In
mathematical field of
representation theory, a symplectic representation is a
representation
Representation may refer to:
Law and politics
*Representation (politics), political activities undertaken by elected representatives, as well as other theories
** Representative democracy, type of democracy in which elected officials represent a ...
of a
group or a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
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
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
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
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
), 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
*.
Representation theory
Symplectic geometry
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