mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of

representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

, 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 ide ...

or a Lie algebra on a

symplectic vector space In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument ...

(''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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

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 Lie algebra 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 gro ...

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), 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