Quaternionic vector space

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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, ... 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[edit]


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