these diﬃculties: the special unitary group is simply connected.

- The universal covering group of SU(n) can be identiﬁed with SU(n) itself, - in particular, the universal covering group of SU(n) is compact. For this reason, it has the complete reducibility property.

Weyl can easily conclude that su(n) and sl(n,C) have also the complete reducibility property. introduction ﬁrst part second part third part / conclusion

The complete reducibility theorem for SL(n,C)

Weyl proves the following lemma in order to overcome these diﬃculties: the special unitary group is simply connected.

Weyl can easily conclude that su(n) and sl(n,C) have also the complete reducibility property. Thus, a topological property — i.e. the simple connectivity of SU(n) — plays a central role in Weyl’s proof of an algebraic theorem: the complete reducibility theorem in the case of SL(n,C). - He doesn’t have in mind an abstract theory of covering spaces which could be indifferently applied to Riemann surfaces and continuous groups. - In fact, he uses exactly the same terminology in Die Idee der Riemannschen Fläche: "Aus einer Darstellung der infinitesimalen Gruppe [su(n)] (...) erhält man durch Integration nach Lie die zugeordnete Matrix T für alle diejenigen t von [SU(n)], welche einer gewissen Umgebung des Einheitselements e angehören. Aber wählt man ein t0 in dieser Umgebung, so kann man die Darstellung fortsetzen aud diejenige Umgebung von t0, in welche die erste Umgebung durch die Translation von e nach t0 übergeht. Der zu iterierende Prozeß der Fortsetzung stößt offenbach niemals gegen eine Grenze".

introduction ﬁrst part second part third part / conclusion

The complete reducibility theorem for SL(n,C)

Weyl introduces progressively the concepts of simple connectivity and universal covering group in a pragmatic way, to solve problems in the course of his reasoning. "Aus einer Darstellung der inﬁnitesimalen Gruppe [su(n)] (...) erhält man durch Integration nach Lie die zugeordnete Matrix T für alle diejenigen t von [SU(n)], welche einer gewissen Umgebung des Einheitselements e angehören. Aber wählt man ein t0 in dieser Umgebung, so kann man die Darstellung fortsetzen aud diejenige Umgebung von t0, in welche die erste Umgebung durch die Translation von e nach t0 übergeht. Der zu iterierende Prozeß der Fortsetzung stößt oﬀenbach niemals gegen eine Grenze".

introduction ﬁrst part second part third part / conclusion

The complete reducibility theorem for SL(n,C)

Weyl introduces progressively the concepts of simple connectivity and universal covering group in a pragmatic way, to solve problems in the course of his reasoning. - He doesn’t have in mind an abstract theory of covering spaces which could be indifferently applied to Riemann surfaces and continuous groups. - In fact, he uses exactly the same terminology in Die Idee der Riemannschen Fläche: "Aus einer Darstellung der infinitesimalen Gruppe [su(n)] (...) erhält man durch Integration nach Lie die zugeordnete Matrix T für alle diejenigen t von [SU(n)], welche einer gewissen Umgebung des Einheitselements e angehören. Aber wählt man ein t0 in dieser Umgebung, so kann man die Darstellung fortsetzen aud diejenige Umgebung von t0, in welche die erste Umgebung durch die Translation von e nach t0 übergeht. Der zu iterierende Prozeß der Fortsetzung stößt offenbach niemals gegen eine Grenze". Die Idee der Riemannschen Fläche and Weyl’s paper on semisimple groups are very close for two main reasons: (i) The theory of the covering surfaces is a source of inspiration in the proof of the complete reducibility theorem, (ii) in these two writings, we can find a dialectic between different methods - Weierstrass’s algebraic and local method / Riemanns’s geometrical and topological method (Die Idee der Riemannschen Fläche) - Cartan’s algebraic and infinitesimal method / Hurwitz’s integral and topological method (Weyl’s paper on semisimple groups)

introduction ﬁrst part second part third part / conclusion

The complete reducibility theorem for SL(n,C)

Thus, Weyl constructs new connections between Riemann’s ideas and group theory by proving the complete reducibility theorem in the case of SL(n,C). (ii) in these two writings, we can find a dialectic between different methods - Weierstrass’s algebraic and local method / Riemanns’s geometrical and topological method (Die Idee der Riemannschen Fläche) - Cartan’s algebraic and infinitesimal method / Hurwitz’s integral and topological method (Weyl’s paper on semisimple groups)

introduction ﬁrst part second part third part / conclusion

The complete reducibility theorem for SL(n,C)

