Lie Group

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 GroupOverview, 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

Other articles related to "lie group, lie groups, groups, group, lie":

Infinite Dimensional Lie Groups
... Lie groups are often defined to be finite dimensional, but there are many groups that resemble Lie groups, except for being infinite dimensional ... The simplest way to define infinite dimensional Lie groups is to model them on Banach spaces, and in this case much of the basic theory is similar to that of finite ... natural examples of infinite dimensional Lie groups are not Banach manifolds ...
Cartan's Equivalence Method - Overview of Cartan's Method
... a pair of manifolds each carrying a G-structure for a structure group G ... g is a function on M taking values in the Lie group G ... For example, if M and N are Riemannian manifolds, then G=O(n) is the orthogonal group and θi and γi are orthonormal coframes of M and N respectively ...
Lie Bracket Of Vector Fields - Examples
... For a Lie group, the Lie algebra is the tangent space at the identity, which can be identified with the left invariant vector fields ... The Lie bracket of the Lie algebra is then the Lie bracket of the left invariant vector fields, which is also left invariant ... For a matrix Lie group, smooth vector fields can be locally represented in the corresponding Lie algebra ...
Lie's Third Theorem
... In mathematics, Lie's third theorem often means the result that states that any finite-dimensional Lie algebra g, over the real numbers, is the Lie algebra associated to some Lie group G ... There were (naturally) two other preceding theorems, of Sophus Lie ... Those relate to the infinitesimal transformations of a transformation group acting on a smooth manifold ...
An Exceptionally Simple Theory Of Everything - Overview - Non-technical Overview
... elementary particles and their interactions is usually through a gauge theory based on a Lie group ... A Lie group is a mathematical structure with many complex symmetries, which can be described as an object with a complex geometry ... with each other according to the geometry of the group and how the particles are related to the group representation ...

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