分裂四元数

✍ dations ◷ 2025-08-02 21:21:51 #狭义相对论,双曲几何,四元数

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(英语:James Cockle)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、幂零元和幂等元(英语:Idempotent_(ring_theory))。(例如, 1 2 ( 1 + j ) {\displaystyle {1 \over 2}(1+j)} 是其范数。 任何满足 q 0 {\displaystyle q\neq 0} 有倒数,即 q N ( q ) {\displaystyle q^{*} \over N(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

减号的出现将反四元数与使用了加号的四元数 H {\displaystyle \mathbb {H} } 和是双曲复数,分裂四元数 q = ( a , b ) = ( ( w + z j ) , ( y + x j ) ) {\displaystyle q=(a,b)=((w+zj),(y+xj))} ∗ is positive definite on the planes and . Consider the counter-sphere {: ∗ = −1}.

Take = + i + where = j cos() + k sin(). Fix and suppose

Since points on the counter-sphere must line on the conjugate of the unit hyperbola in some plane ⊂ P, can be written, for some ∈

Let φ be the angle between the hyperbolas from to and . This angle can be viewed, in the plane tangent to the counter-sphere at , by projection:

as in the expression of angle of parallelism in the hyperbolic plane H2 . The parameter determining the meridian varies over the 1. Thus the counter-sphere appears as the manifold 1 × H2.

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 (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 ∗ = −1, then there is a = i sinh() + cosh() ∈ such that ∈ , and a ∈ such that = exp(). Then if = exp(), = i cosh() + sinh(), and = i, the set {, , , } is a pan-orthogonal basis stemming from , and the orthogonalities persist through applications of the ordinary or hyperbolic rotations.

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 of the Quaternion Society. Alexander Macfarlane called the structure of coquaternion vectors an 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 .

相关

  • 二手烟二手烟,亦称非自愿性吸烟,是指在吸取燃点烟草时随着烟雾释放出来的物质,是一种被动吸烟(Passive smoking)方式有研究指二手烟有焦油、阿摩尼亚、尼古丁、悬浮微粒、超细悬浮微粒
  • 并发症并发症(complication)指在疾病发展过程中之续发性反应所造成的结果,是医学、病理,可能是因为病患在医疗或护理过程中,因为一种疾病合并引发其他的另一种或几种疾病。
  • 语音在语言学中,语音(英语:phone)可以被认为是用来表示语言的声音符号(即语言的物质外壳),也可以被定义为是人的发音器官所发出来的具有一定意义的声音。在语音学与音韵学好中,语音一词
  • 下颚骨下颌骨又称下颚骨,是最大,最强的颜面骨,也是颅骨中唯一可以动的骨头,与上颌骨形成口腔。侧视图。前视图。下颌骨下颌骨外部侧视图。下颌骨内部侧视图。
  • 男男性接触者男男性行为者(英文:men who have sex with men,缩写MSM,又称男男性接触者、男男性行为人群)是指与同性发生性关系的男性,而不管他们自我认定为何种性向。这个术语主要用于美国,用以
  • NHK松江放送局NHK松江放送局,是日本放送协会位于岛根县松江市的地方放送局,也是负责主管当地事务的放送局。
  • 101 (组合)101(韩语:원오원)是在2016年推出的八人企划女团,成员包括仁善、姜允、秀妍、艺智、书馨、娥英、海泳、美笑,成员都曾参加大型女团选秀节目《PRODUCE 101》。注:角标示
  • 王个簃王个.mw-parser-output ruby>rt,.mw-parser-output ruby>rtc{font-feature-settings:"ruby"1}.mw-parser-output ruby.large{font-size:250%}.mw-parser-output ruby.larger
  • 哈维·贝内特哈维·贝内特(英文:Harve Bennett,1930年8月17日-2015年2月25日),生名哈维·贝内特·菲施曼(英文:Harve Bennett Fischman),于1930年8月17日出生于伊利诺伊州芝加哥,是美国的一位电视、
  • 阿尔弗雷德·盖斯利阿尔弗雷德·盖斯利(英语:Sir Alfred Gaselee;1844年6月3日-1918年3月29日),是英国中将,参加过第二次英国-阿富汗战争。1900年中国爆发了义和团运动,出任英国远征军司令,与其他国家的