Editing Lie group (section)

== Overview ==
[[File:Circle as Lie group.svg|right|thumb|The set of all [[complex number]]s with [[absolute value]] 1 (corresponding to points on the [[circle]] of center 0 and radius 1 in the [[complex plane]]) is a Lie group under complex multiplication:  the [[circle group]].]]
Lie groups are [[smoothness|smooth]] [[differentiable manifold]]s and as such can be studied using [[differential calculus]], in contrast with the case of more general [[topological group]]s. One of the key ideas in the theory of Lie groups is to replace the ''global'' object, the group, with its ''local'' or linearized version, which Lie himself called its "infinitesimal group" and which has since become known as its [[Lie algebra]].

Lie groups play an enormous role in modern [[geometry]], on several different levels. [[Felix Klein]] argued in his [[Erlangen program]] that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties [[Invariant (mathematics)|invariant]]. Thus [[Euclidean geometry]] corresponds to the choice of the group [[Euclidean group|E(3)]] of distance-preserving transformations of the Euclidean space {{tmath|1= \mathbb{R}^3 }}, [[conformal geometry]] corresponds to enlarging the group to the [[conformal group]], whereas in [[projective geometry]] one is interested in the properties invariant under the [[projective group]]. This idea later led to the notion of a [[G-structure|''G''-structure]], where ''G'' is a Lie group of "local" symmetries of a manifold.

Lie groups (and their associated Lie algebras) play a major role in modern physics, with the Lie group typically playing the role of a symmetry of a physical system. Here, the [[Representation of a Lie group|representations]] of the Lie group (or of its [[Lie algebra representation|Lie algebra]]) are especially important. Representation theory [[Particle physics and representation theory|is used extensively in particle physics]]. Groups whose representations are of particular importance include [[3D_rotation_group|the rotation group SO(3)]] (or its [[3D_rotation_group#Connection_between_SO(3)_and_SU(2)|double cover SU(2)]]),  [[Clebsch–Gordan coefficients for SU(3)#Representations of the SU.283.29 group|the special unitary group SU(3)]] and the [[Representation theory of the Poincaré group|Poincaré group]].

On a "global" level, whenever a Lie group [[Group action (mathematics)|acts]] on a geometric object, such as a [[Riemannian manifold|Riemannian]] or a [[symplectic manifold]], this action provides a measure of rigidity and yields a rich algebraic structure. The presence of continuous symmetries expressed via a [[Lie group action]] on a manifold places strong constraints on its geometry and facilitates [[global analysis|analysis]] on the manifold. Linear actions of Lie groups are especially important, and are studied in [[representation theory]].

In the 1940s–1950s, [[Ellis Kolchin]], [[Armand Borel]], and [[Claude Chevalley]] realised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of [[algebraic group]]s defined over an arbitrary [[field (mathematics)|field]]. This insight opened new possibilities in pure algebra, by providing a uniform construction for most [[finite simple group]]s, as well as in [[algebraic geometry]]. The theory of [[automorphic form]]s, an important branch of modern [[number theory]], deals extensively with analogues of Lie groups over [[adele ring]]s; [[p-adic number|''p''-adic]] Lie groups play an important role, via their connections with Galois representations in number theory.