physics - Biquaternion

In mathematics, a biquaternion is a numeric and geometric concept developed by William Kingdon Clifford, William Rowan Hamilton, and Alexander MacAuley in the nineteenth century.

Let {1, i, j, k} 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.

The collection of all biquaternions forms a vector space of four complex dimensions or eight real dimensions. Considered with the operations of component-wise addition and multiplication according to the Quaternion group, this collection forms a non-commutative but associative ring.

Linear Representation

Note the matrix product

$\begin{pmatrix}i & 0\\0 & -i\end{pmatrix}\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}$ = $\begin{pmatrix}0 & i\\i & 0\end{pmatrix}$

where each of these three arrays has a square equal to the negative of the identity matrix. When the matrix product is interpreted as i j = k, then one obtains a subgroup of the group matrices that is isomorphic to the Quaternion group.Consequently

$\begin{pmatrix}u+iv & w+ix\\-w+ix & u-iv\end{pmatrix}$ represents biquaternion q.

Given any 2x2 complex matrix, there are complex values u, v, w, and x to put it in this form so that the matrix ring is isomorphic to the biquaternion ring.

Alternative Complex Plane

Suppose we take w to be purely imaginary, w = b ι, where ι ι = - 1. (Here one uses iota instead of i for the complex imaginary to be distinct from quaternion i.) Now when r = w j, then its square is

r r = (w j )(w j ) = (w w)(j j ) = b b (-1)(-1) = b²

In particular, when b = 1 or –1, then r ² = + 1. This development shows that the biquaternions are a source of "algebraic motors" like r that square to +1. Then {a + b ι j : a, b ∈ R } is a subring of biquaternions isomorphic to the split-complex number ring.

Categories: Ring theory