simply a bivector ¬eld. We then add ˜J to the Poisson tensor Πg— , de¬ning a new

tensor Πg— = Πg— + ˜J , with respect to which J is a Poisson map.

Exercise 19

Show that δ˜J = 0 implies that Πg— is again a Poisson tensor and that with

this Poisson structure on g— the map J is Poisson.

™¦

7.6 Flat Connections for Poisson Maps with Symplectic Tar-

get

We will classify complete Poisson maps • : M ’ S, where M is a Poisson manifold

and S is a connected symplectic manifold. The structure of these maps turns out

to be remarkably simple and rigid.

Claim. Any Poisson map • : M ’ S is a submersion.

Proof. If not, then (Tx •)(Tx M ) is a proper subspace V of T•(x) S, and (Tx •)(Π(x)) ⊆

V § V , contradicting the fact that the image of Π under Tx • is symplectic. 2

We can say even more if we assume that • is complete:

Claim. Any complete Poisson map • : M ’ S is surjective.

Exercise 20

Prove this claim.

Let F be any Poisson manifold and let p1 : S — F ’ S be the

Example.

™¦

projection onto the ¬rst factor. This is clearly a complete Poisson map.

Inspired by this example, the claims above indicate that a complete Poisson

map should be a kind of ¬bration over S. To formalize this idea, we de¬ne a ¬‚at

connection for any submersion • : M ’ S between manifolds to be a subbundle

E ⊆ T M such that

7.6 Flat Connections for Poisson Maps with Symplectic Target 45

1. T M = E • ker T • ,

2. [E, E] ⊆ E (that is, sections of E are closed under [·, ·], and so by the Frobe-

nius theorem E is integrable),

3. every path in S has a horizontal lift through each lift of one of its points.

A subbundle E ⊆ T M satisfying conditions 1 and 3, or sometimes just 1, is

called an Ehresmann connection [52]. Conditions 1 and 3 imply that • : M ’ S

is a locally trivial ¬bration. Condition 2 is the ¬‚atness property, which implies

that the ¬bration has a discrete structure group.

Theorem 7.2 A complete Poisson map • : M ’ S to a symplectic manifold has

a natural ¬‚at connection.

Proof. Let s = •(x) for some x ∈ M and choose a v ∈ Ts S. We want to lift v to

Tx M in a canonical way. Because S is symplectic, Π’1 (v) is a well-de¬ned covector

S

at s. De¬ne a horizontal lift

Hx (v) = ΠM (Tx •)— Π’1 (v) ∈ Tx M .

S

The fact that • is a Poisson map implies that (Tx •)(Hx (v)) = v (see Section 6.1).

We need to check that the bracket of two horizontal lifts is again horizontal. On S,

choose canonical coordinates q1 , . . . , qn , p1 , . . . , pn , and lift their hamiltonian vector

¬elds

‚ ‚ ‚ ‚

’ ,...,’ , ,..., .

‚p1 ‚pn ‚q1 ‚qn

The lifts are closed under commutators, hence span an integrable subbundle. Mul-

tiplying these vector ¬elds on S by compactly supported functions if necessary to

make them complete, we obtain a local trivialization of •, because • is complete.

Any path on S lifts to M because any path lifts locally. 2

In particular, if S is simply connected, then there is a Poisson manifold F

such that M and S — F are di¬eomorphic as Poisson manifolds. In general, • is

determined up to isomorphism by its holonomy

π1 (S) ’’ Aut(F )

on a typical ¬ber F of the map.

46 7 HAMILTONIAN ACTIONS

F

s M

s E

•

c

S

s E

We thus found a functor from the category of complete Poisson maps M ’ S

to the category of actions of π1 (S) by Poisson automorphisms on Poisson manifolds

F.

We also have a functor going in the other direction. Let S be the universal

cover of the symplectic manifold S; S is a symplectic manifold. Let F be a Poisson

manifold with a π1 (S)-action by Poisson automorphisms. On the product S — F

there is an induced diagonal action

γ · (˜, f ) = (γ · s, γ · f ) for γ ∈ π1 (S) .

s ˜

If we form the quotient by this action, we still get a projection

S—F •

’’ S .

π1 (S)

This is a complete Poisson map with ¬ber F .

Exercise 21

Show that this actually de¬nes a functor from the category of actions of π1 (S)

by Poisson automorphisms on Poisson manifolds to the category of complete

Poisson maps from Poisson manifolds to S.

Remark. Comparing the results of this section with the theory of hamiltonian

group actions, it is tempting to think of any symplectic manifold S as the “dual of

™¦

the Lie algebra of π1 (S)”!

Part IV

Dual Pairs

8 Operator Algebras

In this chapter, we introduce terminology and quote results leading to the double

commutant theorem (Theorem 8.3) proved by von Neumann [127]. Chapter 9 will

be devoted to analogous results in Poisson geometry.

In the following discussion, we denote the algebra of bounded operators on a

complex Hilbert space H by B(H). There are several topologies worth considering

on B(H).

Norm Topology and C — -Algebras

8.1

The norm of a bounded operator L ∈ B(H) is by de¬nition

||Lu||H

||L|| = sup .

u∈H\{0} ||u||H

Exercise 22

Check that || · || satis¬es the axioms for a norm:

(a) ||» · L|| = |»| · ||L||, » ∈ C,

(b) ||L + M || ¤ ||L|| + ||M ||, and

(c) ||L|| > 0 if L = 0 .

This induces a (complete) metric

d(M, L) = ||L ’ M ||

and thus a topology on B(H), called the norm topology.

On B(H) there is an adjoint operation — de¬ned uniquely by

L— u, v = u, Lv

which has the properties

• L—— = L,

• (LM )— = M — L— , and

• ||LL— || = ||L||2 .

We say that B(H) equipped with this *-operation is a C — -algebra. In general, a C — -

algebra is an algebra with a norm such that the algebra is complete with respect

to the topology induced by the norm and possesses a *-operation satisfying the

properties above. As general references on C — -algebras, we recommend [7, 36, 45].

Any norm-closed *-subalgebra of B(H) inherits the properties above and thus is

a C — -algebra. If A ⊆ B(H) is any *-subalgebra, its norm-closure A is a C — -algebra.

