Quaternionic vector space

In mathematics, a left (or right) quaternionic vector space is a left (or right) $\mathbb {H}$-module where $\mathbb {H}$ is the division ring of quaternions. One must distinguish between left and right quaternionic vector spaces since $\mathbb {H}$ is non-commutative. Further, $\mathbb {H}$ is not a field, so quaternionic vector spaces are not vector spaces, but merely modules.

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

$q(r_{1},\ldots ,r_{n})=(qr_{1},\ldots ,qr_{n}),$

$(r_{1},\ldots ,r_{n})q=(r_{1}q,\ldots ,r_{n}q).$

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

See also

• Vector space
• General linear group
• Special linear group
• SL(n,H)
• Symplectic group

References

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