Editing Conformal geometry (section)

====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}}