Symplectic representation
- 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 ("V", "ω") 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("V","ω") or its Lie algebra sp("V","ω")
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.
References
*Fulton-Harris.
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