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BiquaternionIn 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 - =
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 matricies that is isomorphic to the Quaternion group.Consequently - 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.
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