Conversely, we have the following theorem:

47

48 8 OPERATOR ALGEBRAS

Theorem 8.1 (Gel™fand-Naimark [64]) Any C — -algebra is isomorphic as a normed

*-algebra to a norm-closed subalgebra of B(H).

Example. The collection of all ¬nite rank operators is a *-subalgebra; its closure

is the C — -subalgebra of compact operators on H “ that is, operators L such that

L applied to a bounded subset has compact closure. The identity operator I is

not compact if H is in¬nite dimensional, as the closed unit ball in H is bounded

but not compact. (For instance, the sequence ai = (0, . . . , 0, 1, 0, . . .), where the 1

is in the ith slot, has no convergent subsequences.) For diagonalizable operators,

™¦

compactness amounts to convergence of the eigenvalues to 0.

Let X be any compact Hausdor¬ topological space, and let C(X) be the al-

gebra of complex-valued continuous functions on X equipped with the sup norm.

Then pointwise addition and multiplication together with the *-operation de¬ned

by f — (x) = f (x) give C(X) the structure of a C — -algebra. The following theorem

demonstrates how general this example is:

Theorem 8.2 [63, 65, 64] Any commutative C — -algebra A with identity is iso-

metrically *-isomorphic to C(X) for some compact Hausdor¬ space X. One can

take X to be the space of non-zero *-homomorphisms from A to C. (X is then

called the spectrum of A.)

Recalling Theorem 8.1, how can C(X) be regarded as an algebra of operators

on a Hilbert space? Because X is compact, we can ¬nd a Borel measure on X

which is positive on any non-empty open set. C(X) is then realized as an algebra

of multiplication operators on L2 (X). For any function u ∈ C(X), de¬ne the

multiplication operator mu by mu (g) = ug for g ∈ L2 (X).

Exercise 23

Show that

||mu ||B(L2 (X)) = ||u||C(X) .

8.2 Strong and Weak Topologies

A second topology on B(H) is the topology of pointwise convergence, or the

strong topology. For each u ∈ H, de¬ne a semi-norm

||L||u = ||Lu||H .

A semi-norm is essentially the same as a norm except for the positivity requirement:

non-zero elements may have 0 semi-norm. We de¬ne the strong topology on B(H)

by declaring a sequence {Li } to converge if and only if the sequence converges in

the semi-norms || · ||u for all choices of u ∈ H.

Example. The sequence of operators Li on L2 (N) =: l2 de¬ned by

Li (a0 , a1 , a2 , . . .) = (0, . . . , 0, ai , 0, . . .)

™¦

converges to 0 in the strong topology, though each Li has norm 1.

8.3 Commutants 49

Example. Let Mi be the operator on L2 (N)

Mi (a0 , a1 , a2 , . . .) = (0, . . . , 0, a0 , 0, . . .) ,

where the a0 on the right is the ith entry. The sequence of the Mi ™s does not

converge in the strong topology, yet its adjoint

Mi— (a0 , a1 , a2 , . . .) = (ai , 0, 0, . . .)

™¦

does converge strongly (exercise).

Another topology on B(H) is the weak topology, or the topology of con-

vergence of matrix elements. For u, v ∈ H, de¬ne a semi-norm

||L||u,v = | Lu, v | .

We say that a sequence {Li }i∈I converges in the weak topology, if ||Li ||u,v converges

for each choice of u, v.

The sequences Li , Mi and Mi— in the examples above converge in the weak

topology. In general, any strongly convergent sequence is weakly convergent, and

any norm convergent sequence is strongly convergent, so the weak topology is in

fact weaker than the strong topology, which is still weaker than the norm topology.

By Exercise 23, the inclusion C(X) ’ B(L2 (X)) given by u ’ mu is an

isometry; this implies that C(X) is norm-closed when considered as a subalgebra

of B(L2 (X)), which illustrates Theorem 8.1. However, if we use a weaker topology

(say the strong or weak topology), then C(X) is no longer closed.

Exercise 24

Construct a sequence of functions in C(X) converging to (multiplication by) a

step function in the strong (or weak) topology. Show that this sequence does

not converge in the norm topology.

The weak (or strong) closure of C(X) in B(L2 (X)) is, in fact, L∞ (X). Keep

in mind that elements of L∞ (X) cannot be strictly considered as functions on X,

since two functions which di¬er on a set of measure 0 on X correspond to the same

element of B(L2 (X)).

8.3 Commutants

A subalgebra A of B(H) is called unital if it contains the identity operator of B(H).

For a subset A ⊆ B(H) closed under the *-operator, we de¬ne the commutant of

A to be

A = {L ∈ B(H) | ∀a ∈ A, La = aL} .

Exercise 25

Show that A is a weakly closed *-subalgebra.

A weakly closed unital *-subalgebra of B(H) is called a von Neumann alge-

bra. [47, 74, 156, 157, 158] are general references on von Neumann algebras. There

is a remarkable connection between algebraic and topological properties of these

algebras, as shown by the following theorem.

50 8 OPERATOR ALGEBRAS

Theorem 8.3 (von Neumann [127]) For a unital *-subalgebra A ⊆ B(H), the

following are equivalent:

1. A = A,

2. A is weakly closed,

3. A is strongly closed.

Corollary 8.4 If A is any subset of B(H), then A =A.

For an arbitrary unital *-subalgebra A ⊆ B(H), the double commutant A

coincides with the weak closure of A.

The center of A is

Z(A) = A © A .

If A is a von Neumann algebra with Z(A) = C · 1, then A is called a factor.

These are the building blocks for von Neumann algebras. Von Neumann showed

that every von Neumann algebra is a direct integral (generalized direct sum) of

factors [129, 130].

Example. We have already seen some classes of von Neumann algebras:

• B(H), which is a factor.

• L∞ (X) (with respect to a given measure class on X), which is a generalized

direct integral of copies of C (which are factors):

L∞ (X) = • a copy of C for each point of X .

X

• The commutant of any subset of B(H), for instance the collection of operators

which commute with the action of a group on H.

™¦

8.4 Dual Pairs

A dual pair (A, A ) is a pair of unital *-subalgebras A and A of B(H) which are

the commutants of one another. By Theorem 8.3, A and A are then von Neumann

algebras.

