Quaternionic vector space

Since quaternion algebra is division ring, then module over quaternion algebra is called vector space. Because quaternion algebra is non-commutative, we distinguish left and right vector spaces. In left vector space, linear composition of vectors $v$ and $w$ has form $av+bw$ where $a$, $b\in H$. In right vector space, linear composition of vectors $v$ and $w$ has form $va+wb$.

If quaternionic vector space has finite dimension $n$, then it is isomorphic to direct sum $H^{n}$ of $n$ copies of quaternion algebra $H$. In such case we can use basis which has form

$e_{1}=(1,0,\ldots ,0)$

$\ldots$

$e_{n}=(0,\ldots ,0,1)$

In left quaternionic vector space $H^{n}$ we use componentwise sum of vectors and product of vector over scalar

$(p_{1},\ldots ,p_{n})+(r_{1},\ldots ,r_{n})=(p_{1}+r_{1},\ldots ,p_{n}+r_{n})$

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

In right quaternionic vector space $H^{n}$ we use componentwise sum of vectors and product of vector over scalar

$(p_{1},\ldots ,p_{n})+(r_{1},\ldots ,r_{n})=(p_{1}+r_{1},\ldots ,p_{n}+r_{n})$

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

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.