Examples
A common example involves the quaternionic representation of rotations in three dimensions. Each (proper) rotation is represented by a quaternion with 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 ρ: Spin(3) → GL(1,H) also happens to be a unitary quaternionic representation because
for all g in Spin(3).
Another unitary example is the spin representation of Spin(5). An example of a nonunitary 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 + 8k, 4 + 8k, and 5 + 8k dimensions, where k is an integer. In physics, one often encounters the spinors of Spin(d, 1). These representations have the same type of real or quaternionic structure as the spinors of Spin(d − 1).
Among the compact real forms of the simple Lie groups, irreducible quaternionic representations only exist for the Lie groups of type A4k+1, B4k+1, B4k+2, Ck, D4k+2, and E7.
Read more about this topic: Quaternionic Representation
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