Lie's principal tool, and one of his greatest achievements, was the discovery that continuous transformation groups (now called after him Lie groups) 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, now called the commutator bracket, and have the structure of what is today called a Lie algebra.
He died at the age of 56, suffering from pernicious anemia, a disease that today would be both preventable and treatable. Pernicious anemia occurs if the body does not receive the essential B12 vitamin, usually due to a hereditary lack of the intrinsic factor needed for the body to absorb B12 vitamin through the diet.
Hermann Weyl used Lie's work on group theory in his papers from 1922 and 1923, and Lie's groups today play a role in both quantum physics and quantum mechanics.
However, as noted by Helgason, "to see Lie's work in perspective it is important to realize that his first mathematical love was geometry and that this attachment remained with him all his life" . Indeed Lie himself was particularly pleased with the geometry of spheres and the line-sphere correspondence which bear his name .
Seminar Sophus Lie is an international seminar of mathematicians interested in the theory of Lie groups and their wider horizon. It was founded around 1989-90 when, during the Volkskammer Government of the German Democratic Republic in 1989, open contacts between mathematicians in East- and Westgermany became a reality for the first time since 1961. Several mathematicians located at the Technische Hochschule Darmstadt, the University of Erlangen, the University of Greifswald, and the University of Leipzig organized informally the Seminar with financial support by the Deutsche Forschungsgemeinschaft and met for the first seminar session at the University of Leipzig in January 1991. The Seminar usually meets once a semester since that time. Over the years the seminar had participants from more and more countries, e.g. Austria, France, Hungary, Poland, and Romania. Also the meeting places went beyond Germany, and by now it is a European event.