4-52. Sophus Lie

Marius Sophus Lie (1842–1899) was born in Nordfjordeide, Norway, the son of Johan and Mette (Stabell) Lie. Johan was a former headmaster, who became a vicar in 1835. Tragedy struck the family in 1852 with the death of Mette, followed by that of Sophus’ younger brother Ludvig, two years later. In the fall of 1859 Lie matriculated at the University of Christiania (Oslo, after 1924), where he studied scientific subjects. Lie heard lectures by the mathematician Ludvig Sylow, that touched on the works of Abel and Galois; he also took in lectures by the mathematician and politician Ole Jacob Broch and the physicist C.A. Bjerknes. Upon graduation Lie applied unsuccessfully for both an assistantship at Christiania Observatory and a lectureship at the university, while continuing to read mathematics on his own and attending lectures, including one held in July, 1868, by H.G. Zeuthen on the discoveries of Möbius, Plücker and Michel Chasles, his own former teacher. The next fall, having published a paper in Crelle’s Journal, Lie went to Berlin, where he befriended a former student of Plücker, Felix Klein, and made the acquaintance of Kummer, Kronecker and Weierstrass.

In early 1870, Lie left Berlin for Paris, where he met Gaston Darboux and Camille Jordan. Klein joined him in the summer, and the two men shared their ideas, which they soon published together. In the meantime, war broke out, and Lie was imprisoned in a French jail on suspicion of espionage. His condition soon became a cause célèbre in his native Norway, where the press regaled readers with items about the Norwegian scientist jailed as a German spy. Lie, after a month spent working out his new ideas and reading novels by Walter Scott, was released, thanks to the intervention of Darboux, who was able to convince the authorities that Lie was not, in fact, a German spy, but a Norwegian mathematician.

Lie returned to Christiania a celebrity of sorts, and took up a teaching position in 1871 at the high school he had frequented as a student. He defended his doctoral dissertation “on a class of geometric transformations” in June. The next year, a lectureship was created for Lie at the university. With Sylow, Lie took on the edition of Abel’s papers, and in 1876, he founded the journal Archiv for Mathematik og Naturvidenskab, where he published many of his own works. These writings did not reach a large readership, and Lie found no promising students in Christiania. In 1886, when Lie succeeded Klein as professor of mathematics in Leipzig, his student following increased considerably in quantity and quality. One of his students from Leipzig, Georg Scheffers co-authored textbooks on differential equations, transformation groups, and contact transformations (Lie and Scheffers, 1891, 1893, 1896). With the help of Adolph Mayer’s doctoral student Friedrich Engel, who was like himself the son of a pastor, Lie published his theory of transformation groups (Lie and Engel, 1888, 1890, 1893). Engel went on, after Lie’s death to edit the latter’s collected works, including Engel and Heegaard (1922, 1924, 1927, 1929, 1934, 1935, 1937); Engel et al. (1960).

In November, 1889, Lie was admitted to a clinic in Hanover, where he stayed for the next seven months. He appears to have suffered from depression. Although his position in Leipzig was quite favorable, Lie made it known that he missed his home country. By the 1890s, Lie’s international renown was such that a chair in Christiania was created for him in 1894. Yet Lie did not resign his chair in Leipzig until 1898, when he moved to Christiania with his family and doctoral student E. O. Lovett. He was then suffering from an illness later diagnosed as pernicious anemia, for which there was no treatment at the time, and to which he succumbed on 18 February, 1899.

Lie and Poincaré

Lie and Poincaré first met in Paris in November, 1882. Lie gave a talk to the Mathematical Society of France on the third of November, and met with mathematicians at the Paris Academy of Sciences. Poincaré invited Lie for dinner at his flat on the eighteenth of November (§ 4-52-1). There ensued one of the more compelling mathematical exchanges in Poincaré’s Nachlass. In all, four letters from Poincaré to Lie have been preserved, and seven from Lie to Poincaré; all are transcribed here, along with Poincaré’s official report to the Paris Academy of Sciences on Lie’s work, written in 1892 (§ 4-52-2).

From the outset, Lie considered Poincaré as a skilled ally in his quest to promote the theory of transformation groups, along with Poincaré’s colleagues Émile Picard, Gaston Darboux, and Georges Halphen. In fact, in the 1880s Lie’s theory was appreciated more in Paris than in Berlin, as David Rowe (1989) has observed. The geometry section of the Paris Academy was well-disposed toward Lie, and when the position of Correspondent became available in December, 1891, Lie was their favorite candidate. It fell to Poincaré to write the report on Lie’s accomplishments, spurring him to sollicit a publication list (§ 4-52-8). Lie’s reply has survived, but not the “offscale” (unverhältnissmässig gross) summary he wrote in longhand for the occasion (§ 4-52-9). The report Poincaré delivered appears not to have relied on Lie’s response; he affirms, in essence, the fundamental importance of Lie’s work on continuous transformation groups to the theory of both partial and ordinary differential equations (§ 4-52-12). The concluding paragraphs of Poincaré’s report touch briefly on subjects less directly related to Lie’s theory of transformation groups, including that of minimal surfaces, about which Poincaré later discussed with Lie (§ 4-52-11). Another such topic, and one which had already engaged Poincaré’s attention, was that of the foundations of geometry. Following Hermann Helmholtz’s famous essay “On the facts underlying geometry”, which upheld the principle of free mobility of ideal solids, Lie used his theory of continuous groups to give a more rigorous expression to Helmholtz’s axioms. Lie recalled the latter result for Poincaré’s benefit by letter in March, 1892 (§ 4-52-7), but Poincaré was surely aware of Lie’s theorem, as Lie had related his theory to that of Helmholtz in a note presented to the Paris Academy on 29 February (Lie, 1892).

