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
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 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
1
2
(
1
+
j
)
{\displaystyle {1 \over 2}(1+j)}
i
−
j
{\displaystyle i-j}
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 。
集合
{
1
,
i
,
j
,
k
}
{\displaystyle \left\{1,i,j,k\right\}}
基 。 這些元素的積由
i
j
=
k
=
−
j
i
{\displaystyle ij=k=-ji}
j
k
=
−
i
=
−
k
j
{\displaystyle jk=-i=-kj}
k
i
=
j
=
−
i
k
{\displaystyle ki=j=-ik}
i
2
=
−
1
{\displaystyle i^{2}=-1}
j
2
=
+
1
{\displaystyle j^{2}=+1}
k
2
=
+
1
{\displaystyle k^{2}=+1}
給出。因此
i
j
k
=
1
{\displaystyle ijk=1}
{
1
,
i
,
j
,
k
,
−
1
,
−
i
,
−
j
,
−
k
}
{\displaystyle \left\{1,i,j,k,-1,-i,-j,-k\right\}}
群 ，與二面體群
D
4
{\displaystyle D_{4}}
同構 ，稱為正方形的對稱群。
分裂四元數
q
=
w
+
x
i
+
y
j
+
z
k
{\displaystyle q=w+xi+yj+zk}
共軛
q
∗
=
w
−
x
i
−
y
j
−
z
k
{\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
N
(
q
)
=
q
q
∗
=
w
2
+
x
2
−
y
2
−
z
2
{\displaystyle N(q)=qq^{*}=w^{2}+x^{2}-y^{2}-z^{2}}
給定兩個反四元數
p
{\displaystyle p}
q
{\displaystyle q}
N
(
p
q
)
=
N
(
p
)
N
(
q
)
{\displaystyle N(pq)=N(p)N(q)}
N
{\displaystyle N}
N
{\displaystyle N}
範數 。 任何滿足
q
≠
0
{\displaystyle q\neq 0}
N
(
q
)
=
0
{\displaystyle N(q)=0}
分裂四元數 。
當範數非零時，
q
{\displaystyle q}
倒數 ，即
q
∗
N
(
q
)
{\displaystyle q^{*} \over N(q)}
U
=
{
q
:
q
q
∗
≠
0
}
{\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
P
{\displaystyle \mathbb {P} }
環
(
P
,
+
,
⋅
)
{\displaystyle (\mathbb {P} ,+,\cdot )}
單位群 為
(
U
,
⋅
)
{\displaystyle (U,\cdot )}
N
(
q
)
=
1
{\displaystyle N(q)=1}
緊緻 的拓撲群
S
U
(
1
,
1
)
{\displaystyle SU(1,1)}
S
L
(
2
,
R
)
{\displaystyle SL(2,\mathbb {R} )}
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。
令
q
=
w
+
x
i
+
y
j
+
z
k
{\displaystyle q=w+xi+yj+zk}
u
=
w
+
x
i
{\displaystyle u=w+xi}
v
=
y
+
z
i
{\displaystyle v=y+zi}
u
=
w
+
x
i
{\displaystyle u=w+xi}
v
=
y
+
z
i
{\displaystyle v=y+zi}
(
u
v
v
∗
u
∗
)
{\displaystyle {\begin{pmatrix}u&v\\v^{*}&u^{*}\end{pmatrix}}}
將
q
{\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
u
u
∗
−
v
v
∗
=
q
q
∗
{\displaystyle uu^{*}-vv^{*}=qq^{*}}
減號的出現將反四元數與使用了加號的四元數
H
{\displaystyle \mathbb {H} }
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
(
0
1
1
0
)
(
1
0
0
−
1
)
=
(
0
−
1
1
0
)
{\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
M
(
2
,
R
)
{\displaystyle M(2,\mathbb {R} )}
j
k
=
−
i
