查看“︁分裂四元数”︁的源代码

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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[[結合代數|结合代数]]的元素，由{{Tsl|en|James Cockle}}在1849年引入，当时称为反四元数。 类似于[[威廉·哈密頓|汉密尔顿]]1843年引入的[[四元數|四元数]] ，它们组成了一个四[[維度|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{{Tsl|en|Idempotent_(ring_theory)|幂等元}}。（例如， <math>{1\over2}(1+j)</math> 是幂等的零因子，而<math>i-j</math> 是幂零元。)作为一种[[数学结构]]，分裂四元数形成了域代数，且与2 × 2的[[实数|实]][[矩阵]][[同构]]。

[[集合 (数学)|集合]] <math>\left\{1,i,j,k\right\}</math>组成一个[[基 (線性代數)|基]]。 这些元素的积由
:<math>ij=k=-ji</math>,
:<math>jk=-i=-kj</math>,
:<math>ki=j=-ik</math>,
:<math>i^2=-1</math>,
:<math>j^2=+1</math>,
:<math>k^2=+1</math>
给出。因此<math>ijk=1</math>。 由以上定义可得，集合<math>\left\{1,i,j,k,-1,-i,-j,-k\right\}</math>在分裂四元数乘法的定义下是一个[[群]]，与[[二面體群|二面体群]]<math>D_4</math>[[同构]]，称为正方形的对称群。

分裂四元数<math>q=w+xi+yj+zk</math>的'''共轭'''<math>q^*=w-xi-yj-zk</math>。

由于其基向量的[[反交換律|反交换性]]，分裂四元数与其共轭的积由其[[迷向二次型]]

:<math>N(q)=qq^*=w^2+x^2-y^2-z^2</math>


给出。

给定两个反四元数<math>p</math>和<math>q</math>，有<math>N(pq)=N(p)N(q)</math>，意味着<math>N</math> 是可合成的二次型。 其上的代数是一种合成代数，''<math>N</math>'' 是其'''范数'''。 任何满足<math>q\neq0</math>，<math>N(q)=0</math>的反四元数q称为零向量（Null vector而非Zero vector)，它的存在意味着反四元数形成"分裂的合成代数"，因此反四元数也被称为'''分裂四元数'''。

当范数非零时，''<math>q</math>''有[[倒数]]，即 <math>q^*\over N(q)</math>. 集合
:<math>U=\left\{q:qq^*\neq0\right\}</math>

是[[可逆元|单位元]]的集合。 全体分裂四元数的集合<math>\mathbb{P}</math>组成[[环 (代数)|环]] <math>(\mathbb{P},+,\cdot)</math> ，其[[单位群]]为<math>(U,\cdot)</math>。全体<math>N(q)=1</math>的分裂四元数组成一个非[[紧空间|紧致]]的[[拓扑群]] <math>SU(1,1)</math>，且与[[SL₂(ℝ)|<math>SL(2,\mathbb{R})</math>]]同构（见下）。

历史上讲，分裂四元数早于[[阿瑟·凱萊|凯莱]]的矩阵代数；分裂四元数(及四元数和[[雙複數|双复数]])引发了对[[线性代数]]的深入研究。

== 矩阵表示 ==
令<math>q=w+xi+yj+zk</math>，考虑普通复数<math>u=w+xi</math>, <math>v=y+zi</math>，它们的共轭复数为<math>u=w+xi</math>, <math>v=y+zi</math>。然后

: <math>\begin{pmatrix}u & v \\ v^* & u^* \end{pmatrix}</math>

将''<math>q</math>''[[表示论|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

: <math>uu^*-vv^*=qq^*</math>

减号的出现将反四元数与使用了加号的四元数<math>\mathbb{H}</math>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{{Tsl|en|2_×_2_real_matrices|2×2实矩阵}}联系起来。这种同构可以明确如下：首先注意到积

: <math>\begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0\end{pmatrix}</math>

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<math>M(2,\mathbb{R})</math>的基。可以使上述矩阵乘积对应于反四元数环中的<math>jk=-i</math>。然后，对于任意矩阵有一个[[双射]]

: <math>\begin{pmatrix} a & c \\ b & d\end{pmatrix} \leftrightarrow q = \frac{(a+d) + (c-b)i + (b+c)j + (a-d)k}{2},</math>

