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, ÃÂ) which preserves the symplectic form ÃÂ. Here ÃÂ is a nondegenerate skew symmetric bilinear form
where F is the field of scalars. A representation of a group G preserves ÃÂ if
for all g in G and v, w in V, whereas a representation of a Lie algebra g preserves ÃÂ if
for all þ 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(V,ÃÂ) or its Lie algebra sp(V,ÃÂ)
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 FrobeniusâÂÂSchur indicator.