Quaternionic representation

In mathematical field of representation theory, a quaternionic representation is a representation on a complex vector space V with an invariant quaternionic structure, i.e., an antilinear equivariant map

${\displaystyle j\colon V\to V}$

which satisfies

${\displaystyle j^{2}=-1.}$

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.