Editing Lie group (section)

== More examples of Lie groups ==
{{see also|Table of Lie groups|List of simple Lie groups}}

Lie groups occur in abundance throughout mathematics and physics. [[Matrix group]]s or [[algebraic group]]s are (roughly) groups of matrices (for example, [[orthogonal group|orthogonal]] and [[symplectic group]]s), and these give most of the more common examples of Lie groups.

=== Dimensions one and two ===
The only connected Lie groups with dimension one are the real line <math>\mathbb{R}</math> (with the group operation being addition) and the [[circle group]] <math>S^1</math> of complex numbers with absolute value one (with the group operation being multiplication). The <math>S^1</math> group is often denoted as {{tmath|1= \operatorname{U}(1) }}, the group of <math>1\times 1</math> unitary matrices.

In two dimensions, if we restrict attention to simply connected groups, then they are classified by their Lie algebras. There are (up to isomorphism) only two Lie algebras of dimension two. The associated simply connected Lie groups are <math>\mathbb{R}^2</math> (with the group operation being vector addition) and the affine group in dimension one, described in the previous subsection under "first examples".

=== Additional examples ===
* The [[Special unitary group#The group SU(2)|group SU(2)]] is the group of <math>2\times 2</math> unitary matrices with determinant {{tmath|1= 1 }}. Topologically, <math>\text{SU}(2)</math> is the {{tmath|1= 3 }}-sphere {{tmath|1= S^3 }}; as a group, it may be identified with the group of [[unit quaternion]]s.
* The [[Heisenberg group]] is a connected [[nilpotent group|nilpotent]] Lie group of dimension {{tmath|1= 3 }}, playing a key role in [[quantum mechanics]].
* The [[Lorentz group]] is a 6-dimensional Lie group of linear [[isometry|isometries]] of the [[Minkowski space]].
* The [[Poincaré group]] is a 10-dimensional Lie group of [[affine transformation|affine]] isometries of the Minkowski space.
* The [[exceptional Lie group]]s of types [[G2 (mathematics)|G<sub>2</sub>]], [[F4 (mathematics)|F<sub>4</sub>]], [[E6 (mathematics)|E<sub>6</sub>]], [[E7 (mathematics)|E<sub>7</sub>]], [[E8 (mathematics)|E<sub>8</sub>]] have dimensions 14, 52, 78, 133, and 248. Along with the A–B–C–D series of [[simple Lie group]]s, the exceptional groups complete the list of simple Lie groups. 
*The [[symplectic group]] <math>\text{Sp}(2n,\mathbb{R})</math> consists of all <math>2n \times 2n</math> matrices preserving a ''[[symplectic form]]'' on {{tmath|1= \mathbb{R}^{2n} }}. It is a connected Lie group of dimension {{tmath|1= 2n^2 + n }}.

=== Constructions ===
There are several standard ways to form new Lie groups from old ones:
* The product of two Lie groups is a Lie group.
* Any [[Closed set|topologically closed]] subgroup of a Lie group is a Lie group. This is known as the [[closed subgroup theorem]] or '''Cartan's theorem'''.
* The quotient of a Lie group by a closed normal subgroup is a Lie group.
* The [[universal cover]] of a connected Lie group is a Lie group. For example, the group <math>\mathbb{R}</math> is the universal cover of the circle group {{tmath|1= S^1 }}. In fact any covering of a differentiable manifold is also a differentiable manifold, but by specifying ''universal'' cover, one guarantees a group structure (compatible with its other structures).

=== Related notions ===
Some examples of groups that are ''not'' Lie groups (except in the trivial sense that any group having at most countably many elements<!-- by convention, a manifold is second countable so we need to exclude an uncountable set --> can be viewed as a 0-dimensional Lie group, with the [[discrete topology]]), are:
* Infinite-dimensional groups, such as the additive group of an infinite-dimensional real vector space, or the space of smooth functions from a manifold <math>X</math> to a Lie group {{tmath|1= G }}, {{tmath|1= C^\infty(X,G) }}. These are not Lie groups as they are not ''finite-dimensional'' manifolds.
* Some [[totally disconnected group]]s, such as the [[Galois group]] of an infinite extension of fields,  or the additive group of the ''p''-adic numbers. These are not Lie groups because their underlying spaces are not real manifolds. (Some of these groups are "''p''-adic Lie groups".) In general, only topological groups having similar [[local property|local properties]] to '''R'''<sup>''n''</sup> for some positive integer ''n'' can be Lie groups (of course they must also have a differentiable structure).