View source for 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.

Let consider 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>
They 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 proof that this representation is an algebra homomorphism is straightforward but requires some boring computations, which can be avoided by starting from the expression of split-quaternions as {{math|2×2}} real matrices, and using [[matrix similarity]]. Let {{mvar|S}} be the matrix 
: <math>S=\begin{pmatrix}1 & i \\i &1 \end{pmatrix}.</math>
Then, applied to the representation of split-quaternions as {{math|2×2}} real matrices, the above algebra homomorphism is the matrix similarity.
: <math>M\mapsto S^{-1}MS.</math>

It follows almost immediately 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 quaternions of norm {{math|1}} are exactly the elements of the special unitary group [[SU(1,1)]]. This is used for in [[hyperbolic geometry]] for describing  [[hyperbolic motion#Disk model motions|hyperbolic motions]] of the [[Poincaré disk model]].<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]]