# Sophus Lie

**Marius Sophus Lie** (/liː/ *LEE*; Norwegian: [liː]; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.

Sophus Lie | |
---|---|

Born | Nordfjordeid, Norway | 17 December 1842

Died | 18 February 1899 56) Kristiania, Norway | (aged

Nationality | Norwegian |

Alma mater | University of Christiania |

Known for | One-parameter group See full list |

Awards | Lobachevsky Medal (1897) |

Scientific career | |

Fields | Mathematics |

Institutions | University of Christiania University of Leipzig |

Doctoral advisor | Carl Anton Bjerknes Cato Maximilian Guldberg |

Doctoral students | Hans Blichfeldt Lucjan Emil Böttcher Gerhard Kowalewski Kazimierz Żorawski |

## Biography

His first mathematical work, *Repräsentation der Imaginären der Plangeometrie*, was published in 1869 by the Academy of Sciences in Christiania and also by *Crelle's Journal*. That same year he received a scholarship and traveled to Berlin, where he stayed from September to February 1870. There, he met Felix Klein and they became close friends. When he left Berlin, Lie traveled to Paris, where he was joined by Klein two months later. There, they met Camille Jordan and Gaston Darboux. But on 19 July 1870 the Franco-Prussian War began and Klein (who was Prussian) had to leave France very quickly. Lie left for Fontainebleau where after a while he was arrested under suspicion of being a German spy, an event that made him famous in Norway. He was released from prison after a month, thanks to the intervention of Darboux.[1]

Lie obtained his PhD at the University of Christiania (present day Oslo) in 1871 with a thesis entitled *Over en Classe geometriske Transformationer* (On a Class of Geometric Transformations).[2]

It would be described by Darboux as “one of the most handsome discoveries of modern Geometry”. The next year, the Norwegian Parliament established an extraordinary professorship for him. That same year, Lie visited Klein, who was then at Erlangen and working on the Erlangen program.

At the end of 1872, Sophus Lie proposed to Anna Birch, then eighteen years old, and they were married in 1874. The couple had three children: Marie (b. 1877), Dagny (b. 1880) and Herman (b. 1884).

In 1884, Friedrich Engel arrived at Christiania to help him, with the support of Klein and Adolph Mayer (who were both professors at Leipzig, by then). Engel would help Lie to write his most important treatise, *Theorie der Transformationsgruppen*, published in Leipzig in three volumes from 1888 to 1893. Decades later, Engel would also be one of the two editors of Lie's collected works.

In 1886 Lie became professor at Leipzig, replacing Klein, who had moved to Göttingen. In November 1889, Lie suffered a mental breakdown and had to be hospitalized until June 1890. After that, he returned to his post, but over the years his anaemia progressed to the point where he decided to return to his homeland. Consequently, in 1898 he tendered his resignation in May, and left for home (for good) in September the same year. He died the following year, 1899.

He was made Honorary Member of the London Mathematical Society in 1878, Member of the French Academy of Sciences in 1892, Foreign Member of the Royal Society of London in 1895 and foreign associate of the National Academy of Sciences of the United States of America in 1895.

Sophus Lie died at the age of 56, due to pernicious anemia, a disease caused by impaired absorption of vitamin B12.

## Legacy

Lie's principal tool, and one of his greatest achievements, was the discovery that continuous transformation groups (now called, after him, Lie groups) could be better understood by "linearizing" them, and studying the corresponding generating vector fields (the so-called infinitesimal generators). The generators are subject to a linearized version of the group law, now called the commutator bracket, and have the structure of what is today called a Lie algebra.[3][4]

Hermann Weyl used Lie's work on group theory in his papers from 1922 and 1923, and Lie groups today play a role in quantum mechanics.[4]
However, the subject of Lie groups as it is studied today is vastly different from what the research by Sophus Lie was about and “among the 19th century masters, Lie's work is *in detail* certainly the least known today”.[5]

