Quaternionic vector space

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In noncommutative algebra, a branch of mathematics, a quaternionic vector space is a module over the quaternions. Since the quaternion algebra is division ring, these modules are referred to as "vector spaces". However, the quaternion algebra is noncommutative so we must distinguish left and right vector spaces. In left vector spaces, linear compositions of vectors and have the form where , . In right vector spaces, linear compositions of vectors and have the form .

Similar to vector spaces over a field, if a quaternionic vector space has finite dimension , then it is isomorphic to the direct sum of copies of the quaternion algebra . In this case we can use a standard basis which has the form

In a left quaternionic vector space we use componentwise sum of vectors and product of vectors over scalars

In a right quaternionic vector space we also use componentwise sum of vectors and product of vectors over scalars


See also

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References

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  • Harvey, F. Reese (1990). Spinors and Calibrations. San Diego: Academic Press. ISBN 0-12-329650-1.