where S(x, y, z) is 4 times the symplectic area of the triangle with vertices x, y and

z.

The von Neumann integral formula gives a well-de¬ned product on various

spaces of functions, including Schwartz functions, smooth functions whose par-

tial derivatives of all orders are bounded, and Λ-periodic smooth functions on E

152 20 WEYL ALGEBRAS

where Λ is a lattice, i.e. smooth functions on a torus E/Λ. This product does not

extend to continuous functions on E/Λ, but it is possible to complete C ∞ (E/Λ)

to a noncommutative C — algebra called “the continuous functions on a quantum

™¦

torus” [144].

20.3 A¬ne Invariance of the Weyl Product

The Weyl product on a Poisson vector space (E, Π) is invariant under a¬ne Poisson

maps, i.e. if A : E ’ E is an a¬ne symplectic map, then the induced pull-back

map

A— : C ∞ (E)[[ ]] ’’ C ∞ (E)[[ ]]

is an algebra automorphism for the Weyl product.

By a¬ne invariance, the Weyl product (on the Weyl algebra or any of the

other related spaces of functions mentioned at the end of the previous section)

passes to any Poisson manifold locally modeled on E, as long as we only allow

a¬ne coordinate changes. This condition on E amounts to the existence of a

¬‚at connection without torsion, for which parallel transport preserves the Poisson

structure.

The in¬nitesimal counterpart of a¬ne invariance is that, for every polynomial

function k on E of degree less than or equal to 2,

{f {g, k} + {f, k}

g, k} = f g.

In words, {·, k} is a derivation not just of the pointwise product (Leibniz identity)

and of the Poisson bracket (Jacobi identity), but of the whole -product.

Remark. Dirac™s [44] quantum Poisson bracket

g’g

f f

[f, g] :=

i

satis¬es the derivation law

[f g, k] = f [g, k] + [f, k] g

just as a consequence of associativity. The similar property for {·, k} is explained

by the fact that, for a polynomial k on E of degree ¤ 2, we have [·, k] = {·, k}.

In particular, for k1 and k2 polynomials of degree ¤ 2, we have [k1 , k2 ] =

{k1 , k2 }, which shows that polynomials of degree ¤ 2 form a Lie algebra. ™¦

20.4 Derivations of Formal Weyl Algebras

Let F (E)[[ ]] be the space of formal power series on the vector space E, thought

as an algebra over C[[ ]].

A theorem of E. Borel states that every formal power series is the Taylor ex-

pansion of some function. This implies that the space F (E) of formal power series

on the vector space E is isomorphic to C ∞ (E) modulo the functions which vanish

to in¬nite order at 0.

20.5 Weyl Algebra Bundles 153

Theorem 20.1 Suppose that Π is a non-degenerate Poisson structure on E. Then

every derivation D of F (E)[[ ]] such that D = 0 is of the form [·, f ] for some

f ∈ F (E)[[ ]].

Exercise 78

Prove this theorem. Hints:

A derivation D is determined by its e¬ect on generators of the algebra

q1 , . . . , qn , p1 , . . . , pn . Notice that qi , pi have degree ¤ 2. Suppose that

D = [·, f ] were a inner derivation. Then

‚f

{qi , f }

Dqi = [qi , f ] = = ‚pi

‚f

{pi , f } ’ ‚q

Dpi = [pi , f ] = =

i

To ¬nd the element f , we must solve

df = (Dqi )dpi ’ (Dpi )dqi

for f . If the right-hand side is closed, then the left-hand side will be determined

up to an element in the center C[[ ]] of C ∞ (E)[[ ]]. Let us check that the right-

hand side is closed:

‚ ‚

’{pj , Dqi } + {qi , Dpj }

(Dqi ) + (Dpj ) =

‚qj ‚pi

= [Dqi , pj ] + [qi , Dpj ]

= D[qi , pj ] = D(δi,j ) = 0 .

To ¬nish the proof that D = [·, f ] , consider the ¬ltration of F (E) by ideals

Ak generated by the homogeneous polynomials of degree k. Show that, if D is

a derivation, then DAk ⊆ Ak’1 [[ ]].

Let (E, Π) be a Poisson vector space, and let • be an automorphism of the Weyl

algebra C ∞ (E)[[ ]] as a C[[ ]]-algebra..

The term in • of 0-th order in shows that • induces an automorphism of

∞

C (E), hence a di¬eomorphism of E.

The term in • of ¬rst order in shows that this di¬eomorphism is a Poisson

automorphism of (E, Π).

We hence obtain an exact sequence

E Aut(C ∞ (E)[[ ]]) E P(E, Π)

EI

1 E1

where P(E, Π) is the set of Poisson automorphisms of (E, Π). The kernel I of the

third arrow is the group of inner automorphisms of C ∞ (E)[[ ]] corresponding to

invertible elements of C ∞ (E)[[ ]] [59].

20.5 Weyl Algebra Bundles

Let (E, ρ, [·, ·]E ) be a Lie algebroid over a manifold M , with symplectic structure ω ∈

“(§2 E — ). The symplectic E-2-form ω is non-degenerate and dE ω = 0; it determines

an E-Poisson structure Π (see Section 18.3) by Π = ω ’1 , and an ordinary Poisson

structure ρ(Π) on T M .

Let W E be the Weyl algebra bundle over M whose ¬ber at x ∈ M is the

formal Weyl algebra of the symplectic (hence Poisson) vector space Ex . The smooth

sections of W E are those for which the coe¬cient of each term is a smooth function

on M ; they form an algebra under ¬berwise multiplication. We think of “(W E)

as “functions on the quantized E”. Locally, we write a typical section as f (x, y, ),

where x ∈ M , y is a formal variable in Ex , and is another formal parameter. (The

154 20 WEYL ALGEBRAS

constant is taken the same on each ¬ber, just as Planck™s constant is a universal

constant.)

From now on, to simplify, we will analyze the case where E = T M is the tangent

bundle of M . Everything works for the general Lie algebroid case [126].

Interpret “(W T M ) as the space of smooth functions on the “quantized tangent

bundle”

“(W T M ) = C ∞ (Q T M ) .

The zero section is the map

C ∞ (Q T M ) E C ∞ (M )[[ ]]

given by evaluation at y = 0. We may think of Q T M as an in¬nitesimal neighbor-

hood of the zero section.

In the next chapter, we will describe the quantization method of Fedosov, in

which C ∞ (M )[[ ]] is identi¬ed with a subalgebra of “(W T M ). The Weyl product

is then carried back to C ∞ (M )[[ ]] to give a deformation quantization.

Geometrically, a subalgebra of “(W T M ) annihilated by a Lie algebra of deriva-

tions corresponds to a “foliation” of Q T M . The foliation is transverse to the ¬bers

when the derivations are of the form X as X ranges over the vector ¬elds on M ,

de¬ning a ¬‚at connection on the bundle W T M itself.

