分裂四元数

令${\displaystyle q=w+xi+yj+zk}$ ，考虑普通复数${\displaystyle u=w+xi}$ , ${\displaystyle v=y+zi}$ ，它们的共轭复数为${\displaystyle u=w+xi}$ , ${\displaystyle v=y+zi}$ 。然后

${\displaystyle {\begin{pmatrix}u&v\\v^{*}&u^{*}\end{pmatrix}}}$ 

将${\displaystyle q}$ 表示为矩阵环，其中的分裂四元数的乘法与矩阵乘法的行为相同。例如，这个矩阵的行列式是

${\displaystyle uu^{*}-vv^{*}=qq^{*}}$ 

减号的出现将反四元数与使用了加号的四元数${\displaystyle \mathbb {H} }$ 区分开来。双曲几何中，庞加莱圆盘模型上范数为1的分裂四元数代表多重引导的使用是代数最重要的运用之一。

除了复矩阵表示，另一种线性表示将反四元数与2×2实矩阵（英语：2_×_2_real_matrices）联系起来。这种同构可以明确如下：首先注意到积

${\displaystyle {\begin{pmatrix}0&1\\1&0\end{pmatrix}}{\begin{pmatrix}1&0\\0&-1\end{pmatrix}}={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}}$ 

左边每个因子的平方是单位矩阵，而右边的平方是单位矩阵的负数。此外，注意这三个矩阵，连同单位矩阵，构成了${\displaystyle M(2,\mathbb {R} )}$ 的基。可以使上述矩阵乘积对应于反四元数环中的${\displaystyle jk=-i}$ 。然后，对于任意矩阵有一个双射

${\displaystyle {\begin{pmatrix}a&c\\b&d\end{pmatrix}}\leftrightarrow q={\frac {(a+d)+(c-b)i+(b+c)j+(a-d)k}{2}},}$ 

这实际上形成了环同构。此外，计算各项的平方和表明${\displaystyle qq^{*}=ad-bc}$ ，矩阵的行列式。因此，反四元数的单位拟球（英语：Quasi-sphere）与${\displaystyle SL(2,\mathbb {R} )=\{g\in M(2,\mathbb {R} ):\det g=1\}}$ 群同构，因此与${\displaystyle SU(1,1)}$ 也群同构，后者可以从上面的复表示中得到。

例如，用2×2实矩阵表示双曲运动群，见Karzel和Kist。^[1]

在这两种线性表示中，范数由行列式给出。由于行列式是乘法映射，两个反四元数积的范数等于范数的积。这样反四元数就形成了合成代数。作为实数域上的代数，它是仅有的七个这样的代数之一。

Kevin McCrimon展示了如何按照L. E. Dickson和Adrian Albert为${\displaystyle \mathbb {C} }$ 、${\displaystyle \mathbb {H} }$ 和${\displaystyle \mathbb {O} }$ 给出的除法构造所有的合成代数。^[2]实际上，他给出了real-split的doubled product的乘法法则

${\displaystyle (a,b)(c,d)\ =\ (ac+d^{*}b,\ da+bc^{*})}$ 

如前所述，双共轭 ${\displaystyle (a,b)^{*}\ =\ (a^{*},-b),}$  因此

${\displaystyle N(a,b)\ =\ (a,b)(a,b)^{*}\ =\ (aa^{*}-bb^{*},0).}$ 

如果a和b是双曲复数，分裂四元数${\displaystyle q=(a,b)=((w+zj),(y+xj))}$ 那么

${\displaystyle N(q)=aa^{*}-bb^{*}=w^{2}-z^{2}-(y^{2}-x^{2})=w^{2}+x^{2}-z^{2}-y^{2}}$ .