If A is a von Neumann subalgebra of B(H), then there are inclusions

B(H)

s

d

d

d

d

d

A A

which form a dual pair. The centers of A and A coincide:

Z(A) = A © A = A © A = Z(A ) ,

so that A is a factor if and only if A is.

We next turn to geometric counterparts of dual pairs in the context of Poisson

geometry.

9 Dual Pairs in Poisson Geometry

We will discuss a geometric version of dual pairs for Poisson algebras associated to

Poisson manifolds.

9.1 Commutants in Poisson Geometry

We have seen that a Poisson manifold (M, {·, ·}) determines a Poisson algebra

(C ∞ (M ), {·, ·}) and that a Poisson map • : M ’ N induces a Poisson-algebra

homomorphism •— : C ∞ (N ) ’ C ∞ (M ).

Suppose that N is a Poisson quotient of M . Then there is a map C ∞ (N ) ’

C ∞ (M ) identifying C ∞ (N ) as a Poisson subalgebra of C ∞ (M ) consisting of func-

tions constant along the equivalence classes of M determined by the quotient map.

In the converse direction, we might choose an arbitrary Poisson subalgebra of

C ∞ (M ) and search for a corresponding quotient map. In general this is not possi-

ble. To understand the tie between Poisson quotients and Poisson subalgebras, we

examine examples of commutants in (C ∞ (M ), {·, ·}).

‚ ‚

Example. Let M = R2n , with the standard Poisson structure Π = ‚qi § ‚pi .

The Poisson subalgebra, Pol(R2n ), of polynomial functions does not correspond to

any Poisson quotient manifold. Since Pol(R2n ) separates any two points of R2n ,

the “quotient map” would have to be the identity map on R2n .

On the other hand, the Poisson subalgebra

π — C ∞ (Rn1 ,...,qn ) ⊆ C ∞ (R2n,...,qn ,p1 ,...,pn )

q q1

does correspond to the quotient π : R2n ’ Rn . ™¦

The di¬erent behavior of the subalgebras Pol(R2n ) and C ∞ (Rn1 ,...,qn ) of C ∞ (R2n )

q

can be interpreted in the following manner.

Let A be a Poisson algebra and B ⊆ A a Poisson subalgebra. De¬ne the

commutant of B in A to be

B = {f ∈ A | {f, B} = 0} .

Example. For A = C ∞ (R2n ) we have:

Pol(R2n ) = constant functions

C ∞ (R2n )

(constant functions) =

C ∞ (Rn1 ,...,qn ) C ∞ (Rn1 ,...,qn )

=

q q

C ∞ (Rq1 ) C ∞ (R2n’1 n ,p2 ,...,pn ) .

= q1 ,...,q

The double commutants of these subalgebras are:

C ∞ (R2n )

Pol(R2n ) =

C ∞ (Rn1 ,...,qn ) C ∞ (Rn1 ,...,qn )

=

q q

C ∞ (Rq1 ) C ∞ (Rq1 ) .

=

51

52 9 DUAL PAIRS IN POISSON GEOMETRY

Since Pol(R2n ) does not correspond to a Poisson quotient while the other two

subalgebras do, this seems to indicate that the Poisson subalgebras that correspond

™¦

to quotient maps are those which are their own double commutants.

Question. (R. Conti) Is A = A for every subset of a Poisson algebra? (See

Corollary 8.4.)

9.2 Pairs of Symplectically Complete Foliations

Suppose that M and N are Poisson manifolds and that J : M ’ N is a Poisson map

with a dense image in N . Then the pull-back J — is an injection. The commutant

of the Poisson subalgebra

A = J — (C ∞ (N )) ⊆ C ∞ (M )

is

{f ∈ C ∞ (M ) | {f, A} = 0}

A =

{f ∈ C ∞ (M ) | ∀g ∈ A, Xg f = 0}

=

{f ∈ C ∞ (M ) | ∀g ∈ A, df annihilates Π(dg)} .

=

At a point x of M , we have

—

{values of hamiltonian vector ¬elds Π(dg) at x | g ∈ A} = Π(image Tx J)

Π((ker Tx J)—¦ ) ,

=

where (ker Tx J)—¦ is the subspace of covectors that annihilate ker Tx J ⊆ Tx M . When

M happens to be symplectic,

Π((ker Tx J)—¦ ) = (ker Tx J)⊥ ,

where W ⊥ is the symplectic orthogonal to the subspace W inside the tangent

space. (In the symplectic case, taking orthogonals twice returns the same subspace:

(W ⊥ )⊥ = W .) In the symplectic case, we have

{values of hamiltonian vector ¬elds Π(dg) at x | g ∈ A} = (ker Tx J)⊥ .

Exercise 26

Show that

{values of hamiltonian vector ¬elds Π(dg) at x | g ∈ A } = ker Tx J .

Suppose now that J : M ’ N is a constant-rank map from a symplectic

manifold M to a Poisson manifold N . The kernel

ker T J

forms an integrable subbundle of T M , de¬ning a foliation of M . The symplectic

orthogonal distribution

(ker T J)⊥

is generated by a family of vector ¬elds closed under the bracket operation, since

they are lifts of hamiltonian vector ¬elds on N . Hence, it is an integrable distribu-

tion which de¬nes another foliation. This is a particular instance of the following

lemma.

9.3 Symplectic Dual Pairs 53

Lemma 9.1 Let M be a symplectic manifold and F ⊆ T M an integrable subbun-

dle. Then F ⊥ is integrable if and only if the set of functions on open sets of M

annihilated by vectors in F is closed under the Poisson bracket.

A foliation F de¬ned by a subbundle F ⊆ T M as in this lemma (i.e. integrable,

with the set of functions on open sets of M annihilated by vectors in F closed

under the Poisson bracket) is called a symplectically complete foliation [104].

Symplectically complete foliations come in orthogonal pairs, since (F ⊥ )⊥ = F .

9.3 Symplectic Dual Pairs

Example. Suppose that M is symplectic and J : M ’ g— is a constant-rank

Poisson map. The symplectic orthogonal to the foliation by the level sets of J is

exactly the foliation determined by hamiltonian vector ¬elds generated by functions

on g— , which is the same as the foliation determined by the hamiltonian vector ¬elds

generated by linear functions on g— (since the di¬erentials of linear functions span

the cotangent spaces of g— ). The leaves of this foliation are simply the orbits of

the induced G-action on M . We could hence consider the “dual” to J to be the

projection of M to the orbit space, and write

M

d

J dp

d

d

© ‚

d

g— M/G .

