Élie Cartan

How to use this course

Élie Cartan is a mathematician whose name appears everywhere: Cartan subalgebras, the Cartan matrix, Cartan’s criterion, the Cartan decomposition, Cartan connections, the Maurer–Cartan form, the Cartan–Kähler theorem. For most of his career few people could read him. His papers were dense, geometric and ahead of their time, and recognition came mostly after he turned sixty. This course follows his work from the structure of Lie algebras in 1894 to the global geometry of Lie groups in the 1930s.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04–06 cover Cartan’s algebra (1894–1927); Lessons 07–09 his differential geometry (1899–1926); Lesson 10 the global theory of the 1930s. Lie algebras are over \(\mathbb C\) unless stated otherwise.

Timeline

Dolomieu to the École Normale, 1869–1891

Élie Joseph Cartan was born on 9 April 1869 in Dolomieu, a village in the Isère between Lyon and Chambéry. His father, Joseph Cartan, was the village blacksmith; his mother, Anne Florentine Cottaz, ran the household. Élie had an older sister, Jeanne-Marie, a younger brother, Léon, who later took over the forge, and a younger sister, Anna, whom he encouraged to study and who became a mathematics teacher.

The new republican school system of the 1870s and 1880s looked for talent in villages. His ability was noticed at the Dolomieu primary school, and a state scholarship took him to the Collège de Vienne from 1880 to 1885 and then to the lycée in Grenoble from 1885 to 1887. He spent 1887–88 in the special mathematics class of the Lycée Janson-de-Sailly in Paris, preparing for the entrance competitions.

He entered the École Normale Supérieure in 1888. The teachers he heard there, at the École and the Sorbonne, included Henri Poincaré, Charles Hermite, Jules Tannery, Gaston Darboux, Paul Appell, Émile Picard and Édouard Goursat. Darboux’s geometry of surfaces, with its moving trihedra, would shape his later work. He graduated in 1891 and then served a year in the army, reaching the rank of sergeant.

The thesis, 1891–1894

Back in Paris, Cartan looked for a thesis subject. His friend Arthur Tresse, who had studied with Sophus Lie in Leipzig, told him about the papers of Wilhelm Killing. Between 1888 and 1890 Killing had attempted something astonishing: a list of all simple finite-dimensional complex Lie algebras. His list contained the four infinite families of the classical groups and five unexpected exceptional cases. But his arguments had gaps, and he had not shown that the exceptional algebras actually exist.

Lie spent six months in Paris in 1892, and his presence helped Cartan considerably. In 1893 Cartan published notes constructing algebras for the exceptional types. In the same year he and Friedrich Engel, Lie’s collaborator, independently described the smallest exceptional algebra, \(G_2\), as the infinitesimal symmetries of a system of Pfaffian equations in five variables.

His thesis, Sur la structure des groupes de transformations finis et continus, was defended in Paris in 1894. The Mathematics Genealogy Project lists Darboux and Lie as his advisors. It gave rigorous proofs of the classification, introduced what are now called Cartan’s criteria for solvability and semisimplicity through the Killing form, and established the structure theorem for semisimple algebras. Lesson 04 presents the result. Even so, Cartan was little known outside a small circle for the next thirty years.

Provincial years, 1894–1912

Like most young French mathematicians, Cartan began in the provinces. He was a lecturer at Montpellier from 1894 to 1896 and at Lyon from 1896 to 1903. In Lyon he studied associative algebras, then turned to the integration of systems of partial differential equations. His 1899 paper Sur certaines expressions différentielles et le problème de Pfaff defined exterior differential forms and their calculus, the language of Lesson 07.

In 1903 he married Marie-Louise Bianconi in Lyon, and the same year he became professor at Nancy, where he also taught at the institute of electrical engineering and applied mechanics. Their first son, Henri Cartan, was born in Nancy in 1904 and became one of the leading French mathematicians of the next generation. They had three more children: Jean, a composer; Louis, a physicist; and Hélène, who taught mathematics.

At Nancy he extended Lie’s theory to infinite-dimensional groups of transformations (1904–05), the pseudogroups defined by differential equations, and in 1908 he developed the general method for deciding when two geometric structures are equivalent. In 1909 he moved to Paris as maître de conférences at the Sorbonne, and around 1912 he became professor of differential and integral calculus.

The Killing–Cartan classification

For the general reader. Continuous symmetry groups, such as rotations, are built out of simple pieces in the way numbers are built out of primes. Killing and Cartan found the complete list of simple pieces. It contains four infinite families, the symmetries of familiar geometries, and exactly five exceptions, which nobody had predicted. The list is now a fixed point of mathematics and physics.

