Editing Group (mathematics) (section)

=== Lie groups ===
{{Main|Lie group}}
A ''Lie group'' is a group that also has the structure of a [[differentiable manifold]]; informally, this means that it [[diffeomorphism|looks locally like]] a Euclidean space of some fixed dimension.{{sfn|Warner|1983}} Again, the definition requires the additional structure, here the manifold structure, to be compatible: the multiplication and inverse maps are required to be [[smooth map|smooth]].

A standard example is the general linear group introduced above: it is an [[open subset]] of the space of all <math>n</math>-by-<math>n</math> matrices, because it is given by the inequality
<math display=block>\det (A) \ne 0,</math>
where <math>A</math> denotes an <math>n</math>-by-<math>n</math> matrix.{{sfn|Borel|1991}}

Lie groups are of fundamental importance in modern physics: [[Noether's theorem]] links continuous symmetries to [[conserved quantities]].{{sfn|Goldstein|1980}} [[Rotation]], as well as translations in [[space]] and [[time]], are basic symmetries of the laws of [[mechanics]]. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.{{efn|See [[Schwarzschild metric]] for an example where symmetry greatly reduces the complexity analysis of physical systems.}} Another example is the group of [[Lorentz transformation]]s, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of [[Minkowski space]]. The latter serves—in the absence of significant [[gravitation]]—as a model of [[spacetime]] in [[special relativity]].{{sfn|Weinberg|1972}} The full symmetry group of Minkowski space, i.e., including translations, is known as the [[Poincaré group]]. By the above, it plays a pivotal role in special relativity and, by implication, for [[quantum field theories]].{{sfn|Naber|2003}} [[Local symmetry|Symmetries that vary with location]] are central to the modern description of physical interactions with the help of [[gauge theory]]. An important example of a gauge theory is the [[Standard Model]], which describes three of the four known [[fundamental force]]s and classifies all known [[elementary particle]]s.{{sfn|Zee|2010}}