Quaternionic vector space

In mathematics, a left (or right) quaternionic vector space is a left (or right) H-module where H denotes the noncommutative ring of the quaternions.

The space Hⁿ of n-tuples of quaternions is both a left and right H-module using the componentwise left and right multiplication:

$q (q_1,q_2,\ldots q_n) = (q q_1,q q_2,\ldots q q_n)$

$(q_1,q_2,\ldots q_n) q = (q_1 q, q_2 q,\ldots q_n q)$

for quaternions q and q₁, q₂, ... q_n.

Since H is a division algebra, every finitely generated (left or right) H-module has a basis, and hence is isomorphic to Hⁿ 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.