Quaternionic vector space

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

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

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

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

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

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.

1. Quaternionic representation
2. Quaternionic manifold
3. Symmetric space
4. Quaternion
5. Projective space
6. Spinor
7. Hyperkähler manifold
8. Symplectic group
9. Quaternionic analysis
10. Classical group
11. Complex projective space
12. Real representation
13. Quaternionic polytope
14. Rotation matrix
15. Hypertoric variety
16. Complex hyperbolic space
17. Serre–Swan theorem
18. Frobenius–Schur indicator
19. Stiefel manifold
20. Quaternions and spatial rotation