Sophus Lie

How to use this course

Every physicist and most mathematicians use Lie groups and Lie algebras, usually with no idea of where they came from. Sophus Lie did not set out to found a theory of groups. He wanted to integrate differential equations, and he found that the key was the family of transformations that carry solutions to solutions. He reached that idea through the geometry of lines and spheres, and he developed it largely alone in Christiania, far from the centres of mathematics.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons follow Lie’s own path: line and sphere geometry (1869–1872), contact transformations and partial differential equations (1870s), continuous groups and their infinitesimal generators (from 1873–74), and the foundations of geometry (from 1886). Statements are given in the local, analytic setting Lie worked in; modern global versions are noted where they differ.

Timeline

A pastor’s son, 1842–1868

Marius Sophus Lie was born on 17 December 1842 at Nordfjordeid, a village at the head of a fjord in western Norway. His father, Johann Herman Lie, was a Lutheran pastor; his mother, Mette Stabell, came from a Trondheim family. Sophus was the youngest of six children. In 1851 his father was posted to Moss, on the Kristiania fjord, where Lie went to school.

In 1857 he entered Nissen’s private Latin school in Christiania, now Oslo. He hoped for a military career, but the army rejected him because of poor eyesight. In 1859 he enrolled at the university in Christiania to study science. Mathematics did not yet grip him. He heard Carl Anton Bjerknes and, in 1862, Ludwig Sylow, who lectured on the theory of equations including Galois’s work.

He graduated in 1865 and spent the next years uncertain what to do. The turn came in 1868, when he read the geometry of Julius Plücker and Jean-Victor Poncelet. Plücker’s idea that a line, and not only a point, can serve as the basic element of space became the starting point for all of Lie’s early work.

Berlin, Paris, Fontainebleau, 1869–1870

In 1869 Lie printed at his own expense a short paper on representing the imaginary points of plane geometry by real lines in space. A fuller version was accepted by Crelle’s Journal, and on its strength he won a travel grant. He went first to Göttingen and then to Berlin, where he stayed from September 1869 to February 1870. He heard Ernst Kummer, Leopold Kronecker and Karl Weierstrass, and he found errors in Kummer’s work on line congruences. Above all he met Felix Klein, seven years younger and also a student of Plücker’s line geometry. Both were outsiders to the analytic Berlin school, and they became close friends.

In spring 1870 Lie and Klein went to Paris. They met Camille Jordan, whose treatise on permutation groups had just appeared, and Gaston Darboux. Jordan showed them how central the group concept was. Lie and Klein wrote joint papers on curves that are invariant under one-parameter families of projective transformations. In Paris Lie also found his most striking geometric discovery, a transformation that turns lines into spheres (Lesson 04).

France declared war on Prussia on 19 July 1870. Klein, a Prussian, left at once. Lie stayed in Paris for some weeks, then left for Fontainebleau. In August he was arrested there as a suspected German spy. He spent about a month in prison until Darboux secured his release. He went on to Italy and returned to Norway through Germany, where he saw Klein again.

The Christiania years, 1871–1886

Back in Christiania Lie became an assistant at the university. His doctoral thesis, Over en Classe geometriske Transformationer (On a class of geometric transformations), developed the line–sphere correspondence; his advisors are recorded as Bjerknes and Cato Guldberg. The thesis is dated 1871; sources differ on whether the degree was conferred in 1871 or 1872. On 1 July 1872 the Norwegian parliament created an extraordinary professorship for him. At the end of 1872 he became engaged to Anna Birch, and they married in 1874. They had three children, Marie, Dagny and Herman.

In 1872 he spent months with Sylow preparing a new edition of the works of Niels Henrik Abel; the edition appeared in 1881. Lie insisted that Sylow had done most of the work. In the same years Lie was studying partial differential equations. He saw that the contact transformations of his geometric work move solutions of such equations to other solutions, and he asked whether there was a theory for differential equations analogous to Galois theory for polynomial equations. In the winter of 1873–74 he took the decisive step: he considered families of transformations depending continuously on parameters and closed under composition, and he studied them through their infinitesimal transformations. This is the beginning of the theory of Lie groups (Lessons 06–08).

Lie published mainly in Norwegian, much of it in the Archiv for Mathematik og Naturvidenskab, which he co-founded in 1876 with colleagues including the zoologist Georg Ossian Sars. Few readers outside Norway followed him. He had a priority dispute with Georges Halphen over differential invariants in 1882, and around 1883 Émile Picard became aware of his work in France. Klein, anxious that Lie’s ideas would be lost, sent his young assistant Friedrich Engel to Christiania in September 1884. Engel stayed about nine months and began, with Lie, the systematic exposition that became Theorie der Transformationsgruppen.