When the foliation is transverse to the zero section as well, parallel sections of

W T M are in one-to-one correspondence with elements of C ∞ (M )[[ ]]. Notice that

a ¬‚at linear connection on T M would not work: parallel sections of a ¬‚at connection

on C ∞ (T M ) correspond to functions on a tangent ¬ber, not C ∞ (M )[[ ]] as we need.

Example. Let (M, ω) be a symplectic vector space with coordinates x. De¬ne

the connection by

‚ ‚

’

= ,

‚

‚xi ‚yi

‚xi

where y are the tangent coordinates induced by x. Lift functions u(x) and v(x) on

M to u(x + y) and v(x + y) on T M . To evaluate (u v)(x0 ), freeze the x variable

at x0 , take the Weyl product with respect to y, and then set y = 0 to obtain a

™¦

function on M . This recipe reproduces the usual Weyl product.

21 Deformation Quantization

On a general Poisson manifold, if the rank of the Poisson tensor Π is constant, then

by a theorem of Lie the Poisson manifold is locally isomorphic to a vector space

with constant Poisson structure (see Section 3.4). Such Poisson manifolds, which

are called regular, are always locally deformation quantizable using the Moyal-

Weyl product in canonical coordinates; the problem is to patch together the local

deformations to produce a global -product.

21.1 Fedosov™s Connection

There is one case in which the patching together of local quantizations is easy.

Since the Moyal-Weyl product on a vector space V with constant Poisson structure

is invariant under all the a¬ne automorphisms of V , we can construct a global

quantization of any Poisson manifold (M, Π) covered by canonical coordinate sys-

tems in the general case for which the transition maps are a¬ne. Such a covering

exists when M admits a ¬‚at torsionless linear connection for which the covariant

derivative of Π is zero.

Fedosov overcomes the di¬culty of patching together local Weyl structures by

making the canonical coordinate neighborhoods “in¬nitely small”. To understand

his idea, we should ¬rst think of elements of the deformed algebra C ∞ (T M )[[ ]] as

sections of the bundle W T M over M whose ¬ber at x ∈ M is W Tx M .

Of course we are most interested in dealing with the case where (M, Π) does

not admit a ¬‚at Poisson connection, and this is where the most interesting part of

Fedosov™s proof comes in. He shows (in other terms) that the tangent bundle of

every symplectic (or regular Poisson) manifold does admit a ¬‚at Poisson connection,

if one gives the appropriate extended meaning to that concept, namely admitting

“nonlinear quantum maps” as the structure group.

Fedosov™s connection is constructed on the bundle W T M of Weyl algebras.

The “structure Lie algebra” of this connection, in which the connection forms take

values, is W R2n acting on itself by the adjoint representation of its Lie algebra

structure. Since the full Weyl algebra is used, and not just the quadratic functions

which generate linear symplectic transformations, the structure group allows non-

linear transformations of the (quantized) tangent spaces. Since linear generating

functions are included, the structure group even allows translations.

In fact (this idea was also used in [133]), it is not the full Weyl algebra of

R which serves as the typical ¬ber, but only the formal Weyl algebra F (W R2n ),

2n

consisting of formal Taylor expansions at the origin. Geometrically, one can think

of this step as the replacement of the (quantized) tangent bundle by a formal

Q

neighborhood of the zero section T M .

Q

Remark. Since T M is an in¬nitesimal neighborhood of the zero section, parallel

transport does not go anywhere. This step may hence appear to be inconsistent

with the inclusion of translations in the structure group, since these do not leave

the origin ¬xed. In fact, the e¬ect is to force us to forget the group and to work only

with the structure Lie algebra. A bene¬cial, and somewhat surprising, result of this

e¬ect is that a parallel section with respect to a ¬‚at connection is not determined by

its value at a single point. This situation is very close to that in formal di¬erential

geometry, where the bundle of in¬nite jets of functions on a manifold M has a ¬‚at

155

156 21 DEFORMATION QUANTIZATION

connection whose sections are the lifts of functions on M . (See [160, Section 1] for

™¦

a nice exposition with references.)

Fedosov uses an iterative method for “¬‚attening” a connection which is similar

to that used in many di¬erential geometric problems. (See [119] for an example,

and [147] for a recent survey.) Over the domain of a local trivialization of a principal

G-bundle, a connection is given by a 1-form φ with values in the Lie algebra g; the

curvature of the connection is the Lie algebra valued 2-form

1

„¦φ = dφ + [φ, φ] .

2

If the curvature is not zero, we may try to “improve” the connection by adding

another Lie algebra valued 1-form ±. The curvature zero condition for φ + ± is the

quadratic equation

1

d± + [φ, ±] = ’„¦φ ’ [±, ±] .

2

Rather than trying to solve this equation exactly, we linearize it by dropping the

term ’ 1 [±, ±]. The operator d + [φ, ·] is the covariant exterior derivative Dφ , so

2

our linearized equation has the form

Dφ ± = ’„¦φ .

From the Bianchi identity, Dφ „¦φ = 0, it appears that the obstruction to solving

the equation above for ± lies in a cohomology space. This is not quite correct, since

2

Dφ = [„¦φ , ·], which is not zero because the connection φ is not yet ¬‚at.

Up to now, we have essentially been following Newton™s method for solving

nonlinear equations. At this point, we add an idea similar to one often attributed

to Nash and Moser. (See [155, Section III.6] for an exposition of this method with

original references.) Since the linear di¬erential equation we are trying to solve

is only an approximation to the nonlinear one which we really want to solve, we

do not have to solve it precisely. It is enough to solve it approximately and to

compensate for the error in the later iterations which will in any case be necessary

1

to take care of the neglected quadratic term ’ 2 [±, ±]. Such approximate solutions

are constructed by some version of the Hodge decomposition. In the di¬erential

geometric applications mentioned above, the full story involves elliptic di¬erential

operators, Sobolev spaces, and so on, but in the case at hand, it turns out that the

“Hodge theory” is purely algebraic and quite trivial.

21.2 Preparing the Connection

We now start the construction of a ¬‚at connection on the bundle of Weyl algebras

by an iteration procedure. All the constructions are intrinsic, but for simplicity we

will describe them in local canonical coordinates.

Step 1 We begin with an arbitrary (linear) Poisson connection on the tangent

bundle of the symplectic manifold M .

The connection induces a covariant di¬erentiation operator on the dual

bundle, i.e. on the linear functions on ¬bers. In coordinates (x1 , . . . , xm )

on M :

‚ ‚

= “ijk ωk .

‚

‚xi ‚x ‚x

j

21.2 Preparing the Connection 157

We introduce the coe¬cients (ωk ) of the symplectic form to lower the

last index. For convenience, we assume that ωk is constant (i.e. the xi ™s

are Darboux coordinates).

If the connection has torsion, we can make it torsion-free by symmetriza-

tion [59]

“ijk + “jik

“ijk .