The Killing form

For a finite-dimensional Lie algebra \(\mathfrak g\) over \(\mathbb C\), let \(\operatorname{ad}X(Y)=[X,Y]\). The Killing form is \(B(X,Y)=\operatorname{tr}(\operatorname{ad}X\circ\operatorname{ad}Y)\). It is symmetric and invariant: \(B([X,Y],Z)=B(X,[Y,Z])\). An algebra is semisimple if it has no non-zero solvable ideal, and simple if it is non-abelian with no ideals other than 0 and itself.

Roots

A Cartan subalgebra \(\mathfrak h\) of a semisimple \(\mathfrak g\) is a maximal abelian subalgebra consisting of elements \(H\) with \(\operatorname{ad}H\) diagonalizable. All are conjugate, and their common dimension is the rank. Simultaneously diagonalizing \(\operatorname{ad}\mathfrak h\) gives the root decomposition

The finite set \(\Phi\subset\mathfrak h^*\) of roots, with the inner product induced by \(B\), is a root system: it spans, \(\Phi=-\Phi\), the only multiples of \(\alpha\) in \(\Phi\) are \(\pm\alpha\), each reflection \(s_\alpha(\beta)=\beta-\langle\beta,\alpha^\vee\rangle\alpha\) preserves \(\Phi\), and \(\langle\beta,\alpha^\vee\rangle=2(\beta,\alpha)/(\alpha,\alpha)\) is an integer. A choice of simple roots \(\alpha_1,\dots,\alpha_r\) gives the Cartan matrix \(a_{ij}=\langle\alpha_j,\alpha_i^\vee\rangle\), which determines \(\mathfrak g\) up to isomorphism.

Proof outline. Semisimplicity and Cartan’s criterion make \(B\) non-degenerate on \(\mathfrak h\), so roots live in a Euclidean space. For two roots, \(\langle\beta,\alpha^\vee\rangle\langle\alpha,\beta^\vee\rangle=4\cos^2\theta\in\{0,1,2,3\}\), so angles between simple roots are 90°, 120°, 135° or 150°. Positive definiteness of the matrix \((\alpha_i,\alpha_j)\) then severely restricts which configurations of simple roots can occur, and a combinatorial analysis leaves exactly the nine types. Existence requires a construction for each type: the classical ones are matrix algebras, and the exceptional ones were Cartan’s contribution.

Representations and spinors

For the general reader. A representation lets a group act by matrices. In 1913 Cartan determined every irreducible representation of every simple Lie algebra. Among them he found, for the rotation groups, representations that do not come from vectors or tensors at all. Physicists rediscovered them in 1928 as the electron’s wave function in Dirac’s equation, and they are now called spinors.

Highest weights

Let \(V\) be a finite-dimensional representation of semisimple \(\mathfrak g\). Then \(\mathfrak h\) acts diagonalizably, \(V=\bigoplus_\mu V_\mu\), and the \(\mu\in\mathfrak h^*\) with \(V_\mu\ne0\) are the weights. Root vectors move weights: \(\mathfrak g_\alpha V_\mu\subseteq V_{\mu+\alpha}\). A weight \(\lambda\) is dominant integral if \(\langle\lambda,\alpha_i^\vee\rangle\in\mathbb Z_{\ge0}\) for every simple root.

Hermann Weyl, using the compact groups and integration, proved complete reducibility and gave the character and dimension formulas in 1925–26.

Spinors

For \(\mathfrak{so}(n)\), the representations that appear in tensor powers of the standard \(n\)-dimensional representation do not exhaust the list. Cartan found the rest in 1913.

The modern construction, due to Richard Brauer and Weyl in 1935, uses the Clifford algebra generated by \(\gamma_1,\dots,\gamma_n\) with \(\gamma_i\gamma_j+\gamma_j\gamma_i=2\delta_{ij}\). The elements \(\tfrac14[\gamma_i,\gamma_j]\) satisfy the commutation relations of \(\mathfrak{so}(n)\), and the Clifford algebra acts irreducibly on a space of dimension \(2^{\lfloor n/2\rfloor}\). Cartan’s Leçons sur la théorie des spineurs (1938) gave a geometric account, with spinors described through isotropic subspaces.

Real forms and symmetric spaces

For the general reader. The same complex algebra can have several real versions: the rotations of a sphere and the motions of the hyperbolic plane are two real forms of one complex object. In 1914 Cartan classified all real forms. Twelve years later he saw that the same classification describes the most symmetric curved spaces, in which every point is a centre of symmetry.

