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
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