分裂四元數

在抽象代數中，分裂四元數是一種四維的結合代數的元素，由在1849年引入，當時稱為反四元數。 類似於漢密爾頓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、冪零元和。（例如， formula_1 是冪等的零因子，而formula_2 是冪零元。)作為一種數學結構，分裂四元數形成了域代數，且與2 × 2的實矩陣同構。集合 formula_3組成一個基。 這些元素的積由formula_4,formula_5,formula_6,formula_7,formula_8,formula_9給出。因此formula_10。 由以上定義可得，集合formula_11在分裂四元數乘法的定義下是一個群，與二面體群formula_12同構，稱為正方形的對稱群。分裂四元數formula_13的共軛formula_14。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formula_15給出。給定兩個反四元數formula_16和formula_17，有formula_18，意味著formula_19 是可合成的二次型。 其上的代數是一種合成代數，"formula_19" 是其範數。 任何滿足formula_21，formula_22的反四元數q稱為零向量（Null vector而非Zero vector)，它的存在意味著反四元數形成"分裂的合成代數"，因此反四元數也被稱為分裂四元數。當範數非零時，"formula_17"有倒數，即 formula_24. 集合formula_25是單位元的集合。 全體分裂四元數的集合formula_26組成環 formula_27 ，其單位群為formula_28。全體formula_29的分裂四元陣列成一個非緊緻的拓撲群 formula_30，且與formula_31同構。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。矩陣表示.令formula_13，考慮普通複數formula_33, formula_34，它們的共軛複數為formula_33, formula_34。然後formula_37將"formula_17"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formula_39減號的出現將反四元數與使用了加號的四元數formula_40區分開來。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。除了復矩陣表示，另一種線性表示將反四元數與聯絡起來。這種同構可以明確如下：首先注意到積formula_41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formula_42的基。可以使上述矩陣乘積對應於反四元數環中的formula_43。然後，對於任意矩陣有一個雙射formula_44這實際上形成了環同構。此外，計算各項的平方和表明formula_45，矩陣的行列式。因此，反四元數的單位與formula_46群同構，因此與formula_30也群同構，後者可以從上面的復表示中得到。例如，用2×2實矩陣表示雙曲運動群，見Karzel和Kist。在這兩種線性表示中，範數由行列式給出。由於行列式是乘法對映，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。由雙曲複數生成.Kevin McCrimon展示瞭如何按照L. E. Dickson和Adrian Albert為formula_48、formula_40和formula_50給出的除法構造所有的合成代數。實際上，他給出了real-split的doubled product的乘法法則formula_51如前所述，雙共軛 formula_52 因此formula_53如果"a"和"b"是雙曲複數，分裂四元數formula_54那麼formula_55.性質.可以透過formula_26的子空間formula_57來了解其子代數。令formula_58引數formula_59和formula_60是此子空間中圓柱座標系的基。引數formula_61表示方位角。接下來令a表示任意實數，並考慮反四元數formula_62formula_63這正是和Carmody的等邊雙曲面座標。接下來，在環的向量子空間中構造三個基礎集合:formula_64formula_65, 單葉雙曲面formula_66, 雙葉雙曲面現在很容易驗證formula_67及formula_68這些集合相等意味著當formula_69時，平面formula_70是formula_26的一個與雙曲複數平面同構的子環，就像對formula_72中的任意formula_73，formula_74是與普通複平面formula_48同構的formula_26的平面子環。注意對於所有formula_77， formula_78，因此formula_79和formula_80是冪零元。平面formula_81是formula_26的一個與二元數同構的子環。由於每個反四元數都必須位於某個formula_83、formula_84或formula_19平面上，所以這些平面組成了formula_26，例如，單位擬球formula_87包含了formula_26的構成平面上的“單位圓”：在formula_83中是一個單位雙曲線，在formula_19中是一對平行線，而在formula_84中確實是一個圓。泛正交性.反四元數formula_13的標量部分為w。定義 對於非零反四元數formula_17和formula_94，formula_95當且僅當乘積formula_96的標量部分為零。證明：因向量外積的反交換性，formula_114，因此formula_115。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"formula_116.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:formula_117. Thenformula_118as in the expression of angle of parallelism in the hyperbolic plane H2 . The parameter "θ" determining the meridian varies over the "S"1. Thus the counter-sphere appears as the manifold "S"1 × H2.Application to kinematics.By using the foundations given above, one can show that the mappingformula_119is an ordinary or hyperbolic rotation according asformula_120.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 "tt"∗ = −1, then there is a "p" = i sinh("a") + "r" cosh("a") ∈ "J" such that "t" ∈ "D""p", and a "b" ∈ "R" such that "t" = "p" exp("bp"). Then if "u" = exp("bp"), "v" = i cosh("a") + "r" sinh("a"), and "s" = i"r", the set {"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) 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".