## Books

- Lie, Sophus (1888),
*Theorie der Transformationsgruppen I*(in German), Leipzig: B. G. Teubner. Written with the help of Friedrich Engel. English translation available: Edited and translated from the German and with a foreword by Joël Merker, see ISBN 978-3-662-46210-2 and arXiv:1003.3202 - Lie, Sophus (1890),
*Theorie der Transformationsgruppen II*(in German), Leipzig: B. G. Teubner. Written with the help of Friedrich Engel. - Lie, Sophus (1891),
*Vorlesungen über differentialgleichungen mit bekannten infinitesimalen transformationen*(in German), Leipzig: B. G. Teubner. Written with the help of Georg Scheffers.[6] - Lie, Sophus (1893),
*Vorlesungen über continuierliche Gruppen*(in German), Leipzig: B. G. Teubner. Written with the help of Georg Scheffers.[7] - Lie, Sophus (1893),
*Theorie der Transformationsgruppen III*(in German), Leipzig: B. G. Teubner. Written with the help of Friedrich Engel. - Lie, Sophus (1896),
*Geometrie der Berührungstransformationen*(in German), Leipzig: B. G. Teubner. Written with the help of Georg Scheffers.[8]

## See also

## Notes

- Darboux, Gaston (1899). "Sophus Lie".
*Bull. Amer. Math. Soc*.**5**(7): 367–370. doi:10.1090/s0002-9904-1899-00628-1. - Lie, Sophus (1871).
*Over en classe geometriske Transformationer*(PhD). University of Christiania. -
Helgason, Sigurdur (1994), "Sophus Lie, the Mathematician" (PDF),
*Proceedings of the Sophus Lie Memorial Conference, Oslo, August, 1992*, Oslo: Scandinavian University Press, pp. 3–21. - Gale, Thomson. "Marius Sophus Lie Biography". World of Mathematics. Retrieved 23 January 2009.
- Hermann, Robert, ed. (1975),
*Sophus Lie's 1880 transformation group paper*, Lie groups: History, frontiers and applications,**1**, Math Sci Press, p. iii, ISBN 0-915692-10-4 - Lovett, E. O. (1898). "Review:
*Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen*".*Bull. Amer. Math. Soc*.**4**(4): 155–167. doi:10.1090/s0002-9904-1898-00476-7. - Brooks, J. M. (1895). "Review:
*Vorlesungen über continuerliche Gruppen mit geometrischen und anderen Anwendungen*".*Bull. Amer. Math. Soc*.**1**(10): 241–248. doi:10.1090/s0002-9904-1895-00283-9. - Lovett, E. O. (1897). "Review:
*Geometrie der Berührungstransformationen*".*Bull. Amer. Math. Soc*.**3**(9): 321–350. doi:10.1090/s0002-9904-1897-00430-x.

## References

- Fritzsche, Bernd (1999), "Sophus Lie: A Sketch of his Life and Work",
*Journal of Lie Theory*,**9**(1), pp. 1–38, ISSN 0949-5932, MR 1680023, Zbl 0927.01029, retrieved 2 December 2010 - Freudenthal, Hans (1970–1980), "Lie, Marius Sophus",
*Dictionary of Scientific Biography*, Charles Scribner's Sons - Stubhaug, Arild (2002),
*The mathematician Sophus Lie: It was the audacity of my thinking*, Springer-Verlag, ISBN 3-540-42137-8 - Yaglom, Isaak Moiseevich (1988), Grant, Hardy; Shenitzer, Abe (eds.),
*Felix Klein and Sophus Lie: Evolution of the idea of symmetry in the nineteenth century*, Birkhäuser, ISBN 3-7643-3316-2

## External links

- Chisholm, Hugh, ed. (1911).
*Encyclopædia Britannica*(11th ed.). Cambridge University Press.
. - O'Connor, John J.; Robertson, Edmund F. (February 2000), "Sophus Lie",
*MacTutor History of Mathematics archive*, University of St Andrews. - Sophus Lie at the Mathematics Genealogy Project
- Works by Sophus Lie at Project Gutenberg
- Works by or about Sophus Lie at Internet Archive
- "The foundations of the theory of infinite continuous transformation groups - I" An English translation of a key paper by Lie (Part I)
- "The foundations of the theory of infinite continuous transformation groups - II" An English translation of a key paper by Lie (Part II)
- "On complexes - in particular, line and sphere complexes - with applications to the theory of partial differential equations" An English translation of a key paper by Lie
- "Foundations of an invariant theory of contact transformations" An English translation of a key paper by Lie
- "The infinitesimal contact transformations of mechanics" An English translation of a key paper by Lie
- U. Amaldi, "On the principal results obtained in the theory of continuous groups since the death of Sophus Lie (1898-1907)" English translation of a survey paper that followed his death