Some conditions are required for this diagram to make sense as a pair of Poisson

maps between manifolds, in particular, for M/G to exist as a manifold:

1. J must have constant rank so that the momentum levels form a foliation.

2. The G-orbits must form a ¬bration (i.e. the G-action must be regular).

In this situation, the subalgebras J — (C ∞ (g— )) and p— (C ∞ (M/G)) of C ∞ (M ) are

commutants of one another, and hence their centers are isomorphic. Furthermore,

when J is a submersion, the transverse structures to corresponding leaves in g— and

M/G are anti-isomorphic [163]. So the Poisson geometry of the orbit space M/G

is “modulo symplectic manifolds” very similar to the Poisson geometry of g— . This

construction depends on J being surjective or, equivalently, on the G-action being

locally free. When J is not surjective, we should simply throw out the part of g—

™¦

not in the image of J.

In general, given a symplectic manifold M and Poisson manifolds P1 and P2 , a

symplectic dual pair is a diagram

M

d

J1 d J2

d

d

© d

‚

P1 P2

54 9 DUAL PAIRS IN POISSON GEOMETRY

of Poisson maps with symplectically orthogonal ¬bers. Orthogonality implies

{J1 (C ∞ (P1 )), J2 (C ∞ (P2 ))} = 0 .

— —

Sometimes, this relation is written as {J1 , J2 } = 0.

Remark. For a pair of Poisson maps J1 : M ’ P1 and J2 : M ’ P2 , imposing

{J1 , J2 } = 0 is equivalent to imposing that the product map

J1 — J2

P1 — P2

M E

™¦

be a Poisson map.

9.4 Morita Equivalence

Let J1 , J2 be surjective Poisson submersions from a symplectic manifold M to

Poisson manifolds P1 , P2 . If

J1 (C ∞ (P1 )) = J2 (C ∞ (P2 ))

— —

J2 (C ∞ (P2 )) = J1 (C ∞ (P1 )) ,

— —

and

then the J1 -¬bers are symplectic orthogonals to the J2 -¬bers:

ker T J1 = (ker T J2 )⊥ .

The reverse implication is not true unless we assume that the ¬bers are connected,

essentially because ker T J1 = (ker T J2 )⊥ is a local condition while the hypothesis

was a global condition. If J1 and J2 have connected ¬bers, then the two conditions

above are equivalent. To require that ¬bers be connected is appropriate because of

the following property for such dual pairs.

Proposition 9.2 Let J1 , J2 be a pair of complete surjective Poisson submersions

M

d

J1 d J2

d

d

© ‚

d

P1 P2

from a symplectic manifold M . Assume that the J1 -¬bers are symplectically orthog-

onal to the J2 -¬bers, and that all ¬bers are connected. Then there is a one-to-one

correspondence between the symplectic leaves of P1 and the symplectic leaves of P2 .

Let Fj ⊆ T M be the distribution spanned by the hamiltonian vector

Proof.

¬elds of functions in Jj (C ∞ (Pj )). The assumption says that, at each point, the

—

distribution F1 (respectively F2 ) gives the subspace tangent to the ¬bers of J2

(respectively J1 ); this clearly shows that each of F1 and F2 is integrable. To see

that F1 + F2 is also integrable, note that F1 + F2 is spanned by hamiltonian vector

¬elds, and that the vector ¬elds from J1 commute with those from J2 . So we can

integrate F1 + F2 to a foliation of M .

A leaf L of the foliation de¬ned by F1 + F2 projects by each map Ji to a set

Ji (L), which is in fact a symplectic leaf of Pi (i = 1, 2) for the following two reasons.

9.5 Representation Equivalence 55

First, by completeness, we can move anywhere within a symplectic leaf of Pi by

moving in the Fi direction in L. Secondly, if we move in the F2 (respectively F1 )

direction in L, then nothing happens in the projection to P1 (respectively P2 ).

Therefore, there is a map from the leaf space of F1 + F2 to the product of the

leaf spaces of P1 and P2 . The image R of this map gives a relation between the leaf

space of P1 and the leaf space of P2 . Additionally, the projection of R to either

factor of the product is surjective. Because the ¬bers of J1 , J2 are connected, it

follows that R is the graph of a bijection. 2

We say that two Poisson manifolds P1 , P2 are Morita equivalent [176, 177] if

there is a symplectic manifold M and surjective submersions J1 , J2

M

d

J1 d J2

d

d

© ‚

d

P1 P2

satisfying the following conditions:

• J1 is a Poisson map and J2 is an anti-Poisson map (anti in the sense of being

an anti-homomorphism for the bracket).

• each Ji is complete and has constant rank,

• each Ji has connected, simply connected ¬bers,

• the ¬bers of J1 , J2 are symplectically orthogonal to one another. Equivalently,

J1 (C ∞ (P1 )) and J2 (C ∞ (P2 )) are commutants of one another.

— —

Remark. The map J2 in the Morita equivalence is sometimes denoted as a

Poisson map J2 : M ’ P 2 , where P 2 is the manifold P2 with Poisson bracket

de¬ned by {·, ·}P 2 = ’{·, ·}P2 . ™¦

Remark. In spite of the name, Morita equivalence is not an equivalence relation,

™¦

as it fails to be re¬‚exive [176, 177].

9.5 Representation Equivalence

The Morita equivalence of Poisson manifolds provides a classical analogue to the

Morita equivalence of algebras. Let A1 , A2 be algebras over a ¬eld K. De¬ne an

(A1 , A2 )-bimodule E to be an abelian group E with a left action of A1 and a right

action of A2 such that for a1 ∈ A1 , a2 ∈ A2 , e ∈ E

(a1 e)a2 = a1 (ea2 ) .

So we have injective maps

A1 ’’ EndK (E)

opp

A2 ’’ EndK (E) .

