Editing Split-quaternion

{{Short description|Four-dimensional associative algebra over the reals}}
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In [[abstract algebra]], the '''split-quaternions'''  or '''coquaternions''' form an [[algebraic structure]] introduced by [[James Cockle (lawyer)|James Cockle]] in 1849 under the latter name. They form an [[associative algebra]] of dimension four over the [[real number]]s.

After introduction in the 20th century of coordinate-free definitions of [[ring (mathematics)|rings]] and [[algebra over a field|algebras]], it was proved that the algebra of split-quaternions is [[isomorphism|isomorphic]] to the [[ring (mathematics)|ring]] of the {{math|2×2}} [[real matrix|real matrices]]. So the study of split-quaternions can be reduced to the study of real matrices, and this may explain why there are few mentions of split-quaternions in the mathematical literature of the 20th and 21st centuries.

== Definition ==

The ''split-quaternions'' are the [[linear combination]]s (with real coefficients) of four basis elements {{math|1, i, j, k}} that satisfy the following product rules:
: {{math|1=i<sup>2</sup> = −1}}, 
: {{math|1=j<sup>2</sup> = 1}}, 
: {{math|1=k<sup>2</sup> = 1}},
: {{math|1=ij = k = −ji}}.
By [[associative property|associativity]], these relations imply
: {{math|1=jk = −i = −kj}}, 
: {{math|1=ki = j = −ik}},
and also {{math|1=ijk = 1}}.

So, the split-quaternions form a [[real vector space]] of dimension four with {{math|{{mset|1, i, j, k}}}} as a [[basis (linear algebra)|basis]]. They form also a [[noncommutative ring]], by extending the above product rules by [[distributivity]] to all split-quaternions.

The square matrices
: <math>\begin{align}
\boldsymbol{1} =\begin{pmatrix}1&0\\0&1\end{pmatrix},\qquad&\boldsymbol{i} =\begin{pmatrix}0&1\\-1&0\end{pmatrix},\\
\boldsymbol{j} =\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad&\boldsymbol{k} =\begin{pmatrix}1&0\\0&-1\end{pmatrix}.
\end{align}</math>
satisfy the same multiplication table as the corresponding split-quaternions. As these matrices form a basis of the two-by-two matrices, the unique linear [[function (mathematics)|function]] that maps {{math|1, i, j, k}} to <math>\boldsymbol{1}, \boldsymbol{i}, \boldsymbol{j}, \boldsymbol{k}</math> (respectively) induces an [[algebra isomorphism]] from the split-quaternions to the two-by-two real matrices. 

The above multiplication rules imply that the eight elements {{math|1, i, j, k, −1, −i, −j, −k}} form a [[group (mathematics)|group]] under this multiplication, which is [[isomorphic]] to the [[dihedral group]] D<sub>4</sub>, the [[Examples of groups#The symmetry group of a square: dihedral group of order 8|symmetry group of a square]]. In fact, if one considers a square whose vertices are the points whose coordinates are {{math|0}} or {{math|1}}, the matrix <math>\boldsymbol{i}</math> is the clockwise rotation of the quarter of a turn, <math>\boldsymbol{j}</math> is the symmetry around the first diagonal, and <math>\boldsymbol{k}</math> is the symmetry around the {{mvar|x}} axis.

== Properties ==

Like the [[quaternion]]s introduced by [[William Rowan Hamilton|Hamilton]] in 1843, they form a four [[dimension (vector space)|dimensional]] real [[associative algebra]]. But like the real algebra of 2×2 matrices – and unlike the real algebra of quaternions – the split-quaternions contain nontrivial [[zero divisor]]s, [[nilpotent]] elements, and [[idempotent element (ring theory)|idempotent]]s. (For example, {{nowrap|{{sfrac|1|2}}(1 + j)}} is an idempotent zero-divisor, and {{nowrap|i − j}} is nilpotent.)  As an [[algebra over a field|algebra over the real numbers]], the algebra of split-quaternions is [[algebra isomorphism|isomorphic]] to the algebra of 2×2 real matrices by the above defined isomorphism. 