Thus, Weyl constructs new connections between Riemann’s ideas and group theory by proving the complete reducibility theorem in the case of SL(n,C). Die Idee der Riemannschen Fläche and Weyl’s paper on semisimple groups are very close for two main reasons: (i) The theory of the covering surfaces is a source of inspiration in the proof of the complete reducibility theorem, (ii) in these two writings, we can find a dialectic between different methods - Weierstrass’s algebraic and local method / Riemanns’s geometrical and topological method (Die Idee der Riemannschen Fläche) - Cartan’s algebraic and infinitesimal method / Hurwitz’s integral and topological method (Weyl’s paper on semisimple groups) The first part and the two last parts of Weyl’s article have exactly the same structure: (i) theory of weights, (ii) unitarian trick and proof of the complete reducibility theorem, (iii) theory of characters. If Weyl starts with a particular case, a suitable generalization of his methods will apply to the general case. For instance, he constructs the analogue of SU(n) for all semisimple groups, so as to apply the unitarian trick with full generality.

introduction ﬁrst part second part third part / conclusion The questions of generality and unity

In Weyl’s paper, SL(n,C) and its representations is a paradigmatic example, i.e. a particular case which expresses generality. More precisely, Weyl has to generalize - a method (the unitarian trick) - a theorem (the complete reducibility theorem) - theories (Cartan’s theory of weights and Frobenius’s theory of Characters) to all semisimple complex Lie groups. The ﬁrst part and the two last parts of Weyl’s article have exactly the same structure: (i) theory of weights, (ii) unitarian trick and proof of the complete reducibility theorem, (iii) theory of characters. If Weyl starts with a particular case, a suitable generalization of his methods will apply to the general case. For instance, he constructs the analogue of SU(n) for all semisimple groups, so as to apply the unitarian trick with full generality. Two procedures - "constructive" procedures which are based on the description of particular cases regarded as paradigmatic examples, - axiomatic procedures which are necessary to clarify and to generalize mathematical concepts. These axiomatic procedures have no heuristic functions in Weyl’s paper. For Weyl, we can’t establish a general theory without referring at ﬁrst to concrete situations. - His paper on semisimple groups is fully in accordance with this general thesis on mathematics. - On the other hand, Weyl criticizes implicitly "abstract algebra".

introduction ﬁrst part second part third part / conclusion The questions of generality and unity

In Weyl’s paper, general theorems are not merely deduced from implicit definitions or definitions by axioms. Weyl constructs general results by referring systematically to specific cases which have an heuristic function. For Weyl, we can’t establish a general theory without referring at first to concrete situations. - His paper on semisimple groups is fully in accordance with this general thesis on mathematics. - On the other hand, Weyl criticizes implicitly "abstract algebra".

introduction ﬁrst part second part third part / conclusion The questions of generality and unity

In Weyl’s paper, general theorems are not merely deduced from implicit definitions or definitions by axioms. Weyl constructs general results by referring systematically to specific cases which have an heuristic function. Two procedures - "constructive" procedures which are based on the description of particular cases regarded as paradigmatic examples, - axiomatic procedures which are necessary to clarify and to generalize mathematical concepts. These axiomatic procedures have no heuristic functions in Weyl’s paper. For Weyl, we can’t establish a general theory without referring at first to concrete situations. - His paper on semisimple groups is fully in accordance with this general thesis on mathematics. - On the other hand, Weyl criticizes implicitly "abstract algebra". "Before we can generalize, formalize and axiomatize there must be mathematical substance. I think that the mathematical substance on which we have practiced formalization in the last few decades is near to exhaustion and I predict that the next generation will face in mathematics a tough time".[ibid.]

introduction ﬁrst part second part third part / conclusion Topology versus Algebra

Weyl: "In recent years mathematicians have had to focus on the general and on formalization to such an extent that, predictably, there have turned up many instances of cheap and easy generalizing for its own sake. Pólya has called it generalizing by dilution. It does not increase the essential mathematical substance. It is much like stretching a meal by thinning the soup".[Topology and abstract algebra] "Before we can generalize, formalize and axiomatize there must be mathematical substance. I think that the mathematical substance on which we have practiced formalization in the last few decades is near to exhaustion and I predict that the next generation will face in mathematics a tough time".[ibid.] - Weyl refuses to consider algebra as a self-suﬃcient domain, - He rejects a so-called "hegemony" of algebra, - unity of mathematics is not based on the "model" of algebraic methods, but on the connection between diﬀerent mathematical domains.