这实际上形成了环同构。此外，计算各项的平方和表明<math>qq^*=ad-bc</math>，矩阵的行列式。因此，反四元数的单位{{Tsl|en|Quasi-sphere|拟球}}与<math>SL(2,\mathbb{R})= \{g\in M(2,\mathbb{R}): \det g = 1\}</math>群同构，因此与<math>SU(1,1)</math>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。<ref>Karzel, Helmut & Günter Kist (1985) "Kinematic Algebras and their Geometries", in ''Rings and Geometry'', R. Kaya, P. Plaumann, and K. Strambach editors, pp. 437&#x2013;509, esp 449,50, [[D. Reidel]] </ref>

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]][[体 (数学)|域]]上的代数，它是仅有的七个这样的代数之一。

== 由双曲复数生成 ==
Kevin McCrimon展示了如何按照L. E. Dickson和Adrian Albert为<math>\mathbb{C}</math>、<math>\mathbb{H}</math>和<math>\mathbb{O}</math>给出的除法构造所有的合成代数。<ref>Kevin McCrimmon (2004) ''A Taste of Jordan Algebras'', page 64, Universitext, Springer {{ISBN|0-387-95447-3}} {{MathSciNet}}</ref>实际上，他给出了real-split的doubled product的乘法法则

: <math>(a,b)(c,d)\ = \ (ac + d^* b, \ da + bc^* )</math>

如前所述，双共轭 <math>(a,b)^* \ =\ (a^*, - b), </math> 因此

: <math>N(a,b) \ = \ (a,b)(a,b)^* \ = \ (a a^* - b b^*, 0).</math>

如果''a''和''b''是[[雙曲複數|双曲复数]]，分裂四元数<math>q=(a,b)=((w+zj),(y+xj))</math>那么

:<math>N(q)=aa^*-bb^*=w^2-z^2-(y^2-x^2)=w^2+x^2-z^2-y^2</math>.

== 性质 ==
[[File:HyperboloidOfOneSheet.PNG|右|缩略图|圆E在平面 ''z'' =0中。<br /><br /><br /><br />''J'' 的元素是+1的[[平方根]]。]]
[[File:HyperboloidOfTwoSheets.svg|右|缩略图|<br /><br /><br /><br />I的元素是−1的平方根。]]

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

令

: <math>r(\theta)=j\cos\theta+k\sin\theta</math>

参数<math>z</math>和<math>r(\theta)</math>是此子空间中[[圓柱坐標系|圆柱坐标系]]的基。参数<math>\theta</math>表示[[方位角]]。接下来令a表示任意实数，并考虑反四元数

: <math>p(a,r)=i\sinh a+r\cosh a</math>
: <math>v(a,r)=i\cosh a+r\sinh a</math>

这正是{{Tsl|en|Alexander Macfarlane}}和Carmody的等边双曲面坐标。<ref>Carmody, Kevin (1997) "Circular and hyperbolic quaternions, octonions, sedionions", ''Applied Mathematics and Computation'' 84(1):27&#x2013;47, esp. 38</ref>

接下来，在环的向量子空间中构造三个基础集合:

: <math>E=\{r\in\mathbb{P}:r=r(\theta),0\leq\theta<2\pi\}</math>
: <math>J=\{p(a,r)\in\mathbb{P}:a\in\mathbb{R},r\in E\}</math>, 单叶[[双曲面]]
: <math>I=\{v(a,r)\in\mathbb{P}:a\in\mathbb{R},r\in E\}</math>, 双叶双曲面

现在很容易验证

: <math>\{q\in\mathbb{P}:q^2=1\}=J\cup\{1,-1\}</math>

及

: <math>\{q\in\mathbb{P}:q^2=-1\}=I</math>

这些集合相等意味着当<math>p\in J</math>时，平面

: <math>\{x+yp:x,y\in\mathbb{R}\}=D_p</math>

是<math>\mathbb{P}</math>的一个与双曲复数平面同构的[[子環|子环]]，就像对<math>I</math>中的任意<math>v</math>，

: <math>\{x+yv:x,y\in\mathbb{R}\}=C_v</math>

是与普通[[复平面]]<math>\mathbb{C}</math>同构的<math>\mathbb{P}</math>的平面子环。

注意对于所有<math>r\in E</math>， <math>(r+i)^2=0=(r-i)^2</math>，因此<math>r+i</math>和<math>r-i</math>是幂零元。平面<math>N=\{x+y(r+i):x,y\in\mathbb{R}\}</math>是<math>\mathbb{P}</math>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<math>D_p</math>、<math>C_v</math>或<math>N</math>平面上，所以这些平面组成了<math>\mathbb{P}</math>，例如，单位拟球

