Editing Sophus Lie

{{short description|Norwegian mathematician (1842–1899)}}
{{Use dmy dates|date=September 2020}}
{{Infobox scientist
| name              = Sophus Lie
| image             = Portrett_av_Sophus_Lie.jpg
| image_size        = 
| caption           = Lie in 1896
| birth_name        = Marius Sophus Lie
| birth_date        = {{birth date|1842|12|17|df=y}}
| birth_place       = [[Nordfjordeid]], Norway
| death_date        = {{death date and age|1899|2|18|1842|12|17|df=y}}
| death_place       = [[Oslo|Kristiania]], Norway
| residence         = 
| citizenship       = 
| nationality       = [[Norwegian people|Norwegian]]
| ethnicity         = 
| field             = [[Mathematics]]
| work_institution  = [[University of Oslo|University of Christiania]]<br>[[University of Leipzig]]
| alma_mater        = [[University of Oslo|University of Christiania]]
| doctoral_advisor  = [[Carl Anton Bjerknes]]<br>[[Cato Maximilian Guldberg]]
| doctoral_students = [[Hans Blichfeldt]]<br>[[Lucjan Emil Böttcher]]<br>[[Gerhard Kowalewski]]<br>[[Kazimierz Żorawski]]<br>[[Élie Cartan]]<br>[[Elling Holst]]<br>[[Edgar Odell Lovett]]
| known_for         = [[One-parameter group]]<br>[[Differential invariant]]<br>[[Contact_geometry#Some_historical_remarks|Contact transformation]]<br>[[Infinitesimal transformation]]<br>[[W-curve]]<br>[[Carathéodory-Jacobi-Lie theorem|Carathéodory–Jacobi–Lie theorem]]<br>[[Lie algebra]]<br>[[Lie bracket]]<br>[[Lie group]]<br>[[Lie product formula]]<br>[[Lie sphere geometry]]<br>[[Lie theory]]<br>[[Squeeze_mapping#Lie_transform|Lie transform]]<br>[[Lie's theorem]]<br>[[Lie's third theorem]]<br>[[Lie–Kolchin theorem]]<br>[[List of topics named after Sophus Lie|See full list]]
| author_abbreviation_bot = 
| author_abbreviation_zoo = 
| prizes            = [[Lobachevsky Medal]] (1897)<br>[[List_of_fellows_of_the_Royal_Society_J,_K,_L|ForMemRS]] (1895)
| religion          = 
| footnotes         = 
}}
'''Marius Sophus Lie''' ({{IPAc-en|l|iː}} {{respell|LEE|'}}; {{IPA|no|liː|lang}}; 17 December 1842 – 18 February 1899) was a [[Norway|Norwegian]] [[mathematician]]. He largely created the theory of [[continuous symmetry]] and applied it to the study of [[geometry]] and [[differential equations]]. He also made substantial contributions to the development of algebra.

== Life and career ==
Marius Sophus Lie was born on 17 December 1842 in the small town of [[Nordfjordeid]]. He was the youngest of six children born to Lutheran pastor Johann Herman Lie and his wife, who came from a well-known [[Trondheim]] family.<ref> {{cite book |last=James |first=Ioan |author-link= |date=2002 |title=Remarkable Mathematicians |url= |location= |publisher=Cambridge University Press |page= 201 |isbn=978-0-521-52094-2}}</ref>

He had his primary education in the south-eastern coast of Moss, before attending high school in [[Oslo]] (known then as Christiania). After graduating from high school, his ambition towards a military career was dashed when the army rejected him due to poor eyesight. He then enrolled at the [[University of Christiania]].

Sophus Lie's first mathematical work, ''Repräsentation der Imaginären der Plangeometrie'', was published in 1869 by the Academy of Sciences in [[Oslo|Christiania]] and also by ''[[Crelle's Journal]]''. That same year he received a scholarship and travelled 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 travelled to [[Paris]], where he was joined by Klein two months later. There, they met [[Camille Jordan]] and [[Jean Gaston Darboux|Gaston Darboux]]. But on 19 July 1870 the [[Franco-Prussian War]] began and Klein (who was [[Kingdom of Prussia|Prussian]]) had to leave France very quickly. Lie left for [[Fontainebleau]] where he was arrested, suspected of being a German spy, garnering him fame in Norway. He was released from prison after a month, thanks to the intervention of Darboux.<ref>{{cite journal|author=Darboux, Gaston|author-link=Gaston Darboux|title=Sophus Lie|journal=Bull. Amer. Math. Soc.|year=1899|volume=5|issue=7|pages=367–370|doi=10.1090/s0002-9904-1899-00628-1|doi-access=free}}</ref>

