分裂四元數

在抽象代數中，分裂四元數（split-quaternions）或反四元數（coquaternions）是一種四維的結合代數的元素，由James Cockle（英語：James Cockle）在1849年引入，當時稱為反四元數。 類似於漢密爾頓1843年引入的四元數 ，它們組成了一個四維的實向量空間,且有乘法運算。 與四元數不同，分裂四元數包含非平凡的零因子、冪零元素和冪等元（英語：Idempotent_(ring_theory)）。（例如， ${\displaystyle {1 \over 2}(1+j)}$ 是冪等的零因子，而${\displaystyle i-j}$ 是冪零元素。)作為一種數學結構，分裂四元數形成了域代數，且與2 × 2的實矩陣同構。

分裂四元數乘法

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

集合 ${\displaystyle \left\{1,i,j,k\right\}}$組成一個基。 這些元素的積由

${\displaystyle ij=k=-ji}$,

${\displaystyle jk=-i=-kj}$,

${\displaystyle ki=j=-ik}$,

${\displaystyle i^{2}=-1}$,

${\displaystyle j^{2}=+1}$,

${\displaystyle k^{2}=+1}$

給出。因此${\displaystyle ijk=1}$。 由以上定義可得，集合${\displaystyle \left\{1,i,j,k,-1,-i,-j,-k\right\}}$在分裂四元數乘法的定義下是一個群，與二面體群${\displaystyle D_{4}}$同構，稱為正方形的對稱群。

分裂四元數${\displaystyle q=w+xi+yj+zk}$的共軛${\displaystyle q^{*}=w-xi-yj-zk}$。

由於其基向量的反交換性，分裂四元數與其共軛的積由其迷向二次型

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

給出。

給定兩個反四元數${\displaystyle p}$和${\displaystyle q}$，有${\displaystyle N(pq)=N(p)N(q)}$，意味著${\displaystyle N}$ 是可合成的二次型。 其上的代數是一種合成代數，${\displaystyle N}$ 是其範數。 任何滿足${\displaystyle q\neq 0}$，${\displaystyle N(q)=0}$的反四元數q稱為零向量（Null vector而非Zero vector)，它的存在意味著反四元數形成"分裂的合成代數"，因此反四元數也被稱為分裂四元數。

當範數非零時，${\displaystyle q}$有倒數，即 ${\displaystyle q^{*} \over N(q)}$. 集合

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

是單位元素的集合。 全體分裂四元數的集合${\displaystyle \mathbb {P} }$組成環 ${\displaystyle (\mathbb {P} ,+,\cdot )}$ ，其單位群為${\displaystyle (U,\cdot )}$。全體${\displaystyle N(q)=1}$的分裂四元數組成一個非緊緻的拓撲群 ${\displaystyle SU(1,1)}$，且與${\displaystyle SL(2,\mathbb {R} )}$同構（見下）。

歷史上講，分裂四元數早於凱萊的矩陣代數；分裂四元數(及四元數和雙複數)引發了對線性代數的深入研究。

矩陣表示

令${\displaystyle q=w+xi+yj+zk}$，考慮普通複數${\displaystyle u=w+xi}$, ${\displaystyle v=y+zi}$，它們的共軛複數為${\displaystyle u=w+xi}$, ${\displaystyle v=y+zi}$。然後

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

將${\displaystyle q}$表示為矩陣環，其中的分裂四元數的乘法與矩陣乘法的行為相同。例如，這個矩陣的行列式是

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

減號的出現將反四元數與使用了加號的四元數${\displaystyle \mathbb {H} }$區分開來。雙曲幾何中，龐加萊圓盤模型上範數為1的分裂四元數代表多重引導的使用是代數最重要的運用之一。

除了複矩陣表示，另一種線性表示將反四元數與2×2實矩陣（英語：2_×_2_real_matrices）聯繫起來。這種同構可以明確如下：首先注意到積

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

左邊每個因子的平方是單位矩陣，而右邊的平方是單位矩陣的負數。此外，注意這三個矩陣，連同單位矩陣，構成了${\displaystyle M(2,\mathbb {R} )}$的基。可以使上述矩陣乘積對應於反四元數環中的${\displaystyle jk=-i}$。然後，對於任意矩陣有一個對射

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

這實際上形成了環同構。此外，計算各項的平方和表明${\displaystyle qq^{*}=ad-bc}$，矩陣的行列式。因此，反四元數的單位擬球（英語：Quasi-sphere）與${\displaystyle SL(2,\mathbb {R} )=\{g\in M(2,\mathbb {R} ):\det g=1\}}$群同構，因此與${\displaystyle SU(1,1)}$也群同構，後者可以從上面的復表示中得到。

例如，用2×2實矩陣表示雙曲運動群，見Karzel和Kist。^[1]

在這兩種線性表示中，範數由行列式給出。由於行列式是乘法映射，兩個反四元數積的範數等於範數的積。這樣反四元數就形成了合成代數。作為實數域上的代數，它是僅有的七個這樣的代數之一。

