Quaternionic vector space explained

In mathematics, a left (or right) quaternionic vector space is a left (or right)

H -module where H is the division ring of quaternions. One must distinguish between left and right quaternionic vector spaces since H is non-commutative. Further, H is not a field, so quaternionic vector spaces are not vector spaces, but merely modules.

The space

H ^{n} is both a left and right quaternionic vector space using componentwise multiplication. Namely, for q \in H and (r _{1,} \ldots, r _{n)} \in H ^{n} ,q (r _{1,} \ldots, r _{n)} = (q r _{1,} \ldots, q r _{n),} (r _{1,} \ldots, r _{n)} q = (r _{1} q, \ldots, r _{n} q).

Since

H is a division algebra, every finitely generated (left or right) H -module has a basis, and hence is isomorphic to H ^{n} for some n .

References

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