2

Because this is a symplectic connection, symmetry in the last two indices

comes for free: “ijk = “ikj .

Step 2 The connection form is a 1-form with values in the Lie algebra of the

symplectic group sp(m). The elements of sp(m) may be identi¬ed with

linear hamiltonian vector ¬elds on the manifold and hence with quadratic

functions. Thus the connection form can be written as

1

“ijk yi yj — dxk ,

φ’1 =

2

where (y1 , . . . , ym ) is a basis of linear functions on the ¬bers correspond-

ing to the coordinates (x1 , . . . , xm ) on M .

Step 3 The symplectic connection lifts to the Weyl algebra bundle. A co-

variant di¬erentiation D on the Weyl algebra bundle is described with

respect to a local trivialization by

Du = du + ψu

for a local section u, where ψ is a 1-form with values in Der(W T M ). We

can rewrite this local expression in the form

Du = du + [φ, u] ,

where now φ is a 1-form with values in W T M itself, and [·, ·] is (1/i )

times the commutator bracket; the bracket [·, ·] is the quantum Poisson

bracket of Dirac [44]de¬ned in Section 20.3. The generator φ of this

“inner derivation” is determined up to a 1-form on M with values in the

center C[[ ]] of the Weyl algebra.

Step 4 If we consider the form φ’1 (with the yi ™s now interpreted as formal

variables) as taking values in the bundle F W (T M ),

φ’1 ∈ “(T — M — F W (T M ))

becomes the connection form for the associated connection on that bun-

dle. Even if this connection were ¬‚at, it would not be the correct one to

use for quantization, since its parallel sections would not be identi¬able

in any reasonable way with functions on M . Instead we must use for our

¬rst approximation

1

“ijk yi yj ) — dxk .

φ0 = ( ωkj yj + 2

158 21 DEFORMATION QUANTIZATION

Step 5 To start the recursion, one calculates, using the fact that the connection

is symplectic and torsion free (see [57]), that its curvature is

dφ0 + 1 [φ0 , φ0 ]

„¦0 = 2

’1 1

ωir — dxi § dxr + Rijkl yi yj — dxk § dxl

= 2 4

’1 — ω + R ,

=

where R is the curvature of the original linear symplectic connection,

considered as a 2-form with values in the Lie algebra of quadratic func-

tions. The term linear in y vanishes because the torsion is zero. The

term ’1—ω appears even when the linear connection is ¬‚at, but it causes

no trouble because it is a central element of the Weyl Lie algebra and

therefore acts trivially in the adjoint representation.

21.3 A Derivation and Filtration of the Weyl Algebra

The coe¬cients of the connection forms which we are using are sections of the bun-

dle F W (T M ). Rather than measuring the size of these forms by the usual Sobolev

norms involving derivatives, we shall use a pointwise algebraic measurement.

In the formal Weyl algebra F W (V ) of a Poisson vector space V , we assign

degree 2 to the variable and degree 1 to each linear function on V . We denote

by F Wr (V ) the ideal generated by the monomials of degree r. Because the kth

term in the expansion of the -product involves 2k derivatives and multiplication

by k , we obtain a ¬ltration of the algebra F W (V ). We will also occasionally use

the classical grading, compatible with the commutative multiplication but not with

the -product, which assigns degree 0 to and 1 to each linear function on V .

The Lie algebra structure which we use for the formal Weyl algebra is the

quantum Poisson bracket of Dirac [44] de¬ned in Section 20.3. The factor (1/i )

makes the quantum bracket reduce to the classical one (rather than to zero) when

’ 0. In addition, the quantum and classical brackets are equal when one of the

entries contains only terms linear or quadratic in the variable on V , and they share

the property

[F Wr (V ), F Ws (V )] ⊆ F Wr+s’2 (V ) ,

so that the adjoint action of any element of F W2 (V ) preserves the ¬ltration.

We introduce the algebra

W(V ) = F W (V ) — §— (V )

whose elements may be regarded as di¬erential forms on the “quantum space whose

algebra of functions is F W (V )”. The algebra W(V ) inherits a ¬ltration by sub-

spaces Wr (V ) from the formal Weyl algebra, and a grading from the exterior al-

gebra. We can also consider W(V ) as the algebra of in¬nite jets at the origin of

di¬erential forms on the classical space V , in which case we generally use the classi-

cal grading. In this way, W(V ) inherits the exterior derivative operator, which we

denote by δ. Remarkably, δ is also a derivation for the quantized algebra structure

on W(V ).

We may describe the operator δ in terms of linear coordinates (x1 , . . . , xm )

on V . With an eye toward the case where V is a tangent space, we denote the

corresponding formal generators of F W (V ) by (y1 , ..., ym , ) and the generators of

21.3 A Derivation and Filtration of the Weyl Algebra 159

§— (V ) by (dx1 , . . . , dxm ). Then W(V ) is formally generated by the elements yi — 1,

— 1, and 1 — dxi , and we have

δ(yi — 1) = 1 — dxi , δ( — 1) = 0 , and δ(1 — dxi ) = 0 .

Notice that δ decreases the Weyl algebra ¬ltration degree by 1 while it increases

the exterior algebra grading by 1.

Since δ is essentially the de Rham operator on a contractible space, we expect

the cohomology of the complex which it de¬nes to be trivial. Fedosov makes this

explicit by introducing the dual operator δ — of contraction with the Euler vector

¬eld i yi — ‚xi . More precisely, δ — maps the monomial yi1 · · · yip — dxj1 § · · · § dxjq

‚

to

(’1)k’1 yi1 · · · yip yjk — dxj1 § · · · § dxjk § · · · § dxjq .

k

(This operator is not a derivation for the quantized algebra structure.) A simple

computation (or the Cartan formula for the Lie derivative by the Euler vector ¬eld)

shows that, on the monomial above, we have

δδ — + δ — δ = (p + q)id ,

so that if we de¬ne the operator δ ’1 to be p+q δ — on the monomial above, and 0 on

1

1 — 1, we ¬nd that each element u of W(V ) has the decomposition

u = δδ ’1 u + δ ’1 δu + Hu ,

where the “harmonic” part Hu of u is the part involving only powers of and

no yi ™s or dxi ™s, i.e. the pullback of u by the constant map from V to the origin.

In other words, we have reproduced the usual proof of the Poincar´ lemma via a

e

’1

homotopy operator δ from H to the identity.

When the Poisson vector space V is symplectic, the operator δ has another

description. For any a ∈ F W (V ), [yi , a] = {yi , a} = j πij (‚a/‚yj ). If (ωij ) is the

matrix of the symplectic structure, inverse to (πij ), we get ‚a/‚yi = [ j ωij yj , a],

and hence

δ(a — 1) = (‚a/‚yi ) — dxi = [ ωij yj — dxi , a — 1] .

i ij

It follows from the derivation property that a similar equation holds for any element

of W(V ); i.e. the operator δ is equal to the adjoint action of the element ij ωij yj —

dxi (which is just the symplectic structure itself).