56 9 DUAL PAIRS IN POISSON GEOMETRY

where Aopp denotes A2 acting on the left by inverses. A Morita equivalence from

2

A1 to A2 is an (A1 , A2 )-bimodule E such that A1 and A2 are mutual commutants

in EndK (E). Morita introduced this as a weak equivalence between algebras, and he

showed that it implies that A1 -modules and A2 -modules are equivalent categories.

Xu [176, 177] showed that we can imitate this construction for symplectic real-

izations of Poisson manifolds. In particular, if P1 , P2 are Poisson manifolds, then we

say that they are representation equivalent if the category of complete Poisson

maps to P1 is equivalent to the category of complete Poisson maps to P2 . Xu then

proved the following theorem:

Theorem 9.3 (Xu [176, 177]) If two Poisson manifolds are Morita equivalent,

then they are representation equivalent.

For a survey of Xu™s work and Morita equivalence in general, see the article

by Meyer [118]. For a survey of the relation between Poisson geometry and von

Neumann algebras, see the article by Shlyakhtenko [151].

9.6 Topological Restrictions

The importance of the condition that the ¬bers of Ji be simply connected in the

de¬nition of Morita equivalence between Poisson manifolds is explained by the

following property for the case where P1 and P2 are symplectic.

Proposition 9.4 Let S1 , S2 be symplectic manifolds. Then S1 and S2 are Morita

equivalent if and only if they have isomorphic fundamental groups.

Proof. Suppose that S1 , S2 are Morita equivalent. Then, from the long exact

sequence for homotopy

0 = π1 (¬ber) ’’ π1 (M ) ’’ π1 (Sj ) ’’ π0 (¬ber) = 0 ,

we conclude that

π1 (S1 ) π1 (M ) π1 (S2 ) .

Furthermore, the Morita equivalence induces a speci¬c isomorphism via pull-back

by the maps from S.

Conversely, suppose that π1 (S1 ) π1 (S2 ) π. Let Sj be the universal cover of

Sj , so that Sj is a principal π-bundle over Sj . Because π acts on S1 and S2 , there

is a natural diagonal action of π on S1 — S2 which allows us to de¬ne the dual pair

S1 — S 2

π

d

d

d

d

©

‚

d

S1 = S1 /π S 2 = S 2 /π

Exercise 27

Check that these maps have simply connected ¬bers and that this de¬nes a

Morita equivalence.

9.6 Topological Restrictions 57

2

Isomorphism of fundamental groups implies isomorphism of ¬rst de Rham coho-

mology groups. For symplectic manifolds, the de Rham cohomology is isomorphic

to Poisson cohomology. For general Poisson manifolds, we have the following result.

Theorem 9.5 (Ginzburg-Lu [72]) If P1 , P2 are Morita equivalent Poisson man-

1 1

ifolds, then HΠ (P1 ) HΠ (P2 ).

Since any two simply connected symplectic manifolds are Morita equivalent, we

are not able to say anything about the higher Poisson cohomology groups.

10 Examples of Symplectic Realizations

A symplectic realization of a Poisson manifold P is a Poisson map • from a sym-

plectic manifold M to P .

Injective Realizations of T3

10.1

Let R3 have coordinates (x1 , x2 , x3 ) and (by an abuse of notation) let T3 be the

3-torus with coordinates (x1 , x2 , x3 ) such that xi ∼ xi + 2π. De¬ne a Poisson

structure (on R3 or T3 ) by

‚ ‚ ‚ ‚

§

Π= + ±1 + ±2 .

‚x1 ‚x3 ‚x2 ‚x3

The Poisson bracket relations are:

{x1 , x2 } = 1 , {x2 , x3 } = ’±1 , {x1 , x3 } = ±2 .

On R3 , Π de¬nes a foliation by planes with slope determined by ±1 , ±2 . If

1, ±1 , ±2 are linearly independent over Q, then Π also de¬nes a foliation on T3 by

planes, each of which is dense in T3 . This is called a (fully) irrational foliation.

If both ±1 and ±2 are rational, then the foliation of T3 is by 2-tori, and if exactly

two of 1, ±1 , ±2 are linearly dependent over Q, then the foliation is by cylinders.

In the fully irrational case, the algebra HΠ (T3 ) of Casimir functions is trivial; in

0

fact, the constants are the only L∞ functions constant on symplectic leaves, since

the foliation on T3 is ergodic. There are no proper Poisson ideals. This structure

allows us to regard T3 as being “almost symplectic”. We will see, however, that

its complete symplectic realizations are more interesting than those of a symplectic

manifold.

Exercise 28

If Π de¬nes a foliation by cylinders, are there any (nontrivial) Casimir func-

tions?

First we may de¬ne a realization J by inclusion of a symplectic leaf,

R2 (x1 , x2 )

J

c c

T3 (x1 , x2 , ±1 x1 + ±2 x2 ) (mod 2π)

Although J is not a submersion, it is a complete map. There is such a realization

for each symplectic leaf of T3 , de¬ned by

Jc : (x1 , x2 ) ’’ (x1 , x2 , ±1 x1 + ±2 x2 + c) ,

with c ∈ R. For any integers n0 , n1 , n2 , substituting c + 2π(n0 + ±1 n1 + ±2 n2 ) for

c gives the same leaf. Thus the leaf space of T3 is parametrized by c ∈ R/2π(Z +

±1 Z + ±2 Z).

The leaf space is highly singular; there is not even a sensible way to de¬ne

nonconstant measurable functions. It is better to consider the Poisson manifold T3

59

60 10 EXAMPLES OF SYMPLECTIC REALIZATIONS

itself as a model for the leaf space, just as one uses noncommutative algebras to

model such singular spaces in noncommutative geometry [32].

The map J : R2 ’ T3 has a dense image, and thus the induced pull-back

on functions, J — : C ∞ (T3 ) ’ C ∞ (R2 ), is injective. The following (periodic or

quasi-periodic) functions on R2 ,

eix1 , eix2 , ei(±1 x1 +±2 x2 ) ,

are in the image of J — , and generate such a large class of functions that any function

in C ∞ (R2 ) can be uniformly C ∞ -approximated by them on compact sets. Thus

J — (C ∞ (T3 )) = constants J — (C ∞ (T3 )) = C ∞ (R2 ) .

and

Since the Poisson algebra J — (C ∞ (T3 )) is not its own double commutant, there can

not be another Poisson manifold P which will make

R2

d

J d

d

d

© d

‚

T3 P

into a Morita equivalence. In fact, to form a dual pair, such a “manifold” P would

have to be a single point because each ¬ber of J is a single point and because of

orthogonality of ¬bers. The diagram

R2

d

J d

d

d

©

d

‚

T3 point

satis¬es the conditions that the ¬bers be symplectic orthogonals and that the ¬bers

be all connected and simply connected. However, the function spaces of this pair are

not mutual commutants. Of course, the problem here is that J is not a submersion.

