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In abstract algebra, the biquaternions are the numbers , where ''w'', ''x'', ''y'', and ''z'' are complex numbers and the elements of multiply as in the quaternion group. As there are three types of complex number, so there are three types of biquaternion: * (Ordinary) biquaternions when the coefficients are (ordinary) complex numbers * Split-biquaternions when ''w'', ''x'', ''y'', and ''z'' are split-complex numbers * Dual quaternions when ''w'', ''x'', ''y'', and ''z'' are dual numbers. This article is about the ordinary biquaternions named by William Rowan Hamilton in 1844 (see Proceedings of Royal Irish Academy 1844 & 1850 page 388). Some of the more prominent proponents of these biquaternions include Alexander Macfarlane, Arthur W. Conway, Ludwik Silberstein, and Cornelius Lanczos. As developed below, the unit quasi-sphere of the biquaternions provides a presentation of the Lorentz group, which is the foundation of special relativity. The algebra of biquaternions can be considered as a tensor product (taken over the reals) where C is the field of complex numbers and H is the algebra of (real) quaternions. In other words, the biquaternions are just the complexification of the (real) quaternions. Viewed as a complex algebra, the biquaternions are isomorphic to the algebra of 2×2 complex matrices ''M''2(C). They can be classified as the Clifford algebra . This is also isomorphic to the Pauli algebra Cℓ3,0(R), and the even part of the spacetime algebra Cℓ01,3(R). ==Definition== Let be the basis for the (real) quaternions, and let ''u'', ''v'', ''w'', ''x'' be complex numbers, then :''q'' = ''u'' 1 + ''v'' i + ''w'' j + ''x'' k is a ''biquaternion''.〔Hamilton (1853) page 639〕 To distinguish square roots of minus one in the biquaternions, Hamilton〔Hamilton (1853) page 730〕〔Hamilton (1899) ''Elements of Quaternions'', 2nd edition, page 289〕 and Arthur W. Conway used the convention of representing the square root of minus one in the scalar field C by h since there is an i in the quaternion group. Then : h i = i h, h j = j h, and h k = k h since h is a scalar. Hamilton's primary exposition on biquaternions came in 1853 in his ''Lectures on Quaternions'', now available in the ''Historical Mathematical Monographs'' of Cornell University. The two editions of ''Elements of Quaternions'' (1866 & 1899) reduced the biquaternion coverage in favor of the real quaternions. He introduced the terms bivector, ''biconjugate, bitensor'', and ''biversor''. Considered with the operations of component-wise addition, and multiplication according to the quaternion group, this collection forms a 4-dimensional algebra over the complex numbers. The algebra of biquaternions is associative, but not commutative. A biquaternion is either a unit or a zero divisor. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「biquaternion」の詳細全文を読む スポンサード リンク
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