分裂四元数

title: 矩阵表示.
content: 令formula_13,考虑普通复数formula_33, formula_34,它们的共轭复数为formula_33, formula_34。然后  将"formula_17"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  减号的出现将反四元数与使用了加号的四元数formula_40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联系起来。这种同构可以明确如下:首先注意到积  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formula_42的基。可以使上述矩阵乘积对应于反四元数环中的formula_43。然后,对于任意矩阵有一个双射  这实际上形成了环同构。此外,计算各项的平方和表明formula_45,矩阵的行列式。因此,反四元数的单位与formula_46群同构,因此与formula_30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。  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。
title: 由双曲复数生成.
content: Kevin McCrimon展示了如何按照L. E. Dickson和Adrian Albert为formula_48、formula_40和formula_50给出的除法构造所有的合成代数。实际上,他给出了real-split的doubled product的乘法法则  如前所述,双共轭 formula_52 因此  如果"a"和"b"是双曲复数,分裂四元数formula_54那么
title: 性质.
content: 可以通过formula_26的子空间formula_57来了解其子代数。  令  参数formula_59和formula_60是此子空间中圆柱坐标系的基。参数formula_61表示方位角。接下来令a表示任意实数,并考虑反四元数  这正是和Carmody的等边双曲面坐标。  接下来,在环的向量子空间中构造三个基础集合:  现在很容易验证  及  这些集合相等意味着当formula_69时,平面  是formula_26的一个与双曲复数平面同构的子环,就像对formula_72中的任意formula_73,  是与普通复平面formula_48同构的formula_26的平面子环。  注意对于所有formula_77, formula_78,因此formula_79和formula_80是幂零元。平面formula_81是formula_26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formula_83、formula_84或formula_19平面上,所以这些平面组成了formula_26,例如,单位拟球  包含了formula_26的构成平面上的“单位圆”:在formula_83中是一个单位双曲线,在formula_19中是一对平行线,而在formula_84中确实是一个圆。
title: 泛正交性.
content: 反四元数formula_13的标量部分为w。  定义 对于非零反四元数formula_17和formula_94,formula_95当且仅当乘积formula_96的标量部分为零。  证明:因向量外积的反交换性,formula_114,因此formula_115。
title: Counter-sphere geometry.
content: The quadratic form "qq" is positive definite on the planes "C" 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  Since points on the counter-sphere must line on the conjugate of the unit hyperbola in some plane, "m" can be written, for some  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:  as in the expression of angle of parallelism in the hyperbolic plane H. The parameter "θ" determining the meridian varies over the "S". Thus the counter-sphere appears as the manifold "S" × H.
title: Application to kinematics.
content: By using the foundations given above, one can show that the mapping is an ordinary or hyperbolic rotation according as  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 signature when spacetime is presumed to have signature or. Nevertheless, a transparently relativistic kinematics appears when a point of the counter-sphere is used to represent an inertial frame of reference. Indeed, if, then there is a such that, and a such that. Then if,, and, the set is a pan-orthogonal basis stemming from "t", and the orthogonalities persist through applications of the ordinary or hyperbolic rotations.
title: Historical notes.
content: The coquaternions were initially introduced (under that name) in 1849 by James Cockle in the London–Edinburgh–Dublin Philosophical Magazine. The introductory papers by Cockle were recalled in the 1904 "Bibliography" 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.  The unit sphere was considered in 1910 by Hans Beck. For example, the dihedral group appears on page 419. The coquaternion structure has also been mentioned briefly in the "Annals of Mathematics".

Summary: 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,由在1849年引入,当时称为反四元数。 类似于汉密尔顿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。(例如, formula_1 是幂等的零因子,而formula_2 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的实矩阵同构。

Coquaternion: En mathématiques et en algèbre abstraite, un coquaternion est une idée mise en avant par James Cockle en 1849. Comme les quaternions de Hamilton inventés en 1843, ils forment un espace vectoriel réel à quatre dimensions muni d'une opération multiplicative. À la différence de l'algèbre des quaternions, les coquaternions peuvent avoir des diviseurs de zéro, des éléments idempotents ou nilpotents.