{\displaystyle jk=-i}
對射
(
a
c
b
d
)
↔
q
=
(
a
+
d
)
+
(
c
−
b
)
i
+
(
b
+
c
)
j
+
(
a
−
d
)
k
2
,
{\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
q
q
∗
=
a
d
−
b
c
{\displaystyle qq^{*}=ad-bc}
擬球
S
L
(
2
,
R
)
=
{
g
∈
M
(
2
,
R
)
:
det
g
=
1
}
{\displaystyle SL(2,\mathbb {R} )=\{g\in M(2,\mathbb {R} ):\det g=1\}}
S
U
(
1
,
1
)
{\displaystyle SU(1,1)}
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。
Kevin McCrimon展示了如何按照L. E. Dickson和Adrian Albert為
C
{\displaystyle \mathbb {C} }
H
{\displaystyle \mathbb {H} }
O
{\displaystyle \mathbb {O} }
[ 2]
(
a
,
b
)
(
c
,
d
)
=
(
a
c
+
d
∗
b
,
d
a
+
b
c
∗
)
{\displaystyle (a,b)(c,d)\ =\ (ac+d^{*}b,\ da+bc^{*})}
如前所述，雙共軛
(
a
,
b
)
∗
=
(
a
∗
,
−
b
)
,
{\displaystyle (a,b)^{*}\ =\ (a^{*},-b),}
N
(
a
,
b
)
=
(
a
,
b
)
(
a
,
b
)
∗
=
(
a
a
∗
−
b
b
∗
,
0
)
.
{\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
q
=
(
a
,
b
)
=
(
(
w
+
z
j
)
,
(
y
+
x
j
)
)
{\displaystyle q=(a,b)=((w+zj),(y+xj))}
N
(
q
)
=
a
a
∗
−
b
b
∗
=
w
2
−
z
2
−
(
y
2
−
x
2
)
=
w
2
+
x
2
−
z
2
−
y
2
{\displaystyle N(q)=aa^{*}-bb^{*}=w^{2}-z^{2}-(y^{2}-x^{2})=w^{2}+x^{2}-z^{2}-y^{2}}
圓E在平面 z =0中。J 的元素是+1的平方根 。 可以通過
P
{\displaystyle \mathbb {P} }
{
z
i
+
x
j
+
y
k
:
x
,
y
,
z
∈
R
}
{\displaystyle \{zi+xj+yk:x,y,z\in \mathbb {R} \}}
令
r
(
θ
)
=
j
cos
θ
+
k
sin
θ
{\displaystyle r(\theta )=j\cos \theta +k\sin \theta }
參數
z
{\displaystyle z}
r
(
θ
)
{\displaystyle r(\theta )}
圓柱坐標系 的基。參數
θ
{\displaystyle \theta }
方位角 。接下來令a表示任意實數，並考慮反四元數
p
(
a
,
r
)
=
i
sinh
a
+
r
cosh
a
{\displaystyle p(a,r)=i\sinh a+r\cosh a}
v
(
a
,
r
)
=
i
cosh
a
+
r
sinh
a
{\displaystyle v(a,r)=i\cosh a+r\sinh a}
這正是Alexander Macfarlane [ 3]
接下來，在環的向量子空間中構造三個基礎集合:
E
=
{
r
∈
P
:
r
=
r
(
θ
)
,
0
≤
θ
<
2
π
}
{\displaystyle E=\{r\in \mathbb {P} :r=r(\theta ),0\leq \theta <2\pi \}}
J
=
{
p
(
a
,
r
)
∈
P
:
a
∈
R
,
r
∈
E
}
{\displaystyle J=\{p(a,r)\in \mathbb {P} :a\in \mathbb {R} ,r\in E\}}
雙曲面
I
=
{
v
(
a
,
r
)
∈
P
:
a
∈
R
,
r
∈
E
}
{\displaystyle I=\{v(a,r)\in \mathbb {P} :a\in \mathbb {R} ,r\in E\}}
現在很容易驗證
{
q
∈
P
:
q
2
=
1
}
=
J
∪
{
1
,
−
1
}
{\displaystyle \{q\in \mathbb {P} :q^{2}=1\}=J\cup \{1,-1\}}
及
{
q
∈
P
:
q
2
=
−
1
}
=
I
{\displaystyle \{q\in \mathbb {P} :q^{2}=-1\}=I}
這些集合相等意味著當
p
∈
J
{\displaystyle p\in J}
{
x
+
y
p
:
x
,
y
∈
R
}
=
D
p
{\displaystyle \{x+yp:x,y\in \mathbb {R} \}=D_{p}}
是
P
{\displaystyle \mathbb {P} }
子環 ，就像對
I