可以通过${\displaystyle \mathbb {P} }$ 的子空间${\displaystyle \{zi+xj+yk:x,y,z\in \mathbb {R} \}}$ 来了解其子代数。

令

${\displaystyle r(\theta )=j\cos \theta +k\sin \theta }$ 

参数${\displaystyle z}$ 和${\displaystyle r(\theta )}$ 是此子空间中圆柱坐标系的基。参数${\displaystyle \theta }$ 表示方位角。接下来令a表示任意实数，并考虑反四元数

${\displaystyle p(a,r)=i\sinh a+r\cosh a}$ 

${\displaystyle v(a,r)=i\cosh a+r\sinh a}$ 

这正是Alexander Macfarlane（英语：Alexander Macfarlane）和Carmody的等边双曲面坐标。^[3]

接下来，在环的向量子空间中构造三个基础集合:

${\displaystyle E=\{r\in \mathbb {P} :r=r(\theta ),0\leq \theta <2\pi \}}$ 

${\displaystyle J=\{p(a,r)\in \mathbb {P} :a\in \mathbb {R} ,r\in E\}}$ , 单叶双曲面

${\displaystyle I=\{v(a,r)\in \mathbb {P} :a\in \mathbb {R} ,r\in E\}}$ , 双叶双曲面

现在很容易验证

${\displaystyle \{q\in \mathbb {P} :q^{2}=1\}=J\cup \{1,-1\}}$ 

及

${\displaystyle \{q\in \mathbb {P} :q^{2}=-1\}=I}$ 

这些集合相等意味着当${\displaystyle p\in J}$ 时，平面

${\displaystyle \{x+yp:x,y\in \mathbb {R} \}=D_{p}}$ 

是${\displaystyle \mathbb {P} }$ 的一个与双曲复数平面同构的子环，就像对${\displaystyle I}$ 中的任意${\displaystyle v}$ ，

${\displaystyle \{x+yv:x,y\in \mathbb {R} \}=C_{v}}$ 

是与普通复平面${\displaystyle \mathbb {C} }$ 同构的${\displaystyle \mathbb {P} }$ 的平面子环。

注意对于所有${\displaystyle r\in E}$ ， ${\displaystyle (r+i)^{2}=0=(r-i)^{2}}$ ，因此${\displaystyle r+i}$ 和${\displaystyle r-i}$ 是幂零元。平面${\displaystyle N=\{x+y(r+i):x,y\in \mathbb {R} \}}$ 是${\displaystyle \mathbb {P} }$ 的一个与二元数同构的子环。由于每个反四元数都必须位于某个${\displaystyle D_{p}}$ 、${\displaystyle C_{v}}$ 或${\displaystyle N}$ 平面上，所以这些平面组成了${\displaystyle \mathbb {P} }$ ，例如，单位拟球

${\displaystyle SU(1,1)=\{q\in \mathbb {P} :qq^{*}=1\}}$ 

包含了${\displaystyle \mathbb {P} }$ 的构成平面上的“单位圆”：在${\displaystyle D_{p}}$ 中是一个单位双曲线，在${\displaystyle N}$ 中是一对平行线，而在${\displaystyle C_{v}}$ 中确实是一个圆。

反四元数${\displaystyle q=w+xi+yj+zk}$ 的标量部分为w。

定义 对于非零反四元数${\displaystyle q}$ 和${\displaystyle t}$ ，${\displaystyle q\perp t}$ 当且仅当乘积${\displaystyle qt^{*}}$ 的标量部分为零。

• 对任意的 ${\displaystyle v\in I}$ ，如果 ${\displaystyle q,t\in C_{v}}$ ，那么${\displaystyle q\perp t}$ 意味着从${\displaystyle 0}$ 到${\displaystyle q}$ 和${\displaystyle t}$ 的射线是垂直的。
• 对任意的 ${\displaystyle p\in J}$ ，如果 ${\displaystyle q,t\in D_{p}}$ ，那么${\displaystyle q\perp t}$ 意味着这两点是双曲正交（英语：Hyperbolic_orthogonality）的。
• 对任意的 ${\displaystyle r\in E}$  ， ${\displaystyle a\in \mathbb {R} }$ ，${\displaystyle p(a,r)}$ 和${\displaystyle v(a,r)}$ 满足 ${\displaystyle q\perp t}$ 。
• 如果${\displaystyle u}$ 是反四元数环中的一个单位元，那么${\displaystyle q\perp t}$ 意味着${\displaystyle qu\perp tu}$ 。

证明：因向量外积的反交换性，${\displaystyle (tu)^{*}=u^{*}t^{*}}$ ，因此${\displaystyle (qu)(tu)^{*}=(uu^{*})q(t^{*})}$ 。

The quadratic form qq^∗ is positive definite on the planes C_v and N. Consider the counter-sphere {q: qq^∗ = −1}.