Of course, all the considerations above apply when V is replaced by a symplectic

vector bundle E and W(V ) by the space of sections of the associated bundle

W(E) = F W (E) — §— (E) .

In particular, when E is the tangent bundle of a symplectic manifold M , the op-

erator δ and its relatives act on the algebra of di¬erential forms on M with values

in F W (T M ). These operators are purely algebraic with respect to the variable in

M , with δ being just the adjoint action of the symplectic structure viewed as an

F W (T M )-valued 1-form.

160 21 DEFORMATION QUANTIZATION

21.4 Flattening the Connection

Following Section 21.2, we next try to construct a convergent (with respect to

the ¬ltration) sequence φn of connections whose curvatures „¦n tend to the central

element ’1 — ω. Fedosov calls this central element the Weyl curvature of the

limit connection; to simplify notation, we will write „¦ = „¦ + 1 — ω for the form

which should be zero, and we call this the e¬ective curvature.

Step 6 As suggested in Section 21.1, we let

φn+1 = φn + ±n+1 ,

where ±n+1 is a section of W(T M ).

The corresponding curvature is

dφn+1 + 1 [φn+1 , φn+1 ]

„¦n+1 = 2

„¦n + d±n+1 + [φn , ±n+1 ] + 1 [±n+1 , ±n+1 ]

= 2

Dn ±n+1

where Dn = Dφn = d + [φn , ·]. Instead of solving

1

Dn ±n+1 = ’„¦n ’ 2 [±n+1 , ±n+1 ] ’ 1 — ω ,

we drop the quadratic term and look at the simpler equation

Dn ±n+1 = ’„¦n ’ 1 — ω .

This would solve approximately the linearized equation for zero e¬ective

curvature

Dn ±n+1 + „¦n = 0 .

The operator Dn = Dφn will have the form d + δ + [cn , ·], where cn

Step 7

is an F W (T M )-valued 1-form. We will try to arrange for cn to lie in

F W2 (T M ) so that the operator [cn , ·], like d, is ¬ltration preserving.

Since δ lowers the ¬ltration degree by 1, the principal part of the di¬er-

ential operator Dn will actually be the algebraic operator δ (and not d

as it would be if we measured forms by the size of their derivatives.)

We cannot even solve

δ±n+1 + „¦n = 0

exactly, because the Bianchi identity gives Dn „¦n = 0 instead of δ „¦n = 0.

(The term 1 — ω is killed by both operators.) Nevertheless, we de¬ne

±n+1 = ’δ ’1 („¦n ) ,

and take care of the errors later.

Step 8 From the recursion relation

1

„¦n+1 = „¦n + Dn ±n+1 + 2 [±n+1 , ±n+1 ] ,

we ¬nd after a straightforward calculation using the decompositions

u = δδ ’1 u + δ ’1 δu + Hu

Dn = d + δ + [cn , ·] and

21.5 Classi¬cation of Deformation Quantizations 161

that

δ ’1 δ „¦n

„¦n+1 =

+H„¦n + d±n+1 + [cn , ±n+1 ] + 1 [±n+1 , ±n+1 ] .

2

Using Dn = d + δ + [cn , ·] again, we can rewrite this as

δ ’1 Dn „¦n ’ δ ’1 d„¦n ’ δ ’1 [cn , „¦n ]

„¦n+1 =

+H„¦n + d±n+1 + [cn , ±n+1 ] + 1 [±n+1 , ±n+1 ] .

2

By the Bianchi identity Dn „¦n = 0, we get

„¦n+1 = H„¦n ’δ ’1 d„¦n ’δ ’1 [cn , „¦n ]+d±n+1 +[cn , ±n+1 ]+ 1 [±n+1 , ±n+1 ] .

2

Suppose now that „¦n ∈ Wr (T M ) with r ≥ 1. Then H„¦n = 0 and

±n+1 ∈ Wr+1 (T M ), so that cn ∈ W2 (T M ) and hence all the terms on

the right hand side of the equation above belong to Wr+1 (T M ).

Step 9 Since „¦0 = R has ¬ltration-degree 2, we conclude that „¦n has degree

at least n + 2, and ±n+1 has degree at least n + 3, so the sequence φn

converges to a connection form

φ = φ0 + ±1 + ±2 + . . .

for which the curvature is „¦ = ’1 — ω. This curvature is a central

section, so the connection on F W (T M ) associated to φ by the adjoint

representation F W (T M ) is ¬‚at. Since the adjoint action is by deriva-

tions of the multiplicative structure, the space of parallel sections is a

subalgebra of the space of all sections.

Step 10 The last step in Fedosov™s construction is to show by a recursive con-

struction, similar to the one above, that each element of C ∞ (M )[[ ]]

is the harmonic part of a unique parallel section of F W (T M ), so that

C ∞ (M )[[ ]] is identi¬ed with the space of parallel sections and thus

inherits from it an algebra structure, which is easily shown to be a de-

formation quantization associated with the symplectic structure ω.

21.5 Classi¬cation of Deformation Quantizations

Fedosov [59] showed that his iterative construction of a connection on F W (T M )

j

—ωj , for any sequence of closed

can be modi¬ed so that the curvature becomes

2-forms ωj such that ω0 is the original symplectic structure ω. He also showed that

the isomorphism class of the resulting -product depends precisely on the sequence

of de Rham cohomology classes [ωj ] ∈ H 2 (M, R) and in particular is independent

of the initial choice of connection φ0 .

In summary, the relevant data for an equivalence class of deformation quanti-

zations on a manifold M is

ω , [ω1 ] , [ω2 ] , . . .

A representative of such an equivalence class is called a Fedosov quantization of

M.

162 21 DEFORMATION QUANTIZATION

This left open the question of whether every -product is isomorphic to one ob-

tained by Fedosov™s construction. A positive answer to this question has been given

by Nest and Tsygan. Using a noncommutative version of Gel™fand-Fuks cohomol-

ogy, they construct in [124] for each deformation quantization a characteristic class

in H 2 (M, R)[[ ]] with constant term ω. In [125], they show that this class deter-

mines the -product up to isomorphism and that it agrees with Fedosov™s curvature

for the -products constructed by his method. By Moser™s classi¬cation [121] of

nearby symplectic structures by their cohomology classes, the isomorphism classes

of -products on a symplectic manifold are thus in one-to-one correspondence with

isomorphism classes of formal deformations of the symplectic structure. Other ref-

erences concerning this classi¬cation are Bertelson-Cahen-Gutt [15] Kontsevich [97],

and Weinstein-Xu [173].

One consequence of this classi¬cation is that there is (up to isomorphism) a

unique deformation quantization whose characteristic class is independent of .

Although one might think that this special quantization is somehow the natural

one, there is considerable evidence that the others are important as well. For

instance, [54] suggests that -products with nonconstant characteristic classes may

be related to geometric phases and deformations of symplectic forms which arise in

the analysis of coupled wave equations [107].

