分裂四元数

× 1 i j k
1 1 i j k
i i -1 k −j
j j −k 1 i
k k j i 1

${\displaystyle ij=k=-ji}$,
${\displaystyle jk=-i=-kj}$,
${\displaystyle ki=j=-ik}$,
${\displaystyle i^{2}=-1}$,
${\displaystyle j^{2}=+1}$,
${\displaystyle k^{2}=+1}$

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

${\displaystyle U=\left\{q:qq^{*}\neq 0\right\}}$

矩阵表示

${\displaystyle q=w+xi+yj+zk}$，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

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

${\displaystyle q}$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

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

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

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

由双曲复数生成

Kevin McCrimon展示了如何按照L. E. Dickson和Adrian Albert为${\displaystyle \mathbb {C} }$${\displaystyle \mathbb {H} }$${\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

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

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

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

性质

J 的元素是+1的平方根

I的元素是−1的平方根。

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

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

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

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

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

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

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

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

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

泛正交性

• 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}$${\displaystyle q}$${\displaystyle t}$射线垂直的。
• 对任意的 ${\displaystyle p\in J}$，如果 ${\displaystyle q,t\in D_{p}}$，那么${\displaystyle q\perp t}$意味着这两点是的。
• 对任意的 ${\displaystyle r\in E}$${\displaystyle a\in \mathbb {R} }$${\displaystyle p(a,r)}$${\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${\displaystyle qu\perp tu}$

Counter-sphere geometry

The quadratic form qq is positive definite on the planes Cv 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 DpP, m can be written, for some pJ

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

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.

Application to kinematics

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 tDp, and a bR 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.

Historical notes

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]

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

参考资料

1. ^ Karzel, Helmut & Günter Kist (1985) "Kinematic Algebras and their Geometries", in Rings and Geometry, R. Kaya, P. Plaumann, and K. Strambach editors, pp. 437–509, esp 449,50, D. Reidel
2. ^ Kevin McCrimmon (2004) A Taste of Jordan Algebras, page 64, Universitext, Springer ISBN 0-387-95447-3 MR[1]
3. ^ Carmody, Kevin (1997) "Circular and hyperbolic quaternions, octonions, sedionions", Applied Mathematics and Computation 84(1):27–47, esp. 38
4. ^ James Cockle (1849), On Systems of Algebra involving more than one Imaginary, Philosophical Magazine (series 3) 35: 434,5, link from Biodiversity Heritage Library
5. ^ A. Macfarlane (1904) Bibliography of Quaternions and Allied Systems of Mathematics, from Cornell University Historical Math Monographs, entries for James Cockle, pp. 17–18
6. ^ Alexander Macfarlane (1900) Application of space analysis to curvilinear coordinates 页面存档备份，存于互联网档案馆, Proceedings of the International Congress of Mathematicians, Paris, page 306, from International Mathematical Union
7. ^
8. ^ A. A. Albert (1942), "Quadratic Forms permitting Composition", Annals of Mathematics 43:161 to 77
9. ^
10. ^ Rosenfeld, B.A. (1988) A History of Non-Euclidean Geometry, page 389, Springer-Verlag ISBN 0-387-96458-4
11. ^ Isaak Yaglom (1968) Complex Numbers in Geometry, page 24, Academic Press