Submersive Realizations of T3

10.2

Noticing that T3 is a regular Poisson manifold, we can use the construction for prov-

ing Lie™s theorem (Chapter 4) to form a symplectic realization by adding enough

extra dimensions. Speci¬cally, consider the map

R4 (x1 , x2 , x3 , x4 )

J

c c

T3 (x1 , x2 , x3 )

where R4 has symplectic structure de¬ned by

‚ ‚ ‚ ‚ ‚ ‚

§ §

ΠR4 = + ±1 + ±2 + ,

‚x1 ‚x3 ‚x2 ‚x3 ‚x3 ‚x4

Submersive Realizations of T3

10.2 61

and T3 has the fully irrational Poisson structure as above:

‚ ‚ ‚ ‚

§

Π= + ±1 + ±2 .

‚x1 ‚x3 ‚x2 ‚x3

Exercise 29

Check that ΠR4 de¬nes a non-degenerate 2-form on R4 which is equivalent to

the standard symplectic structure

‚ ‚ ‚ ‚

§ §

Πstd = + .

‚x1 ‚x2 ‚x3 ‚x4

Show that the symplectic structures induced on T4 by ΠR4 and Πstd are not

equivalent, though they both have the same volume element

‚ ‚ ‚ ‚

ΠR4 § ΠR4 = Πstd § Πstd = § § § .

‚x1 ‚x2 ‚x3 ‚x4

1

(Consider times the cohomology class of each symplectic structure.)

2π

To ¬nd the commutant of J — (C ∞ (T3 )) in this case, we examine the symplectic

orthogonals to the ¬bers of J. First, we list the Poisson brackets for the symplectic

structure on R4 :

{x1 , x2 } = 1 {x1 , x4 } = 0

{x2 , x3 } = ’±1 {x2 , x4 } = 0

{x1 , x3 } = ±2 {x3 , x4 } = 1

and the hamiltonian vector ¬elds

‚ ‚

’ ’ ±2

Xx1 = ,

‚x2 ‚x3

‚ ‚

Xx2 = + ±1 ,

‚x1 ‚x3

‚ ‚ ‚

’ ±1 ’

Xx3 = ±2 ,

‚x1 ‚x2 ‚x4

‚

Xx4 = .

‚x3

The commutant of J — (C ∞ (T3 )) consists of the functions killed by Xx1 , Xx2 and Xx3 .

Since these three vector ¬elds are constant, it su¬ces to ¬nd the linear functions

c1 x1 + c2 x2 + c3 x3 + c4 x4 killed by these vector ¬elds, i.e. solve the system

±

’c2 ’ ±2 c3 = 0

c1 + ±1 c3 = 0

±2 c1 ’ ±1 c2 ’ c4 = 0

The linear solutions are the constant multiples of

±1 x1 + ±2 x2 ’ x3 ,

and J — (C ∞ (T3 )) ⊆ C ∞ (R4 ) consists of functions of ±1 x1 + ±2 x2 ’ x3 .

Given the commutant, we can geometrically de¬ne the other leg of a dual pair

to be the map J2 : R4 ’ R given

(x1 , x2 , x3 , x4 ) ’’ ±1 x1 + ±2 x2 ’ x3 .

62 10 EXAMPLES OF SYMPLECTIC REALIZATIONS

Thus we have the diagram

R4

d

J d J2

d

d

© d

‚

T3 R

Although ±1 x1 +±2 x2 ’x3 is not quasi-periodic, it lies in the closure of J — (C ∞ (T3 )).

One can check that J2 (C ∞ (R)) = J — (C ∞ (T3 )), and so this does not de¬ne a

—

Morita equivalence. The obstruction stems from the fact that J does not have

connected ¬bers; a ¬ber of J is an in¬nite collection of parallel lines in R4 .

We can factor J through the quotient R4 ’ T4 , and denote the induced map by

JT4 : T4 ’ T3 . The commutant of JT4 (C ∞ (T3 )) in C ∞ (T4 ) should be generated

—

by the linear function ±1 x1 + ±2 x2 ’ x3 on T4 , but this is not periodic on R4 and

thus is not de¬ned on T4 . Therefore, the commutant of JT4 (C ∞ (T3 )) in C ∞ (T4 )

—

is trivial, and the double commutant must be all of C ∞ (T4 ), which again prevents

Morita equivalence (moreover, ¬bers would fail to be simply connected). As before,

the other leg of the dual pair would have to be a single point rather than R:

T4

d

JT4 d

d

d

©

d

‚

T3 point

The “dual” to T3 thus depends on the choice of realization, but requiring that

the realization have connected ¬bers seems to imply that the dual is “pointlike”.

We close these sections on T3 by mentioning that there is still much to investigate

in the classi¬cation of complete realizations. For instance, it would be interesting

to be able to classify complete Poisson maps from (connected) symplectic manifolds

to

‚ ‚ ‚ ‚

• T3 with the Poisson tensor Π = + ±1 ‚x2 § + ±2 ‚x3 , or

‚x1 ‚x2

• a given manifold M with the zero Poisson tensor.

10.3 Complex Coordinates in Symplectic Geometry

The symplectic vector space R2n can be identi¬ed with the complex space Cn by

the coordinate change

zj = qj + ipj .

In order to study Cn as a (real) manifold, it helps to use the complex valued func-

tions, vector ¬elds, etc., even though the (real) symplectic form is not holomorphic.

On a general manifold M , the complexi¬ed tangent bundle is

TM — C

TC M =

T M • iT M ,

=

10.4 The Harmonic Oscillator 63

and the complexi¬ed cotangent bundle is

—

T —M — C

TC M =

T — M • iT — M

=

= HomC (TC M, C)

= HomR (T M, C).