Line geometry and the line–sphere map

For the general reader. Lie began by treating lines in space as points of a new space. In Paris he found a dictionary that translates lines into spheres, so that two lines meet exactly when the corresponding spheres touch. Problems about one kind of object became problems about the other. The dictionary is a transformation that does not map points to points, and studying such transformations led him to contact transformations and then to groups.

Plücker coordinates

Let \(a=(a_0:a_1:a_2:a_3)\) and \(b\) be distinct points of \(\mathbb P^3\). The line through them has Plücker coordinates \(p_{ij}=a_ib_j-a_jb_i\) for \(0\le i<j\le3\), a point of \(\mathbb P^5\) that does not depend on the choice of \(a,b\) on the line. These six numbers satisfy one quadratic relation, and every solution comes from a line.

The relation is the expansion of \(\det(a,b,a,b)=0\), and \(\Omega(p,q)\) is, up to sign, the \(4\times4\) determinant \(\det(a,b,c,d)\) of two points on each line. It vanishes exactly when the four points are coplanar, which for two lines means they meet.

Lie’s sphere coordinates

Give an oriented sphere in \(\mathbb R^3\) with centre \(c\) and signed radius \(r\) the coordinates

and use the bilinear form \(\langle x,y\rangle=-x_0y_0+x_1y_1+x_2y_2+x_3y_3+x_4y_4-x_5y_5\) of signature \((4,2)\). Points are spheres of radius 0; oriented planes appear as further points of the quadric \(\langle x,x\rangle=0\), the Lie quadric.

The correspondence

Over \(\mathbb C\) every non-degenerate quadric in \(\mathbb P^5\) is projectively equivalent to every other. A linear isomorphism carrying the Klein quadric to the complexified Lie quadric preserves conjugacy, so it carries meeting lines to touching spheres. Lie wrote such a map down explicitly in 1870 and published it in Über Complexe, insbesondere Linien- und Kugel-Complexe (Mathematische Annalen 5, 1872).

The second statement is the one that made Lie’s name. Asymptotic curves are a projective notion and lines of curvature a metric one, so the map exchanges two geometries. It is not a point transformation: it acts on surfaces through their tangent planes and points of contact. The group of transformations of the Lie quadric, now called the Lie sphere group, includes the Möbius transformations and the Laguerre transformations as subgroups.

Contact transformations and first-order PDE

For the general reader. A curve can be described by its points or by its tangent lines. Lie worked with both at once: a point together with a direction through it, called a contact element. Transformations of contact elements that keep touching curves touching are contact transformations. They turned out to be the natural symmetries of first-order partial differential equations, and Lie used them to reorganize the classical theory of Lagrange, Monge, Jacobi and Hamilton.

Contact elements

A contact element of the plane is a triple \((x,y,p)\): a point and a slope. A curve \(y=f(x)\) lifts to the curve \((x,f(x),f'(x))\) on which the contact form \(\theta=dy-p\,dx\) vanishes. A transformation \(\Phi\) of \((x,y,p)\)-space is a contact transformation if \(\Phi^*\theta=\lambda\,\theta\) for some non-vanishing function \(\lambda\). It then maps lifted curves to lifted curves (or to the lifts of points), and preserves tangency.

The basic example is the Legendre transformation \((x,y,p)\mapsto(X,Y,P)=(p,\ xp-y,\ x)\). Here \(dY-P\,dX=p\,dx+x\,dp-dy-x\,dp=-(dy-p\,dx)\).

Derivation. Cartan’s formula gives \(\mathcal L_X\theta=d(\theta(X))+\iota_X d\theta\), with \(d\theta=dx\wedge dp\). For \(X=a\,\partial_x+b\,\partial_y+c\,\partial_p\), \(\iota_Xd\theta=a\,dp-c\,dx\). Writing \(W=b-pa\) and requiring \(dW+a\,dp-c\,dx\) to be a multiple of \(dy-p\,dx\) forces \(a=-W_p\) and \(c=W_x+pW_y\), with multiple \(W_y\). The same holds in any number of variables, with \(\theta=du-\sum_ip_i\,dx_i\). The point transformations are the case where \(W\) is linear in \(p\).

