mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

field of

representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

, a symplectic representation is a representation of a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

or a

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

on a

symplectic vector space In mathematics, a symplectic vector space is a vector space V over a Field (mathematics), field F (for example the real numbers \mathbb) equipped with a symplectic bilinear form. A symplectic bilinear form is a map (mathematics), mapping \omega : ...

(''V'', ''ω'') which preserves the symplectic form ''ω''. Here ''ω'' is a nondegenerate skew symmetric bilinear form :

\omega\colon V\times V \to \mathbb F

where F is the field of scalars. A representation of a group ''G'' preserves ''ω'' if :

\omega(g\cdot v,g\cdot w)= \omega(v,w)

for all ''g'' in ''G'' and ''v'', ''w'' in ''V'', whereas a representation of a

g preserves ''ω'' if :

\omega(\xi\cdot v,w)+\omega(v,\xi\cdot w)=0

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 In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gr ...

Sp(''V'',''ω'') or its Lie algebra sp(''V'',''ω'') If ''G'' is a

compact group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...

(for example, a

finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...

), 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 {{algebra-stub