Editing Split-quaternion (section)

== Stratification ==
{{unreferenced section|date=February 2021}}
In this section, the real [[subalgebra]]s generated by a single split-quaternion are studied and classified.

Let {{math|''p'' {{=}} ''w'' + ''x''i + ''y''j + ''z''k}} be a split-quaternion. Its ''real part'' is {{math|1=''w'' =  {{sfrac|1|2}}(''p'' + ''p''{{sup|*}})}}. Let {{math|1=''q'' = ''p'' – ''w'' = {{sfrac|1|2}}(''p'' – ''p''{{sup|*}})}} be its ''nonreal part''. One has {{math|1=''q''{{sup|*}} = –''q''}}, and therefore <math>p^2=w^2+2wq-N(q).</math> It follows that {{math|''p''<sup>2</sup>}} is a real number if and only {{math|''p''}} is either a real number ({{math|1=''q'' = 0}} and {{math|1=''p'' = ''w''}}) or a ''purely nonreal split quaternion'' ({{math|1=''w'' = 0}} and {{math|1=''p'' = ''q''}}).

The structure of the subalgebra <math>\mathbb R[p]</math> generated by {{mvar|p}} follows straightforwardly. One has 
: <math>\mathbb R[p]=\mathbb R[q]=\{a+bq\mid a,b\in\mathbb R\},</math>
and this is a [[commutative algebra (structure)|commutative algebra]]. Its [[dimension (linear algebra)|dimension]] is two except if {{mvar|p}} is real (in this case, the subalgebra is simply <math>\mathbb R</math>). 

The nonreal elements of <math>\mathbb R[p]</math> whose square is real have the form {{math|''aq''}} with <math>a\in \mathbb R.</math> 

Three cases have to be considered, which are detailed in the next subsections.

=== Nilpotent case ===
With above notation, if <math>q^2=0,</math> (that is, if {{math|''q''}} is [[nilpotent]]), then {{math|1=''N''(''q'') = 0}}, that is, <math>x^2-y^2-z^2=0.</math>  This implies that there exist {{mvar|w}} and {{mvar|t}} in <math>\mathbb R</math> such that {{math|0 ≤ ''t'' < 2{{pi}}}} and
: <math>p=w+a\mathrm i + a\cos(t)\mathrm j + a\sin(t)\mathrm k.</math>

This is a parametrization of all split-quaternions whose nonreal part is nilpotent. 

This is also a parameterization of these subalgebras by the points of a circle: the split-quaternions of the form <math>\mathrm i + \cos(t)\mathrm j + \sin(t)\mathrm k</math> form a [[circle]]; a subalgebra generated by a nilpotent element contains exactly one point of the circle; and the circle does not contain any other point.  

The algebra generated by a nilpotent element is isomorphic to <math>\mathbb R[X]/\langle X^2\rangle</math> and to the plane of [[dual number]]s.

=== Imaginary units ===
[[Image:HyperboloidOfTwoSheets.svg|right|thumb|Hyperboloid of two sheets, source of [[imaginary unit]]s]]
This is the case where {{math|''N''(''q'') > 0}}. Letting <math display="inline">n=\sqrt{N(q)},</math>  one has
: <math>q^2 =-q^*q=N(q)=n^2=x^2-y^2-z^2.</math>
It follows that {{math|{{sfrac|''n''}} ''q''}} belongs to the [[hyperboloid of two sheets]] of equation <math>x^2-y^2-z^2=1.</math> Therefore, there are real numbers {{math|''n'', ''t'', ''u''}} such that {{math|0 ≤ ''t'' < 2{{pi}}}} and 
: <math>p=w+n\cosh(u)\mathrm i + n\sinh(u)\cos(t)\mathrm j + n\sinh(u)\sin(t)\mathrm k.</math>

This is a parametrization of all split-quaternions whose nonreal part has a positive norm.