References

[1] Abraham, R., Marsden, J., and Ratiu, T., Manifolds, Tensor Analysis, and

Applications, second edition, Applied Mathematical Sciences 75, Springer-

Verlag, New York-Berlin, 1988.

[2] Albert, C., and Dazord, P., Groupo¨ ±des de Lie et groupo¨

±des symplectiques,

Symplectic Geometry, Groupoids, and Integrable Systems, S´minaire Sud-

e

Rhodanien de G´om´trie ` Berkeley (1989), P. Dazord and A. Weinstein,

ee a

eds., Springer-MSRI Series (1991), 1-11.

[3] Almeida, R., and Molino, P., Suites d™Atiyah, feuilletages et quanti¬cation

g´om´trique, Universit´ des Sciences et Techniques de Languedoc, Montpel-

ee e

lier, S´minaire de g´om´trie di¬erentielle (1984), 39-59.

e ee

[4] Almeida, R., and Molino, P., Suites d™Atiyah et feuilletages transversalement

complets, C. R. Acad. Sci. Paris 300 (1985), 13-15.

[5] Alvarez-Sanchez, G., Geometric methods of classical mechanics applied to

control theory, Ph.D. thesis, University of California at Berkeley, 1986.

[6] Arnold, V., Mathematical Methods of Classical Mechanics, Graduate Texts in

Math. 60, Springer-Verlag, New York, 1978.

[7] Arveson, W., An Invitation to C — -algebras, Graduate Texts in Mathematics

39, Springer-Verlag, New York-Heidelberg, 1976.

[8] Astashkevich, A., Fedosov™s quantization of semisimple coadjoint orbits,

Ph.D. thesis, M.I.T., 1996.

[9] Atiyah, M., Complex analytic connections in ¬bre bundles, Trans. Amer.

Math. Soc. 85 (1957), 181-207.

[10] Baer, A., Zur Einf¨hrung des Scharbegri¬s, J. Reine Angew. Math. 160

u

(1929), 199-207.

[11] Bates, S., and Weinstein, A., Lectures on the Geometry of Quantization,

Berkeley Mathematical Lecture Notes 8, Amer. Math. Soc., Providence, 1997.

[12] Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sternheimer, D.,

Deformation theory and quantization, I and II, Ann. Phys. 111 (1977), 61-

151.

[13] Berezin, F., Some remarks about the associated envelope of a Lie algebra,

Funct. Anal. Appl. 1 (1967), 91-102.

[14] Bertelson, M., Existence of star products, a brief history, survey article

for the Berkeley Math 277 course taught by A. Weinstein, available at

http://math.berkeley.edu/∼alanw/ (1997).

[15] Bertelson, M., Cahen, M., and Gutt, S., Equivalence of star products, Class.

Quan. Grav. 14 (1997), A93-A107.

[16] Bkouche, R., Id´aux mous d™un anneau commutatif, Applications aux an-

e

neaux de fonctions, C. R. Acad. Sci. Paris 260 (1965), 6496-6498.

163

164 REFERENCES

¨

[17] Brandt, W., Uber eine Verallgemeinerung des Gruppenbegri¬es, Math. Ann.

96 (1926), 360-366.

[18] Br¨cker, T., and tom Dieck, T., Representations of Compact Lie Groups,

o

corrected reprint of the 1985 translation from German, Graduate Texts in

Mathematics 98, Springer-Verlag, New York, 1995.

[19] Brown, R., From groups to groupoids: a brief survey, Bull. London Math.

Soc. 19 (1987), 113-134.

[20] Brown, R., Topology: a Geometric Account of General Topology, Homotopy

Types, and the Fundamental Groupoid, Halsted Press, New York, 1988.

[21] Cahen, M., Gutt, S., and De Wilde, M., Local cohomology of the algebra of

C ∞ functions on a connected manifold, Lett. Math. Phys. 4 (1980), 157-167.

[22] Cahen, M., Gutt, S., and Rawnsley, J., Tangential star products for the

coadjoint Poisson structure, Comm. Math. Phys. 180 (1996), 99-108.

´

[23] Cartan, E., La troisi`me th´or`me fondamental de Lie, C. R. Acad. Sci. Paris

e ee

190 (1930), 914-916, 1005-1007.

[24] Certaine, J., The ternary operation (abc) = ab’1 c of a group, Bull. Amer.

Math. Soc. 49 (1943), 869-877.

[25] Chari, V., and Pressley, A., A Guide to Quantum Groups, Cambridge Uni-

versity Press, Cambridge, 1994.

[26] Coll, V., Gerstenhaber, M., and Schack, S., Universal deformation formulas

and breaking symmetry, J. Pure Appl. Algebra 90 (1993), 201-219.

[27] Conn, J., Normal forms for analytic Poisson structures, Ann. of Math. 119

(1984), 576-601.

[28] Conn, J., Normal forms for smooth Poisson structures, Ann. of Math. 121

(1985), 565-593.

[29] Connes, A., A survey of foliations and operator algebras, Operator Algebras

and Applications, R. Kadison, ed., Proc. Symp. Pure Math. 38, Amer. Math.

Soc., Providence (1982), 521-628.

[30] Connes, A., Cyclic cohomology and noncommutative di¬erential geometry,

Proc. ICM, Berkeley 2 (1986), 879-889.

[31] Connes, A., G´om´trie non Commutative, InterEditions, Paris, 1990.

ee

[32] Connes, A., Noncommutative Geometry, Academic Press, San Diego, 1994.

[33] Coppersmith, D., A family of Lie algebras not extendible to a family of Lie

groups, Proc. Amer. Math. Soc. 66 (1977), 365-366.

[34] Coste, A., Dazord, P., and Weinstein, A., Groupo¨

±des symplectiques, Publi-

cations du D´partement de Math´matiques, Universit´ Claude Bernard-Lyon

e e e

I 2A (1987), 1-62.

[35] Courant, T., Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990), 631-661.

REFERENCES 165

[36] Davidson, K., C — -algebras by Example, Fields Institute Monographs 6, Amer-

ican Mathematical Society, Providence, 1996.

[37] Dazord, P., Holonomie des feuilletages singuliers, C. R. Acad. Sci. Paris 298

(1984), 27-30.

[38] Dazord, P., Feuilletages ` singularit´s, Nederl. Akad. Wetensch. Indag. Math.

a e

47 (1985), 21-39.

[39] Dazord, P., Obstruction ` un troisi`me th´or`me de Lie non lin´aire pour

a e ee e

certaines vari´t´s de Poisson, C. R. Acad. Sci. Paris S´r. I Math. 306 (1988),

ee e

273-278

[40] Dazord, P., Groupo¨ d™holonomie et g´om´trie globale, C. R. Acad. Sci.

