Sophus lie

Marius Sophus Lie was a Norwegian mathematician (December 17, 1842 – February 18, 1899) who created much of the theory of continuous symmetry, and applied it to the study of geometry. and differential equations.

Lie's main tool, and one of its greatest achievements, was the discovery that continuous transformation groups (now called Lie groups), could be better understood by "linearizing" them, and studying the corresponding generator vector fields (the so-called infinitesimal generators). The generators obey a linearized version of the group law called the bracket or commutator, and have the structure of what is now called in his memory a Lie algebra.

The most complicated Lie group, called E8, is a 248-dimensional object that describes a 57-dimensional structure that was conceptualized and designed by a team of 18 mathematicians in four years of work, culminating in early 2007. For For this they used a computer from the University of Washington equipped with the Sage computer algebra system and 64 Gigabytes of RAM to be able to store the resolution matrix in memory.

Biography

His first mathematical paper, "Repräsentation der Imaginären der Plangeometrie", was published in 1869 by the Academy of Sciences in Christiania and also by "Crelles magazine". 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 Klein joined him two months later. There, they met Camille Jordan and Gastón Darboux. But on July 19, 1870, the Franco-Prussian War began and Klein (who was Prussian) had to leave France very quickly. Lie went to Fontainebleau where, after a while, he was arrested on suspicion of being a German spy, an event that made him famous in Norway. He was released from prison after a month, thanks to Darboux's intervention.

Lie earned his doctorate at the University of Christiania (now Oslo) in 1871 with a thesis entitled "Over en Classe geometriske Transformationer" (On a class of geometric transformations).

Darboux would describe it as one of the most beautiful discoveries in modern geometry. The following year, the Norwegian Parliament established an extraordinary professorship for him. That same year, Lie visited Klein, who was in Erlangen working on the Erlangen Program.

In late 1872, Sophus Lie proposed to Anna Birch, then eighteen, and they married in 1874. The couple had three children: Marie (b. 1877), Dagny (b. 1880), and Herman (b. 1884).

In 1884, Friedrich Engel came to Christiania to help him, with the support of Klein and Adolph Mayer (who were then professors in Leipzig). Engel would help Lie 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 two editors of the Lie's Collected Works.

In 1886 he became a professor in 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 anemia progressed to the point where he decided to return to earth. birthplace of him Consequently, in 1898 he tendered his resignation in May and went home (for good) in September of the same year. He died the following year, in 1899.

He was made an Honorary Member of the London Mathematical Society in 1878, a Fellow of the French Academy of Sciences in 1892, a Foreign Fellow of the Royal Society in 1895, and a Foreign Associate of the United States National Academy of Sciences in 1895.

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

Legacy

The main tool of Lie, and one of his greatest achievements, was the discovery that groups of continuous transformations (now called Lie Groups in his memory) could be better understood by "linearizing" and studying the corresponding generator vector fields (the so-called infinitesimal generators). The generators are subject to a group linearized version, now called a two-operator commutator, and have the structure of what is now called Lie algebra.

Hermann Weyl used Lie's work on group theory in his 1922 and 1923 papers, and Lie groups play an important role in quantum mechanics today. However, the subject of Lie groups as studied today is very different from what Sophus Lie's research was and “among the masters of the century XIX, Lie's work is certainly the least well known in detail.