Lie obtained his PhD at the University of Christiania (in present-day [[Oslo]]) in 1871 with a thesis entitled ''Over en Classe geometriske Transformationer'' (On a Class of Geometric Transformations).<ref name=thesis>{{cite thesis|type=PhD|last=Lie|first=Sophus|title=Over en classe geometriske Transformationer|url=http://urn.nb.no/URN:NBN:no-nb_digibok_2010012912009 |date=1871|publisher=University of Christiania}}</ref> 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 [[University of Erlangen-Nuremberg|Erlangen]] and working on the [[Erlangen program]].

In 1872, Lie spent eight months together with [[Peter Ludwig Mejdell Sylow]], editing and publishing the mathematical works of their countryman, [[Niels Henrik Abel]].

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).

From 1876, he co-edited the journal ''[[Archiv for Mathematik og Naturvidenskab]]'', together with the physician Jacob Worm-Müller, and the biologist [[Georg Ossian Sars]].

In 1884, [[Friedrich Engel (mathematician)|Friedrich Engel]] arrived at Christiania to help him, with the support of [[Felix Klein|Klein]] and [[Christian Gustav Adolph Mayer|Adolph Mayer]] (who were both professors at [[University of Leipzig|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 a professor at Leipzig, replacing Klein, who had moved to [[University of Göttingen|Göttingen]]. In November 1889, Lie suffered a mental breakdown and had to be hospitalized until June 1890. Subsequently he returned to his post, but over the years his anaemia progressed to the point where he returned to his homeland. In 1898 he tendered his resignation in May, and left for home in September the same year. He died the following year in 1899 at the age of 56, due to [[pernicious anemia]], a disease caused by impaired absorption of [[vitamin B12|vitamin B<sub>12</sub>]].

He was made Honorary Member of the [[London Mathematical Society]] in 1878, Corresponding Member of the [[French Academy of Sciences]] in 1892, Foreign Member of the [[Royal Society of London]] in 1895 and foreign associate of the [[United States National Academy of Sciences|National Academy of Sciences of the United States of America]] in 1895.<gallery>
File:Lie-1.jpg|1888 copy of "Theorie der Transformationsgruppen," volume I
File:Lie-2.jpg|Title page to "Theorie der Transformationsgruppen"
File:Lie-3.jpg|Preface to "Theorie der Transformationsgruppen"
</gallery>

== Legacy ==

Lie's principal tool, and one of his greatest achievements, was the discovery that continuous [[transformation groups]] (now called, after him, [[Lie group]]s) could be better understood by "linearizing" them, and studying the corresponding generating [[vector field]]s (the so-called [[Lie group#The Lie algebra associated with a Lie group|infinitesimal generator]]s). 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]].<ref name= pdf>
{{Citation|last=Helgason|first=Sigurdur|author-link=Sigurður Helgason (mathematician)| contribution=Sophus Lie, the Mathematician|contribution-url=http://ocw.mit.edu/courses/mathematics/18-755-introduction-to-lie-groups-fall-2004/readings/helga_sopmath3_2.pdf|title= Proceedings of the Sophus Lie Memorial Conference, Oslo, August, 1992| publisher=Scandinavian University Press|place= Oslo| year=1994|pages= 3–21|postscript=.}}</ref><ref name="two" />

[[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]].<ref name=two>{{cite book
  | last = Gale
  | first = Thomson
  | title = Marius Sophus Lie Biography
  | publisher = World of Mathematics
  | url = http://www.bookrags.com/biography/marius-sophus-lie-wom/
  | access-date =23 January 2009 }}</ref> 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".<ref>{{Citation | editor-last = Hermann | editor-first = Robert | editor-link=Robert Hermann (mathematician)| publisher = Math Sci Press | year = 1975 | series = Lie groups: History, frontiers and applications | title = Sophus Lie's 1880 transformation group paper | volume = 1 | isbn = 0-915692-10-4 | page = iii }}</ref>