This is also a parameterization of the corresponding subalgebras by the pairs of opposite points of a hyperboloid of two sheets: the split-quaternions of the form <math>\cosh(u)\mathrm i + \sinh(u)\cos(t)\mathrm j + \sinh(u)\sin(t)\mathrm k</math> form a hyperboloid of two sheets; a subalgebra generated by a split-quaternion with a nonreal part of positive norm contains exactly two opposite points on this hyperboloid, one on each sheet; and the hyperboloid does not contain any other point.

The algebra generated by a split-quaternion with a nonreal part of positive norm is isomorphic to <math>\mathbb R[X]/\langle X^2+1\rangle</math> and to the field <math>\Complex</math> of [[complex number]]s.

=== Hyperbolic units ===
[[Image:HyperboloidOfOneSheet.PNG|right|thumb|Hyperboloid of one sheet, source of [[hyperbolic unit]]s.<br>(the vertical axis is called {{mvar|x}} in the article)]]
This is the case where {{math|''N''(''q'') < 0}}. Letting <math display="inline">n=\sqrt{-N(q)},</math>  one has
: <math>q^2 = -q^*q=N(q)=-n^2=x^2-y^2-z^2.</math>
It follows that {{math|{{sfrac|''n''}} ''q''}} belongs to the [[hyperboloid of one sheet]] of equation {{math|1=''y''<sup>2</sup> + ''z''<sup>2</sup> − ''x''<sup>2</sup> = 1}}. Therefore, there are real numbers {{math|''n'', ''t'', ''u''}} such that {{math|0 ≤ ''t'' < 2{{pi}}}} and 
: <math>p=w+n\sinh(u)\mathrm i + n\cosh(u)\cos(t)\mathrm j + n\cosh(u)\sin(t)\mathrm k.</math>

This is a parametrization of all split-quaternions whose nonreal part has a negative norm.

This is also a parameterization of the corresponding subalgebras by the pairs of opposite points of a hyperboloid of one sheet: the split-quaternions of the form <math>\sinh(u)\mathrm i + \cosh(u)\cos(t)\mathrm j + \cosh(u)\sin(t)\mathrm k</math> form a hyperboloid of one sheet; a subalgebra generated by a split-quaternion with a nonreal part of negative norm contains exactly two opposite points on this hyperboloid; and the hyperboloid does not contain any other point.

The algebra generated by a split-quaternion with a nonreal part of negative norm is isomorphic to <math>\mathbb R[X]/\langle X^2-1\rangle</math> and to the [[ring (mathematics)|ring]] of [[split-complex number]]s. It is also isomorphic (as an algebra) to <math>\mathbb R^2</math> by the mapping defined by
<math display="inline">(1,0)\mapsto \frac{1+X}2, \quad
(0,1)\mapsto \frac{1-X}2.
</math>

=== Stratification by the norm ===
As seen above, the purely nonreal split-quaternions of norm {{math|–1, 1}} and {{math|0}} form respectively a hyperboloid of one sheet, a hyperboloid of two sheets and a [[circular cone]] in the space of non real quaternions.

These surfaces are pairwise [[asymptote]] and do not intersect. Their [[set complement|complement]] consist of six connected regions:
* the two regions located on the concave side of the hyperboloid of two sheets, where <math>N(q)>1</math>
* the two regions between the hyperboloid of two sheets and the cone, where <math>0<N(q)<1</math>
* the region between the cone and the hyperboloid of one sheet where <math>-1<N(q)<0</math>
* the region outside the hyperboloid of one sheet, where <math>N(q)<-1</math>

This stratification can be refined by considering split-quaternions of a fixed norm: for every real number {{math|''n'' ≠ 0}} the purely nonreal split-quaternions of norm {{math|''n''}} form an hyperboloid. All these hyperboloids are asymptote to the above cone, and none of these surfaces intersect any other. As the set of the purely nonreal split-quaternions is the [[disjoint union]] of these surfaces, this provides the desired stratification.