nLab split quaternion

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Contents

Contents

Definition

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

q=a 0+a 1i+a 2j+a 3k q = a_0 + a_1 i + a_2 j + a_3k

where the basis elements satisfy the following products:

×\timesijk
i-1k-j
j-k+1-i
kji+1

and conjugation t *=a 0a 1ia 2ja 3kt^* = 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.

References

See also

On projective spaces over split-quaternions:

Last revised on November 3, 2023 at 05:16:33. See the history of this page for a list of all contributions to it.