This isomorphism allows identifying each split-quaternion with a 2×2 matrix. So every property of split-quaternions corresponds to a similar property of matrices, which is often named differently. 

The ''conjugate'' of a split-quaternion
{{math|1=''q'' = ''w'' + ''x''i + ''y''j + ''z''k}}, is {{math|1=''q''<sup>∗</sup> = ''w'' − ''x''i − ''y''j − ''z''k}}. In term of matrices, the conjugate is the [[cofactor matrix]] obtained by exchanging the diagonal entries and changing the sign of the other two entries.

The product of a split-quaternion with its conjugate is the [[isotropic quadratic form]]:
: <math>N(q) = q q^* = w^2 + x^2 - y^2 - z^2,</math>
which is called the [[Norm (mathematics)#Composition algebras|''norm'']] of the split-quaternion or the [[determinant]] of the associated matrix.

The real part of a split-quaternion {{math|1=''q'' = ''w'' + ''x''i + ''y''j + ''z''k}} is {{math|1=''w'' = (''q''<sup>∗</sup> + ''q'')/2}}. It equals the [[trace (linear algebra)|trace]] of associated matrix.

The norm of a product of two split-quaternions is the product of their norms. Equivalently, the determinant of a product of matrices is the product of their determinants. This property means that split-quaternions form a [[composition algebra]]. As there are nonzero split-quaternions having a zero norm, split-quaternions form a "split composition algebra" – hence their name.

A split-quaternion with a nonzero norm has a [[multiplicative inverse]], namely {{math|''q''<sup>∗</sup>/''N''(''q'')}}. In terms of matrices, this is equivalent to the [[Cramer rule]] that asserts that a matrix is [[invertible matrix|invertible]] if and only its determinant is nonzero, and, in this case, the inverse of the matrix is the quotient of the cofactor matrix by the determinant.

The isomorphism between split-quaternions and 2×2 real matrices shows that the multiplicative group of split-quaternions with a nonzero norm is isomorphic with <math>\operatorname{GL}(2, \mathbb R),</math> and the group of split quaternions of norm {{math|1}} is isomorphic with <math>\operatorname{SL}(2, \mathbb R).</math>

Geometrically, the split-quaternions can be compared to Hamilton's quaternions as [[pencil of planes|pencils of planes]]. In both cases the real numbers form the axis of a pencil. In Hamilton quaternions there is a sphere of imaginary units, and any pair of antipodal imaginary units generates a complex plane with the real line. For split-quaternions there are hyperboloids of hyperbolic and imaginary units that generate split-complex or ordinary complex planes, as described below in [[#Stratification|§ Stratification]].

== Representation as complex matrices ==
There is a representation of the split-quaternions as a [[unital associative algebra|unital associative subalgebra]] of the {{math|2×2}} matrices with [[complex number|complex]] entries. This representation can be defined by the [[algebra homomorphism]] that maps a split-quaternion {{math|''w'' + ''x''i + ''y''j + ''z''k}} to the matrix
: <math>\begin{pmatrix}w+xi& y+zi\\y-zi&w-xi\end{pmatrix}.</math>
Here, {{mvar|i}} ([[italic type|italic]]) is the [[imaginary unit]], not to be confused with the split quaternion basis element {{math|i}} ([[roman type|upright roman]]).

The image of this homomorphism is the [[matrix ring]] formed by the matrices of the form
: <math>\begin{pmatrix}u & v \\ v^* & u^* \end{pmatrix},</math>
where the superscript <math>^*</math> denotes a [[complex conjugate]].

This homomorphism maps respectively the split-quaternions {{math|i, j, k}} on the matrices
: <math>\begin{pmatrix}i & 0 \\0 &-i \end{pmatrix}, \quad\begin{pmatrix}0 & 1 \\1 &0 \end{pmatrix},\quad \begin{pmatrix}0 & i \\-i &0 \end{pmatrix}.</math>

The isomorphism of algebras is completed by use of [[matrix multiplication]] to verify the identities involving i, j, and k. For instance,
:<math>j k = \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix}0 & i \\ -i & 0 \end{pmatrix} = \begin{pmatrix}-i & 0 \\ 0 & i \end{pmatrix} = - i .</math>

It follows that for a split quaternion represented as a complex matrix, the conjugate is the matrix of the cofactors, and the norm is the determinant.

