VI.53 Sophus Lie
b. Nordfjordeid (western Norway), 1842; d. Oslo, 1899
Transformation groups; Lie groups; partial differential equations
Lie was twenty-six when he discovered that, in his own words, he “harbored a mathematician.” Before then he had primarily wanted to be an observational astronomer. Later in life, looking back on his career, he said that it was the “audacity of his thinking” more than any formal knowledge and education that had given him a position among the foremost of mathematicians. During a career spanning more than thirty years, Lie produced almost eight thousand pages of mathematics, making him one of the most productive mathematicians of his time.
Lie graduated in general science from the university in Oslo in 1865 but without showing any special aptitude for mathematics. It was not until 1868, when he attended a lecture by the Danish geometer Hieronymus Zeuthen on the work of Chasles, MÖBIUS [VI.30], and Plücker, that he became inspired by modern geometry. He studied the works of Poncelet (projective geometry) and Plücker (line geometry), and wrote a dissertation on “imaginary geometry,” that is, geometry based on complex numbers. In the fall of 1869 he traveled to Berlin, Göttingen, and Paris, where he met mathematicians who would remain friends and colleagues for the rest of his life. In Berlin he met KLEIN [VI.57], in Göttingen he met Clebsch, and in Paris, where he was joined by Klein, he met Darboux and JORDAN [VI.52]. These two had a particular influence on him—Darboux through his theory of surfaces and Jordan through his knowledge of group theory and the work of GALOIS [VI.41]—with the result that he (and Klein) began to recognize the value of group theory for the study of geometry. Lie and Klein published three joint papers on geometrical topics, including one on the so-called Lie line-sphere transformation (the contact transformation, which is a transformation that maps straight lines into spheres and principal tangent curves into curvature lines; and then the study of the geometrical entities that are invariant under such transformations).
When Klein prepared what was to become his famous “Erlanger Programm” (his characterization of geometry as properties invariant under a group action), Lie was with him. This work later created a deep rift between them. (Friendship turned into aloofness and hostility and culminated in the following statement by Lie in 1893: “I am no pupil of Klein’s, nor is the reverse the case, although this would be nearer to the truth.”)
Lie returned (after his first trip abroad) to Oslo and, in 1872, a chair of mathematics at the university was created especially for him. During the early 1870s Lie worked on turning his line-sphere transformation into a general theory of contact transformations. From 1873 he worked on a systematic study of continuous transformation groups (today known as LIE GROUPS [III.48 §1]), his aim being to classify LIE ALGEBRAS [III.48 §§2, 3] and apply the results to the solution of differential equations. He also published studies on MINIMAL SURFACES [III.94 §3.1]. In Norway, however, there was no scientific milieu, and he felt very isolated. In 1884 Klein and his friend Adolf Mayer in Leipzig tried to help him by sending their student Friedrich Engel to study with him and to help him with the formulation and writing of his new ideas. The work that Engel and Lie started together resulted in three volumes, Theorie der Transformationsgruppen (1888-93). In 1886 Lie accepted the professorship in Leipzig (in succession to Klein, who had moved to Göttingen). In Leipzig he became a leading mathematician and a central figure in the European community of mathematicians. Promising new students from both France and the United States were sent to study with him. Besides teaching he continued his research on transformation groups and differential equations, and he solved the so-called Helmholtz space problem (characterizing the geometry of space in terms of groups of transformations). In 1898, the year before he died, Lie returned to Oslo to take up a position created especially for him.
The theory of transformation groups, which Lie initiated and developed in the study of differential equations, has grown into a field of its own, the theory of Lie groups and Lie algebras, which today permeates large parts of mathematics and mathematical physics.
Further Reading
Borel, A. 2001. Essays in the History of Lie Groups and Algebraic Groups. Providence, RI: American Mathematical Society.
Hawkins, T. 2000. Emergence of the Theory of Lie Groups. New York: Springer.
Laudal, O. A., and B. Jahrien, eds. 1994. Proceedings, Sophus Lie Memorial Conference. Oslo: Scandinavian University Press.
Stubhaug, A. 2002. The Mathematician Sophus Lie. Berlin: Springer.