Editing Quaternionic representation (section)

== Examples ==

A common example involves the quaternionic representation of [[rotation]]s in three dimensions. Each (proper) rotation is represented by a quaternion with [[Circle group|unit norm]]. There is an obvious one-dimensional quaternionic vector space, namely the space '''H''' of quaternions themselves under left multiplication. By restricting this to the unit quaternions, we obtain a quaternionic representation of the [[spinor group]] Spin(3).

This representation ''&rho;'': Spin(3) &rarr; GL(1,'''H''') also happens to be a unitary quaternionic representation because

:<math>\rho(g)^\dagger \rho(g)=\mathbf{1}</math>

for all ''g'' in Spin(3).

Another unitary example is the [[spin representation]] of Spin(5). An example of a non-unitary quaternionic representation would be the two dimensional irreducible representation of Spin(5,1).

More generally, the spin representations of Spin(''d'') are quaternionic when ''d'' equals 3&nbsp;+&nbsp;8''k'',&nbsp;4&nbsp;+&nbsp;8''k'', and 5&nbsp;+&nbsp;8''k'' dimensions, where ''k'' is an [[integer]].  In physics, one often encounters the [[spinor]]s of Spin(''d'',&nbsp;1).  These representations have the same type of real or quaternionic structure as the spinors of Spin(''d''&nbsp;&minus;&nbsp;1).

Among the compact real forms of the simple Lie groups, irreducible quaternionic representations only exist for the Lie groups of type ''A''<sub>4''k''+1</sub>, ''B''<sub>4''k''+1</sub>, ''B''<sub>4''k''+2</sub>, ''C''<sub>''k''</sub>, ''D''<sub>4''k''+2</sub>, and ''E''<sub>7</sub>.