Algebraic Structure
Like the quaternions, the set of hyperbolic quaternions form a vector space over the real numbers of dimension 4. A linear combination
is a hyperbolic quaternion when and are real numbers and the basis set has these products:
Unlike the ordinary quaternions, the hyperbolic quaternions are not associative. For example, while . The first three relations show that products of the (non-real) basis elements are anti-commutative. Although this basis set does not form a group, the set
forms a quasigroup. One also notes that any subplane of the set M of hyperbolic quaternions that contains the real axis forms a plane of split-complex numbers. If
is the conjugate of, then the product
is the quadratic form used in spacetime theory. In fact, the bilinear form called the Minkowski inner product arises as the negative of the real part of the hyperbolic quaternion product pq* :
- .
Note that the set of units U = {q : qq* ≠ 0 } is not closed under multiplication. See the references (external link) for details.
Read more about this topic: Hyperbolic Quaternion
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