Sophus Lie
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Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a
Norwegian Norwegian, Norwayan, or Norsk may refer to: *Something of, from, or related to Norway, a country in northwestern Europe *Norwegians, both a nation and an ethnic group native to Norway *Demographics of Norway *The Norwegian language, including the ...
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
. He largely created the theory of
continuous symmetry In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some Symmetry in mathematics, symmetries as Motion (physics), motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant u ...
and applied it to the study of
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and
differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
.


Life and career

Marius Sophus Lie was born on 17 December 1842 in the small town of
Nordfjordeid Nordfjordeid is the administrative centre of the municipality of Stad in Vestland county, western Norway. It is located at the end of the Eidsfjorden, an arm off of the main Nordfjorden, west of the large lake Hornindalsvatnet. The village of S ...
. He was the youngest of six children born to a Lutheran pastor named Johann Herman Lie, and his wife who came from a well-known
Trondheim Trondheim ( , , ; sma, Tråante), historically Kaupangen, Nidaros and Trondhjem (), is a city and municipality in Trøndelag county, Norway. As of 2020, it had a population of 205,332, was the third most populous municipality in Norway, and ...
family. He had his primary education in the south-eastern coast of Moss, before attending high school at
Oslo Oslo ( , , or ; sma, Oslove) is the capital and most populous city of Norway. It constitutes both a county and a municipality. The municipality of Oslo had a population of in 2022, while the city's greater urban area had a population of ...
(known then as Christiania). After graduating from high school, his ambition towards a military career was dashed when the army rejected him due to his poor eyesight. It was then that he decided to enrol at the
University of Christiania The University of Oslo ( no, Universitetet i Oslo; la, Universitas Osloensis) is a public research university located in Oslo, Norway. It is the highest ranked and oldest university in Norway. It is consistently ranked among the top universit ...
. Sophus Lie's 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 ''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics''). History The journal was founded by Augus ...
''. That same year he received a scholarship and travelled to
Berlin Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constitue ...
, where he stayed from September to February 1870. There, he met
Felix Klein Christian Felix Klein (; 25 April 1849 â€“ 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
and they became close friends. When he left Berlin, Lie travelled to
Paris Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. S ...
, where he was joined by Klein two months later. There, they met
Camille Jordan Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''. Biography Jordan was born in Lyon and educated at ...
and
Gaston Darboux Jean-Gaston Darboux FAS MIF FRS FRSE (14 August 1842 â€“ 23 February 1917) was a French mathematician. Life According this birth certificate he was born in Nîmes in France on 14 August 1842, at 1 am. However, probably due to the midni ...
. But on 19 July 1870 the Franco-Prussian War began and Klein (who was
Prussian Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a German state on the southeast coast of the Baltic Sea. It formed the German Empire under Prussian rule when it united the German states in 1871. It was ''de facto'' dissolved by an e ...
) had to leave France very quickly. Lie left for
Fontainebleau Fontainebleau (; ) is a commune in the metropolitan area of Paris, France. It is located south-southeast of the centre of Paris. Fontainebleau is a sub-prefecture of the Seine-et-Marne department, and it is the seat of the ''arrondissement ...
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. Lie obtained his PhD at the University of Christiania (in present-day
Oslo Oslo ( , , or ; sma, Oslove) is the capital and most populous city of Norway. It constitutes both a county and a municipality. The municipality of Oslo had a population of in 2022, while the city's greater urban area had a population of ...
) in 1871 with a thesis entitled ''Over en Classe geometriske Transformationer'' (On a Class of Geometric Transformations). 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 Erlangen (; East Franconian German, East Franconian: ''Erlang'', Bavarian language, Bavarian: ''Erlanga'') is a Middle Franconian city in Bavaria, Germany. It is the seat of the administrative district Erlangen-Höchstadt (former administrative d ...
and working on the
Erlangen program In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is nam ...
. In 1872, Lie spent eight years together with
Peter Ludwig Mejdell Sylow Peter Ludwig Mejdell Sylow () (12 December 1832 – 7 September 1918) was a Norwegian mathematician who proved foundational results in group theory. Biography He was born and died in Christiania (now Oslo). Sylow was a son of government minister ...
, editing and publishing the mathematical works of their countryman,
Niels Henrik Abel Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
. 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). Since 1876, he co-edited the journal '' Archiv for Mathematik og Naturvidenskab'', together with the physician Jacob Worm-Müller, and the biologist
Georg Ossian Sars Prof Georg Ossian Sars HFRSE (20 April 1837 – 9 April 1927) was a Norwegian marine and freshwater biologist. Life Georg Ossian Sars was born on 20 April 1837 in Kinn, Norway (now part of Flora), the son of Pastor Michael Sars and Maren Sa ...
. In 1884, Friedrich Engel arrived at Christiania to help him, with the support of Klein and Adolph Mayer (who were both professors at
Leipzig Leipzig ( , ; Upper Saxon: ) is the most populous city in the German state of Saxony. Leipzig's population of 605,407 inhabitants (1.1 million in the larger urban zone) as of 2021 places the city as Germany's eighth most populous, as wel ...
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
Göttingen Göttingen (, , ; nds, Chöttingen) is a college town, university city in Lower Saxony, central Germany, the Capital (political), capital of Göttingen (district), the eponymous district. The River Leine runs through it. At the end of 2019, t ...
. 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 in September the same year. He died the following year in 1899 at the age of 56, due to
pernicious anemia Pernicious anemia is a type of vitamin B12 deficiency anemia, a disease in which not enough red blood cells are produced due to the malabsorption of vitamin B12. Malabsorption in pernicious anemia results from the lack or loss of intrinsic fa ...
, a disease caused by impaired absorption of vitamin B12. He was made Honorary Member of the
London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical S ...
in 1878, Member of the
French Academy of Sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV of France, Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific me ...
in 1892, Foreign Member of the
Royal Society of London The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
in 1895 and foreign associate of the
National Academy of Sciences of the United States of America The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nati ...
in 1895. 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"