{\displaystyle I}
v
{\displaystyle v}
{
x
+
y
v
:
x
,
y
∈
R
}
=
C
v
{\displaystyle \{x+yv:x,y\in \mathbb {R} \}=C_{v}}
是與普通複數平面
C
{\displaystyle \mathbb {C} }
P
{\displaystyle \mathbb {P} }
注意對於所有
r
∈
E
{\displaystyle r\in E}
(
r
+
i
)
2
=
0
=
(
r
−
i
)
2
{\displaystyle (r+i)^{2}=0=(r-i)^{2}}
r
+
i
{\displaystyle r+i}
r
−
i
{\displaystyle r-i}
N
=
{
x
+
y
(
r
+
i
)
:
x
,
y
∈
R
}
{\displaystyle N=\{x+y(r+i):x,y\in \mathbb {R} \}}
P
{\displaystyle \mathbb {P} }
二元數 同構的子環。由於每個反四元數都必須位於某個
D
p
{\displaystyle D_{p}}
C
v
{\displaystyle C_{v}}
N
{\displaystyle N}
P
{\displaystyle \mathbb {P} }
S
U
(
1
,
1
)
=
{
q
∈
P
:
q
q
∗
=
1
}
{\displaystyle SU(1,1)=\{q\in \mathbb {P} :qq^{*}=1\}}
包含了
P
{\displaystyle \mathbb {P} }
D
p
{\displaystyle D_{p}}
單位雙曲線 ，在
N
{\displaystyle N}
C
v
{\displaystyle C_{v}}
反四元數
q
=
w
+
x
i
+
y
j
+
z
k
{\displaystyle q=w+xi+yj+zk}
純量 部分為w。
定義 對於非零反四元數
q
{\displaystyle q}
t
{\displaystyle t}
q
⊥
t
{\displaystyle q\perp t}
q
t
∗
{\displaystyle qt^{*}}
對任意的
v
∈
I
{\displaystyle v\in I}
q
,
t
∈
C
v
{\displaystyle q,t\in C_{v}}
q
⊥
t
{\displaystyle q\perp t}
0
{\displaystyle 0}
q
{\displaystyle q}
t
{\displaystyle t}
射線 是垂直 的。
對任意的
p
∈
J
{\displaystyle p\in J}
q
,
t
∈
D
p
{\displaystyle q,t\in D_{p}}
q
⊥
t
{\displaystyle q\perp t}
雙曲正交
對任意的
r
∈
E
{\displaystyle r\in E}
a
∈
R
{\displaystyle a\in \mathbb {R} }
p
(
a
,
r
)
{\displaystyle p(a,r)}
v
(
a
,
r
)
{\displaystyle v(a,r)}
q
⊥
t
{\displaystyle q\perp t}
如果
u
{\displaystyle u}
q
⊥
t
{\displaystyle q\perp t}
q
u
⊥
t
u
{\displaystyle qu\perp tu}
證明 ：因向量外積 的反交換性，
(
t
u
)
∗
=
u
∗
t
∗
{\displaystyle (tu)^{*}=u^{*}t^{*}}
(
q
u
)
(
t
u
)
∗
=
(
u
u
∗
)
q
(
t
∗
)
{\displaystyle (qu)(tu)^{*}=(uu^{*})q(t^{*})}
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 + y i + zr where r = j cos(θ ) + k sin(θ ). Fix θ and suppose
mm ∗ = −1 = x 2 + y2 − z2 .Since points on the counter-sphere must line on the conjugate of the unit hyperbola in some plane Dp ⊂ P m can be written, for some p ∈ J
m
=
p
exp
(
b
p
)
=
sinh
b
+
p
cosh
b
=
sinh
b
+
i
sinh
a
cosh
b
+
r
cosh
a
cosh
b
{\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:
tan
ϕ
=
x
y
=
sinh
b
sinh
a
cosh
b
=
tanh
b
sinh
a
{\displaystyle \tan \phi ={\frac {x}{y}}={\frac {\sinh b}{\sinh a~\cosh b}}={\frac {\tanh b}{\sinh a}}}
lim
b
→
∞
tan
ϕ
=
1
sinh
a
,
{\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 S 1 manifold S 1 × H2 .
