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j
1 в‰¤ i в‰¤ n.
B(ei , w) = Оґi ,

j
We know that these n equations are linearly independent, so there exists a solution f0 . Of
j
course, once one particular solution is found, any other solution is of the form f j = f0 +aji ei
for some n2 numbers aji . Thus, we have found the general solutions f j to the equations
j
B(ei , f j ) = Оґi .

L.5.2 81
We now show that we can choose the aij so as to satisfy the last remaining set of
j
conditions, B(f i , f j ) = 0. If we set bij = B(f0 , f0 ) = в€’bji , then we can compute
i

j j
B(f i , f j ) = B(f0 , f0 ) + B(aik ek , f0 ) + B(f0 , ajl el ) + B(aik ek , ajl el )
i i

= bij + aij в€’ aji + 0.

Thus, it suп¬ѓces to set aij = в€’bij /2. (This is where the hypothesis that the characteristic
of R is not 2 is used.)
Finally, it remains to show that the vectors e1 , . . . , en , f 1 . . . , f n form a basis of V .
Since we already know that dim(V ) = 2n, it is enough to show that these vectors are
linearly independent. However, any linear relation of the form

ai ei + bj f j = 0,

implies bk = B(ek , ai ei + bj f j ) = 0 and ak = в€’B(f k , ai ei + bj f j ) = 0.

We often say that a basis of the form found in Proposition 1 is a symplectic or standard
basis of the symplectic space (V, B).

Symplectic Reduction of Vector Spaces. If B: V Г— V в†’ R is a skew-symmetric
bilinear form which is not necessarily non-degenerate, then we deп¬Ѓne the null space of B
to be the subspace
NB = {v в€€ V | B(v, w) = 0 for all w в€€ V } .
On the quotient vector space V = V /NB , there is a well-deп¬Ѓned skew-symmetric bilinear
form B: V Г— V в†’ R given by
B(x, y) = B(x, y)
where x and y are the cosets in V of x and y in V . It is easy to see that (V , B) is a
symplectic space.
Deп¬Ѓnition 2: If B is a skew-symmetric bilinear form on a vector space V , then the
symplectic space (V , B) is called the symplectic reduction of (V, B).

Here is an application of the symplectic reduction idea: Using the identiп¬Ѓcation of
A2 (V ) with О›2 (V в€— ) mentioned earlier, Proposition 1 allows us to write down a normal
form for any alternating 2-form on any п¬Ѓnite dimensional vector space.

Proposition 2: For any non-zero ОІ в€€ О›2 (V в€— ), there exist an integer n в‰¤ 1
dim(V ) and
2
linearly independent 1-forms П‰ 1 , П‰ 2 , . . . , П‰ 2n в€€ V в€— for which

ОІ = П‰ 1 в€§ П‰ 2 + П‰ 3 в€§ П‰ 4 . . . + П‰ 2nв€’1 в€§ П‰ 2n.

Thus, n is the largest integer so that ОІ n = 0.

L.5.3 82
Proof: Regard ОІ as a skew-symmetric bilinear form B on V in the usual way. Let
(V , B) be the symplectic reduction of (V, B). Since B = 0, we known that V = {0}. Let
1 n
dim(V ) = 2n в‰Ґ 2 and let e1 , . . . , en , f 1 . . . , f n be elements of V so that e1 , . . . , en , f . . . , f
forms a symplectic basis of V with respect to B. Let p = dim(V ) в€’ 2n, and let b1 , . . . , bp
be a basis of NB .
It is easy to see that

b = e1 f 1 e2 f 2 В· В· В· en f n b1 В· В· В· bp

forms a basis of V . Let
П‰ 1 В· В· В· П‰ 2n+p
denote the dual basis of V в€— . Then, as the reader can easily check, the 2-form

в„¦ = П‰ 1 в€§ П‰ 2 + П‰ 3 в€§ П‰ 4 . . . + П‰ 2nв€’1 в€§ П‰ 2n

has the same values as ОІ does on all pairs of elements of b. Of course this implies that
ОІ = в„¦. The rest of the Proposition also follows easily since, for example, we have

