Symplectic representation
In mathematical field of representation theory, a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space which preserves the symplectic form ω. Here ω is a nondegenerate skew symmetric bilinear form
where F is the field of scalars. A representation of a group G preserves ω if
for all g in G and v, w in V, whereas a representation of a Lie algebra g preserves ω if
for all ξ in g and v, w in V. Thus a representation of G or g is equivalently a group or Lie algebra homomorphism from G or g to the symplectic group Sp or its Lie algebra sp
If G is a compact group, and F is the field of complex numbers, then by introducing a compatible unitary structure, 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.