Application to kinematics
By using the foundations given above, one can show that the mapping
q
↦
u
−
1
q
u
{\displaystyle q\mapsto u^{-1}qu}
is an ordinary or hyperbolic rotation according as
u
=
e
a
v
,
v
∈
I
or
u
=
e
a
p
,
p
∈
J
{\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 quaternion s.
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 ∗ = −1p = i sinh(a ) + r cosh(a ) ∈ J t ∈ D p b ∈ R t = p exp(bp )u = exp(bp )v = i cosh(a ) + r sinh(a )s = ir {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] James Cockle in the London–Edinburgh–Dublin Philosophical Magazine . The introductory papers by Cockle were recalled in the 1904 Bibliography [ 5] 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] 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]
^ 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
^ Kevin McCrimmon (2004) A Taste of Jordan Algebras , page 64, Universitext, Springer ISBN 0-387-95447-3 MR [1]
^ Carmody, Kevin (1997) "Circular and hyperbolic quaternions, octonions, sedionions", Applied Mathematics and Computation 84(1):27–47, esp. 38
^ 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 Biodiversity Heritage Library ^ A. Macfarlane (1904) Bibliography of Quaternions and Allied Systems of Mathematics , from Cornell University Historical Math Monographs , entries for James Cockle, pp. 17–18
^ 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
^ 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
^ A. A. Albert (1942), "Quadratic Forms permitting Composition", Annals of Mathematics ^ Valentine Bargmann (1947), "Irreducible unitary representations of the Lorentz Group" , Annals of Mathematics ^ Rosenfeld, B.A. (1988) A History of Non-Euclidean Geometry , page 389, Springer-Verlag ISBN 0-387-96458-4
^ Isaak Yaglom (1968) Complex Numbers in Geometry , page 24, Academic Press
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, , MR 2158044 .
Mohaupt, Thomas (2006), "New developments in special geometry", .
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
可數集
自然數 (
N
{\displaystyle \mathbb {N} }
整數 (
Z
{\displaystyle \mathbb {Z} }
有理數 (
Q
{\displaystyle \mathbb {Q} }
規矩數 代數數 (
A
{\displaystyle \mathbb {A} }
週期 可計算數 可定義數 高斯整數 (
Z
[
i
]
{\displaystyle \mathbb {Z} [i]}
艾森斯坦整數 合成代數
可除代數 實數 (
R
{\displaystyle \mathbb {R} }
複數 (
C
{\displaystyle \mathbb {C} }
四元數 (
H
{\displaystyle \mathbb {H} }
八元數 (
O
{\displaystyle \mathbb {O} }
凱萊－迪克森結構
實數 (
R
{\displaystyle \mathbb {R} }
複數 (
C
{\displaystyle \mathbb {C} }
四元數 (
H
{\displaystyle \mathbb {H} }
八元數 (
O
{\displaystyle \mathbb {O} }
十六元數 (
S
{\displaystyle \mathbb {S} }
三十二元數 六十四元數 一百二十八元數 二百五十六元數…… 分裂 其他超複數 其他系統
![{\displaystyle \mathbb {Z} [i]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ffa94e9e2e6d9e5e5373d5fafb954b902743fde)