ОІ n = n! П‰ 1 в€§ В· В· В· 2n
в€§П‰ = 0,

although ОІ n+1 clearly vanishes.
If we regard ОІ as an element of A2 (V ), then n is one-half the dimension of V . Some
sources call the integer n the half-rank of ОІ and others call n the rank. I use вЂњhalf-rankвЂќ.
Note that, unlike the case of symmetric bilinear forms, there is no notion of signature
type or вЂњpositive deп¬ЃnitenessвЂќ for skew-symmetric forms.
It follows from Proposition 2 that for ОІ in A2 (V ), where V is п¬Ѓnite dimensional, the
pair (V, ОІ) is a symplectic space if and only if V has dimension 2n for some n and ОІ n = 0.

Subspaces of Symplectic Vector Spaces. Let в„¦ be a symplectic form on a vector
space V . For any subspace W вЉ‚ V , we deп¬Ѓne the в„¦-complement to W to be the subspace

W вЉҐ = {v в€€ V | в„¦(v, w) = 0 for all w в€€ W }.

The в„¦-complement of a subspace W is sometimes called its skew-complement. It is an
вЉҐ
exercise for the reader to check that, because в„¦ is non-degenerate, W вЉҐ = W and that,
when V is п¬Ѓnite-dimensional,

dim W + dim W вЉҐ = dim V.

However, unlike the case of an orthogonal with respect to a positive deп¬Ѓnite inner product,
the intersection W в€© W вЉҐ does not have to be the zero subspace. For example, in an
в„¦-standard basis for V , the vectors e1 , . . . , en obviously span a subspace L which satisп¬Ѓes
LвЉҐ = L.

L.5.4 83
If V is п¬Ѓnite dimensional, it turns out (see the Exercises) that, up to symplectic linear
transformations of V , a subspace W вЉ‚ V is characterized by the numbers d = dim W and
ОЅ = dim (W в€© W вЉҐ ) в‰¤ d. If ОЅ = 0 we say that W is a symplectic subspace of V . This
corresponds to the case that в„¦ restricts to W to deп¬Ѓne a symplectic structure on W . At
the other extreme is when ОЅ = d, for then we have W в€© W вЉҐ = W . Such a subspace is
called Lagrangian.

Symplectic Manifolds.
We are now ready to return to the study of manifolds.
Deп¬Ѓnition 3: A symplectic structure on a smooth manifold M is a non-degenerate, closed
2-form в„¦ в€€ A2 (M). The pair (M, в„¦) is called a symplectic manifold. If в„¦ is a symplectic
structure on M and ОҐ is a symplectic structure on N , then a smooth map П†: M в†’ N
satisfying П†в€— (ОҐ) = в„¦ is called a symplectic map. If, in addition, П† is a diп¬Ђeomorphism, we
say that П† is a symplectomorphism.
Before developing any of the theory, it is helpful to see a few examples.

Surfaces with Area Forms. If S is an orientable smooth surface, then there exists
a volume form Вµ on S. By deп¬Ѓnition, Вµ is a non-degenerate closed 2-form on S and hence
deп¬Ѓnes a symplectic structure on S.

Lagrangian Structures on T M. From Lecture 4, any non-degenerate Lagrangian
L: T M в†’ R deп¬Ѓnes the 2-form dП‰L , which is a symplectic structure on T M.