Characteristics

A first-order equation \(F(x,u,p)=0\), with \(x\in\mathbb R^n\) and \(p=\nabla u\), is a hypersurface in the \((2n+1)\)-dimensional space of contact elements. A solution is an \(n\)-dimensional surface in it on which \(\theta=0\). The classical characteristic system

is, up to sign, the contact vector field with characteristic function \(F\). It is tangent to \(F=0\), and every solution surface is a union of its integral curves. Lie’s point of view explains why: the characteristic field is the infinitesimal contact transformation determined by the equation itself. He used this to give a unified account of Jacobi’s methods for integrating systems of first-order equations, through the Poisson bracket \(\{F,G\}\) and his theory of function groups.

One-parameter groups

For the general reader. A rotation by any angle can be built up from very small rotations. Lie’s central idea was to replace a continuous family of transformations by its infinitesimal transformation, a velocity field that says where each point moves in the first instant. The whole family can be recovered from this field by solving a differential equation.

Groups and generators

A (local) one-parameter group of transformations of \(\mathbb R^n\) is a smooth family \(\varphi_t\) with \(\varphi_0=\mathrm{id}\) and \(\varphi_s\circ\varphi_t=\varphi_{s+t}\). Its infinitesimal generator is the vector field

Conversely, the flow of any smooth vector field is a local one-parameter group. Lie wrote \(X f=\sum\xi_i\,\partial f/\partial x_i\) and thought of \(X\) as the transformation \(x\mapsto x+\varepsilon\,\xi(x)\) for infinitely small \(\varepsilon\).

Proof sketch. Let \(g(t)=f(\varphi_t(x))\). The group law gives \(g'(t)=(Xf)(\varphi_t(x))\), and by induction \(g^{(k)}(t)=(X^kf)(\varphi_t(x))\). Taylor’s theorem at \(t=0\) gives the series.

The \(r_j\) are \(n-1\) independent invariants of the group, found by solving \(dx_1/\xi_1=\cdots=dx_n/\xi_n\). In canonical coordinates a problem invariant under the group no longer depends on \(s\), and that is the source of every reduction in Lesson 07.

Symmetries of differential equations

For the general reader. Textbooks teach a collection of tricks for solving differential equations: separate the variables, try \(y=vx\), find an integrating factor. Lie showed that almost all of them work for one reason. The equation is unchanged by some continuous family of transformations, and that symmetry lowers its order by one.

First-order equations

A one-parameter group with generator \(X=\xi(x,y)\,\partial_x+\eta(x,y)\,\partial_y\) is a symmetry of \(y'=f(x,y)\) if it maps solution curves to solution curves. Prolonging \(X\) to act on slopes and requiring the equation to be preserved gives the linearized symmetry condition

Proof sketch. In canonical coordinates \((r,s)\) for \(X\), the equation is invariant under \(s\mapsto s+t\), so it takes the form \(ds=g(r)\,dr\), which is solved by one quadrature. Writing \(ds-g(r)\,dr\) back in the original variables and using \(Xr=0,\ Xs=1\) shows that it equals \(\mu\,(P\,dx+Q\,dy)\) with \(\mu\) as stated. The quantity \(\eta-\xi f\), the characteristic of \(X\), vanishes exactly where \(X\) is tangent to the solutions; there the symmetry is trivial and gives no information.

Higher order

For an equation of order \(n\), a one-parameter symmetry reduces the order by one. An \(r\)-dimensional symmetry algebra can reduce the order by \(r\), and if the algebra is solvable the reductions can be carried out by quadratures (Lesson 09).

The solutions of \(y''=0\) are the lines, and these eight fields generate the projective transformations of the plane, which permute lines.

The three fundamental theorems

For the general reader. A group with several parameters, such as the rotations of space, has several independent infinitesimal transformations. Lie found that everything about the group near the identity is coded in one table of numbers: how these infinitesimal transformations fail to commute. That table is what we now call a Lie algebra.

The bracket

For vector fields \(X,Y\), the commutator \([X,Y]f=X(Yf)-Y(Xf)\) is again a vector field, with components \([X,Y]^i=X(Y^i)-Y(X^i)\). It measures the failure of the flows to commute: if \(\varphi_t,\psi_t\) are the flows of \(X,Y\), the commutator \(\psi_{-t}\circ\varphi_{-t}\circ\psi_t\circ\varphi_t\) moves each point by \(t^2[X,Y]+O(t^3)\), up to a sign fixed by the conventions.

The second relation is the Jacobi identity \([[X,Y],Z]+[[Y,Z],X]+[[Z,X],Y]=0\) written in a basis. For vector fields it holds automatically, because the commutator of operators always satisfies it. The converse in the third theorem is the deep part: it says that every abstract Lie algebra comes from a group. Lie proved the local statement. The global version, that every finite-dimensional real Lie algebra is the Lie algebra of a simply connected Lie group, was proved by Élie Cartan in 1930.

