Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Biquaternion
Summary
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 (taken over the reals) where C or is the field of complex numbers and H or 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.
Let {1, i, j, k} be the basis for the (real) quaternions H, and let u, v, w, x be complex numbers, then
is a biquaternion. To distinguish square roots of minus one in the biquaternions, Hamilton and Arthur W. Conway used the convention of representing the square root of minus one in the scalar field C by h to avoid confusion with the i in the quaternion group. Commutativity of the scalar field with the quaternion group is assumed:
Hamilton introduced the terms bivector, biconjugate, bitensor, and biversor to extend notions used with real quaternions H.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.