With the representation of split quaternions as complex matrices, the matrices of determinant {{math|1}} form the special unitary group [[SU(1,1)]], which is used to describe  [[hyperbolic motion#Disk model motions|hyperbolic motions]] of the [[Poincaré disk model]]  in [[hyperbolic geometry]].<ref>Karzel, Helmut & Günter Kist (1985) "Kinematic Algebras and their Geometries", in ''Rings and Geometry'', R. Kaya, P. Plaumann, and K. Strambach editors, pp. 437&ndash;509, esp 449,50, [[D. Reidel]] {{isbn|90-277-2112-2}}</ref>

== Generation from split-complex numbers ==

Split-quaternions may be generated by [[Cayley%E2%80%93Dickson_construction#Modified_Cayley%E2%80%93Dickson_construction|modified Cayley–Dickson construction]]<ref>[[Kevin McCrimmon]] (2004) ''A Taste of Jordan Algebras'', page 64, Universitext, Springer {{isbn|0-387-95447-3}} {{mr|id=2014924}}</ref> similar to the method of [[L. E. Dickson]] and [[Adrian Albert]]. for the division algebras '''C''', '''H''', and '''O'''. The multiplication rule
<math display="block">(a,b)(c,d)\ = \ (ac + d^* b, \ da + bc^* )</math>
is used when producing the doubled product in the real-split cases. The doubled conjugate <math>(a,b)^* = (a^*, - b), </math> so that
<math display="block">N(a,b) \ = \ (a,b)(a,b)^* \ = \ (a a^* - b b^* , 0).</math>
If ''a'' and ''b'' are [[split-complex number]]s and split-quaternion <math>q =  (a,b) = ((w + z j), (y + xj)), </math>

then <math display="block">N(q) = a a^* - b b^* = w^2 - z^2 - (y^2 - x^2) = w^2 + x^2 - y^2 - z^2 .</math>

== 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.

== Colour space ==
Split quaternions have been applied to [[colour balance]]<ref>Michel Berthier, Nicoletta Prencipe & Edouardo Provenzi (2023) [https://hal.science/hal-04149289 Split quaternions for perceptual white balance] @ [[HAL (open archive)|HAL]]</ref> The model refers to the [[Jordan algebra]] of [[symmetric matrix|symmetric matrices]] representing the algebra. The model reconciles [[trichromacy]] with [[opponent process|Hering's opponency]] and uses the [[Cayley–Klein model]] of [[hyperbolic geometry]] for chromatic distances.

== Historical notes ==

The coquaternions were initially introduced (under that name)<ref>[[James Cockle]] (1849), [https://www.biodiversitylibrary.org/item/20114#page/448/mode/1up On Systems of Algebra involving more than one Imaginary], ''[[Philosophical Magazine]]'' (series 3) 35: 434,5, link from [[Biodiversity Heritage Library]]</ref> in 1849 by [[James Cockle]] in the London&ndash;Edinburgh&ndash;Dublin [[Philosophical Magazine]]. The introductory papers by Cockle were recalled in the 1904 ''Bibliography''<ref>A. Macfarlane (1904) [http://dlxs2.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=03030001 Bibliography of Quaternions and Allied Systems of Mathematics], from [[Cornell University]] ''Historical Math Monographs'', entries for James Cockle, pp.&nbsp;17–18</ref> of the [[Quaternion Society]].

[[Alexander Macfarlane]] called the structure of split-quaternion vectors an ''exspherical system'' when he was speaking at the [[International Congress of Mathematicians]] in Paris in 1900.<ref>A. Macfarlane (1900) [http://www.mathunion.org/ICM/ICM1900/Main/icm1900.0305.0312.ocr.pdf Application of space analysis to curvilinear coordinates] {{Webarchive|url=https://web.archive.org/web/20140810042126/http://www.mathunion.org/ICM/ICM1900/Main/icm1900.0305.0312.ocr.pdf |date=2014-08-10 }}, ''Proceedings of the ''[[International Congress of Mathematicians]], Paris, page 306, from [[International Mathematical Union]]</ref> Macfarlane considered the "hyperboloidal counterpart to spherical analysis" in a 1910 article "Unification and Development of the Principles of the Algebra of Space" in the ''Bulletin'' of the [[Quaternion Society]].<ref>A. Macfarlane (1910) [https://archive.org/details/proceedingsfifth00hobs/page/266/mode/2up "Unification and Development of the Principles of the Algebra of Space"] via Internet Archive.</ref>

