Biquaternion

Biquaternion

In abstract algebra, the biquaternions are the numbers where w, x, y, and z are complex numbers and the elements of {1, i, j, k} 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
• Study biquaternions or dual quaternions when w, x, y, and z are dual numbers.

The following 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 C⊗H (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 M₂(C). In terms of Clifford algebra they can be classified as Cℓ₂(C) = Cℓ0₃(C). This is also isomorphic to the Pauli algebra Cℓ_3,0(R), and the even part of the space-time algebra Cℓ0_1,3(R).