Sophus Lie

Encyclopedia

**Marius Sophus Lie**(17 December 1842 – 18 February 1899) was a Norwegian

Norway

Norway , officially the Kingdom of Norway, is a Nordic unitary constitutional monarchy whose territory comprises the western portion of the Scandinavian Peninsula, Jan Mayen, and the Arctic archipelago of Svalbard and Bouvet Island. Norway has a total area of and a population of about 4.9 million...

mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

. He largely created the theory of continuous symmetry

Continuous symmetry

In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to e.g. reflection symmetry, which is invariance under a kind of flip from one state to another. It has largely and successfully been formalised in the...

, and applied it to the study of geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

and differential equations.

## Biography

His first mathematical work,*Repräsentation der Imaginären der Plangeometrie*, was published, in 1869, by the Academy of Sciences in Christiania

Oslo

Oslo is a municipality, as well as the capital and most populous city in Norway. As a municipality , it was established on 1 January 1838. Founded around 1048 by King Harald III of Norway, the city was largely destroyed by fire in 1624. The city was moved under the reign of Denmark–Norway's King...

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

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 .- History :...

Felix Klein

Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...

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

Camille Jordan

Marie Ennemond Camille Jordan was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse. He was born in Lyon and educated at the École polytechnique...

and Gaston Darboux

Jean Gaston Darboux

Jean-Gaston Darboux was a French mathematician.-Life:Darboux made several important contributions to geometry and mathematical analysis . He was a biographer of Henri Poincaré and he edited the Selected Works of Joseph Fourier.Darboux received his Ph.D...

. But on 19 July 1870 the Franco-Prussian War

Franco-Prussian War

The Franco-Prussian War or Franco-German War, often referred to in France as the 1870 War was a conflict between the Second French Empire and the Kingdom of Prussia. Prussia was aided by the North German Confederation, of which it was a member, and the South German states of Baden, Württemberg and...

began and Klein (who was Prussian

Kingdom of Prussia

The Kingdom of Prussia was a German kingdom from 1701 to 1918. Until the defeat of Germany in World War I, it comprised almost two-thirds of the area of the German Empire...

) had to leave France very quickly. Lie decided then to visit Luigi Cremona

Luigi Cremona

Luigi Cremona was an Italian mathematician. His life was devoted to the study of geometry and reforming advanced mathematical teaching in Italy. His reputation mainly rests on his Introduzione ad una teoria geometrica delle curve piane...

in Milan but he was arrested at Fontainebleau

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 of Fontainebleau...

under suspicion of being a German spy, an event which 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 (present day Oslo

Oslo

Oslo is a municipality, as well as the capital and most populous city in Norway. As a municipality , it was established on 1 January 1838. Founded around 1048 by King Harald III of Norway, the city was largely destroyed by fire in 1624. The city was moved under the reign of Denmark–Norway's King...

), in 1871, with a thesis entitled

*On a class of geometric transformations*. It would be described by Darboux

Jean Gaston Darboux

Jean-Gaston Darboux was a French mathematician.-Life:Darboux made several important contributions to geometry and mathematical analysis . He was a biographer of Henri Poincaré and he edited the Selected Works of Joseph Fourier.Darboux received his Ph.D...

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

Felix Klein

Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...

, who was then at Erlangen and working on the Erlangen program

Erlangen program

An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen...

.

At the end of 1872, Sophus Lie proposed to Anna Birch, then eighteen-year-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

Friedrich Engel (mathematician)

Friedrich Engel was a German mathematician.Engel was born in Lugau, Saxony, as the son of a Lutheran pastor. He attended the Universities of both Leipzig and Berlin, before receiving his doctorate from Leipzig in 1883.Engel studied under Felix Klein at Leipzig, and collaborated with Sophus Lie for...

arrived at Christiania to help him, with the support of Klein

Felix Klein

Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...

and Adolph Mayer

Christian Gustav Adolph Mayer

Christian Gustav Adolph Mayer was a German mathematician.Mayer was born on February 15, 1839 in Leipzig, Germany. His father was a businessman from Leipzig. He studied at the University of Leipzig, University of Göttingen, University of Heidelberg and University of Königsberg...