introduction ﬁrst part second part third part / conclusion Topology versus Algebra

The axiomatic method is not suﬃcient in order to product new mathematical knowledge. It must be associated with constructive procedures. Weyl can’t be considered as an algebraist or as a formalist. His mathematical practice seems to be here in accordance with his conceptions of mathematics. Nevertheless, his criticisms against abstract algebra are too strong. In Gruppentheorie und Quantenmechanik and in his algebraic theory of numbers, Weyl formulates theorems due to Noether and Artin which belong to abstract algebra.

introduction ﬁrst part second part third part / conclusion Topology versus Algebra

The axiomatic method is not suﬃcient in order to product new mathematical knowledge. It must be associated with constructive procedures.

- Weyl refuses to consider algebra as a self-suﬃcient domain, - He rejects a so-called "hegemony" of algebra, - unity of mathematics is not based on the "model" of algebraic methods, but on the connection between diﬀerent mathematical domains. Nevertheless, his criticisms against abstract algebra are too strong. In Gruppentheorie und Quantenmechanik and in his algebraic theory of numbers, Weyl formulates theorems due to Noether and Artin which belong to abstract algebra.

introduction ﬁrst part second part third part / conclusion Topology versus Algebra

The axiomatic method is not suﬃcient in order to product new mathematical knowledge. It must be associated with constructive procedures.

Weyl can’t be considered as an algebraist or as a formalist. His mathematical practice seems to be here in accordance with his conceptions of mathematics. introduction ﬁrst part second part third part / conclusion Topology versus Algebra

The axiomatic method is not suﬃcient in order to product new mathematical knowledge. It must be associated with constructive procedures.

Weyl can’t be considered as an algebraist or as a formalist. His mathematical practice seems to be here in accordance with his conceptions of mathematics. Nevertheless, his criticisms against abstract algebra are too strong. In Gruppentheorie und Quantenmechanik and in his algebraic theory of numbers, Weyl formulates theorems due to Noether and Artin which belong to abstract algebra. For instance, in a testimonial "signed by Weyl on july 12, 1933 and sent by Hasse to the Ministerium in Berlin together with 13 other testimonials" [P. Roquette], we can read: "Quantum Theory has made Abstract Algebra the area of Mathematics most closely related to physics". Weyl admits the eﬀectiveness of "abstract algebra" — which he considers as the purest expression of conceptual mathematics — in the domain of quantum mechanics. This argument can be illustrated by refering to the third chapter of Weyl’s own monography on Quantummechanics.

introduction ﬁrst part second part third part / conclusion Topology versus Algebra

In other words, we shall not over-estimate Weyl’s arguments on abstract algebra which are formulated in his conference entitled Topologie und abstrakte Algebra (1931). Weyl admits the eﬀectiveness of "abstract algebra" — which he considers as the purest expression of conceptual mathematics — in the domain of quantum mechanics. This argument can be illustrated by refering to the third chapter of Weyl’s own monography on Quantummechanics.

introduction ﬁrst part second part third part / conclusion Topology versus Algebra

In other words, we shall not over-estimate Weyl’s arguments on abstract algebra which are formulated in his conference entitled Topologie und abstrakte Algebra (1931). For instance, in a testimonial "signed by Weyl on july 12, 1933 and sent by Hasse to the Ministerium in Berlin together with 13 other testimonials" [P. Roquette], we can read: "Quantum Theory has made Abstract Algebra the area of Mathematics most closely related to physics". Weyl admits the eﬀectiveness of "abstract algebra" — which he considers as the purest expression of conceptual mathematics — in the domain of quantum mechanics. This argument can be illustrated by refering to the third chapter of Weyl’s own monography on Quantummechanics. introduction ﬁrst part second part third part / conclusion

third part / conclusion According to Cartan, it is "always delicate" to use considerations which belong to analysis situs. In fact, this opinion is provisional: already in 1927, Cartan will study Lie groups from a global and topological point of view. Cartan’s book La théorie des groupes finis et continus et l’analysis situs (1930) confirms this argument. He is fully aware of the effectiveness of topological methods in the theory of Lie groups. - Thus, Weyl’s paper will have a deep impact on Cartan’s own research in the framework of Lie groups.