: <math>SU(1,1)=\{q\in\mathbb{P}:qq^*=1\}</math>

包含了<math>\mathbb{P}</math>的构成平面上的“单位圆”：在<math>D_p</math>中是一个[[单位双曲线]]，在<math>N</math>中是一对平行线，而在<math>C_v</math>中确实是一个圆。

== 泛正交性 ==
反四元数<math>q=w+xi+yj+zk</math>的[[标量 (数学)|标量]]部分为w。

'''定义'''  对于非零反四元数<math>q</math>和<math>t</math>，<math>q\perp t</math>当且仅当乘积<math>qt^*</math>的标量部分为零。

* 对任意的 <math>v\in I</math>，如果 <math>q,t\in C_v</math>，那么<math>q\perp t</math>意味着从<math>0</math>到<math>q</math>和<math>t</math>的[[射线]]是[[垂直]]的。
* 对任意的 <math>p\in J</math>，如果 <math>q,t\in D_p</math>，那么<math>q\perp t</math>意味着这两点是{{Tsl|en|Hyperbolic_orthogonality|双曲正交}}的。
* 对任意的 <math>r\in E</math> ， <math>a\in\mathbb{R}</math>，<math>p(a,r)</math>和<math>v(a,r)</math>满足 <math>q\perp t</math>。
* 如果<math>u</math>是反四元数环中的一个单位元，那么<math>q\perp t</math>意味着<math>qu\perp tu</math>。

<blockquote>'''证明'''：因向量[[外积]]的反交换性，<math>(tu)^*=u^*t^*</math>，因此<math>(qu)(tu)^*=(uu^*)q(t^*)</math>。</blockquote>

==Counter-sphere geometry==
The quadratic form ''qq''<sup>∗</sup> is positive definite on the planes ''C<sub>v</sub>'' and ''N''. Consider the '''counter-sphere''' {''q'': ''qq''<sup>∗</sup> = −1}.

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

:''mm''<sup>∗</sup> = −1 = ''x''<sup>2</sup> + y<sup>2</sup> − z<sup>2</sup>.

Since points on the counter-sphere must line on the conjugate of the [[unit hyperbola]] in some plane {{nowrap|''D<sub>p</sub>'' ⊂ '''P'''}}, ''m'' can be written, for some {{nowrap|''p'' ∈ ''J''}}

:<math>m~= p \exp{(bp)} = \sinh b + p \cosh b = \sinh b + i \sinh a~\cosh b + r \cosh a~\cosh b</math>.

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:

:<math>\tan \phi = \frac{x}{y} = \frac{\sinh b}{\sinh a ~\cosh b} = \frac{\tanh b}{\sinh a}</math>. Then

:<math>\lim_{b \to \infin}\tan \phi  = \frac 1 {\sinh a},</math>

as in the expression of [[angle of parallelism]] in the [[hyperbolic geometry|hyperbolic plane]] H<sup>2</sup> . The parameter ''θ'' determining the meridian varies over the [[unit circle|''S''<sup>1</sup>]]. Thus the counter-sphere appears as the [[manifold]] ''S''<sup>1</sup> × H<sup>2</sup>.

==Application to kinematics==

By using the foundations given above, one can show that the mapping
:<math>q \mapsto u^{-1}q u </math>
is an ordinary or hyperbolic rotation according as
:<math>u = e^{av} ,\quad v \in I \quad \text{or}\quad u = e^{ap} ,\quad p \in J </math>.