A вЂњStandardвЂќ Structure on R2n . Think of R2n as a smooth manifold and let в„¦
be the 2-form with constant coeп¬ѓcients

dx = dx1 в€§ dxn+1 + В· В· В· + dxn в€§ dx2n .
1t
в„¦= 2 dx Jn

Symplectic Submanifolds. Let (M 2m , в„¦) be a symplectic manifold. Suppose that
P 2p вЉ‚ M 2m be any submanifold to which the form в„¦ pulls back to be a non-degenerate
2-form в„¦P . Then (P, в„¦P ) is a symplectic manifold. We say that P is a symplectic sub-
manifold of M.
It is not obvious just how to п¬Ѓnd symplectic submanifolds of M. Even though being
a symplectic submanifold is an вЂњopenвЂќ condition on submanifolds of M, is is not вЂњdenseвЂќ.
One cannot hope to perturb an arbitrary even dimensional submanifold of M slightly so
as to make it symplectic. There are even restrictions on the topology of the submanifolds
of M on which a symplectic form restricts to be non-degenerate.
For example, no symplectic submanifold of R2n (with any symplectic structure on
R2n ) could be compact for the following simple reason: Since R2n is contractible, its
second deRham cohomology group vanishes. In particular, for any symplectic form в„¦
on R2n , there must be a 1-form П‰ so that в„¦ = dП‰ which implies that в„¦m = d П‰в€§в„¦mв€’1 .
Thus, for all m > 0, the 2m-form в„¦m is exact on R2n (and every submanifold of R2n ).

L.5.5 84
By Proposition 2, if M 2m were a submanifold of R2n on which в„¦ restricted to be non-
degenerate, then в„¦m would be a volume form on M. However, on a compact manifold the
volume form is never exact (just apply StokesвЂ™ Theorem).

Example. Complex Submanifolds. Nevertheless, there are many symplectic subman-
ifolds of R2n . One way to construct them is to regard R2n as Cn in such a way that
the linear map J : R2n в†’ R2n represented by Jn becomes complex multiplication. (For
example, just deп¬Ѓne the complex coordinates by z k = xk + ixk+n .) Then, for any non-zero
vector v в€€ R2n , we have в„¦(v, J v) = в€’|v|2 = 0. In particular, в„¦ is non-degenerate on every
complex subspace S вЉ‚ Cn . Thus, if M 2m вЉ‚ Cn is any complex submanifold (i.e., all of
its tangent spaces are m-dimensional complex subspaces of Cm ), then в„¦ restricts to be
non-degenerate on M.

The Cotangent Bundle. Let M be any smooth manifold and let T в€— M be its
cotangent bundle. As we saw in Lecture 4, there is a canonical 2-form on T в€— M which
can be deп¬Ѓned as follows: Let ПЂ: T в€— M в†’ M be the basepoint projection. Then, for every
v в€€ TО± (T в€— M), deп¬Ѓne
П‰(v) = О± ПЂ (v) .

I claim that П‰ is a smooth 1-form on T в€— M and that в„¦ = dП‰ is a symplectic form on T в€— M.
To see this, let us compute П‰ in local coordinates. Let x: U в†’ Rn be a local coordinate
chart. Since the 1-forms dx1 , . . . , dxn are linearly independent at every point of U, it follows
that there are unique functions Оѕi on T в€— U so that, for О± в€€ Ta U, в€—

О± = Оѕ1 (О±) dx1 |a + В· В· В· + Оѕn (О±) dxn |a .

The functions x1 , . . . , xn , Оѕ1 , . . . , Оѕn then form a smooth coordinate system on T в€— U in which
the projection mapping ПЂ is given by

ПЂ(x, p) = x.

It is then straightforward to compute that, in this coordinate system,

П‰ = Оѕi dxi .

Hence, в„¦ = dОѕi в€§dxi and so is non-degenerate.

Symplectic Products. If (M, в„¦) and (N, ОҐ) are symplectic manifolds, then M Г— N
carries a natural symplectic structure, called the product symplectic structure в„¦ вЉ• ОҐ,
deп¬Ѓned by
в€— в€—
в„¦ вЉ• ОҐ = ПЂ1 (в„¦) + ПЂ2 (ОҐ).

Thus, for example, n-fold products of compact surfaces endowed with area forms give
examples of compact symplectic 2n-manifolds.