The theorems turn a question about nonlinear transformations into linear algebra. Two local groups with the same structure constants in suitable bases are locally isomorphic as abstract groups. This is why Wilhelm Killing and then Cartan could classify the simple groups by classifying finite-dimensional algebras.

Solvability, Lie’s theorem, groups on the line

For the general reader. Galois called a polynomial equation solvable by radicals when its symmetry group can be broken into a chain of commutative steps. Lie found the same condition for differential equations: when the symmetry algebra can be broken into commutative steps, the equation can be solved by integrations alone. The algebraic condition, now called solvability, has a striking consequence in linear algebra.

Solvable algebras

For a Lie algebra \(\mathfrak g\), the derived series is \(\mathfrak g^{(0)}=\mathfrak g,\ \mathfrak g^{(k+1)}=[\mathfrak g^{(k)},\mathfrak g^{(k)}]\). The algebra is solvable if \(\mathfrak g^{(k)}=0\) for some \(k\). The lower central series \(\mathfrak g_1=\mathfrak g,\ \mathfrak g_{k+1}=[\mathfrak g,\mathfrak g_k]\) defines nilpotent algebras in the same way.

Proof idea for Lie’s theorem. Induct on \(\dim\mathfrak g\). Solvability gives an ideal \(\mathfrak h\) of codimension one. By induction \(\mathfrak h\) has a common eigenvector with eigenvalue \(\lambda(h)\). The key lemma is that the weight space \(W=\{v: hv=\lambda(h)v\ \forall h\in\mathfrak h\}\) is preserved by all of \(\mathfrak g\); this uses a trace argument and needs characteristic 0. An eigenvector in \(W\) of an element \(x\in\mathfrak g\setminus\mathfrak h\) is then common to all of \(\mathfrak g\).

Integration by quadratures

One integrates step by step along the derived series, using at each stage a symmetry that remains after the previous reduction. This is the differential analogue of Galois’s criterion, and the aim with which Lie began in 1873.

Groups acting on the line

Lie went on to classify the finite-dimensional transformation groups of the complex plane, a long list that he used to classify second-order equations by their symmetries.

The Helmholtz–Lie space problem

For the general reader. Why is space Euclidean, or nearly so? In 1868 the physicist Hermann von Helmholtz argued that the possibility of moving rigid bodies freely already restricts geometry to three types: Euclidean, spherical and hyperbolic. Lie found gaps in the argument and rebuilt it with his theory of groups. This work was the main reason he received the first Lobachevsky Prize.

From Riemann and Helmholtz to Lie

Bernhard Riemann’s 1854 lecture on the hypotheses of geometry allowed any metric \(ds^2=\sum g_{ij}\,dx_i\,dx_j\) and singled out constant curvature as the case where figures can move without stretching. Helmholtz, in Über die Thatsachen, die der Geometrie zum Grunde liegen (1868), started instead from the motions: a space admits rigid motions with as much freedom as our experience of solid bodies suggests. He concluded that the metric must be Riemannian of constant curvature. Lie, from 1886, recast the axioms as conditions on a finite-dimensional group of transformations and their infinitesimal generators, identified where Helmholtz’s reasoning was incomplete, and solved the problem in three dimensions. He published the solution in Über die Grundlagen der Geometrie (1890) and in the third volume of Theorie der Transformationsgruppen.

Proof sketch. Averaging over the compact stabilizer gives an invariant inner product at \(o\), and transitivity spreads it to an invariant metric. Because the stabilizer is transitive on 2-planes in \(T_oM\), the sectional curvature at \(o\) is the same for all planes, and homogeneity makes it the same at all points. A simply connected complete space of constant curvature is one of the three model spaces, whose full motion groups have dimension \(n+\dim\mathrm{SO}(n)=\tfrac12n(n+1)\).

Leipzig, 1886–1898

In 1886 Klein moved to Göttingen, and Lie accepted the chair at Leipzig that Klein left, with Engel already on the staff. For the first time he had a large audience. Students came from many countries, and Leipzig became the centre of the theory of transformation groups. His doctoral students there included Hans Blichfeldt, Lucjan Böttcher, Gerhard Kowalewski, Kazimierz Żorawski and Edgar Odell Lovett. Students praised his research lectures; his compulsory courses were less successful. Georg Scheffers edited his lectures into books on differential equations (1891), continuous groups (1893) and the geometry of contact transformations (1896). His ideas also took root in France, where Élie Cartan’s thesis of 1894 built on them.