Introducing complex conjugate coordinates z j = qj ’ ipj , we ¬nd dzj = dqj +

idpj , dz j = dqj ’ idpj as linear functionals on TC M , and

dzj § dz j (dqj + idpj ) § (dqj ’ idpj )

=

’2i (dqj § dpj ) .

=

—

Thus the standard symplectic structure on TC M can be written in complex coor-

dinates as

i

dzj § dz j .

ω=

2j

{·, ·} to complex valued functions and

We linearly extend the Poisson bracket

compute

{zk , zj } = 0

{z k , z j } = 0

{zk , z j } = ’2iδkj .

By these formulas, the Poisson tensor becomes

‚ ‚

ΠR2n = ’2i § ,

‚zj ‚z j

j

‚ ‚

where ‚zj , ‚z j form the dual basis to dzj , dz j , and hence satisfy

‚ 1 ‚ ‚ ‚ 1 ‚ ‚

’i

= , = +i .

‚zj 2 ‚qj ‚pj ‚z j 2 ‚qj ‚pj

10.4 The Harmonic Oscillator

The harmonic oscillator is a system of n simple harmonic oscillators without

coupling, modeled by (R2n , ΠR2n ) with hamiltonian function

1

±j (qj + p2 ) .

2

h± = j

2 j

The coe¬cients ±j are the n frequencies of oscillation. Using complex coordinates,

we rewrite h± as

1

h± = ±j zj z j .

2j

To compute the ¬‚ow of h± , we work out the hamiltonian equations:

dzk 1 1

= {zk , h± } = ±j {zk , zj z j } = ±k zk (’2i) = ’i±k zk ,

dt 2 2

j

dz k

= i±k z k . The solution is thus zk (t) = zk (0)e’i±k t .

and similarly, dt

64 10 EXAMPLES OF SYMPLECTIC REALIZATIONS

If ±k = 1 for all k, the ¬‚ow is the standard action of S 1 on Cn , which is free on

Cn \ {0}.

If all the ±k are rationally related, then we can assume after a change of time

scale that ±k ∈ Z and see that we still have an action of S 1 . This action on Cn \ {0}

will generally not be free, but rather will have discrete stabilizers.

If the ±k are not rationally related, then this de¬nes an R-action, as the typical

orbits will not be closed, but will be dense on a torus. From now on, we will

concentrate on this case.

To study the orbit space of the R-action, we start by calculating the commu-

tant. Speci¬cally, we want to ¬nd the polynomial functions commuting with the

hamiltonian.

Exercise 30

For a typical monomial z z m = z11 · · · znn z m1 · · · z mn , compute:

n

1

m m m

{zj j z j j , zj z j } = j zj j z j j (2i) + j zj j z j j (’2i) ,

(a)

{z z m , 1 ±j zj z j } = i ±j (mj ’ j )z z m .

(b) 2

Thus the monomials in zj and z j are eigenvectors of the hamiltonian vector ¬eld

of the oscillator h± . The corresponding eigenvalues are

±j (mj ’

i j) .

j

The commutant of h± in Pol(Cn— ) is spanned by the monomials z z m with ±j (mj ’

j ) = 0.

Example. Suppose that the ±j ™s are linearly independent over Q. Then the only

monomials in the commutant are those with mj = j for all j, that is, monomials of

the form z z = (zz) . In this case, the functions invariant under the hamiltonian

action are just polynomials in Ij = zj z j = |zj |2 . Then we can see this roughly as a

pair

Cn

d

Ij d h±

d

d

© ‚

d

Rn R

Of course, Ij has a singularity at 0, and its image is only in the positive orthant of

Rn . This also could not be a dual pair of symplectic realizations, as the dimensions

of the ¬bers do not match up properly unless we delete the origin. Even so, this

™¦

example provides some intuition toward our study of dual pairs.

If ±j ∈ Z for all j, then calculating the commutant of h± is equivalent to solving

the system of linear equations

±j (mj ’ j) =0

j

over the integers. What makes this problem non-trivial is that we are only interested

in non-negative integer solutions for j , mj , in order to study the ring of invariant

functions de¬ned on all of Cn .

10.5 A Dual Pair from Complex Geometry 65

To avoid this di¬culty, we look ¬rst at the case ±j = 1 for all j. Thus our

equation reduces to (mj ’ j ) = 0, or mj = j . The set of solutions for

this system of equations is spanned by the monomials zj z k . In fact, the set {zj z k }

forms a basis for the subring of solutions.

Remark. The real part of zj z k is invariant under the hamiltonian action since

it can be expressed as zj z k + zk z j . Similarly, the imaginary part zj z k ’ zk z j is

™¦

invariant under the hamiltonian ¬‚ow.

The most general linear combination of the basis elements (that is, the most

general quadratic solution) is

ajk ∈ C ,

ha = ajk zj z k ,

j,k

and any function of this form is invariant under the hamiltonian ¬‚ow. Furthermore,

these are all the quadratic invariants. The invariant functions will not commute

with one another, as the basis elements themselves did not commute.

10.5 A Dual Pair from Complex Geometry

To summarize the previous section: on Cn , the hamiltonian h = 1 zj z j generates

2

a ¬‚ow, which is just multiplication by unit complex numbers. The invariant func-

tions ha = j,k ajk zj z k generate complex linear ¬‚ows (i.e. ¬‚ows by transformations

commuting with multiplication by complex constants), which preserve h as well as

the symplectic form ω.

Hence, transformations generated by ha are unitary. The group of all linear

transformations leaving h invariant is the unitary group U(n). We would like to

show that the ¬‚ows of the ha ™s give a basis for the unitary Lie algebra u(n).

Remark. The function ha is real valued if and only if ajk = akj , i.e. the matrix

(ajk ) is hermitian. Thus the set of real valued quadratic solutions corresponds to

the set of hermitian matrices.

Recall that the Poisson bracket of two invariant functions is again invariant

under the hamiltonian ¬‚ow. Moreover, the bracket of two quadratics is again

quadratic, and thus we can use the correspondence above to de¬ne a bracket on

the group of hermitian matrices.

Exercise 31

Check that

{ha , hb } = hi[a,b] ,

where [a, b] is the standard commutator bracket of matrices.

