Quaternionic vector space

Quaternionic vector space

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 H"n" 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 "q"1, "q"2, ... "q""n".

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

ee also

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

References

*cite book
first = F. Reese
last = Harvey
year = 1990
title = Spinors and Calibrations
publisher = Academic Press
location = San Diego
id = ISBN 0-12-329650-1


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