Legacy

Lie's principal tool, and one of his greatest achievements, was the discovery that continuous
transformation groups In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the ...
(now called, after him,
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s) 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 In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. The ...
, now called the
commutator bracket In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
, and have the structure of what is today called a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
.
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
used Lie's work on group theory in his papers from 1922 and 1923, and Lie groups today play a role in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. 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". Sophus Lie was an eager proponent in the establishment of the
Abel Prize The Abel Prize ( ; no, Abelprisen ) is awarded annually by the King of Norway to one or more outstanding mathematicians. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and directly modeled after the Nobel Prizes. ...
. Inspired by the Nansen fund named after
Fridtjof Nansen Fridtjof Wedel-Jarlsberg Nansen (; 10 October 186113 May 1930) was a Norwegian polymath and Nobel Peace Prize laureate. He gained prominence at various points in his life as an explorer, scientist, diplomat, and humanitarian. He led the team t ...
, and the lack of a prize for mathematics in the
Nobel Prize The Nobel Prizes ( ; sv, Nobelpriset ; no, Nobelprisen ) are five separate prizes that, according to Alfred Nobel's will of 1895, are awarded to "those who, during the preceding year, have conferred the greatest benefit to humankind." Alfr ...
. He gathered support for the establishment of an award for outstanding work in pure mathematics. Lie advised many doctoral students who went on to become successful mathematicians.
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. ...
became widely regarded as one of the greatest mathematicians of the twentieth century.
Kazimierz Żorawski Kazimierz Żorawski (June 22, 1866 – January 23, 1953) was a Polish mathematician. His work earned him an honored place in mathematics alongside such Polish mathematicians as Wojciech Brudzewski, Jan Brożek (Broscius), Nicolas Copernicus, S ...
's work was proved to be of importance to a variety of fields.
Hans Frederick Blichfeldt Hans Frederick Blichfeldt (1873–1945) was a Danish-American mathematician at Stanford University, known for his contributions to group theory, the representation theory of finite groups, the geometry of numbers, sphere packing, and quadratic ...
made great contributions to various fields of mathematics.


Books

*. Written with the help of Friedrich Engel. English translation available: Edited and translated from the German and with a foreword by Joël Merker, see and *. Written with the help of Friedrich Engel. *. Written with the help of
Georg Scheffers 250px Georg Scheffers (21 November 1866 – 12 August 1945) was a German mathematician specializing in differential geometry. Life Scheffers was born on 21 November 1866 in the village of Altendorf near Holzminden (today incorporated into Holzm ...
. *. Written with the help of Georg Scheffers. *. Written with the help of Friedrich Engel. *. Written with the help of Georg Scheffers. * (with links to 1923 review of Vol. III, 1925 review of Vol. V, & 1928 review of Vol. VI)


See also

*
Lie derivative In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
*
List of simple Lie groups In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symme ...
* List of things named after Sophus Lie


Notes


References

* * * *


External links

* * * *
"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 {{DEFAULTSORT:Lie, Marius Sophus 1842 births 1899 deaths People from Eid, Norway 19th-century Norwegian mathematicians Group theorists Members of the French Academy of Sciences Corresponding members of the Saint Petersburg Academy of Sciences University of Oslo alumni Foreign Members of the Royal Society Foreign associates of the National Academy of Sciences Leipzig University faculty