Editing Möbius transformation (section)

=== Parabolic transforms ===
A non-identity Möbius transformation defined by a matrix <math>\mathfrak{H}</math> of determinant one is said to be ''parabolic'' if
<math display="block">\operatorname{tr}^2\mathfrak{H} = (a+d)^2 = 4</math>
(so the trace is plus or minus 2; either can occur for a given transformation since <math>\mathfrak{H}</math> is determined only up to sign). In fact one of the choices for <math>\mathfrak{H}</math> has the same [[characteristic polynomial]] {{nowrap|''X''<sup>2</sup> − 2''X'' + 1}} as the identity matrix, and is therefore [[unipotent]]. A Möbius transform is parabolic if and only if it has exactly one fixed point in the [[Riemann sphere|extended complex plane]] <math>\widehat{\Complex} = \Complex\cup\{\infty\}</math>, which happens if and only if it can be defined by a matrix [[conjugacy class|conjugate to]]
<math display="block">\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}</math>
which describes a translation in the complex plane.

The set of all parabolic Möbius transformations with a ''given'' fixed point in <math>\widehat{\Complex}</math>, together with the identity, forms a [[group (mathematics)|subgroup]] isomorphic to the group of matrices
<math display="block">\left\{\begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix} \mid b\in\Complex\right\};</math>
this is an example of the [[unipotent|unipotent radical]] of a [[Borel subgroup]] (of the Möbius group, or of {{nowrap|SL(2, '''C''')}} for the matrix group; the notion is defined for any [[reductive Lie group]]).