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 ${\displaystyle v}$ and ${\displaystyle w}$ has form ${\displaystyle av+bw}$ where ${\displaystyle a}$, ${\displaystyle b\in H}$. In right vector space, linear composition of vectors ${\displaystyle v}$ and ${\displaystyle w}$ has form ${\displaystyle va+wb}$.

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

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

${\displaystyle \ldots }$

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

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

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

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

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

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

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

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.