introduction ﬁrst part second part third part / conclusion Weyl and Cartan

In a letter to Weyl (march 1925), Cartan sketches a proof of the complete reducibility theorem which is also based on "the unitarian trick", but he avoids "Weyl’s idea of introducing the universal covering group and proving its compactness" (Hawkins). In fact, this opinion is provisional: already in 1927, Cartan will study Lie groups from a global and topological point of view. Cartan’s book La théorie des groupes finis et continus et l’analysis situs (1930) confirms this argument. He is fully aware of the effectiveness of topological methods in the theory of Lie groups. - Thus, Weyl’s paper will have a deep impact on Cartan’s own research in the framework of Lie groups.

introduction ﬁrst part second part third part / conclusion Weyl and Cartan

In a letter to Weyl (march 1925), Cartan sketches a proof of the complete reducibility theorem which is also based on "the unitarian trick", but he avoids "Weyl’s idea of introducing the universal covering group and proving its compactness" (Hawkins). According to Cartan, it is "always delicate" to use considerations which belong to analysis situs. In fact, this opinion is provisional: already in 1927, Cartan will study Lie groups from a global and topological point of view. Cartan’s book La théorie des groupes finis et continus et l’analysis situs (1930) confirms this argument. He is fully aware of the effectiveness of topological methods in the theory of Lie groups. - Thus, Weyl’s paper will have a deep impact on Cartan’s own research in the framework of Lie groups. "I am not so well-versed in Lie theory as to dare to take more from it than the [fact] that form the infinitesimal group, by integration, the neighborhood of th identity element of the identity element of the continuous group can be constructed; the entire group I obtain first by a process of "continuation" and I orient myself about its connectivity relations by means of topological consideration. Incidentally, this consideration of analysis situs is very simple and applies to all semisimple groups without distinguishing cases. This approach lies closer to my whole way of thinking than your more algebraic method, which at the moment I only half understand".

introduction ﬁrst part second part third part / conclusion Weyl and Cartan

In his reply (22 march 1925), Weyl considers Cartan as an algebraist. For Weyl topological methods are all the more simple in order to prove the complete reducibility theorem directly for all semisimple groups. "I am not so well-versed in Lie theory as to dare to take more from it than the [fact] that form the inﬁnitesimal group, by integration, the neighborhood of th identity element of the identity element of the continuous group can be constructed; the entire group I obtain ﬁrst by a process of "continuation" and I orient myself about its connectivity relations by means of topological consideration. Incidentally, this consideration of analysis situs is very simple and applies to all semisimple groups without distinguishing cases. This approach lies closer to my whole way of thinking than your more algebraic method, which at the moment I only half understand". (b) "[Early in 1927, Wigner] enriched the study of invariance by including rotations of the state space of electrons in an outer atomic shell".[Scholz] Von Neumann advises Wigner to read Schur’s and Weyl’s articles (1924) on the representations of the special orthogonal group and the orthogonal group. (c) "Between December 1927 and June 1928, Wigner and von Neumann submitted a series of three papers on spectra and the "quantum mechanics of the spinning electron" to the Zeitschrift für Physik". In these papers, they refer to the second part of Weyl’s article on semisimple groups.

introduction ﬁrst part second part third part / conclusion Wigner and von Neumann

(a) Thanks to his colleague and friend von Neumann, Wigner becomes aware that the representations of the symmetric group can play a central role in the formalization of a quantic systems consisting of n particles (nov. 1926). Reference to Frobenius’s papers (1900, 1903). (b) "[Early in 1927, Wigner] enriched the study of invariance by including rotations of the state space of electrons in an outer atomic shell".[Scholz] Von Neumann advises Wigner to read Schur’s and Weyl’s articles (1924) on the representations of the special orthogonal group and the orthogonal group. (c) "Between December 1927 and June 1928, Wigner and von Neumann submitted a series of three papers on spectra and the "quantum mechanics of the spinning electron" to the Zeitschrift für Physik". In these papers, they refer to the second part of Weyl’s article on semisimple groups. We have to criticize Mackey’s argument following which "Weyl’s idea diﬀered from that of Wigner in that he wanted to apply group representations to get a better understanding of the foundations of quantum mechanics in general and not so much to gain insight into particular problems". In fact, Wigner and Weyl use indiﬀerently the framework of representation theory so as to clarify the foundations of quantum mechanics and to describe atomic spectra.

