Editing Special unitary group (section)

== SU(1, 1) ==
<math>\mathrm{SU}(1,1) = \left \{ \begin{pmatrix}u & v \\ v^* & u^* \end{pmatrix} \in M(2,\mathbb{C}): u u^* - v v^* = 1 \right \},</math> where <math>~u^*~</math> denotes the [[complex conjugate]] of the complex number {{mvar|u}}.

This group is isomorphic to {{math|SL(2,ℝ)}} and {{math|Spin(2,1)}}<ref>{{cite book |first=Robert |last=Gilmore |year=1974 |title=Lie Groups, Lie Algebras and some of their Applications |pages=52, 201−205 |publisher=[[John Wiley & Sons]] |mr=1275599}}</ref> where the numbers separated by a comma refer to the [[signature (quadratic form)|signature]] of the [[quadratic form]] preserved by the group. The expression <math>~u u^* - v v^*~</math> in the definition of {{math|SU(1,1)}} is an [[Hermitian form]] which becomes an [[isotropic quadratic form]] when {{mvar|u}} and {{math|''v''}} are expanded with their real components.

An early appearance of this group was as the "unit sphere" of [[coquaternion]]s, introduced by [[James Cockle]] in 1852. Let

<math display="block">
  j = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\,, \quad
  k = \begin{bmatrix} 1 & \;~0 \\ 0 & -1 \end{bmatrix}\,, \quad
  i = \begin{bmatrix} \;~0 & 1 \\ -1 & 0 \end{bmatrix}~.
</math>

Then <math>~j\,k = \begin{bmatrix} 0 & -1 \\ 1 & \;~0 \end{bmatrix} = -i ~,~</math> <math>~ i\,j\,k = I_2 \equiv \begin{bmatrix} 1 & 0 \\ 0 &  1 \end{bmatrix}~,~</math> the 2×2 identity matrix, <math>~k\,i = j ~,</math> and <math>\;i\,j = k \;,</math> and the elements {{mvar|i, j,}} and {{mvar|k}} all [[anticommutative property|anticommute]], as in [[quaternion]]s. Also <math>i</math> is still a square root of {{math|−''I''{{sub|2}}}} (negative of the identity matrix), whereas <math>~j^2 = k^2 = +I_2~</math> are not, unlike in quaternions. For both quaternions and [[coquaternion]]s, all scalar quantities are treated as implicit multiples of {{mvar|I}}{{sub|2}} and notated as {{math|'''1'''}}.

The coquaternion <math>~q = w + x\,i + y\,j + z\,k~</math> with scalar {{mvar|w}}, has conjugate <math>~q = w - x\,i - y\,j - z\,k~</math> similar to Hamilton's quaternions. The quadratic form is <math>~q\,q^* = w^2 + x^2 - y^2 - z^2.</math>

Note that the 2-sheet [[hyperboloid]] <math>\left\{ x i + y j + z k : x^2 - y^2 - z^2 = 1 \right\}</math> corresponds to the [[imaginary unit]]s in the algebra so that any point {{mvar|p}} on this hyperboloid can be used as a '''pole''' of a sinusoidal wave according to [[Euler's formula]].

The hyperboloid is stable under {{math|SU(1, 1)}}, illustrating the isomorphism with {{math|Spin(2, 1)}}. The variability of the pole of a wave, as noted in studies of [[polarization (waves)|polarization]], might view [[elliptical polarization]] as an exhibit of the elliptical shape of a wave with {{nowrap|pole <math>~p \ne \pm i~</math>.}} The [[Poincaré sphere (optics)|Poincaré sphere]] model used since 1892 has been compared to a 2-sheet hyperboloid model,<ref>{{cite journal |first1=R.D. |last1=Mota |first2=D. |last2=Ojeda-Guillén |first3=M. |last3=Salazar-Ramírez |first4=V.D. |last4=Granados |year=2016 |title=SU(1,1) approach to Stokes parameters and the theory of light polarization |journal=Journal of the Optical Society of America B |volume=33 |issue=8 |pages=1696–1701 |arxiv=1602.03223 |doi=10.1364/JOSAB.33.001696|bibcode=2016JOSAB..33.1696M |s2cid=119146980 }}</ref> and the practice of [[SU(1,1) interferometry|{{math|SU(1, 1)}} interferometry]] has been introduced.

When an element of {{math|SU(1, 1)}} is interpreted as a [[Möbius transformation]], it leaves the [[unit disk]] stable, so this group represents the [[motion (geometry)|motion]]s of the [[Poincaré disk model]] of hyperbolic plane geometry. Indeed, for a point {{math|<big>[</big>z, 1<big>]</big>}} in the [[complex projective line]], the action of {{math|SU(1,1)}} is given by

<math display="block">\begin{pmatrix}u & v \\ v^* & u^* \end{pmatrix}\,\bigl[\;z,\;1\;\bigr] = [\;u\,z + v, \, v^*\,z +u^*\;] \, = \, \left[\;\frac{uz + v}{v^*z +u^*}, \, 1 \;\right]</math>

since in [[projective coordinates]] <math>(\;u\,z + v, \; v^*\,z +u^*\;) \thicksim \left(\;\frac{\,u\,z + v\,}{v^*\,z +u^*}, \; 1 \;\right).</math>

Writing <math>\;suv + \overline{suv} = 2\,\Re\mathord\bigl(\,suv\,\bigr)\;,</math> complex number arithmetic shows

<math display="block">\bigl|u\,z + v\bigr|^2 = S + z\,z^* \quad \text{ and } \quad \bigl|v^*\,z + u^*\bigr|^2 = S + 1~,</math>
where <math>S = v\,v^* \left(z\,z^* + 1\right) + 2\,\Re\mathord\bigl(\,uvz\,\bigr).</math>

Therefore, <math>z\,z^* < 1 \implies \bigl|uz + v\bigr| < \bigl|\,v^*\,z + u^*\,\bigr|</math> so that their ratio lies in the open disk.<ref>{{cite book |author-link=C. L. Siegel |last=Siegel |first=C. L. |year=1971 |title=Topics in Complex Function Theory |volume=2 |pages=13–15 |translator1-last=Shenitzer |translator1-first=A. |translator2-last=Tretkoff |translator2-first=M. |publisher=Wiley-Interscience |isbn=0-471-79080 X}}</ref>