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In [[mathematics]], a '''left''' (or '''right''') '''quaternionic vector space''' is a left (or right) '''H'''-[[module (mathematics)|module]] where '''H''' denotes the [[ring (mathematics)|noncommutative ring]] of the [[quaternion]]s.
 
The space '''H'''<sup>''n''</sup> of ''n''-tuples of quaternions is both a left and right '''H'''-module using the componentwise left and right multiplication:
:<math> q (q_1,q_2,\ldots q_n) = (q q_1,q q_2,\ldots q q_n) </math>
:<math> (q_1,q_2,\ldots q_n) q = (q_1 q, q_2 q,\ldots q_n q)</math>
for quaternions ''q'' and ''q''<sub>1</sub>, ''q''<sub>2</sub>, ... ''q''<sub>''n''</sub>.
 
Since '''H''' is a [[division algebra]], every [[Finitely generated module|finitely generated]] (left or right) '''H'''-module has a [[basis (linear algebra)|basis]], and hence is isomorphic to '''H'''<sup>''n''</sup> for some ''n''.
 
==See also==
* [[Vector space]] <!-- Doesn't mention the quaternions, but should!-->
* [[General linear group]] <!-- Doesn't mention the quaternions, but should!-->
* [[Special linear group]] <!-- Doesn't mention the quaternions, but should!-->
* [[List of simple Lie groups#A2n.E2.88.921_II_.28n_.E2.89.A5_2.29|SL(n,H)]] <!-- No definitions here!-->
* [[Symplectic group]] <!-- Hoorah, it mentions Sp(n) and Sp(p,q)-->
 
==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
}}
 
{{algebra-stub}}
 
[[Category:Quaternions]]
[[Category:Linear algebra]]

Revision as of 16:35, 19 March 2013

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(q1,q2,qn)=(qq1,qq2,qqn)
(q1,q2,qn)q=(q1q,q2q,qnq)

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

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

References

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