由雙曲複數生成

Kevin McCrimon展示了如何按照L. E. Dickson和Adrian Albert為${\displaystyle \mathbb {C} }$、${\displaystyle \mathbb {H} }$和${\displaystyle \mathbb {O} }$給出的除法構造所有的合成代數。^[2]實際上，他給出了real-split的doubled product的乘法法則

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

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

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

如果a和b是雙曲複數，分裂四元數${\displaystyle q=(a,b)=((w+zj),(y+xj))}$那麼

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

性質

圓E在平面 z =0中。



J 的元素是+1的平方根。

I的元素是−1的平方根。

可以通過${\displaystyle \mathbb {P} }$的子空間${\displaystyle \{zi+xj+yk:x,y,z\in \mathbb {R} \}}$來了解其子代數。

令

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

參數${\displaystyle z}$和${\displaystyle r(\theta )}$是此子空間中圓柱坐標系的基。參數${\displaystyle \theta }$表示方位角。接下來令a表示任意實數，並考慮反四元數

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

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

這正是Alexander Macfarlane（英語：Alexander Macfarlane）和Carmody的等邊雙曲面坐標。^[3]

接下來，在環的向量子空間中構造三個基礎集合:

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

${\displaystyle J=\{p(a,r)\in \mathbb {P} :a\in \mathbb {R} ,r\in E\}}$, 單葉雙曲面

${\displaystyle I=\{v(a,r)\in \mathbb {P} :a\in \mathbb {R} ,r\in E\}}$, 雙葉雙曲面

現在很容易驗證

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

及

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

這些集合相等意味著當${\displaystyle p\in J}$時，平面

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

是${\displaystyle \mathbb {P} }$的一個與雙曲複數平面同構的子環，就像對${\displaystyle I}$中的任意${\displaystyle v}$，

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

是與普通複數平面${\displaystyle \mathbb {C} }$同構的${\displaystyle \mathbb {P} }$的平面子環。

注意對於所有${\displaystyle r\in E}$， ${\displaystyle (r+i)^{2}=0=(r-i)^{2}}$，因此${\displaystyle r+i}$和${\displaystyle r-i}$是冪零元素。平面${\displaystyle N=\{x+y(r+i):x,y\in \mathbb {R} \}}$是${\displaystyle \mathbb {P} }$的一個與二元數同構的子環。由於每個反四元數都必須位於某個${\displaystyle D_{p}}$、${\displaystyle C_{v}}$或${\displaystyle N}$平面上，所以這些平面組成了${\displaystyle \mathbb {P} }$，例如，單位擬球

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

包含了${\displaystyle \mathbb {P} }$的構成平面上的「單位圓」：在${\displaystyle D_{p}}$中是一個單位雙曲線，在${\displaystyle N}$中是一對平行線，而在${\displaystyle C_{v}}$中確實是一個圓。

泛正交性

反四元數${\displaystyle q=w+xi+yj+zk}$的純量部分為w。

定義 對於非零反四元數${\displaystyle q}$和${\displaystyle t}$，${\displaystyle q\perp t}$若且唯若乘積${\displaystyle qt^{*}}$的純量部分為零。

• 對任意的 ${\displaystyle v\in I}$，如果 ${\displaystyle q,t\in C_{v}}$，那麼${\displaystyle q\perp t}$意味著從${\displaystyle 0}$到${\displaystyle q}$和${\displaystyle t}$的射線是垂直的。
• 對任意的 ${\displaystyle p\in J}$，如果 ${\displaystyle q,t\in D_{p}}$，那麼${\displaystyle q\perp t}$意味著這兩點是雙曲正交（英語：Hyperbolic_orthogonality）的。
• 對任意的 ${\displaystyle r\in E}$ ， ${\displaystyle a\in \mathbb {R} }$，${\displaystyle p(a,r)}$和${\displaystyle v(a,r)}$滿足 ${\displaystyle q\perp t}$。
• 如果${\displaystyle u}$是反四元數環中的一個單位元素，那麼${\displaystyle q\perp t}$意味著${\displaystyle qu\perp tu}$。

證明：因向量外積的反交換性，${\displaystyle (tu)^{*}=u^{*}t^{*}}$，因此${\displaystyle (qu)(tu)^{*}=(uu^{*})q(t^{*})}$。

Counter-sphere geometry

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.

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 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.

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)

參見

• Split-biquaternions
• Split-octonions
• Hypercomplex numbers

參考資料

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 （頁面存檔備份，存於網際網路檔案館）, 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 （頁面存檔備份，存於網際網路檔案館）, Proceedings of the International Congress of Mathematicians, Paris, page 306, from International Mathematical Union
7. Hans Beck (1910) Ein Seitenstück zur Mobius'schen Geometrie der Kreisverwandschaften （頁面存檔備份，存於網際網路檔案館）, Transactions of the American Mathematical Society 11
8. A. A. Albert (1942), "Quadratic Forms permitting Composition", Annals of Mathematics 43:161 to 77
9. Valentine Bargmann (1947), "Irreducible unitary representations of the Lorentz Group", Annals of Mathematics 48: 568–640
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

延伸閱讀

• 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^{[永久失效連結]}, Advances in Applied Clifford Algebras.