In mathematical field of representation theory, a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space (V, ω) which preserves the symplectic form ω. Here ω is a nondegenerate skew symmetric bilinear form
\omega\colonV x V\toF
\omega(g ⋅ v,g ⋅ w)=\omega(v,w)
\omega(\xi ⋅ v,w)+\omega(v,\xi ⋅ w)=0
If G is a compact group (for example, a finite group), and F is the field of complex numbers, then by introducing a compatible unitary structure (which exists by an averaging argument), one can show that any complex symplectic representation is a quaternionic representation. Quaternionic representations of finite or compact groups are often called symplectic representations, and may be identified using the Frobenius–Schur indicator.