Quaternionic representation

In

Together with the imaginary unit i and the antilinear map k := ij, j equips V with the structure of a quaternionic vector space (i.e., V becomes a module over the division algebra of quaternions). From this point of view, quaternionic representation of a group G is a group homomorphism φ: G → GL(V, H), the group of invertible quaternion-linear transformations of V. In particular, a quaternionic matrix representation of g assigns a square matrix of quaternions ρ(g) to each element g of G such that ρ(e) is the identity matrix and

${\displaystyle \rho (gh)=\rho (g)\rho (h){\text{ for all }}g,h\in G.}$

Quaternionic representations of associative and Lie algebras can be defined in a similar way.

Properties and related concepts

If V is a

Quaternionic representations are similar to

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 + 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 A_4k+1, B_4k+1, B_4k+2, C_k, D_4k+2, and E₇.

References

• OCLC 246650103
  ..
• Serre, Jean-Pierre (1977), Linear Representations of Finite Groups, Springer-Verlag,
  ISBN 978-0-387-90190-9
  .

• Symplectic vector space