±de ee

Paris S´r. I Math. 324 (1997), 77-80.

e

[41] Dazord, P., and Weinstein, A., eds., Symplectic Geometry, Groupoids, and In-

tegrable Systems, S´minaire Sud-Rhodanien de G´om´trie ` Berkeley (1989),

e ee a

Springer-MSRI Series, 1991.

[42] de Rham, G., Di¬erentiable Manifolds. Forms, Currents, Harmonic Forms,

Grundlehren der Mathematischen Wissenschaften 266, Springer-Verlag,

Berlin-New York, 1984.

[43] De Wilde, M., and Lecomte, P., Existence of star-products and of formal

deformations of the Poisson Lie algebra of arbitrary symplectic manifolds,

Lett. Math. Phys. 7 (1983), 487-496.

[44] Dirac, P., The Principles of Quantum Mechanics, Clarenden Press, Oxford,

1930.

[45] Dixmier, J., C — -algebras, translated from the French by F. Jellett,

North-Holland Mathematical Library 15, North-Holland Publishing Co.,

Amsterdam-New York-Oxford, 1977.

[46] Dixmier, J., Enveloping Algebras, North-Holland Math. Library 14, North-

Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[47] Dixmier, J., Von Neumann Algebras, with a preface by E. Lance, translated

from the second French edition by F. Jellett, North-Holland Mathematical

Library 27, North-Holland Publishing Co., Amsterdam-New York, 1981.

[48] Douady, A., and Lazard, M., Espaces ¬br´s en alg`bres de Lie et en groupes,

e e

Invent. Math. 1 (1966), 133-151.

[49] Dufour, J.-P., Quadratisation de structures de Poisson ` partie quadra-

a

tique diagonale, S´minaire Gaston Darboux de G´om´trie et Topologie

e ee

Di¬´rentielle, 1992-1993 (Montpellier), iii, 10-13, Univ. Montpellier II, Mont-

e

pellier, 1994.

[50] Dufour, J.-P., and Haraki, A., Rotationnels et structures de Poisson quadra-

tiques, C.R.A.S. Paris 312 (1991), 137-140.

[51] Egilsson, A., On embedding the 1:1:2 resonance space in a Poisson manifold,

Electronic Research Announcements of the Amer. Math. Soc. 1 (1995), 48-56.

166 REFERENCES

[52] Ehresmann, C., Les connexions in¬nit´simales dans un espace ¬br´

e e

di¬´rentiable, Colloque de Topologie (Espaces Fibr´s) tenu a Bruxelles du

e e

5 au 8 juin, 1950, Georges Thone, Li`ge and Masson et Cie., Paris (1951),

e

29-55.

[53] Ehresmann, C., Oeuvres compl`tes et comment´es, A. Ehresmann, ed., Suppl.

e e

Cahiers Top. G´om. Di¬., Amiens, 1980-1984.

e

[54] Emmrich, C., and Weinstein, A., Geometry of the transport equation in mul-

ticomponent WKB approximations, Comm. Math. Phys. 176 (1996), 701-711.

[55] Emmrich, C., and Weinstein, A., The di¬erential geometry of Fedosov™s quan-

tization, Lie Theory and Geometry, 217-239, Progr. Math., 123, Birkh¨user

a

Boston, Boston, 1994.

[56] Etingof, P., and Kazhdan, D., Quantization of Lie bialgebras, I, Selecta Math.

(N.S.) 2 (1996), 1-41.

[57] Fedosov, B., A simple geometrical construction of deformation quantization,

J. Di¬. Geom. 40 (1994), 213-238.

[58] Fedosov, B., Reduction and eigenstates in deformation quantization, Pseudo-

di¬erential Calculus and Mathematical Physics, 277-297, Mathematical Top-

ics 5, Akademie Verlag, Berlin, 1994.

[59] Fedosov, B., Deformation Quantization and Index Theory, Mathematical

Topics 9, Akademie Verlag, Berlin, 1996.

[60] Flato, M., and Sternheimer, D., Closedness of star products and cohomolo-

gies, Lie Theory and Geometry, in Honor of B. Kostant, J.-L. Brylinski and R.

Brylinski, eds., Progress in Mathematics, Birkh¨user Boston, Boston, 1994.

a

[61] Fr¨licher, A., and Nijenhuis, A., Theory of vector-valued di¬erential forms, I,

o

Derivations of the graded ring of di¬erential forms, Nederl. Akad. Wetensch.

Proc. Ser. A 59 = Indag. Math. 18 (1956), 338-359.

[62] Fuchssteiner, B., The Lie algebra structure of degenerate Hamiltonian and

bi-Hamiltonian systems, Progr. Theoret. Phys. 68 (1982), 1082-1104.

[63] Gel™fand, I., Normierte Ringe, Rec. Math. [Mat. Sbornik] N. S. 9 (51) (1941),

3-24.

[64] Gel™fand, I., and Naimark, M., On the imbedding of normed rings into the

ring of operators in Hilbert space, Mat. Sbornik 12 (1943), 197-213. Corrected

reprint of the 1943 original in Contemp. Math. 167, C — -algebras: 1943-1993

(San Antonio, Texas, 1993), Amer. Math. Soc., Providence (1994), 2-19.

ˇ

[65] Gel™fand, I., Ra˜

±kov, D., and Silov, G., Commutative normed rings, (Russian),

Uspehi Matem. Nauk (N. S.) 1 (1946) 2 (12), 48-146.

[66] Gerstenhaber, M., On the cohomology structure of an associative ring, Ann.

of Math., 78 (1963), 267-288.

[67] Gerstenhaber, M., On the deformations of rings and algebras, Ann. of Math.

79 (1964), 59-103.

REFERENCES 167

[68] Gerstenhaber, M., On the deformation of rings and algebras: II, Annals of

Math., 84 (1966), 1-19.

[69] Gerstenhaber, M., and Schack, S., Algebraic cohomology and deformation

theory, Deformation Theory of Algebras and Structures and Applications (pa-

pers from the NATO Advanced Study Institute held in Il Ciocco, June 1-

14, 1986), M. Hazewinkel and M. Gerstenhaber, eds., NATO Advanced Sci-

ence Institutes Series C: Mathematical and Physical Sciences 247, Kluwer

Academic Publishers Group, Dordrecht, Part A “ Deformations of Algebras

(1988), 11-264.

[70] Ginzburg, V.L., Momentum mappings and Poisson cohomology, Internat. J.

Math. 7 (1996), 329-358.

[71] Ginzburg, V.L., Equivariant Poisson cohomology and spectral sequences,

preprint (1996).

[72] Ginzburg, V.L., and Lu, J.-H., Poisson cohomology of Morita-equivalent Pois-

son manifolds, Duke Math. J. 68 (1992), A199-A205.

[73] Gompf, R., A new construction of symplectic manifolds, Ann. of Math. (2)

142 (1995), 527-595.

[74] Goodman, F., de la Harpe, P., and Jones, V., Coxeter Graphs and Tow-

ers of Algebras, Mathematical Sciences Research Institute Publications 14,

Springer-Verlag, New York-Berlin, 1989.