Real forms

A real form of a complex Lie algebra \(\mathfrak g\) is a real subalgebra \(\mathfrak g_0\) with \(\mathfrak g=\mathfrak g_0\oplus i\mathfrak g_0\). For example \(\mathfrak{sl}(n,\mathbb R)\) and \(\mathfrak{su}(n)\) are real forms of \(\mathfrak{sl}(n,\mathbb C)\). A real semisimple algebra is compact if its Killing form is negative definite.

For \(\mathfrak{sl}(n,\mathbb R)\), \(\theta(X)=-X^{\mathsf T}\): \(\mathfrak k=\mathfrak{so}(n)\) and \(\mathfrak p\) is the symmetric traceless matrices. At the group level this is the polar decomposition \(g=k\,e^{P}\).

Symmetric spaces

A connected Riemannian manifold \(M\) is a symmetric space if for every \(p\in M\) there is an isometry \(s_p\) fixing \(p\) with \(ds_p=-\mathrm{id}\) on \(T_pM\): it reverses geodesics through \(p\).

The curvature is given purely by brackets: \(R(X,Y)Z=-[[X,Y],Z]\) for \(X,Y,Z\in\mathfrak p\). Spheres, projective spaces, Grassmannians, hyperbolic spaces and the space \(\mathrm{SL}(n,\mathbb R)/\mathrm{SO}(n)\) of positive definite matrices of determinant 1 are all symmetric spaces.

Exterior differential forms and Pfaff systems

For the general reader. Cartan gave analysis a new grammar. Expressions like \(dx\wedge dy\), which measure oriented areas, can be multiplied and differentiated by simple rules that work in any coordinates. With them, questions about whether a system of differential equations has solutions become algebraic computations.

The calculus

A \(k\)-form on \(\mathbb R^n\) is \(\omega=\sum_{I}a_I\,dx_{i_1}\wedge\cdots\wedge dx_{i_k}\), with \(dx_i\wedge dx_j=-dx_j\wedge dx_i\). The exterior derivative is \(d\omega=\sum_I da_I\wedge dx_{i_1}\wedge\cdots\wedge dx_{i_k}\). It satisfies \(d\circ d=0\), the Leibniz rule \(d(\alpha\wedge\beta)=d\alpha\wedge\beta+(-1)^k\alpha\wedge d\beta\) for a \(k\)-form \(\alpha\), and commutes with pull-back by smooth maps. Cartan defined these operations in 1899 to study the Pfaff problem: bring a 1-form to the simplest possible shape by a change of variables.

This is the contact form of Lie’s theory. In general a system of partial differential equations can be written as an exterior differential system: an ideal \(\mathcal I\) of forms closed under \(d\), whose integral manifolds are the solutions. Cartan defined characters \(s_1,s_2,\dots\) measuring the freedom in extending integral elements, and a test for when a system is in involution.

Cartan proved the case of Pfaffian systems in 1901; Erich Kähler extended it to general systems in 1934. The theorem rests on the Cauchy–Kovalevskaya theorem, applied one dimension at a time.

Moving frames and structure equations

For the general reader. To study a curve or surface, Cartan attached to each point a set of perpendicular axes adapted to the shape, and watched how the axes turn as the point moves. The rates of turning are differential forms, and a few identities among them contain the whole local geometry. Darboux had used moving trihedra for surfaces; Cartan made the method work for any geometry defined by a group.

Frames in Euclidean space

A frame in \(\mathbb R^3\) is a point \(x\) with an orthonormal basis \(e_1,e_2,e_3\). On the space of frames, write

Derivation. Apply \(d\) to \(dx=\sum\omega^ie_i\): \(0=\sum_i d\omega^i\,e_i-\sum_i\omega^i\wedge de_i=\sum_j\big(d\omega^j-\sum_i\omega^i\wedge\omega_i^{\,j}\big)e_j\). Apply \(d\) to \(de_i\) similarly. The sign comes from moving a 1-form past \(d\).

Surfaces

For a surface, choose frames with \(e_3\) normal, so that \(\omega^3=0\) on the surface. Then \(\omega^1,\omega^2\) form an orthonormal coframe, and the structure equations give

The first two equations determine \(\omega_1^{\,2}\) from the metric alone. The third then expresses the Gaussian curvature \(K\) intrinsically: this is Gauss’s Theorema Egregium in three lines.

