• Source: 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




ω
:
V
×
V
→

F



{\displaystyle \omega \colon V\times V\to \mathbb {F} }


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




ω
(
g
⋅
v
,
g
⋅
w
)
=
ω
(
v
,
w
)


{\displaystyle \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




ω
(
ξ
⋅
v
,
w
)
+
ω
(
v
,
ξ
⋅
w
)
=
0


{\displaystyle \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 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, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103..

Kata Kunci Pencarian:

• Vladimir Arnold
• Symplectic representation
• Symplectic
• Symplectic vector space
• Symplectic matrix
• Quaternionic representation
• Symplectic group
• Representation theory
• Metaplectic group
• Representation theory of the Poincaré group
• Poisson manifold