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| | In [[mathematics|mathematical]] field of [[representation theory]], a '''symplectic representation''' is a [[group representation|representation]] of a [[group (mathematics)|group]] or a [[Lie algebra representation|Lie algebra]] on a [[symplectic vector space]] (''V'', ''ω'') which preserves the symplectic form ''ω''. Here ''ω'' is a nondegenerate skew symmetric bilinear form |
| | :<math>\omega\colon V\times V \to \mathbb F</math> |
| | where '''F''' is the [[field (mathematics)|field]] of scalars. A representation of a group ''G'' preserves ''ω'' if |
| | :<math>\omega(g\cdot v,g\cdot w)= \omega(v,w)</math> |
| | for all ''g'' in ''G'' and ''v'', ''w'' in ''V'', whereas a representation of a [[Lie algebra]] '''g''' preserves ''ω'' if |
| | :<math>\omega(\xi\cdot v,w)+\omega(v,\xi\cdot w)=0</math> |
| | 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'') |
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| | 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]]. |
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| | ==References== |
| | *{{Fulton-Harris}}. |
| | |
| | [[Category:Representation theory]] |
| | [[Category:Symplectic geometry]] |
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| | {{algebra-stub}} |
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
![{\displaystyle \omega \colon V\times V\to \mathbb {F} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca4d18b529ae297501d3aaf4125c72bb58de3735)
where F is the field of scalars. A representation of a group G preserves ω if
![{\displaystyle \omega (g\cdot v,g\cdot w)=\omega (v,w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf24b104e678fe80d6239852242c9baf6983dae7)
for all g in G and v, w in V, whereas a representation of a Lie algebra g preserves ω if
![{\displaystyle \omega (\xi \cdot v,w)+\omega (v,\xi \cdot w)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2898ac65394cd0afd0a81eac86e7667ae3d99652)
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
Template:Algebra-stub