Editing Lie group (section)

== Definitions and examples ==
A '''real Lie group''' is a [[group (mathematics)|group]] that is also a finite-dimensional real [[Differentiable manifold#Definition|smooth manifold]], in which the group operations of [[multiplication]] and inversion are [[smooth map]]s.  Smoothness of the group multiplication
: <math> \mu:G\times G\to G\quad \mu(x,y)=xy</math>

means that ''μ'' is a smooth mapping of the [[Manifold#Cartesian products|product manifold]] {{nowrap|''G'' × ''G''}} into ''G''.  The two requirements can be combined to the single requirement that the mapping
: <math>(x,y)\mapsto x^{-1}y</math>
be a smooth mapping of the product manifold into ''G''.

=== First examples ===
* The 2×2 [[real number|real]] [[invertible matrix|invertible matrices]] form a group under multiplication, called [[general linear group|general linear group of degree 2]] and denoted by <math>\operatorname{GL}(2, \mathbb{R})</math> or by {{tmath|1= \operatorname{GL}_2(\mathbb{R}) }}: <math display="block">\operatorname{GL}(2, \mathbb{R}) = \left\{A = \begin{pmatrix}a & b\\c & d\end{pmatrix} : \det A = ad-bc \ne 0\right\}.</math> This is a four-dimensional [[compact space|noncompact]] real Lie group; it is an open subset of {{tmath|1= \mathbb R^4 }}. This group is [[connected space|disconnected]]; it has two connected components corresponding to the positive and negative values of the [[determinant]].
* The [[rotation (mathematics)|rotation]] matrices form a [[subgroup]] of {{tmath|1= \operatorname{GL}(2, \mathbb{R}) }}, denoted by {{tmath|1= \operatorname{SO}(2, \mathbb{R}) }}. It is a Lie group in its own right: specifically, a one-dimensional compact connected Lie group which is [[diffeomorphic]] to the [[circle]]. Using the rotation angle <math>\varphi</math> as a parameter, this group can be [[parametric equations|parametrized]] as follows: <math display="block">\operatorname{SO}(2, \mathbb{R}) = \left\{\begin{pmatrix} \cos\varphi & -\sin\varphi \\ \sin\varphi & \cos\varphi \end{pmatrix} : \varphi \in \mathbb{R}\ /\ 2\pi\mathbb{Z}\right\}.</math> Addition of the angles corresponds to multiplication of the elements of {{tmath|1= \operatorname{SO}(2, \mathbb{R}) }}, and taking the opposite angle corresponds to inversion. Thus both multiplication and inversion are differentiable maps.
* The [[Affine group#Matrix representation|affine group of one dimension]] is a two-dimensional matrix Lie group, consisting of <math>2 \times 2</math> real, upper-triangular matrices, with the first diagonal entry being positive and the second diagonal entry being 1. Thus, the group consists of matrices of the form <math display="block"> A= \left( \begin{array}{cc} a & b\\ 0 & 1 \end{array}\right),\quad a>0,\, b \in \mathbb{R}.</math>

=== Non-example ===
{{further|Linear flow on the torus}}
We now present an example of a group with an [[uncountable set|uncountable]] number of elements that is not a Lie group under a certain topology.  The group given by 
: <math>H = \left\{\left(\begin{matrix}e^{2\pi i\theta} & 0\\0 & e^{2\pi ia\theta}\end{matrix}\right) :\, \theta \in \mathbb{R}\right\} \subset \mathbb{T}^2 = \left\{\left(\begin{matrix}e^{2\pi i\theta} & 0\\0 & e^{2\pi i\phi}\end{matrix}\right) :\, \theta, \phi \in \mathbb{R}\right\},</math>
with <math>a \in \mathbb R \setminus \mathbb Q</math> a ''fixed'' [[irrational number]], is a subgroup of the [[torus]] <math>\mathbb T^2</math> that is not a Lie group when given the [[subspace topology]].<ref>{{harvnb|Rossmann|2001|loc=Chapter 2}}</ref> If we take any small [[neighborhood (mathematics)|neighborhood]] <math>U</math> of a point <math>h</math> in {{tmath|1= H }}, for example, the portion of <math>H</math> in <math>U</math> is disconnected. The group <math>H</math> winds repeatedly around the torus without ever reaching a previous point of the spiral and thus forms a [[dense set|dense]] subgroup of {{tmath|1= \mathbb T^2 }}.