Take m = x + yi + zr where r = j cos(θ) + k sin(θ). Fix θ and suppose

mm^∗ = −1 = x² + y² − z².

Since points on the counter-sphere must line on the conjugate of the unit hyperbola in some plane D_p ⊂ P, m can be written, for some p ∈ J

${\displaystyle m~=p\exp {(bp)}=\sinh b+p\cosh b=\sinh b+i\sinh a~\cosh b+r\cosh a~\cosh b}$ .

Let φ be the angle between the hyperbolas from r to p and m. This angle can be viewed, in the plane tangent to the counter-sphere at r, by projection:

${\displaystyle \tan \phi ={\frac {x}{y}}={\frac {\sinh b}{\sinh a~\cosh b}}={\frac {\tanh b}{\sinh a}}}$ . Then

${\displaystyle \lim _{b\to \infty }\tan \phi ={\frac {1}{\sinh a}},}$ 

as in the expression of angle of parallelism in the hyperbolic plane H² . The parameter θ determining the meridian varies over the S¹. Thus the counter-sphere appears as the manifold S¹ × H².

By using the foundations given above, one can show that the mapping

${\displaystyle q\mapsto u^{-1}qu}$ 

is an ordinary or hyperbolic rotation according as

${\displaystyle u=e^{av},\quad v\in I\quad {\text{or}}\quad u=e^{ap},\quad p\in J}$ .

The collection of these mappings bears some relation to the Lorentz group since it is also composed of ordinary and hyperbolic rotations. Among the peculiarities of this approach to relativistic kinematic is the anisotropic profile, say as compared to hyperbolic quaternions.

Reluctance to use coquaternions for kinematic models may stem from the (2, 2) signature when spacetime is presumed to have signature (1, 3) or (3, 1). Nevertheless, a transparently relativistic kinematics appears when a point of the counter-sphere is used to represent an inertial frame of reference. Indeed, if tt^∗ = −1, then there is a p = i sinh(a) + r cosh(a) ∈ J such that t ∈ D_p, and a b ∈ R such that t = p exp(bp). Then if u = exp(bp), v = i cosh(a) + r sinh(a), and s = ir, the set {t, u, v, s} is a pan-orthogonal basis stemming from t, and the orthogonalities persist through applications of the ordinary or hyperbolic rotations.

The coquaternions were initially introduced (under that name)^[4] in 1849 by James Cockle in the London–Edinburgh–Dublin Philosophical Magazine. The introductory papers by Cockle were recalled in the 1904 Bibliography^[5] of the Quaternion Society. Alexander Macfarlane called the structure of coquaternion vectors an exspherical system when he was speaking at the International Congress of Mathematicians in Paris in 1900.^[6]

The unit sphere was considered in 1910 by Hans Beck.^[7] For example, the dihedral group appears on page 419. The coquaternion structure has also been mentioned briefly in the Annals of Mathematics.^[8][9]

• 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 k is replaced with −k.
• Exspherical system (Macfarlane 1900)
• Split-quaternions (Rosenfeld 1988)^[10]
• Antiquaternions (Rosenfeld 1988)
• Pseudoquaternions (Yaglom 1968^[11] Rosenfeld 1988)

• Split-biquaternions
• Split-octonions
• Hypercomplex numbers

• 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. 205–234, arXiv:math.DG/0310415, MR2158044.
• Mohaupt, Thomas (2006), "New developments in special geometry", arXiv:hep-th/0602171.
• Özdemir, M. (2009) "The roots of a split quaternion", Applied Mathematics Letters 22:258–63. [2]
• Özdemir, M. & A.A. Ergin (2006) "Rotations with timelike quaternions in Minkowski 3-space", Journal of Geometry and Physics 56: 322–36.[3]
• Pogoruy, Anatoliy & Ramon M Rodrigues-Dagnino (2008) Some algebraic and analytical properties of coquaternion algebra^{[永久失效链接]}, Advances in Applied Clifford Algebras.