introduction ﬁrst part second part third part / conclusion Wigner and von Neumann

Independently from Wigner and von Neumann, Weyl refers to the theory of Lie groups and the representation theory of (ﬁnite and Lie) groups in order

- to clarify Born’s and Jordan’s relation PQ QP ħ 1, − = i - to formalize quantic systems of increasing complexity, - to develop the so-called Schur-Weyl duality. In fact, Wigner and Weyl use indiﬀerently the framework of representation theory so as to clarify the foundations of quantum mechanics and to describe atomic spectra.

introduction ﬁrst part second part third part / conclusion Wigner and von Neumann

Independently from Wigner and von Neumann, Weyl refers to the theory of Lie groups and the representation theory of (ﬁnite and Lie) groups in order

- to clarify Born’s and Jordan’s relation PQ QP ħ 1, − = i - to formalize quantic systems of increasing complexity, - to develop the so-called Schur-Weyl duality.

Independently from Wigner and von Neumann, Weyl refers to the theory of Lie groups and the representation theory of (ﬁnite and Lie) groups in order

- to clarify Born’s and Jordan’s relation PQ QP ħ 1, − = i - to formalize quantic systems of increasing complexity, - to develop the so-called Schur-Weyl duality.

We have to criticize Mackey’s argument following which "Weyl’s idea diﬀered from that of Wigner in that he wanted to apply group representations to get a better understanding of the foundations of quantum mechanics in general and not so much to gain insight into particular problems". In fact, Wigner and Weyl use indiﬀerently the framework of representation theory so as to clarify the foundations of quantum mechanics and to describe atomic spectra. - In his articles "Zum Haarschen Maß in topologischen Gruppen"(1934) and "The uniqueness of Haar’s measure"(1936), von Neumann proves the uniqueness of such a measure (up to a factor) for compact groups.

introduction ﬁrst part second part third part / conclusion The representation theory of topological groups

The representation theory of locally compact and compact topological groups is developed by Haar and von Neumann in 1933-1936. - In his paper "Der Maaßbegriﬀ in der Theorie der kontinuierlichen Gruppen" (1933), Haar proves the existence of a so-called Haar measure for locally compact groups conceived in abstracto, The complete reducibility theorem in the case of compact topological groups derives from these results. We may think that Haar and von Neumann continue in a certain way Weyl’s paper on semisimple groups.

introduction ﬁrst part second part third part / conclusion The representation theory of topological groups

The representation theory of locally compact and compact topological groups is developed by Haar and von Neumann in 1933-1936. - In his paper "Der Maaßbegriﬀ in der Theorie der kontinuierlichen Gruppen" (1933), Haar proves the existence of a so-called Haar measure for locally compact groups conceived in abstracto, - In his articles "Zum Haarschen Maß in topologischen Gruppen"(1934) and "The uniqueness of Haar’s measure"(1936), von Neumann proves the uniqueness of such a measure (up to a factor) for compact groups. introduction ﬁrst part second part third part / conclusion The representation theory of topological groups

The representation theory of locally compact and compact topological groups is developed by Haar and von Neumann in 1933-1936. - In his paper "Der Maaßbegriﬀ in der Theorie der kontinuierlichen Gruppen" (1933), Haar proves the existence of a so-called Haar measure for locally compact groups conceived in abstracto, - In his articles "Zum Haarschen Maß in topologischen Gruppen"(1934) and "The uniqueness of Haar’s measure"(1936), von Neumann proves the uniqueness of such a measure (up to a factor) for compact groups.