[[File:Irrational line on a torus.png|thumb|right|A portion of the group <math>H</math> inside {{tmath|1= \mathbb T^2 }}. Small neighborhoods of the element <math>h\in H</math> are disconnected in the subset topology on {{tmath|1= H }}]]

The group <math>H</math> can, however, be given a different topology, in which the distance between two points <math>h_1,h_2\in H</math> is defined as the length of the shortest path ''in the group'' <math>H</math> joining <math>h_1</math> to {{tmath|1= h_2 }}. In this topology, <math>H</math> is identified homeomorphically with the real line by identifying each element with the number <math>\theta</math> in the definition of {{tmath|1= H }}. With this topology, <math>H</math> is just the group of real numbers under addition and is therefore a Lie group.

The group <math>H</math> is an example of a "[[Lie group#Lie subgroup|Lie subgroup]]" of a Lie group that is not closed. See the discussion below of Lie subgroups in the section on basic concepts.

=== Matrix Lie groups ===

Let <math>\operatorname{GL}(n, \mathbb{C})</math> denote the group of <math>n\times n</math> invertible matrices with entries in {{tmath|1= \mathbb{C} }}. Any [[Closed subgroup theorem|closed subgroup]] of <math>\operatorname{GL}(n, \mathbb{C})</math> is a Lie group;<ref>{{harvnb|Hall|2015}} Corollary 3.45</ref> Lie groups of this sort are called '''matrix Lie groups.''' Since most of the interesting examples of Lie groups can be realized as matrix Lie groups, some textbooks restrict attention to this class, including those of Hall,{{sfn|ps=|Hall|2015}} Rossmann,<ref>{{harvnb|Rossmann|2001}}</ref> and Stillwell.<ref>{{harvnb|Stillwell|2008}}</ref> 
Restricting attention to matrix Lie groups simplifies the definition of the Lie algebra and the exponential map. The following are standard examples of matrix Lie groups.
* The [[special linear group]]s over <math>\mathbb{R}</math> and {{tmath|1= \mathbb{C} }}, <math>\operatorname{SL}(n, \mathbb{R})</math> and {{tmath|1= \operatorname{SL}(n, \mathbb{C}) }}, consisting of <math>n\times n</math> matrices with determinant one and entries in <math>\mathbb{R}</math> or <math>\mathbb{C}</math>
* The [[unitary group]]s and special unitary groups, <math>\operatorname{U}(n,\mathbb{C})</math> and {{tmath|1= \operatorname{SU}(n,\mathbb{C}) }}, consisting of <math>n\times n</math> complex matrices satisfying <math>U^*=U^{-1}</math> (and also <math>\det(U)=1</math> in the case of <math>\operatorname{SU}(n)</math>)
* The [[orthogonal group]]s and special orthogonal groups, <math>\operatorname{O}(n,\mathbb{R})</math> and {{tmath|1= \operatorname{SO}(n,\mathbb{R}) }}, consisting of <math>n\times n</math> real matrices satisfying <math>R^\mathrm{T}=R^{-1}</math> (and also <math>\det(R)=1</math> in the case of <math>\operatorname{SO}(n,\mathbb{R})</math>)
All of the preceding examples fall under the heading of the [[classical group]]s.

=== Related concepts ===
A '''[[complex Lie group]]''' is defined in the same way using [[complex manifold]]s rather than real ones (example: <math>\operatorname{SL}(2, \mathbb{C})</math>), and holomorphic maps. Similarly, using an alternate [[Complete metric space#Completion|metric completion]] of {{tmath|1= \mathbb{Q} }}, one can define a '''''p''-adic Lie group''' over the [[p-adic number|''p''-adic numbers]], a topological group which is also an analytic ''p''-adic manifold, such that the group operations are analytic. In particular, each point has a ''p''-adic neighborhood.

