分裂四元数乘法
×
1
i
j
k
1
1
i
j
k
i
i
-1
k
−j
j
j
−k
1
-i
k
k
j
i
1
在抽象代数 中，分裂四元数 （split-quaternions）或反四元数 （coquaternions）是一种四维的结合代数 的元素，由James Cockle 汉密尔顿 1843年引入的四元数 ，它们组成了一个四维 的实向量空间 ,且有乘法运算。 与四元数不同，分裂四元数包含非平凡的零因子 、幂零元 和幂等元
1
2
(
1
+
j
)
{\displaystyle {1 \over 2}(1+j)}
i
−
j
{\displaystyle i-j}
数学结构 ，分裂四元数形成了域代数，且与2 × 2的实 矩阵 同构 。
集合
{
1
,
i
,
j
,
k
}
{\displaystyle \left\{1,i,j,k\right\}}
基 。 这些元素的积由
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}
给出。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 ，与二面体群
D
4
{\displaystyle D_{4}}
同构 ，称为正方形的对称群。
分裂四元数
q
=
w
+
x
i
+
y
j
+
z
k
{\displaystyle q=w+xi+yj+zk}
共轭
q
∗
=
w
−
x
i
−
y
j
−
z
k
{\displaystyle q^{*}=w-xi-yj-zk}
由于其基向量的反交换性 ，分裂四元数与其共轭的积由其迷向二次型
N
(
q
)
=
q
q
∗
=
w
2
+
x
2
−
y
2
−
z
2
{\displaystyle N(q)=qq^{*}=w^{2}+x^{2}-y^{2}-z^{2}}
给定两个反四元数
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}
范数 。 任何满足
q
≠
0
{\displaystyle q\neq 0}
N
(
q
)
=
0
{\displaystyle N(q)=0}
分裂四元数 。
当范数非零时，
q
{\displaystyle q}
倒数 ，即
q
∗
N
(
q
)
{\displaystyle q^{*} \over N(q)}
U
=
{
q
:
q
q
∗
≠
0
}
{\displaystyle U=\left\{q:qq^{*}\neq 0\right\}}
是单位元 的集合。 全体分裂四元数的集合
P
{\displaystyle \mathbb {P} }
环
(
P
,
+
,
⋅
)
{\displaystyle (\mathbb {P} ,+,\cdot )}
单位群 为
(
U
,
⋅
)
{\displaystyle (U,\cdot )}
N
(
q
)
=
1
{\displaystyle N(q)=1}
紧致 的拓扑群
S
U
(
1
,
1
)
{\displaystyle SU(1,1)}
S
L
(
2
,
R
)
{\displaystyle SL(2,\mathbb {R} )}
历史上讲，分裂四元数早于凯莱 的矩阵代数；分裂四元数(及四元数和双复数 )引发了对线性代数 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}}}
将
q
{\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
u
u
∗
−
v
v
∗
=
q
q
∗
{\displaystyle uu^{*}-vv^{*}=qq^{*}}
减号的出现将反四元数与使用了加号的四元数
H
{\displaystyle \mathbb {H} }
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
(
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
M
(
2
,
R
)
{\displaystyle M(2,\mathbb {R} )}
j
k
=
−
i
{\displaystyle jk=-i}
双射
(
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}},}
这实际上形成了环同构。此外，计算各项的平方和表明
q
q
∗
=
a
d
−
b
c
{\displaystyle qq^{*}=ad-bc}
拟球
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
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，双共轭
(
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 是双曲复数 ，分裂四元数
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}}
圆E在平面 z =0中。J 的元素是+1的平方根 。 可以通过
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 }
参数
z
{\displaystyle z}
r
(
θ
)
{\displaystyle r(\theta )}
圆柱坐标系 的基。参数
θ
{\displaystyle \theta }
方位角 。接下来令a表示任意实数，并考虑反四元数
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}
这正是Alexander Macfarlane [ 3]
接下来，在环的向量子空间中构造三个基础集合:
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\}}
双曲面
I
=
{
v
(
a
,
r
)
∈
P
:
a
∈
R
,
r
∈
E
}
{\displaystyle I=\{v(a,r)\in \mathbb {P} :a\in \mathbb {R} ,r\in E\}}
现在很容易验证
{
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}
这些集合相等意味着当
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} }
子环 ，就像对
I
{\displaystyle I}
v
{\displaystyle v}
{
x
+
y
v
:
x
,
y
∈
R
}
=
C
v
{\displaystyle \{x+yv:x,y\in \mathbb {R} \}=C_{v}}
是与普通复平面
C
{\displaystyle \mathbb {C} }
P
{\displaystyle \mathbb {P} }
注意对于所有
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} }
二元数 同构的子环。由于每个反四元数都必须位于某个
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}}
单位双曲线 ，在
N
{\displaystyle N}
C
v
{\displaystyle C_{v}}
反四元数
q
=
w
+
x
i
+
y
j
+
z
k
{\displaystyle q=w+xi+yj+zk}
标量 部分为w。
定义 对于非零反四元数
q
{\displaystyle q}
t
{\displaystyle t}
q
⊥
t
{\displaystyle q\perp t}
q
t
∗
{\displaystyle qt^{*}}
对任意的
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}
射线 是垂直 的。
对任意的
p
∈
J
{\displaystyle p\in J}
q
,
t
∈
D
p
{\displaystyle q,t\in D_{p}}
q
⊥
t
{\displaystyle q\perp t}
双曲正交
对任意的
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}
证明 ：因向量外积 的反交换性，
(
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 （页面存档备份 ，存于互联网档案馆 ）, 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 （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
^ Hans Beck (1910) Ein Seitenstück zur Mobius'schen Geometrie der Kreisverwandschaften （页面存档备份 ，存于互联网档案馆 ）, 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
自然数 (
N
{\displaystyle \mathbb {N} }
整数 (
Z
{\displaystyle \mathbb {Z} }
有理数 (
Q
{\displaystyle \mathbb {Q} }
規矩數 代數數 (
A
{\displaystyle \mathbb {A} }
周期 可計算數 可定义数 高斯整數 (
Z
[
i
]
{\displaystyle \mathbb {Z} [i]}
艾森斯坦整数 合成代數
可除代數 实数 (
R
{\displaystyle \mathbb {R} }
複數 (
C
{\displaystyle \mathbb {C} }
四元數 (
H
{\displaystyle \mathbb {H} }
八元数 (
O
{\displaystyle \mathbb {O} }
凯莱－迪克森结构
实数 (
R
{\displaystyle \mathbb {R} }
複數 (
C
{\displaystyle \mathbb {C} }
四元數 (
H
{\displaystyle \mathbb {H} }
八元数 (
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)
