High Quality Content by WIKIPEDIA articles! 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, ?) which preserves the symplectic form ?. Here ? is a nondegenerate skew symmetric bilinear formIf 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.

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