Sophus Lie was an eager proponent in the establishment of the [[Abel Prize]]. Inspired by the Nansen fund named after [[Fridtjof Nansen]], and the lack of a prize for mathematics in the [[Nobel Prize]]. He gathered support for the establishment of an award for outstanding work in pure mathematics.<ref>{{Cite web |title=The History of the Abel Prize |url=https://www.abelprize.no/c53679/artikkel/vis.html?tid=53709 |archive-url=https://web.archive.org/web/20180316235024/https://www.abelprize.no/c53679/artikkel/vis.html?tid=53709 |archive-date=2018-03-16 |access-date=2021-02-04 |website=www.abelprize.no}}</ref>

Lie advised many doctoral students who went on to become successful mathematicians. [[Élie Cartan]] became widely regarded as one of the greatest mathematicians of the 20th century. [[Kazimierz Żorawski]]'s work was proved to be of importance to a variety of fields. [[Hans Frederick Blichfeldt]] made  contributions to various fields of mathematics.

== Books ==

*{{Citation | last = Lie | first = Sophus | title = Theorie der Transformationsgruppen I | publisher = B. G. Teubner | year = 1888 | location = Leipzig | language = de }}. Written with the help of [[Friedrich Engel (mathematician)|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}}
*{{Citation | last = Lie | first = Sophus | title = Theorie der Transformationsgruppen II | publisher = B. G. Teubner | year = 1890 | location = Leipzig | language = de }}. Written with the help of Friedrich Engel.
*{{Citation | last = Lie | first = Sophus | title = Vorlesungen über differentialgleichungen mit bekannten infinitesimalen transformationen | publisher = B. G. Teubner | year = 1891 | location = Leipzig | language = de }}. Written with the help of [[Georg Scheffers]].<ref>{{cite journal|author=Lovett, E. O.|author-link=Edgar Odell Lovett|title=Review: ''Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen''|journal=Bull. Amer. Math. Soc.|year=1898|volume=4|issue=4|pages=155–167|url=http://www.ams.org/journals/bull/1898-04-04/S0002-9904-1898-00476-7/|doi=10.1090/s0002-9904-1898-00476-7|doi-access=free}}</ref>
*{{Citation | last = Lie | first = Sophus | title = Vorlesungen über continuierliche Gruppen | publisher = B. G. Teubner | year = 1893 | location = Leipzig | language = de }}. Written with the help of Georg Scheffers.<ref>{{cite journal|author=Brooks, J. M.|title=Review: ''Vorlesungen über continuerliche Gruppen mit geometrischen und anderen Anwendungen''|journal=Bull. Amer. Math. Soc.|year=1895|volume=1|issue=10|pages=241–248|url=http://www.ams.org/journals/bull/1895-01-10/S0002-9904-1895-00283-9/|doi=10.1090/s0002-9904-1895-00283-9|doi-access=free}}</ref>
*{{Citation | last = Lie | first = Sophus | title = Theorie der Transformationsgruppen III | publisher = B. G. Teubner | year = 1893 | location = Leipzig | language = de }}. Written with the help of Friedrich Engel.
*{{Citation | last = Lie | first = Sophus | title = Geometrie der Berührungstransformationen | publisher = B. G. Teubner | year = 1896 | location = Leipzig | language = de }}. Written with the help of Georg Scheffers.<ref>{{cite journal|author=Lovett, E. O.|title=Review: ''Geometrie der Berührungstransformationen''|journal=Bull. Amer. Math. Soc.|year=1897|volume=3|issue=9|pages=321–350|url=http://www.ams.org/journals/bull/1897-03-09/S0002-9904-1897-00430-X/|doi=10.1090/s0002-9904-1897-00430-x|doi-access=free}}</ref>
*{{Citation | last = Lie | first = Sophus | title=Gesammelte Abhandlungen | location=Leipzig | publisher=Teubner | postscript=; 7 vols., 1922–1960 | editor=Engel, Friedrich | editor-link=Friedrich Engel (mathematician) | editor2=Heegaard, Poul | editor-link2=Poul Heegaard | url=https://catalog.hathitrust.org/Record/005830809}}<ref name="Schilling1939">{{cite journal|last1=Schilling|first1=O. F. G.|author-link=Otto Schilling|title=Book Review: Sophus Lie's ''Gesammelte Abhandlungen. Geometrische Abhandlungen'', Volumes I & II|journal=Bulletin of the American Mathematical Society|volume=45|issue=7|year=1939|pages=513–514|issn=0002-9904|doi=10.1090/S0002-9904-1939-07032-8|doi-access=free}}</ref><ref name="Carmichael1930">{{cite journal|last1=Carmichael|first1=R. D.|author-link=Robert Daniel Carmichael|title=Book Review: vol. IV of Sophus Lie's ''Gesammelte Abhandlungen'' (''Samlede Avhandlinger'', Norwegian edition published by Aschehoug)|journal=Bulletin of the American Mathematical Society|volume=36|issue=5|year=1930|pages=337–338|issn=0002-9904|doi=10.1090/S0002-9904-1930-04950-2|doi-access=free}} (with links to 1923 review of Vol. III, 1925 review of Vol. V, & 1928 review of Vol. VI)</ref>