When writing his report on Lie’s candidacy, Poincaré probably knew that the Academy would ratify the geometry section’s ranking of the candidates, and as it happened, the election, which took place on 7 June, saw Lie easily outdistance his rivals Luigi Cremona and H.A. Schwarz (Comptes rendus 114, 1329).

At the invitation of Darboux and J. Tannery, Lie visited Paris, now as a Correspondent of the Paris Academy of Sciences, from 1–20 April, 1893. On this occasion, Lie met with his former student in Leipzig, Ernest Vessiot, and the Norwegian physicist Kristian Birkeland. He also made the acquaintance of Élie Cartan, who was then writing his doctoral thesis on the structure of finite and continuous groups. While in Paris, Lie also met with Poincaré over dinner at Camille Jordan’s flat.

Poincaré may have been inspired by Lie to work on minimal surfaces, a subject they spoke about during Poincaré’s visit to Leipzig in June, 1895. As Poincaré recalled their conversation in his letter to Lie of early October, 1897 (§ 4-52-11), Lie spoke of a means of constructing all surfaces of double translation. At the time, Poincaré also sought a solution, not only for surfaces, but for manifolds of double translation, using an original method. When Lie read Poincaré’s memoir, he felt that the results were but special cases covered by his “general theorems”. Lie continued to seek a solution for the general case, as did Poincaré and others, but a full solution to Lie’s problem remained out of reach until the Lie-Wirtinger theorem of 1938.

In 1897, Lie again suggested that Poincaré had neglected his results, this time, concerning Poincaré’s theory of integral invariants (Poincaré, 1890). As Lie understood it, Poincaré’s theory was a corollary of his own theory of differential invariants; likewise for a recent theorem by Gabriel Koenigs, later known as the Lie-Koenigs theorem, presented to the Paris Academy by Poincaré in 1895. For Lie, there was only one mathematician – besides himself – to have “correctly grasped the state of affairs” with respect to integral invariants and his theory of groups: Élie Cartan.

Lie’s death in 1899 marks the beginning of Poincaré’s engagement with continuous groups. In his contribution to the Stokes Festschift of 1899, Poincaré proved a major theorem in what was later known as Lie algebra, referred to as the Poincaré-Birkhoff-Witt theorem. The very next year, the Paris Academy set for its Grand prix des sciences mathématiques the application of the theory of continuous Lie groups to the study of partial differential equations.

Along with the election of Lie as Correspondent of the geometry section of the Paris Academy, another mark of the esteem in which Lie’s contributions were held in the 1890s is the invitation extended to honor the memory of Evariste Galois on the occasion of the centennial anniversary of the foundation of the École normale supérieure. Lie concluded his contribution with the prediction that transformation groups – known by then as “Lie groups” – would find application in the natural world:

“What do natural phenomena represent, if not a succession of infinitesimal transformations, the invariants of which are the laws of the universe?”

At the time of the 1895 centennial, such an idea was somewhat far-fetched, although the related idea that geometric properties remain invariant under the transformations of a certain group gained significant recognition in the 1880s. The latter, geometric idea was expressed in Klein’s “Erlangen Program” of 1872, but had yet to find a significant echo in mathematical physics, at least beyond crystallography.

A decade later, however, in May, 1905, Poincaré came to understand the laws of physics to be invariant with respect to certain coordinate transformations, which formed a 10-parameter Lie group, known later as the inhomogeneous Lorentz group, or the “Poincaré” group. Newton’s law of universal gravitation is clearly not invariant with respect to these transformations, so Poincaré, following a method outlined by Lie and Scheffers in 1893, readily identified the principal invariants of the Lorentz group, from which he devised a Lorentz-invariant law of gravitation, expressed as a four-vector. This new mathematical view of the universe inspired – alongside Albert Einstein’s theory of relativity – Hermann Minkowski’s discovery of spacetime. Unlike Einstein and Minkowski, Poincaré recommended in a lecture in London of May, 1912, that spacetime be considered not an empirical fact, but a convention, freely imposed by scientists in their search for understanding of the natural world.