[[Hans Beck (mathematician)|Hans Beck]] compared split-quaternion transformations to the circle-permuting property of [[Möbius transformation]]s in 1910.<ref>[[Hans Beck (mathematician)|Hans Beck]] (1910) [http://www.ams.org/journals/tran/1910-011-04/S0002-9947-1910-1500872-0/S0002-9947-1910-1500872-0.pdf Ein Seitenstück zur Mobius'schen Geometrie der Kreisverwandschaften], [[Transactions of the American Mathematical Society]] 11</ref> The split-quaternion structure has also been mentioned briefly in the ''[[Annals of Mathematics]]''.<ref>[[A. A. Albert]] (1942), "Quadratic Forms permitting Composition", ''[[Annals of Mathematics]]'' 43:161 to 77</ref><ref>[[Valentine Bargmann]] (1947), [https://www.jstor.org/discover/10.2307/1969129 "Irreducible unitary representations of the Lorentz Group"], ''[[Annals of Mathematics]]'' 48: 568&ndash;640</ref>

== Synonyms ==
* Para-quaternions (Ivanov and Zamkovoy 2005, Mohaupt 2006) Manifolds with para-quaternionic structures are studied in [[differential geometry]] and [[string theory]]. In the para-quaternionic literature, {{math|k}} is replaced with {{math|&minus;k}}.
* Exspherical system (Macfarlane 1900)
* Split-quaternions (Rosenfeld 1988)<ref>Rosenfeld, B.A. (1988) ''A History of Non-Euclidean Geometry'', page 389, Springer-Verlag {{isbn|0-387-96458-4}}</ref>
* Antiquaternions (Rosenfeld 1988)
* Pseudoquaternions (Yaglom 1968<ref>[[Isaak Yaglom]] (1968) ''Complex Numbers in Geometry'', page 24, [[Academic Press]]</ref> Rosenfeld 1988)

== See also ==
* [[Pauli matrices]]
* [[Split-biquaternion]]s
* [[Split-octonion]]s
* [[Dual quaternion]]s

== References ==
{{reflist}}

== Further reading ==
* [[Brody, Dorje C.]], and [[Eva-Maria Graefe]]. "On complexified mechanics and coquaternions". Journal of Physics A: Mathematical and Theoretical 44.7 (2011): 072001. {{doi|10.1088/1751-8113/44/7/072001}}
* Ivanov, Stefan; Zamkovoy, Simeon (2005), "Parahermitian and paraquaternionic manifolds", ''Differential Geometry and its Applications'' '''23''', pp.&nbsp;205&ndash;234, {{arxiv|math.DG/0310415}}, {{MR|2158044}}.
* Mohaupt, Thomas (2006), "New developments in special geometry", {{arxiv|hep-th/0602171}}.
* Özdemir, M. (2009) "The roots of a split quaternion", ''Applied Mathematics Letters'' 22:258&ndash;63. [https://www.researchgate.net/publication/270760686_The_Roots_of_a_Split_Quaternion]
* Özdemir, M. & A.A. Ergin (2006) "Rotations with timelike quaternions in Minkowski 3-space", ''Journal of Geometry and Physics'' 56: 322&ndash;36.[https://www.researchgate.net/publication/235591460_Rotations_with_unit_timelike_quaternions_in_Minkowski_3-space]
* Pogoruy, Anatoliy & Ramon M Rodrigues-Dagnino (2008) [https://doi.org/10.1007%2Fs00006-008-0142-3 Some algebraic and analytical properties of coquaternion algebra], ''[[Advances in Applied Clifford Algebras]]''.

{{Number systems}}

[[Category:Composition algebras]]
[[Category:Quaternions]]
[[Category:Hyperbolic geometry]]
[[Category:Special relativity]]