Biquaternion
Quaternions with complex number coefficients / From Wikipedia, the free encyclopedia
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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.[1] 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 ⊗R H, 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 C ⊗R H = Cl[0]
3(C) = Cl2(C) = Cl1,2(R),[2] the Pauli algebra Cl3,0(R),[3][4] and the even part Cl[0]
1,3(R) = Cl[0]
3,1(R) of the spacetime algebra.[5]