(who were both professors at Leipzig

University of Leipzig

The University of Leipzig , located in Leipzig in the Free State of Saxony, Germany, is one of the oldest universities in the world and the second-oldest university in Germany...

, 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

University of Leipzig

The University of Leipzig , located in Leipzig in the Free State of Saxony, Germany, is one of the oldest universities in the world and the second-oldest university in Germany...

, replacing Klein, who had moved to Göttingen. In November 1889 he suffered a mental breakdown and had to be hospitalized until June 1890. Lie resigned from his post in May 1898 and returned to Norway in September of that year.

He was made Honorary Member of the London Mathematical Society

London Mathematical Society

-See also:* American Mathematical Society* Edinburgh Mathematical Society* European Mathematical Society* List of Mathematical Societies* Council for the Mathematical Sciences* BCS-FACS Specialist Group-External links:* * *...

in 1878, Member of the French Academy of Sciences

French Academy of Sciences

The French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research...

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

United States National Academy of Sciences

The National Academy of Sciences is a corporation in the United States whose members serve pro bono as "advisers to the nation on science, engineering, and medicine." As a national academy, new members of the organization are elected annually by current members, based on their distinguished and...

in 1895.

Sophus Lie died at the age of 56, due to pernicious anemia

Pernicious anemia

Pernicious anemia is one of many types of the larger family of megaloblastic anemias...

, a disease caused by impaired absorption of vitamin B12

Vitamin B12

Vitamin B12, vitamin B12 or vitamin B-12, also called cobalamin, is a water-soluble vitamin with a key role in the normal functioning of the brain and nervous system, and for the formation of blood. It is one of the eight B vitamins...

.

## Legacy

Lie's principal tool, and one of his greatest achievements, was the discovery that continuous transformation groups (now called, after him, Lie groupLie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

s) could be better understood by "linearizing" them, and studying the corresponding generating vector field

Vector field

In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

s (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

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

.

Hermann Weyl

Hermann Weyl

Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...

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

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

.

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”.

## Books written by Sophus Lie

. Written with the help of Friedrich EngelFriedrich Engel (mathematician)

Friedrich Engel was a German mathematician.Engel was born in Lugau, Saxony, as the son of a Lutheran pastor. He attended the Universities of both Leipzig and Berlin, before receiving his doctorate from Leipzig in 1883.Engel studied under Felix Klein at Leipzig, and collaborated with Sophus Lie for...

.. Written with the help of Friedrich Engel.. Written with the help of Georg Scheffers

Georg Scheffers

Georg Scheffers was a German mathematician specializing in differential geometry. He was born on November 21, 1899 in the village of Altendorf near Holzminden . Scheffers began his university career at the University of Leipzig where he studied with Felix Klein and Sophus Lie...

.. Written with the help of Georg Scheffers.. Written with the help of Friedrich Engel.. Written with the help of Georg Scheffers.

## See also

- Lie bracketLie bracketLie bracket can refer to:*A bilinear binary operation defined on elements of a Lie algebra*Lie bracket of vector fields...
- Lie derivativeLie derivativeIn mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
- Lie sphere geometryLie sphere geometryLie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century...
- Lie's third theoremLie's third theoremIn mathematics, Lie's third theorem often means the result that states that any finite-dimensional Lie algebra g, over the real numbers, is the Lie algebra associated to some Lie group G. The relationship to the history has though become confused....
- Carathéodory-Jacobi-Lie theoremCarathéodory-Jacobi-Lie theoremThe Carathéodory–Jacobi–Lie theorem is a theorem in symplectic geometry which generalizes Darboux's theorem.-Statement:Let M be a 2n-dimensional symplectic manifold with symplectic form ω...
- List of simple Lie groups