The three volumes of Theorie der Transformationsgruppen, written with Engel, appeared between 1888 and 1893. In the same years Wilhelm Killing, working independently from the algebra of infinitesimal transformations, attempted to classify all simple groups. Lie thought little of Killing’s approach, and Cartan later completed and corrected the classification in his 1894 thesis.

Lie missed Norway’s mountains and forests, and the workload weighed on him. In November 1889 he suffered a severe breakdown, diagnosed as neurasthenia, and was treated in a clinic near Hanover until June 1890. He returned to teaching, but friends found him more suspicious. His working relationship with Engel cooled, and in 1892 his friendship with Klein ended. Lie believed that Klein had taken credit for ideas they had developed together, and the Erlangen Program of 1872 was the most visible case. In the preface to the third volume of 1893 he wrote: “I am no pupil of Klein, nor is the opposite the case.”

Recognition came nonetheless. He had been an honorary member of the London Mathematical Society since 1878. He became a corresponding member of the French Academy of Sciences in 1892, a Foreign Member of the Royal Society in 1895, and a foreign associate of the US National Academy of Sciences in the same year. In 1897 the Physico-Mathematical Society of Kazan gave him the first Lobachevsky Prize for his group-theoretic treatment of the foundations of geometry; Killing received the second, in 1900.

Return, death and legacy

In 1898 a chair in the theory of transformation groups was created for him in Christiania. He resigned at Leipzig in May and left for Norway in September 1898. He was already ill with pernicious anaemia, then incurable, and was able to do little in his new post. He died in Kristiania on 18 February 1899, aged 56.

In his last year Lie proposed a mathematical prize named after Abel, for the centenary of Abel’s birth in 1902, after learning that Alfred Nobel’s new prizes would not include mathematics. The plan lapsed after his death and with the dissolution of the union between Norway and Sweden in 1905. The Norwegian government established the Abel Prize in 2002. Lie’s collected papers were edited in seven volumes by Engel and Poul Heegaard between 1922 and 1960.

Legacy

• Lie groups and Lie algebras: through Killing, Cartan and Hermann Weyl, Lie’s infinitesimal groups became the structure theory and representation theory of semisimple groups, central to modern mathematics.
• Physics: rotation and Lorentz groups, gauge groups and the classification of elementary particles by representations are all applications of Lie theory.
• Differential equations: symmetry methods for ODE and PDE, revived in the 20th century by Lev Ovsiannikov, Garrett Birkhoff and Peter Olver, follow Lie’s program directly.
• Geometry: contact geometry, Lie sphere geometry, and the group-theoretic view of geometry he shared with Klein.
• Galois theory of differential equations: Ernest Vessiot and Émile Picard developed the differential Galois theory that Lie had in mind from the start.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Marius Sophus Lie, MacTutor History of Mathematics.
2. Sophus Lie, Wikipedia.
3. Lie, Sophus, Neue Deutsche Biographie and Allgemeine Deutsche Biographie.
4. B. Fritzsche, Sophus Lie: a sketch of his life and work, Journal of Lie Theory 9 (1999).
5. A. Stubhaug, The Mathematician Sophus Lie: It Was the Audacity of My Thinking, Springer, 2002.
6. Lobachevsky Prize, Wikipedia (list of laureates).
7. Abel Prize, Wikipedia (history of Lie’s proposal).

Mathematics

1. S. Lie, Über Complexe, insbesondere Linien- und Kugel-Complexe, mit Anwendung auf die Theorie partieller Differential-Gleichungen, Math. Ann. 5 (1872).
2. S. Lie and F. Engel, Theorie der Transformationsgruppen, 3 vols, Teubner, Leipzig, 1888–1893.
3. S. Lie, Über die Grundlagen der Geometrie, Berichte der Sächsischen Gesellschaft der Wissenschaften, 1890.
4. H. von Helmholtz, Über die Thatsachen, die der Geometrie zum Grunde liegen, Nachr. Ges. Wiss. Göttingen (1868).
5. T. Hawkins, Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics 1869–1926, Springer, 2000.
6. I. M. Yaglom, Felix Klein and Sophus Lie: Evolution of the Idea of Symmetry in the Nineteenth Century, Birkhäuser, 1988.
7. P. J. Olver, Applications of Lie Groups to Differential Equations, Springer GTM 107, 2nd ed. 1993.
8. P. E. Hydon, Symmetry Methods for Differential Equations, Cambridge University Press, 2000.
9. T. E. Cecil, Lie Sphere Geometry, Springer Universitext, 2nd ed. 2008.
10. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer GTM 9, 1972.