L.5.6 85
Coadjoint Orbits. Let Adв€— : G в†’ GL(gв€— ) denote the coadjoint representation of G.
This is the so-called вЂњcontragredientвЂќ representation to the adjoint representation. Thus,
for any a в€€ G and Оѕ в€€ gв€— , the element Adв€— (a)(Оѕ) в€€ gв€— is determined by the rule

Adв€— (a)(Оѕ)(x) = Оѕ Ad(aв€’1 )(x) for all x в€€ g.

в€—
One must be careful not to confuse Adв€— (a) with Ad(a) . Instead, as our deп¬Ѓnition
в€—
shows, Adв€— (a) = Ad(aв€’1 ) .
Note that the induced homomorphism of Lie algebras, adв€— : g в†’ gl(gв€— ) is given by

adв€— (x)(Оѕ)(y) = в€’Оѕ [x, y]

The orbits G В· Оѕ in gв€— are called the coadjoint orbits. Each of them carries a natural
symplectic structure. To see how this is deп¬Ѓned, let Оѕ в€€ gв€— be п¬Ѓxed, and let П†: G в†’ G В· Оѕ
be the usual submersion induced by the Adв€— -action, П†(a) = Adв€— (a)(Оѕ) = a В· Оѕ. Now let П‰Оѕ
be the left-invariant 1-form on G whose value at e is Оѕ. Thus, П‰Оѕ = Оѕ(П‰) where П‰ is the
canonical g-valued 1-form on G.

Proposition 3: There is a unique symplectic form в„¦Оѕ on the orbit GВ·Оѕ = G/GОѕ satisfying
П†в€— (в„¦Оѕ ) = dП‰Оѕ .

Proof: If Proposition 3 is to be true, then в„¦Оѕ must satisfy the rule

в„¦Оѕ П† (v), П† (w) = dП‰Оѕ (v, w) for all v, w в€€ Ta G.

What we must do is show that this rule actually does deп¬Ѓne a symplectic 2-form on G В· Оѕ.
First, note that, for x, y в€€ g = Te G, we may compute via the structure equations that

dП‰Оѕ (x, y) = Оѕ dП‰(x, y) = Оѕ в€’[x, y] = adв€— (x)(Оѕ)(y).

In particular, adв€— (x)(Оѕ) = 0, if and only if x lies in the null space of the 2-form dП‰Оѕ (e). In
other words, the null space of dП‰Оѕ (e) is gОѕ , the Lie algebra of GОѕ . Since dП‰Оѕ is left-invariant,
it follows that the null space of dП‰Оѕ (a) is La (gОѕ ) вЉ‚ Ta G. Of course, this is precisely the
tangent space at a to the left coset aGОѕ . Thus, for each a в€€ G,

NdП‰Оѕ (a) = ker П† (a),

It follows that, TaВ·Оѕ (G В· Оѕ) = П† (a)(Ta G) is naturally isomorphic to the symplectic quotient
space (Ta G)/ La (gОѕ ) for each a в€€ G. Thus, there is a unique, non-degenerate 2-form в„¦a
в€—
on TaВ·Оѕ (G В· Оѕ) so that П† (a) (в„¦a ) = dП‰Оѕ (a).
It remains to show that в„¦a = в„¦b if a В· Оѕ = b В· Оѕ. However, this latter case occurs only
if a = bh where h в€€ GОѕ . Now, for any h в€€ GОѕ , we have

Rв€— (П‰Оѕ ) = Оѕ Rв€— (П‰) = Оѕ Ad(hв€’1 )(П‰) = Adв€— (h)(Оѕ)(П‰) = Оѕ(П‰) = П‰Оѕ .
h h

L.5.7 86
Thus, Rв€— (dП‰Оѕ ) = dП‰Оѕ . Since the following square commutes, it follows that в„¦a = в„¦b .
h
R
в€’в†’
h
Ta G Tb G
пЈ¦ пЈ¦
пЈ¦ пЈ¦
П† (a) П† (b)
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