The Euclidean structure equations are the Maurer–Cartan equation of the Euclidean group. The converse is the fundamental theorem of moving frames: a submanifold is determined up to a rigid motion by its invariants, provided they satisfy the structure equations. Cartan’s method of equivalence (1908) turns this into a general algorithm for deciding when two geometric structures are locally the same.

Connections, torsion and holonomy

For the general reader. On a curved surface there is no single meaning of “the same direction” at two different points. A connection supplies a rule for carrying directions along paths. Carry an arrow around a closed loop and it may come back turned: the turning measures curvature. Cartan saw that connections need not come from a metric, and that they can also have torsion, a twisting in which small parallelograms fail to close.

Structure equations of a connection

On an \(n\)-manifold, let \(\omega=(\omega^i)\) be a coframe and \(\Gamma=(\Gamma^i{}_j)\) a matrix of 1-forms, the connection forms. Define the torsion and curvature forms by Cartan’s structure equations

Proof. \(d\Theta=d\Gamma\wedge\omega-\Gamma\wedge d\omega=d\Gamma\wedge\omega-\Gamma\wedge(\Theta-\Gamma\wedge\omega)=\Omega\wedge\omega-\Gamma\wedge\Theta\). The second is similar, using \(d\Gamma=\Omega-\Gamma\wedge\Gamma\).

The surface formulas of Lesson 08 are the case \(n=2\), with \(\Gamma^2{}_1=\omega_1^{\,2}\). In 1922 Cartan proposed allowing \(\Theta\ne0\) in relativity, and in papers of 1923–25 he developed manifolds with affine connection as models of spacetime. In what is now called Einstein–Cartan theory, torsion is tied algebraically to the spin density of matter, so it vanishes in vacuum. Thomas Kibble (1961) and Dennis Sciama (1964) rediscovered the theory from gauge principles.

Holonomy

Parallel transport around loops based at \(p\) gives linear maps of \(T_pM\). They form the holonomy group \(\mathrm{Hol}_p\), which Cartan introduced in the 1920s. For a Riemannian manifold, \(\mathrm{Hol}_p\subseteq\mathrm{O}(n)\). Its Lie algebra is spanned by the curvature endomorphisms transported back to \(p\) (Ambrose–Singer, 1953). For a symmetric space \(G/K\) the restricted holonomy is the image of the identity component of \(K\) acting on \(\mathfrak p\). Marcel Berger’s 1955 list of possible holonomy groups grew out of this.

Cartan’s most general notion, the Cartan connection, models a manifold infinitesimally on a homogeneous space \(G/H\): Euclidean, affine, projective or conformal. It answered the problem, noted in the Klein course, that a Riemannian manifold has no symmetry group in Klein’s sense. Each tangent space carries a copy of the model, and the connection says how to roll the model along curves.

Global Lie theory and topology

For the general reader. Lie worked near the identity, with small transformations. After Weyl’s work on compact groups in 1925, Cartan turned to the whole group: its shape as a space, its holes and loops. He proved that every Lie algebra comes from a group globally, and showed how the topology of a group is read off from invariant differential forms.

Cartan proved these in the monograph La théorie des groupes finis et continus et l’Analysis situs (1930). John von Neumann had proved the closed subgroup theorem for closed groups of matrices in 1929. Today the first part is usually derived from Ado’s theorem that every Lie algebra has a faithful matrix representation.

Invariant forms and cohomology

On a compact connected Lie group \(G\), averaging over left and right translations shows that every de Rham cohomology class contains a unique bi-invariant form, and bi-invariant forms are closed. So \(H^*(G;\mathbb R)\) is the algebra of \(\mathrm{Ad}\)-invariant elements of \(\Lambda^*\mathfrak g^*\). Cartan used this in the late 1920s to compute Betti numbers, before Georges de Rham proved in 1931 the theorem identifying form cohomology with topological cohomology, a result Cartan had conjectured. He also proved in 1936 that \(\pi_2(G)=0\) for every Lie group \(G\).

The Sorbonne, 1912–1940

From 1915 to 1918 Cartan served in the army in the First World War, again as a sergeant. After the war he held the chair of rational mechanics from 1920 and the chair of higher geometry from 1924 until he retired in 1940. Relativity drew him back to differential geometry: his books Leçons sur les invariants intégraux (1922) and Leçons sur la géométrie des espaces de Riemann (1928) introduced a generation to his methods, though many readers found them hard.