[[Hilbert's fifth problem]] asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples. The answer to this question turned out to be negative: in 1952, [[Andrew Gleason|Gleason]], [[Deane Montgomery|Montgomery]] and [[Leo Zippin|Zippin]] showed that if ''G'' is a topological manifold with continuous group operations, then there exists exactly one analytic structure on ''G'' which turns it into a Lie group (see also [[Hilbert–Smith conjecture]]). If the underlying manifold is allowed to be infinite-dimensional (for example, a [[Hilbert manifold]]), then one arrives at the notion of an infinite-dimensional Lie group.  It is possible to define analogues of many [[group of Lie type|Lie groups over finite fields]], and these give most of the examples of [[finite simple group]]s.

The language of [[category theory]] provides a concise definition for Lie groups: a Lie group is a [[group object]] in the [[category (mathematics)|category]] of smooth manifolds. This is important, because it allows generalization of the notion of a Lie group to [[supergroup (physics)|Lie supergroups]]. This categorical point of view leads also to a different generalization of Lie groups, namely [[Lie groupoid|Lie groupoids]], which are [[Groupoid object|groupoid objects]] in the category of smooth manifolds with a further requirement.

=== Topological definition ===
A Lie group can be defined as a ([[Hausdorff space|Hausdorff]]) [[topological group]] that, near the identity element, looks like a transformation group, with no reference to differentiable manifolds.<ref>{{harvnb|Kobayashi|Oshima|2005|loc=Definition 5.3}}</ref> First, we define an '''immersely linear Lie group''' to be a subgroup ''G'' of the general linear group <math>\operatorname{GL}(n, \mathbb{C})</math> such that
# for some neighborhood ''V'' of the identity element ''e'' in ''G'', the topology on ''V'' is the [[subspace topology]] of <math>\operatorname{GL}(n, \mathbb{C})</math> and ''V'' is closed in {{tmath|1= \operatorname{GL}(n, \mathbb{C}) }}.
# ''G'' has at most [[countable set|countably many]] connected components.
(For example, a closed subgroup of {{tmath|1= \operatorname{GL}(n, \mathbb{C}) }}; that is, a matrix Lie group satisfies the above conditions.)

Then a ''Lie group'' is defined as a topological group that (1) is locally isomorphic near the identities to an immersely linear Lie group and (2) has at most countably many connected components. Showing the topological definition is equivalent to the usual one is technical (and the beginning readers should skip the following) but is done roughly as follows:
# Given a Lie group ''G'' in the usual manifold sense, the [[Lie group–Lie algebra correspondence]] (or a version of [[Lie's third theorem]]) constructs an immersed Lie subgroup <math>G' \subset \operatorname{GL}(n, \mathbb{C})</math> such that <math>G, G'</math> share the same Lie algebra; thus, they are locally isomorphic. Hence, <math>G</math> satisfies the above topological definition.
# Conversely, let <math>G</math> be a topological group that is a Lie group in the above topological sense and choose an immersely linear Lie group <math>G'</math> that is locally isomorphic to {{tmath|1= G }}. Then, by a version of the [[closed subgroup theorem]], <math>G'</math> is a [[real-analytic manifold]] and then, through the local isomorphism, ''G'' acquires a structure of a manifold near the identity element. One then shows that the group law on ''G'' can be given by formal [[power series]];{{efn|This is the statement that a Lie group is a [[formal Lie group]]. For the latter concept, see Bruhat.<ref>{{Cite web |first=F. |last=Bruhat |url=http://www.math.tifr.res.in/~publ/ln/tifr14.pdf |title=Lectures on Lie Groups and Representations of Locally Compact Groups |date=1958 |publisher=Tata Institute of Fundamental Research, Bombay}}</ref>}} so the group operations are real-analytic and <math>G</math> itself is a real-analytic manifold.

The topological definition implies the statement that if two Lie groups are isomorphic as topological groups, then they are isomorphic as Lie groups. In fact, it states the general principle that, to a large extent, ''the topology of a Lie group'' together with the group law determines the geometry of the group.