In the 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 which satisfies 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 Quaternionic representations of associative and Lie algebras can be defined in a similar way.
If V is a unitary representation and the quaternionic structure j is a unitary operator, then V admits an invariant complex symplectic form ω, and hence is a symplectic representation.
Such a representation will be a direct sum of central simple R-algebras, which, by the Artin-Wedderburn theorem, must be matrix algebras over the real numbers or the quaternions.
These representations have the same type of real or quaternionic structure as the spinors of Spin(d − 1).