Standard symplectic space Symplectic vector space

the standard symplectic space r symplectic form given nonsingular, skew-symmetric matrix. typically ω chosen block matrix

ω
=


[



0



i

n






−

i

n




0



]




{\displaystyle \omega ={\begin{bmatrix}0&i_{n}\\-i_{n}&0\end{bmatrix}}}

where in n × n identity matrix. in terms of basis vectors (x1, ..., xn, y1, ..., yn):

ω
(

x

i


,

y

j


)
=
−
ω
(

y

j


,

x

i


)
=

δ

i
j





{\displaystyle \omega (x_{i},y_{j})=-\omega (y_{j},x_{i})=\delta _{ij}\,}






ω
(

x

i


,

x

j


)
=
ω
(

y

i


,

y

j


)
=
0.



{\displaystyle \omega (x_{i},x_{j})=\omega (y_{i},y_{j})=0.\,}

a modified version of gram–schmidt process shows finite-dimensional symplectic vector space has basis such ω takes form, called darboux basis, or symplectic basis.

there way interpret standard symplectic form. since model space r used above carries canonical structure might lead misinterpretation, use anonymous vector spaces instead. let v real vector space of dimension n , v dual space. consider direct sum w = v ⊕ v of these spaces equipped following form:

ω
(
x
⊕
η
,
y
⊕
ξ
)
=
ξ
(
x
)
−
η
(
y
)
.


{\displaystyle \omega (x\oplus \eta ,y\oplus \xi )=\xi (x)-\eta (y).}

now choose basis (v1, ..., vn) of v , consider dual basis

(

v

1


∗


,
…
,

v

n


∗


)
.


{\displaystyle (v_{1}^{*},\ldots ,v_{n}^{*}).}

we can interpret basis vectors lying in w if write xi = (vi, 0) , yi = (0, vi). taken together, these form complete basis of w,

(

x

1


,
…
,

x

n


,

y

1


,
…
,

y

n


)
.


{\displaystyle (x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n}).}

the form ω defined here can shown have same properties in beginning of section. on other hand, every symplectic structure isomorphic 1 of form v ⊕ v. subspace v not unique, , choice of subspace v called polarization. subspaces give such isomorphism called lagrangian subspaces or lagrangians.

explicitly, given lagrangian subspace (as defined below), choice of basis (x1, ..., xn) defines dual basis complement, ω(xi, yj) = δij.

analogy complex structures

just every symplectic structure isomorphic 1 of form v ⊕ v, every complex structure on vector space isomorphic 1 of form v ⊕ v. using these structures, tangent bundle of n-manifold, considered 2n-manifold, has complex structure, , cotangent bundle of n-manifold, considered 2n-manifold, has symplectic structure: t∗(tm)p = tp(m) ⊕ (tp(m)).

the complex analog lagrangian subspace real subspace, subspace complexification whole space: w = v ⊕ j v.