In mathematics, a Lie group ( /ˈliː/) is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous transformation groups.
Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. They provide a natural framework for analysing the continuous symmetries of differential equations (differential Galois theory), in much the same way as permutation groups are used in Galois theory for analysing the discrete symmetries of algebraic equations. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations.
Read more about Lie Group: Overview, Definitions and Examples, More Examples of Lie Groups, Early History, The Concept of A Lie Group, and Possibilities of Classification, Properties, Types of Lie Groups and Structure Theory, The Lie Algebra Associated With A Lie Group, Homomorphisms and Isomorphisms, The Exponential Map, Infinite Dimensional Lie Groups
Famous quotes containing the words lie and/or group:
“Side by side, their faces blurred,
The earl and countess lie in stone....”
—Philip Larkin (1922–1986)
“Belonging to a group can provide the child with a variety of resources that an individual friendship often cannot—a sense of collective participation, experience with organizational roles, and group support in the enterprise of growing up. Groups also pose for the child some of the most acute problems of social life—of inclusion and exclusion, conformity and independence.”
—Zick Rubin (20th century)