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 {{nowrap|(2, 2)}} signature when [[spacetime]] is presumed to have signature {{nowrap|(1, 3)}} or {{nowrap|(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 {{nowrap|1=''tt''<sup>∗</sup> = −1}}, then there is a {{nowrap|1=''p'' = i sinh(''a'') + ''r'' cosh(''a'') ∈ ''J''}} such that {{nowrap|''t'' ∈ ''D''<sub>''p''</sub>}}, and a {{nowrap|''b'' ∈ ''R''}} such that {{nowrap|1=''t'' = ''p'' exp(''bp'')}}. Then if {{nowrap|1=''u'' = exp(''bp'')}}, {{nowrap|1=''v'' = i cosh(''a'') + ''r'' sinh(''a'')}}, and {{nowrap|1=''s'' = i''r''}}, the set {{nowrap|{''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)<ref>[[James Cockle]] (1849), [https://www.biodiversitylibrary.org/item/20114#page/448/mode/1up On Systems of Algebra involving more than one Imaginary] {{Wayback|url=https://www.biodiversitylibrary.org/item/20114#page/448/mode/1up|date=20160308023205}}, ''[[Philosophical Magazine]]'' (series 3) 35: 434,5, link from [[Biodiversity Heritage Library]]</ref> in 1849 by [[James Cockle]] in the London&ndash;Edinburgh&ndash;Dublin [[Philosophical Magazine]]. The introductory papers by Cockle were recalled in the 1904 ''Bibliography''<ref>A. Macfarlane (1904) [http://dlxs2.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=03030001 Bibliography of Quaternions and Allied Systems of Mathematics], from [[Cornell University]] ''Historical Math Monographs'', entries for James Cockle, pp.&nbsp;17–18</ref> 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.<ref>Alexander Macfarlane (1900) [http://www.mathunion.org/ICM/ICM1900/Main/icm1900.0305.0312.ocr.pdf Application of space analysis to curvilinear coordinates] {{Wayback|url=http://www.mathunion.org/ICM/ICM1900/Main/icm1900.0305.0312.ocr.pdf|date=20140810042126}}, ''Proceedings of the ''[[International Congress of Mathematicians]], Paris, page 306, from [[International Mathematical Union]]</ref>

The unit sphere was considered in 1910 by Hans Beck.<ref>Hans Beck (1910) [http://www.ams.org/journals/tran/1910-011-04/S0002-9947-1910-1500872-0/S0002-9947-1910-1500872-0.pdf Ein Seitenstück zur Mobius'schen Geometrie der Kreisverwandschaften] {{Wayback|url=http://www.ams.org/journals/tran/1910-011-04/S0002-9947-1910-1500872-0/S0002-9947-1910-1500872-0.pdf|date=20190503215226}}, [[Transactions of the American Mathematical Society]] 11</ref> For example, the dihedral group appears on page 419. The coquaternion structure has also been mentioned briefly in the ''[[Annals of Mathematics]]''.<ref>[[A. A. Albert]] (1942), "Quadratic Forms permitting Composition", ''[[Annals of Mathematics]]'' 43:161 to 77</ref><ref>[[Valentine Bargmann]] (1947), [https://www.jstor.org/discover/10.2307/1969129 "Irreducible unitary representations of the Lorentz Group"], ''[[Annals of Mathematics]]'' 48: 568&#x2013;640</ref>

==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 &minus;k.
* Exspherical system (Macfarlane 1900)
* Split-quaternions (Rosenfeld 1988)<ref>Rosenfeld, B.A. (1988) ''A History of Non-Euclidean Geometry'', page 389, Springer-Verlag {{isbn|0-387-96458-4}}</ref>
* Antiquaternions (Rosenfeld 1988)
* Pseudoquaternions (Yaglom 1968<ref>[[Isaak Yaglom]] (1968) ''Complex Numbers in Geometry'', page 24, [[Academic Press]]</ref> Rosenfeld 1988)

==参见==
* [[Split-biquaternion]]s
* [[Split-octonion]]s
* [[Hypercomplex numbers]]

==参考资料==
{{reflist}}

==延伸阅读==
* [[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.&nbsp;205&ndash;234, {{arxiv|math.DG/0310415}}, {{MR|2158044}}.
* 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&ndash;63. [https://www.researchgate.net/publication/270760686_The_Roots_of_a_Split_Quaternion]
* Özdemir, M. & A.A. Ergin (2006) "Rotations with timelike quaternions in Minkowski 3-space", ''Journal of Geometry and Physics'' 56: 322&ndash;36.[https://www.researchgate.net/publication/235591460_Rotations_with_unit_timelike_quaternions_in_Minkowski_3-space]
* Pogoruy, Anatoliy & Ramon M Rodrigues-Dagnino (2008) [http://www.springerlink.com/content/r0166812hk163851/ Some algebraic and analytical properties of coquaternion algebra]{{Dead link|date=2020年2月 |bot=InternetArchiveBot |fix-attempted=yes }}, ''[[Advances in Applied Clifford Algebras]]''.

{{數的系統}}
[[Category:狹義相對論]]
[[Category:雙曲幾何]]
[[Category:四元數]]