nLab split quaternion

Definition

The split-quaternions are an algebra over the real numbers. Every split quaternion $q$ may be represented as

$$q = a_0 + a_1 i + a_2 j + a_3k$$

where the basis elements satisfy the following products:

$\times$	i	j	k
i	-1	k	-j
j	-k	+1	-i
k	j	i	+1

and conjugation $t^* = a_0 - a_1 i - a_2 j - a_3k$.

These are closely related to the quaternions, as the generators satisfy similarly-looking relations. They are obtained from the split-complex numbers through the generalization of the Cayley-Dickson construction.

Related concepts

• quaternion

• tessarines

• split-complex numbers

• split octonion

References

See also

• Wikipedia, Split-quaternion

On projective spaces over split-quaternions:

• Konrad Voelkel, Motivic cell structures for projective spaces over split quaternions, 2016 (freidok:11448, pdf)