[75] Grothendieck, A., Techniques de construction et th´or`mes d™existence en

ee

g´om´trie alg´brique III: pr´schemas quotients, S´minaire Bourbaki 13e ann´e

ee e e e e

1960/61, no. 212 (1961).

[76] Guillemin, V., and Sternberg, S., Symplectic Techniques in Physics, Cam-

bridge Univ. Press, Cambridge, 1984.

[77] Guruprasad, K., Huebschmann, J., Je¬rey, L., and Weinstein, A., Group

systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J.

89 (1997), 377-412.

[78] Gutt, S., An explicit — -product on the cotangent bundle of a Lie group, Lett.

Math. Phys 7 (1983), 249-258.

[79] Hae¬‚iger, A., Structures feuillet´es et cohomologie ` valeur dans un faisceau

e a

de groupo¨±des, Comment. Math. Helv. 32 (1958), 248-329.

[80] Haraki, A., Quadratisation de certaines structures de Poisson, J. London

Math. Soc. 56 (1997), 384-394.

[81] Hector, G., Mac´ E., and Saralegi, M., Lemme de Moser feuillet´ et classi-

±as, e

¬cation des vari´t´s de Poisson r´guli`res, Publ. Mat. 33 (1989), 423-430.

ee e e

[82] Hochschild, G., Kostant, B., and Rosenberg, A., Di¬erential forms on regular

a¬ne algebras, Trans. Amer. Math. Soc. 102 (1962), 383-408.

[83] Huebschmann, J., Poisson cohomology and quantization, J. Reine Angew.

Math. 408 (1990), 57-113.

168 REFERENCES

[84] Huebschmann, J., Lie-Rinehart algebras, Gerstenhaber algebras, and Batalin-

Vilkovisky algebras, Ann. Inst. Fourier (Grenoble) 48 (1998), 425-440.

[85] Huebschmann, J., Extensions of Lie-Rinehart algebras and the Chern-Weil

construction, preprint (1998), to appear in Festschrift for J. Stashe¬.

[86] Jacobi, C., Gesammelte Werke, Herausgegeben auf Veranlassung der

K¨niglich Preussischen Akademie der Wissenschaften, Zweite Ausgabe,

o

Chelsea Publishing Co., New York, 1969.

[87] Jacobowitz, H., An Introduction to CR Structures, Mathematical Surveys and

Monographs 32, Amer. Math. Soc., Providence, 1990.

[88] Jimbo, M., A q-di¬erence analogue of U (g) and the Yang-Baxter equation,

Lett. Math. Phys. 10 (1985), 63-69.

[89] Karasev, M., Analogues of objects of the theory of Lie groups for nonlinear

Poisson brackets, Math. USSR Izvestiya 28 (1987), 497-527.

[90] Karasev, M., ed., Coherent Transform, Quantization, and Poisson Geometry,

Moscow Institute of Electronics and Mathematics, Amer. Math. Soc., 1998.

[91] Karasev, M., and Maslov, V., Pseudodi¬erential operators and the canonical

operator in general symplectic manifolds, Math. USSR Izvestia 23 (1984),

277-305 (translation of Izv. Akad. Nauk. SSSR Ser. Mat. 47 (1983), 999-

1029).

[92] Karasev, M., and Maslov, V., Nonlinear Poisson Brackets: Geometry and

Quantization, Translations of Mathematical Monographs 119, Amer. Math.

Soc., Providence, 1993.

[93] Karshon, Y., and Lerman, E., The centralizer of invariant functions and di-

vision properties of the moment map, Ill. J. Math. 41 (1997), 462-487.

[94] Keel, S., and Mori, S., Quotients by groupoids, Ann. of Math. (2) 145 (1997),

193-213.

[95] Kirillov, A., Local Lie algebras, Russian Math. Surveys 31 (1976), 55-75.

[96] Kogan, A., Drinfeld-Sokolov reduction and W-algebras, survey article

for the Berkeley Math 277 course taught by A. Weinstein, available at

http://math.berkeley.edu/∼alanw/ (1997).

[97] Kontsevich, M., Deformation quantization of Poisson manifolds, I, preprint

(1997), q-alg/9709040.

[98] Kosmann-Schwarzbach, Y., Exact Gerstenhaber algebras and Lie bialge-

broids, Acta Appl. Math. 41 (1995), 153-165.

[99] Kostant, B., Quantization and unitary representations, Lecture Notes in

Math. 170 (1970), 87-208.

[100] Lako¬, G., and Nu˜ez, R., The metaphorical structure of mathematics:

n

sketching out cognitive foundations for a mind-based mathematics, Math-

ematical Reasoning, Analogies, Metaphors, and Images, L. English, ed.,

Lawrence Erlbaum Associates, Mahwah, New Jersey, 1997, pp. 21-89.

REFERENCES 169

[101] Landsman, N., Mathematical Topics between Classical and Quantum Mechan-

ics, Springer-Verlag, to appear.

[102] Lasso de la Vega, M., Groupo¨ fondamental et d™holonomie de certains

±de

feuilletages r´guliers, Publicacions Matematiques 33 (1989), 431-443.

e

[103] Lerman, E., On the centralizer of invariant functions on a Hamiltonian G-

space, J. Di¬. Geom. 30 (1989), 805-815.

[104] Libermann, P., Probl`mes d™´quivalence et g´om´trie symplectique, IIIe ren-

e e ee

contre de g´om´trie du Schnepfenried, 10-15 mai 1982, I. Ast´risque 107-108

ee e

(1983), 43-68.

[105] Lichnerowicz, A., Les vari´t´s de Poisson et leurs alg`bres de Lie associ´es,

ee e e

J. Di¬erential Geometry 12 (1977), 253-300.

[106] Lie, S., Theorie der Transformationsgruppen, (Zweiter Abschnitt, unter

Mitwirkung von Prof. Dr. Friedrich Engel), Leipzig, Teubner, 1890.

[107] Littlejohn, R., and Flynn, W., Geometric phases in the asymptotic theory of

coupled wave equations, Phys. Rev. A44 (1991), 5239-5256.

[108] Loewy, A., Neue elementare Begr¨ndung und Erweiterung der Galoisschen

u

Theorie, S.-B. Heidelberger Akad. Wiss. Math. Nat. Kl. 1925 (1927), Abh.

7.

[109] Lu, J.-H., Moment maps at the quantum level, Comm. Math. Phys. 157

(1993), 389-404.

[110] Mackenzie, K., Lie Groupoids and Lie Algebroids in Di¬erential Geometry,

London Math. Soc. Lecture Notes Series 124, Cambridge Univ. Press, 1987.

[111] Mackenzie, K., and Xu, P., Lie bialgebroids and Poisson groupoids, Duke

Math. J. 73 (1994), 415-452.

