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

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

**Lie Group**s

...

**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 ...

... 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 ...

... 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 ...

... 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 ...

... 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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