Editing Conformal geometry (section)

====The conformal Lie algebras====
To describe the groups and algebras involved in the flat model space, fix the following form on {{nowrap|'''R'''<sup>''p''+1,''q''+1</sup>}}:

:<math>
Q=\begin{pmatrix}
0&0&-1\\
0&J&0\\
-1&0&0
\end{pmatrix}
</math>
where ''J'' is a quadratic form of signature {{nowrap|(''p'', ''q'')}}.  Then {{nowrap|1=''G'' = O(''p'' + 1, ''q'' + 1)}} consists of {{nowrap|(''n'' + 2) × (''n'' + 2)}} matrices stabilizing {{nowrap|1=''Q'' : <sup>t</sup>''MQM'' = ''Q''}} (the superscript ''t'' means transpose).  The Lie algebra admits a [[Cartan decomposition]]

:<math>\mathbf{g}=\mathbf{g}_{-1}\oplus\mathbf{g}_0\oplus\mathbf{g}_1</math>

where

:<math>
\mathbf{g}_{-1} = \left\{\left.
\begin{pmatrix}
0&^tp&0\\
0&0&J^{-1}p\\
0&0&0
\end{pmatrix}\right| p\in\mathbb{R}^n\right\},\quad 
\mathbf{g}_{-1} = \left\{\left.
\begin{pmatrix}
0&0&0\\
^tq&0&0\\
0&qJ^{-1}&0
\end{pmatrix}\right| q\in(\mathbb{R}^n)^*\right\}
</math>

:<math>
\mathbf{g}_0 = \left\{\left.
\begin{pmatrix}
-a & 0 & 0\\
0 & A & 0\\
0 & 0 & a
\end{pmatrix} \right| A \in \mathfrak{so} ( p , q ) , a \in \mathbb{R} \right\} .</math>

Alternatively, this decomposition agrees with a natural Lie algebra structure defined on {{nowrap|'''R'''<sup>''n''</sup> ⊕ '''cso'''(''p'', ''q'') ⊕ ('''R'''<sup>''n''</sup>)<sup>∗</sup>}}.

The stabilizer of the null ray pointing up the last coordinate vector is given by the [[Borel subalgebra]]

:'''h''' = '''g'''<sub>0</sub> ⊕ '''g'''<sub>1</sub>.