The complete reducibility theorem in the case of compact topological groups derives from these results. We may think that Haar and von Neumann continue in a certain way Weyl’s paper on semisimple groups. "Cette méthode [l’intégration dans l’espace de groupe] féconde, dont Hurwitz paraît s’être servi le premier en 1897 et qui depuis lors avait permis à I. Schur, H. Weyl et E. Cartan lui-même d’étudier les représentations linéaires des groupes de Lie clos, a reçu dans les dernières années une extension considérable. En 1933, A. Haar a démontré que l’existence d’une mesure invariante dans un espace de groupe est liée à (...) la compacité locale du groupe".

introduction ﬁrst part second part third part / conclusion The representation theory of topological groups

This hypothesis is based on a linear conception of the history of mathematics, see A. Weil’s book L’intégration dans les groupes topologiques et ses applications (1st ed. 1940): - Independently from Weyl, O. Schreier develops a theory of abstract topological groups in 1925, see for instance "Abstrakte kontinulierliche Gruppen" (1925), - Haar’s and von Neumann’s approach is abstract and structural. This is not the case of Weyl’s paper.

introduction ﬁrst part second part third part / conclusion The representation theory of topological groups

This hypothesis is based on a linear conception of the history of mathematics, see A. Weil’s book L’intégration dans les groupes topologiques et ses applications (1st ed. 1940): "Cette méthode [l’intégration dans l’espace de groupe] féconde, dont Hurwitz paraît s’être servi le premier en 1897 et qui depuis lors avait permis à I. Schur, H. Weyl et E. Cartan lui-même d’étudier les représentations linéaires des groupes de Lie clos, a reçu dans les dernières années une extension considérable. En 1933, A. Haar a démontré que l’existence d’une mesure invariante dans un espace de groupe est liée à (...) la compacité locale du groupe". introduction ﬁrst part second part third part / conclusion The representation theory of topological groups

This hypothesis is based on a linear conception of the history of mathematics, see A. Weil’s book L’intégration dans les groupes topologiques et ses applications (1st ed. 1940): "Cette méthode [l’intégration dans l’espace de groupe] féconde, dont Hurwitz paraît s’être servi le premier en 1897 et qui depuis lors avait permis à I. Schur, H. Weyl et E. Cartan lui-même d’étudier les représentations linéaires des groupes de Lie clos, a reçu dans les dernières années une extension considérable. En 1933, A. Haar a démontré que l’existence d’une mesure invariante dans un espace de groupe est liée à (...) la compacité locale du groupe".

- Independently from Weyl, O. Schreier develops a theory of abstract topological groups in 1925, see for instance "Abstrakte kontinulierliche Gruppen" (1925), - Haar’s and von Neumann’s approach is abstract and structural. This is not the case of Weyl’s paper. But in 1935, Casimir and van der Waerden prove this theorem algebraically in their paper "Algebraischer Beweis der vollständigen Reduzibilität der Darstellungen halbeinfacher Liescher Gruppen": "Es ist natürlich erwünscht, den erwähnten algebraischen Satz auch algebraisch zu beweisen".

- According to Casimir and van der Waerden, the proof of an algebraic theorem may be based on purely algebraic methods. - In fact, their reasoning is not simpler than Weyl’s one. Another algebraic proof (much simpler) is given by Brauer in 1936.

introduction ﬁrst part second part third part / conclusion An algebraic proof of the complete reducibility theorem

Weyl’s proof of the complete reducibility theorem for semisimple groups is based on topological considerations. According to Weyl, this fact shows that algebra is not a self-suﬃcient domain. - According to Casimir and van der Waerden, the proof of an algebraic theorem may be based on purely algebraic methods. - In fact, their reasoning is not simpler than Weyl’s one. Another algebraic proof (much simpler) is given by Brauer in 1936.

introduction ﬁrst part second part third part / conclusion An algebraic proof of the complete reducibility theorem

Weyl’s proof of the complete reducibility theorem for semisimple groups is based on topological considerations. According to Weyl, this fact shows that algebra is not a self-suﬃcient domain. But in 1935, Casimir and van der Waerden prove this theorem algebraically in their paper "Algebraischer Beweis der vollständigen Reduzibilität der Darstellungen halbeinfacher Liescher Gruppen": "Es ist natürlich erwünscht, den erwähnten algebraischen Satz auch algebraisch zu beweisen". introduction ﬁrst part second part third part / conclusion An algebraic proof of the complete reducibility theorem

Weyl’s proof of the complete reducibility theorem for semisimple groups is based on topological considerations. According to Weyl, this fact shows that algebra is not a self-suﬃcient domain. But in 1935, Casimir and van der Waerden prove this theorem algebraically in their paper "Algebraischer Beweis der vollständigen Reduzibilität der Darstellungen halbeinfacher Liescher Gruppen": "Es ist natürlich erwünscht, den erwähnten algebraischen Satz auch algebraisch zu beweisen".