Biquaternion | Wikipedia audio article
Channel: wikipedia tts
Date: 24/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Biquaternion


00:01:30 1 Definition
00:01:40 2 Place in ring theory
00:02:00 2.1 Linear representation
00:04:35 2.2 Subalgebras
00:04:44 3 Algebraic properties
00:05:55 3.1 Relation to Lorentz transformations
00:06:46 4 Associated terminology
00:08:34 5 As a composition algebra
00:12:36 6 See also
00:20:06 7 Notes
00:28:57 8 References
00:32:22 See also



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SUMMARY
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In abstract algebra, the biquaternions are the numbers  w + x i + y j + z k, where w, x, y, and z are complex numbers, or variants thereof, and the elements of {1, i, j, k} multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof:

Biquaternions when the coefficients are complex numbers.
Split-biquaternions when the coefficients are split-complex numbers.
Dual quaternions when the coefficients are dual numbers.This article is about the ordinary biquaternions named by William Rowan Hamilton in 1844 (see Proceedings of the 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 representation of the Lorentz group, which is the foundation of special relativity.
The algebra of biquaternions can be considered as a tensor product ℂ ⊗ ℍ (taken over the reals) where ℂ is the field of complex numbers and ℍ is the division algebra of (real) quaternions. In other words, the biquaternions are just the complexification of the quaternions. Viewed as a complex algebra, the biquaternions are isomorphic to the algebra of 2 × 2 complex matrices M2(ℂ).  They are also isomorphic to several Clifford algebras including ℍ(ℂ) = Cℓ03(ℂ) = Cℓ2(ℂ) = Cℓ1,2(ℝ), the Pauli algebra Cℓ3,0(ℝ), and the even part Cℓ01,3(ℝ) = Cℓ03,1(ℝ) of the spacetime algebra.