Editing Conformal geometry (section)

===Two dimensions===

====Minkowski plane====
The [[conformal group]] for the Minkowski quadratic form {{nowrap|1=''q''(''x'', ''y'') = 2''xy''}} in the plane is the [[abelian group|abelian]] [[Lie group]]

:<math> \operatorname{CSO}(1,1) = \left\{ \left. \begin{pmatrix}
e^a&0\\
0&e^b
\end{pmatrix} \right| a , b \in \mathbb{R} \right\} ,</math>

with [[Lie algebra]] {{nowrap|'''cso'''(1, 1)}} consisting of all real diagonal {{nowrap|2 × 2}} matrices.

Consider now the Minkowski plane, <math>\mathbb{R}^2</math> equipped with the metric
: <math> g = 2 \, dx \, dy ~ .</math>

A 1-parameter group of conformal transformations gives rise to a vector field ''X'' with the property that the [[Lie derivative]] of ''g'' along ''X'' is proportional to ''g''.  Symbolically,
:{{math|1='''L'''<sub>''X''</sub> ''g'' = ''λg''}}  &nbsp; for some ''λ''.

In particular, using the above description of the Lie algebra {{nowrap|'''cso'''(1, 1)}}, this implies that
#  '''L'''<sub>''X''</sub>&nbsp; {{nowrap|1=''dx'' = ''a''(''x'') ''dx''}}
#  '''L'''<sub>''X''</sub>&nbsp; {{nowrap|1=''dy'' = ''b''(''y'') ''dy''}} 
for some real-valued functions ''a'' and ''b'' depending, respectively, on ''x'' and ''y''.

Conversely, given any such pair of real-valued functions, there exists a vector field ''X'' satisfying 1. and 2.  Hence the [[Lie algebra]] of infinitesimal symmetries of the conformal structure, the [[Witt algebra]],  is [[Conformal field theory#Two dimensions|infinite-dimensional]].

The conformal compactification of the Minkowski plane is a Cartesian product of two circles {{nowrap|''S''<sup>1</sup> × ''S''<sup>1</sup>}}.  On the [[universal cover]], there is no obstruction to integrating the infinitesimal symmetries, and so the group of conformal transformations is the infinite-dimensional Lie group
:<math>(\mathbb{Z}\rtimes\mathrm{Diff}(S^1))\times(\mathbb{Z}\rtimes\mathrm{Diff}(S^1)) ,</math>
where Diff(''S''<sup>1</sup>) is the [[diffeomorphism group]] of the circle.<ref>[[Paul Ginsparg]] (1989), ''Applied Conformal Field Theory''. {{arxiv|hep-th/9108028}}. Published in ''Ecole d'Eté de Physique Théorique: Champs, cordes et phénomènes critiques/Fields, strings and critical phenomena'' (Les Houches), ed. by E. Brézin and J. Zinn-Justin,  Elsevier Science Publishers B.V.</ref>

The conformal group {{nowrap|CSO(1, 1)}} and its Lie algebra are of current interest in [[two-dimensional conformal field theory]].

{{see also |Virasoro algebra}}

====Euclidean space====
[[Image:Conformal grid before Möbius transformation.svg|thumb|right|A coordinate grid prior to a Möbius transformation]]
[[Image:Conformal grid after Möbius transformation.svg|thumb|right|The same grid after a Möbius transformation]]
The group of conformal symmetries of the quadratic form

:<math>q(z,\bar{z}) = z\bar{z} </math>

is the group {{nowrap|1=GL<sub>1</sub>('''C''') = '''C'''<sup>×</sup>}}, the [[multiplicative group]] of the complex numbers.  Its Lie algebra is {{nowrap|1='''gl'''<sub>1</sub>('''C''') = '''C'''}}.

Consider the (Euclidean) [[complex plane]] equipped with the metric

:<math>g = dz \, d\bar{z}.</math>

The infinitesimal conformal symmetries satisfy
#<math>\mathbf{L}_X \, dz = f(z) \, dz</math>
#<math>\mathbf{L}_X \, d\bar{z} = f(\bar{z}) \, d\bar{z} ,</math>
where ''f'' satisfies the [[Cauchy–Riemann equation]], and so is [[Holomorphic function|holomorphic]] over its domain.  (See [[Witt algebra]].)

The conformal isometries of a domain therefore consist of holomorphic self-maps.  In particular, on the conformal compactification &ndash; the [[Riemann sphere]] &ndash; the conformal transformations are given by the [[Möbius transformation]]s

:<math>z \mapsto \frac{az+b}{cz+d}</math>

where {{nowrap|''ad'' &minus; ''bc''}} is nonzero.