™¦

The algebra u(n) is the Lie algebra of skew-hermitian matrices. Denoting the

space of hermitian matrices by hn , we identify

←’

hn u(n)

»

’’

a ia .

For a, b ∈ hn , it is easy to check that

[»a, »b] = »(i[a, b]) ,

66 10 EXAMPLES OF SYMPLECTIC REALIZATIONS

and thus the bilinear map hn — hn ’ hn taking (a, b) to i[a, b] is the usual commu-

tator bracket on u(n) pulled back by » to hn . With this identi¬cation of invariant

¬‚ows as unitary matrices, we see that the map

u(n) ’’ C ∞ (Cn )

hn

is a Lie algebra homomorphism. From our discussion in Section 7.2, we conclude

that there is a complete momentum map J : Cn ’ u(n)— h— corresponding to

n

an action of U(n) on Cn . This is the standard action of the unitary group on Cn .

We may view J as a map J : z ’ z — z (zj z k ). The value of the function ha at

(zj z k ) ∈ u(n)— is the inner product of the matrix (ajk ) with the matrix (zj z k ).

Therefore, we have a pair

Cn

d

J dh

d

d

© ‚

d

h— u(n)— R u(1)—

n

Removing the origin in Cn , we get a dual pair for which the image of the left leg

is the collection of rank-one skew-hermitian, positive semi-de¬nite matrices, and

the image of the right leg is R+ . A function which commutes with J is invariant

on the concentric spheres centered at 0 and is thus a function of |zz| “ the square

of the radius. On the other hand, even though there is a singularity at 0 ∈ Cn , any

function on Cn commuting with h is in fact a pull-back of a function on u(n) by

the map J. In general, functions which are pull-backs by the momentum map J

are called collective functions.

Conjecture 10.1 (Guillemin-Sternberg [76]) Suppose that a symplectic torus

Tk acts linearly on Cn with quadratic momentum map J : Cn ’ (tk )— . If the map

Cn ’ Cn /Tk corresponds to the invariant functions under the torus action, then

Cn

d

J dp

d

d

© d

‚

(tk )— Cn /Tk

is a dual pair, in the sense that the images of J — and p— are mutual commutants in

C ∞ (Cn ).

Guillemin and Sternberg [76] almost proved this as stated for tori and conjec-

tured that it held for any compact connected Lie group acting symplectically on

Cn . Lerman [103] gave a counterexample and, with Karshon [93], provided a proof

of the conjecture for (tk )— as well as an understanding of when this conjecture does

and does not hold for arbitrary compact groups.

Example. Lerman™s counterexample for the more general conjecture is the group

SU(2) acting on C2 (see [93, 103] for more information). As for the case of u(2)

studied above, the invariant functions corresponding to the collective functions are

functions of the square of the radius. The commutator of these functions are pulled

10.5 A Dual Pair from Complex Geometry 67

back from u(2)— , not su(2)— . For instance, the function z1 z 1 + z2 z 2 cannot be the

pull-back of a smooth function on su(2)— , although the function (z1 z 1 + z2 z 2 )2 can

be so expressed. Thus the pair of maps

C2

d

J d

d

d

© ‚

d

su(2)— R3 R

is not a dual pair.

Exercise 32

What happens when we remove the origin from each space?

™¦

Part V

Generalized Functions

11 Group Algebras

Multiplication on a (locally compact) group G can be coded into a coproduct struc-

ture on the algebra C(G) of continuous real functions on G, making it into a com-

mutative Hopf algebra. Conversely, the algebra C(G) determines the multiplication

on G. Noncommutative analogues of C(G) are studied as if they were algebras of

functions on so-called quantum groups.

11.1 Hopf Algebras

Example. Let G be a ¬nite set, and let C(G) be its algebra of real func-

tions. The tensor product C(G) — C(G) is naturally isomorphic as an algebra to

C(G — G) via the map

• — ψ ’’ ((g, h) ’ •(g)ψ(h)) .

Now suppose that G is a group. Besides the pointwise product of functions,

m : C(G) — C(G) ’’ C(G) , m(• — ψ) = •ψ ,

m

we can use the group multiplication G — G ’ G to de¬ne a coproduct on C(G)

m— : C(G) ’’ C(G — G) = C(G) — C(G) , m— (•)(g, h) = •(gh) .

It is an easy exercise to check that this is a homomorphism with respect to the

pointwise products on C(G) and C(G — G). With this product and coproduct,

™¦

C(G) becomes a Hopf algebra.

In general, a Hopf algebra is a vector space A equipped with the following

operations:

1. a multiplication

m

A — A ’’ A , m(•, ψ) = • · ψ ,

also denoted

2. a comultiplication

∆

A ’’ A — A ,

3. a unit (or identity),

i : C ’’ A ,

4. a co-unit (or coidentity),

µ : A ’’ C , and

5. an antipode map

± : A ’’ A ,

69

70 11 GROUP ALGEBRAS

satisfying the following axioms:

1. the multiplication is associative, i.e.

m1,2 — E

id

A—A—A A—A

id — m2,3 m

m

c c

A—A EA

commutes, where m1,2 — id : • — ψ — ρ ’ m(•, ψ) — ρ, and similarly for other

indexed maps on tensor product spaces,

2. the comultiplication is coassociative, i.e.

∆ — id

A—A—A ' A—A

T T

id — ∆ ∆

∆

A—A ' A

commutes,

3. the comultiplication ∆ is a homomorphism of algebras, i.e.

∆

A—A ' A

T

T

m1,3 — m2,4 m

∆—∆

A—A—A—A ' A—A

commutes, (that is, ∆(• · ψ) = ∆(•) · ∆(ψ) where the multiplication on the

right hand side is m — m),

4. the unit is an identity for multiplication, i.e.

id — i

A—C C—A A—A

E

d

d id

i — id m

d

d

‚

d

mEc

c

A—A A

commutes,

5. the co-unit is a co-identity for comultiplication, i.e.

id — µ

A' A—A

Ts T

d

d id

µ — id ∆

d

d

d

∆

A—A ' A

11.1 Hopf Algebras 71

commutes,

6. the unit is a homomorphism of coalgebras, i.e.

i EA

C