== See also ==
* [[Lie derivative]]
* [[List of simple Lie groups]]
* [[List of things named after Sophus Lie]]

== Notes ==
{{Reflist|30em}}

== References ==
* {{Citation | last = Fritzsche | first = Bernd | periodical = Journal of Lie Theory | title = Sophus Lie: A Sketch of his Life and Work | url = http://www.emis.de/journals/JLT/vol.9_no.1/1.html | year = 1999 | volume = 9 | pages = 1–38 | mr = 1680023 | zbl = 0927.01029 | access-date =2 December 2010 | issn = 0949-5932 | issue =1}}
* {{Citation | last = Freudenthal | first = Hans | author-link = Hans Freudenthal | title = Dictionary of Scientific Biography | chapter = Lie, Marius Sophus | chapter-url=http://www.encyclopedia.com/people/science-and-technology/mathematics-biographies/marius-sophus-lie | year = 1970–1980 | publisher = [[Charles Scribner's Sons]]| title-link = Dictionary of Scientific Biography }}
* {{Citation | last = Stubhaug | first = Arild | author-link = Arild Stubhaug | title = The mathematician Sophus Lie: It was the audacity of my thinking | publisher = [[Springer Science+Business Media|Springer-Verlag]] | year = 2002 | isbn = 3-540-42137-8 | url-access = registration | url = https://archive.org/details/mathematiciansop0000stub }}
* {{Citation | last = Yaglom | first = Isaak Moiseevich | author-link = Isaak Yaglom | title = Felix Klein and Sophus Lie: Evolution of the idea of symmetry in the nineteenth century | publisher = [[Birkhäuser Verlag|Birkhäuser]] | editor1-last = Grant | editor1-first = Hardy | editor2-last = Shenitzer | editor2-first = Abe | year = 1988 | isbn = 3-7643-3316-2 | title-link = Isaak Yaglom#Felix Klein and Sophus Lie (1988) }}

== External links ==
* {{Cite EB1911|wstitle=Lie, Marius Sophus}}
* {{MacTutor | id = Lie | date = February 2000}}
* {{Gutenberg author | id=31111| name=Sophus Lie}}
* {{Internet Archive author |sname=Sophus Lie}}
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/lie-_infinite_continuous_groups_-_i.pdf "The foundations of the theory of infinite continuous transformation groups – I"]  An English translation of a key paper by Lie (Part I)
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/lie-_infinite_continuous_groups_-_ii.pdf "The foundations of the theory of infinite continuous transformation groups – II"] An English translation of a key paper by Lie (Part II)
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/lie_-_line_and_sphere_complexes.pdf "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
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/lie_-_contact_transformations.pdf "Foundations of an invariant theory of contact transformations"] An English translation of a key paper by Lie
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/lie_-_infinitesimal_contact_transformations_in_mechanics.pdf "The infinitesimal contact transformations of mechanics"] An English translation of a key paper by Lie
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/amaldi_-_results_since_the_death_of_lie.pdf 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

{{Authority control}}

{{DEFAULTSORT:Lie, Marius Sophus}}
[[Category:1842 births]]
[[Category:1899 deaths]]
[[Category:People from Eid, Norway]]
[[Category:19th-century Norwegian mathematicians]]
[[Category:Group theorists]]
[[Category:Members of the French Academy of Sciences]]
[[Category:Corresponding members of the Saint Petersburg Academy of Sciences]]
[[Category:University of Christiania alumni]]
[[Category:Foreign members of the Royal Society]]
[[Category:Foreign associates of the National Academy of Sciences]]
[[Category:Academic staff of Leipzig University]]