## 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).

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.^{1}^{1}endnote:
^{1}
On Lie’s life and career, see
Freudenthal (1973); Stubhaug (2002).

### 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).^{2}^{2}endnote:
^{2}
Poincaré (1895b, a, 1901b).
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.^{3}^{3}endnote:
^{3}
Helmholtz (1868); Lie (1890a, b); Merker (2010).
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).

Lie’s first visit to Paris after becoming Correspondent of the Academy
of Sciences took place from 1–20 April, 1893, at the invitation of
Darboux and J. Tannery.^{4}^{4}endnote:
^{4}
Stubhaug (2002, 374). 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.^{5}^{5}endnote:
^{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). While in Paris,
Lie also met with Poincaré over dinner at Camille Jordan’s
flat.^{6}^{6}endnote:
^{6}
Stubhaug (2002, 390).

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”.^{7}^{7}endnote:
^{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). 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.^{8}^{8}endnote:
^{8}
Lie (1897a); Poincaré (1901b); Wirtinger (1938). For details of the
history of surfaces and manifolds of double translation, see
Little (1983); Chern (1994).

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.^{9}^{9}endnote:
^{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
mechanics due to Hill and Kerner (1966). 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.^{10}^{10}endnote:
^{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).

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.^{11}^{11}endnote:
^{11}
Gray (2013, 492);
Poincaré (1899b, c). Subsequent papers on continuous groups
include Poincaré (1901a, 1908).

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?”

^{12}^{12}endnote:^{12}“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)

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.^{13}^{13}endnote:
^{13}
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).

A decade later, however, in May, 1905, Poincaré came to understand the laws
of the universe 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.^{14}^{14}endnote:
^{14}
See Poincaré to Lorentz, ca. May 1905
(§ 2-38-4). 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.^{15}^{15}endnote:
^{15}
Walter (2007).
This new mathematical view of the universe inspired – alongside
Albert Einstein’s theory of relativity – Hermann Minkowski’s
discovery of spacetime.^{16}^{16}endnote:
^{16}
Minkowski (1908). Unlike Einstein
and Minkowski, Poincaré recommended that spacetime be considered not
an empirical fact, but a *convention*, freely imposed by
scientists in their search for understanding of the natural
world.^{17}^{17}endnote:
^{17}
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).

Time-stamp: "18.03.2024 19:47"

### 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 mechanics due to 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 “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)
- 13 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).
- 14 See Poincaré to Lorentz, ca. May 1905 (§ 2-38-4).
- 15 Walter (2007).
- 16 Minkowski (1908).
- 17 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).

## References

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