For years only a few contemporaries, Poincaré and later Weyl among them, understood what Cartan had done. Weyl’s work on compact groups (1925–26) built on Cartan’s classification and in turn pushed Cartan toward global questions. From 1929 to 1932 Cartan exchanged letters with Albert Einstein about Einstein’s unified field theory based on distant parallelism, a connection with torsion but no curvature, which Cartan had already studied in 1922. He spoke at the International Congresses in Toronto (1924), Zürich (1932, plenary) and Oslo (1936).

Students came. The Mathematics Genealogy Project lists 29 doctoral students, among them Charles Ehresmann, André Lichnerowicz, Kentaro Yano and Mohsen Hashtroodi. Shiing-Shen Chern spent 1936–37 in Paris working with him, and recalled receiving letters after their meetings full of new results and questions. Recognition followed late. He was elected to the Académie des Sciences on 9 March 1931, received the Lobachevsky Prize in 1937, and was honoured at a jubilee at the Sorbonne on 18 May 1939 for his seventieth birthday.

There was grief as well. His son Jean, a promising composer, died of tuberculosis in 1932 at the age of 25.

Last years and legacy

Cartan retired from the Sorbonne in 1940 and continued to teach at the École Normale Supérieure for girls. His son Louis, a physicist at Poitiers and a member of the Resistance, was arrested in February 1943 and beheaded by the Nazis in December 1943; the family learned of his death only in May 1945. In 1945 Cartan published Les systèmes différentiels extérieurs et leurs applications géométriques, based on his Sorbonne lectures of 1936–37. He was vice-president of the Académie des Sciences in 1945 and president in 1946, and became a Foreign Member of the Royal Society on 1 May 1947. He received honorary degrees from Liège, Harvard, Berlin, Bucharest, Louvain and Pisa. He died in Paris on 6 May 1951 after a long illness, aged 82.

Legacy

• Lie theory: the classification of simple algebras, real forms and representations by highest weight is the backbone of the theory as taught today, and of the Standard Model’s use of \(\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)\).
• Spinors: the mathematics of fermions, of Dirac operators and of the Atiyah–Singer index theorem.
• Differential forms: the language of modern geometry, topology and physics, from de Rham cohomology to Maxwell’s equations written as \(dF=0,\ d{\star}F=J\).
• Connections and holonomy: through Ehresmann’s principal bundles they became the gauge fields of Yang–Mills theory; Chern’s characteristic classes are built from Cartan’s curvature forms.
• Symmetric spaces: central in geometry, harmonic analysis and number theory, where locally symmetric spaces carry automorphic forms.
• Family: his son Henri Cartan became a founder of Bourbaki and a leader of homological algebra and sheaf theory.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Élie Joseph Cartan, MacTutor History of Mathematics.
2. Élie Cartan, Wikipedia.
3. Élie Cartan, Mathematics Genealogy Project.
4. S.-S. Chern and C. Chevalley, Élie Cartan and his mathematical work, Bull. Amer. Math. Soc. 58 (1952).
5. R. Debever (ed.), Élie Cartan and Albert Einstein: Letters on Absolute Parallelism, 1929–1932, Princeton University Press, 1979.
6. M. A. Akivis and B. A. Rosenfeld, Élie Cartan (1869–1951), American Mathematical Society, 1993.
7. Lobachevsky Prize, Wikipedia (list of laureates).

Mathematics

1. É. Cartan, Sur la structure des groupes de transformations finis et continus, thesis, Paris, 1894.
2. É. Cartan, Sur certaines expressions différentielles et le problème de Pfaff, Ann. Sci. ENS 16 (1899).
3. É. Cartan, Les groupes projectifs qui ne laissent invariante aucune multiplicité plane, Bull. SMF 41 (1913).
4. É. Cartan, Les groupes réels simples finis et continus, Ann. Sci. ENS 31 (1914).
5. É. Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée, Ann. Sci. ENS 40–42 (1923–25).
6. É. Cartan, Sur une classe remarquable d’espaces de Riemann, Bull. SMF 54 (1926), 55 (1927).
7. É. Cartan, La théorie des groupes finis et continus et l’Analysis situs, Mémorial des sciences mathématiques 42 (1930; reprinted 1952).
8. É. Cartan, Leçons sur la théorie des spineurs, Hermann, 1938.
9. T. Hawkins, Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics 1869–1926, Springer, 2000.
10. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, 1978.
11. R. W. Sharpe, Differential Geometry: Cartan’s Generalization of Klein’s Erlangen Program, Springer GTM 166, 1997.
12. R. Bryant, S.-S. Chern et al., Exterior Differential Systems, MSRI Publications 18, Springer, 1991.
13. Einstein–Cartan theory, Wikipedia.