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

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

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

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

( p 1 , … , p n ) + ( r 1 , … , r n ) = ( p 1 + r 1 , … , p n + r n ) {\displaystyle (p_{1},\ldots ,p_{n})+(r_{1},\ldots ,r_{n})=(p_{1}+r_{1},\ldots ,p_{n}+r_{n})}
q ( r 1 , … , r n ) = ( q r 1 , … , q r n ) {\displaystyle q(r_{1},\ldots ,r_{n})=(qr_{1},\ldots ,qr_{n})}

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

( p 1 , … , p n ) + ( r 1 , … , r n ) = ( p 1 + r 1 , … , p n + r n ) {\displaystyle (p_{1},\ldots ,p_{n})+(r_{1},\ldots ,r_{n})=(p_{1}+r_{1},\ldots ,p_{n}+r_{n})}
( r 1 , … , r n ) q = ( r 1 q , … , r n q ) {\displaystyle (r_{1},\ldots ,r_{n})q=(r_{1}q,\ldots ,r_{n}q)}

## See also

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

## References

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

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