Poincaré commented further on the nature of the Lorentz group in the conclusion of his recommendation of Élie Cartan’s candidacy for a Sorbonne chair in mathematics:

We know the importance of what we called the Lorentz group in mathematical physics: it is upon this group that rest our new ideas about the principle of relativity and the dynamics of the electron. A while back, from another direction Laguerre introduced in geometry a group of transformations that change spheres into spheres. These two groups are isomorphic, such that mathematically these two theories, one physical, the other geometric, present no essential difference.

Such associations arrive in droves to those who study carefully the works of Lie and Mr. Cartan. (Poincaré, 1921, 145)

While the first part of this passage echoes Cartan’s Notice sur les travaux (1912a), the second is clearly self-referential, as Poincaré was among the many mathematicians who studied the theory of Lie groups at the turn of the twentieth century.

Time-stamp: "24.09.2024 17:53"

Notes

• 1 On Lie’s life and career, see Freudenthal (1973); Stubhaug (2002).
• 2 Poincaré (1895b, a, 1901b).
• 3 Helmholtz (1868); Lie (1890a, b); Merker (2010).
• 4 Stubhaug (2002, 374).
• 5 Cartan (1894). In 1912, Poincaré wrote a favorable report on Cartan’s candidacy for a chair at the Sorbonne, commenting that “group theory, in a manner of speaking, is all of mathematics, stripped of its substance and reduced to a pure form” (Poincaré, 1921). The chair of differential and integral calculus went to Cartan, who dominated the field of differential geometry for decades (Cogliati, 2018).
• 6 Stubhaug (2002, 390).
• 7 Poincaré (1895b, a); Lie and Scheffers (1896, 143). Lie was known for unbalanced attacks since 1893, when he published harsh criticism of his old friend Felix Klein for having neglected his priority (Tobies, 2019, 312).
• 8 Lie (1897a); Poincaré (1901b); Wirtinger (1938). For details of the history of surfaces and manifolds of double translation, see Little (1983); Chern (1994).
• 9 Koenigs (1895). Poincaré (1899a) developed the theory of integral invariants without referring explicitly to group theory. On the Lie-Koenigs theorem (also known as the Lie-Königs theorem), see Whittaker (1917, 275). The latter theorem later found an interpretation in relativistic particle dynamics (Hill and Kerner, 1966).
• 10 (Lie, 1897b, 342). Cartan later explained that the essential difference between his approach to integral invariants and that of Poincaré as follows: instead of considering a multiple integral that is invariant with respect to a system of differential equations, one may consider its invariance with respect to a transformation group (Cartan, 1922, ix). Cartan’s work on the calculus of differential forms in this context is discussed by Hawkins (2000, 282).
• 11 Gray (2013, 492); Poincaré (1899b, c). Subsequent papers on continuous groups include Poincaré (1901a, 1908).
• 12 Poincaré was a member of the prize jury, along with Camille Jordan, Émile Picard, Paul Appell and Paul Painlevé (Comptes rendus hebdomadaires de l’Académie des sciences de Paris 135(25), 1902, 1154). The jury awarded the prize to Ernest Vessiot (ibid., 1161). For an analysis of Vessiot’s prize paper, see Cogliati (2018, chap. 3).
• 13 “Que nous représentent en effet les phénomènes naturels, si ce n’est une succession de transformations infinitésimales, dont les lois de l’univers sont les invariants ?” (Lie, 1895, 489)
• 14 The Göttingen Privatdozent Arthur Schönflies employed group theory in the classification of crystallographic structures; see Schönflies (1891) and Scholz (1989a, b). Lie’s remark may have been inspired by the geometrization of dynamics, reflected in the work of Rudolf Lipschitz, Gaston Darboux, Wilhelm Killing, Paul Stäckel, and Heinrich Hertz; for an overview, see Lützen (1995). Traces of Klein’s program may be discerned in modern physics, following important contributions from Poincaré, Minkowski, Weyl, Élie Cartan and others, as noted by Goenner (2015).
• 15 See Poincaré to Lorentz, ca. May 1905 (§ 2-38-4).
• 16 Walter (2007).
• 17 Minkowski (1908).
• 18 In 1912, Poincaré promoted Galilei spacetime as an alternative to Minkowski spacetime (Walter, 2009); on Poincaré’s earlier understanding of the principle of relativity, see Darrigol (2022, § 6).
• 19 “On connaît l’importance en Physique Mathématique de ce qu’on a appelé le groupe de Lorentz; c’est sur ce groupe que reposent nos idées nouvelles sur le principe de relativité et sur la Dynamique de l’Electron. D’un autre coté, Laguerre a autrefois introduit en géométrie un groupe de transformations qui changent les sphères en sphères. Ces deux groupes sont isomorphes, de sorte que mathématiquement ces deux théories, l’une physique, l’autre géométrique, ne présentent pas de différence essentielle. Les rapprochements de ce genre se présenteront en foule à ceux qui étudieront avec soin les travaux de Lie et de M. Cartan.” Poincaré comments here on the paper Cartan read to the Société mathématique de France on 12 Jan., 1912 (Cartan, 1912b), and which Cartan glossed in the Notice presented in support of his candidacy (Cartan, 1912a). See also note 5.