[112] Mackey, G., The Mathematical Foundations of Quantum Mechanics, W. A.

Benjamin, New York, 1963.

[113] Mackey, G., Ergodic theory and virtual groups, Math. Ann. 166 (1966), 187-

207.

[114] MacLane, S., Categories for the Working Mathematician, Graduate Texts in

Mathematics 5, Springer-Verlag, New York-Berlin, 1971.

[115] Maeda, Y., Omori, H., and Weinstein, A., eds., Symplectic Geometry and

Quantization: Two Symposia on Symplectic Geometry and Quantization

Problems, July 1993, Japan, Contemporary Mathematics 179, Amer. Math.

Soc., Providence, 1994.

[116] Marsden, J., and Ratiu, T., Introduction to Mechanics and Symmetry: A

Basic Exposition of Classical Mechanical Systems, Texts in Applied Math.

17, Springer-Verlag, 1994.

[117] Melrose, R., The Atiyah-Patodi-Singer Index Theorem, Research Notes in

Mathematics 4, A K Peters, Ltd., Wellesley, 1993.

170 REFERENCES

[118] Meyer, R., Morita equivalence in algebra and geometry, survey article

for the Berkeley Math 277 course taught by A. Weinstein, available at

http://math.berkeley.edu/∼alanw/ (1997).

[119] Min-Oo, and Ruh, E., Comparison theorems for compact symmetric spaces,

´

Ann. Sci. Ecole Norm. Sup. 12 (1979), 335-353.

[120] Moore, C., and Schochet, C., Global analysis on foliated spaces, with appen-

dices by S. Hurder, Moore, Schochet and R. J. Zimmer, Mathematical Sci-

ences Research Institute Publications, 9, Springer-Verlag, New York-Berlin,

1988.

[121] Moser, J., On the volume elements on a manifold, Trans. Amer. Math. Soc.

120 (1965), 280-296.

[122] Moyal, J., Quantum mechanics as a statistical theory, Proc. Cambridge Phi-

los. Soc. 45 (1949), 99-124.

[123] Nakanishi, N., Poisson cohomology of plane quadratic Poisson structures,

Publ. Res. Inst. Math. Sci. 33 (1997), 73-89.

[124] Nest, R., and Tsygan, B., Algebraic index theorem, Comm. Math. Phys. 172

(1995), 223-262.

[125] Nest, R., and Tsygan, B., Algebraic index theorem for families, Advances in

Math. 113 (1995), 151-205.

[126] Nest, R., and Tsygan, B., Formal deformations of symplectic Lie algebroids,

preprint (1997).

[127] von Neumann, J., Zur Algebra der Funktionaloperationen und Theorie der

Normalen Operatoren, Math. Ann. 102 (1929/30), 370-427.

[128] von Neumann, J., Die Eindeutigkeit der Schr¨dingerschen Operatoren Math.

o

Ann. 104 (1931), 570-578.

[129] von Neumann, J., On rings of operators, III, Ann. of Math. 41 (1940), 94-161.

[130] von Neumann, J., On rings of operators, reduction theory, Ann. of Math. (2)

50 (1949), 401-485.

[131] Newlander, A., and Nirenberg, L., Complex analytic coordinates in almost

complex manifolds, Ann. of Math. (2) 65 (1957), 391-404.

[132] Novikov, S., The topology of foliations, (Russian), Trudy Moskov. Mat. Obˇˇ

sc

14 (1965), 248-278.

[133] Omori, H., Maeda, Y., and Yoshioka, A., Weyl manifolds and deformation

quantization, Advances in Math. 85 (1991), 224-255.

[134] Palais, R., A Global Formulation of the Lie Theory of Transformation Groups,

Mem. Amer. Math. Soc. 22, Providence, 1957.

[135] Paterson, A., Groupoids, Inverse Semigroups and Their Operator Algebras,

Birkh¨user, Boston, to appear.

a

REFERENCES 171

[136] Etingof, P., and Varchenko, A., Geometry and classi¬cation of solutions of the

classical dynamical Yang-Baxter equation, Comm. Math. Phys. 192 (1998),

77-120.

[137] Phillips, J., The holonomic imperative and the homotopy groupoid of a foli-

ated manifold, Rocky Mountain J. Math. 17 (1987), 151-165.

[138] Poisson, S.-D., Sur la variation des constantes arbitraires dans les questions

´

de m´canique, J. Ecole Polytechnique 8 (1809), 266-344.

e

[139] Pradines, J., Troisi`me th´or`me de Lie sur les groupo¨

e ee ±des di¬´rentiables, C.

e

R. Acad. Sci. Paris, 267 (1968), 21 - 23.

[140] Pukanszky, L., Symplectic structure on generalized orbits of solvable Lie

groups, J. Reine Angew. Math. 347 (1984), 33-68.

[141] Pukanszky, L., Quantization and Hamiltonian G-foliations, Trans. Amer.

Math. Soc. 295 (1986), 811-847.

[142] Reeb, G., Sur certaines propri´t´s topologiques des vari´t´s feuillet´es, Publ.

ee ee e

Inst. Math. Univ. Strasbourg 11, 5-89, 155-156, Actualit´s Sci. Ind. 1183,

e

Hermann & Cie., Paris, 1952.

[143] Renault, J., A Groupoid Approach to C — Algebras, Lecture Notes in Math.

793, Springer-Verlag, 1980.

[144] Rie¬el, M., Deformation quantization of Heisenberg manifolds, Comm. Math.

Phys. 122 (1989), 531-562.

[145] Rinehart, G., Di¬erential forms on general commutative algebras, Trans.

Amer. Math. Soc. 108 (1963), 195-222.

[146] Rudin, W., Functional Analysis, second edition, International Series in Pure

and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.

[147] Ruh, E., Cartan connections, Proc. Symp. Pure Math. 54 (1993), 505-519.

[148] Schwartz, L., Th´orie des Distributions, Publications de l™Institut de

e

Math´matique de l™Universit´ de Strasbourg, Hermann, Paris, 1966.

e e

[149] Segal, I., and Kunze, R., Integrals and Operators, second revised and enlarged

edition, Grundlehren der Mathematischen Wissenschaften 228, Springer-

Verlag, Berlin-New York, 1978.

[150] Serre, J.-P., Lie Algebras and Lie Groups, Lecture Notes in Math. 1500,

Springer-Verlag, 1992.

[151] Shlyakhtenko, D., Von Neumann algebras and Poisson manifolds, survey ar-

ticle for the Berkeley Math 277 course taught by A. Weinstein, available at

http://math.berkeley.edu/∼alanw/ (1997).

[152] Smale, S., Topology and mechanics I & II, Invent. Math. 10 (1970), 305-331

& 11 (1970), 45-64.

[153] Souriau, J.-M., Structure des Syst`mes Dynamiques, Dunod, Paris, 1970.

e

172 REFERENCES

[154] Souriau, J.-M., Groupes di¬´rentiels, Lecture Notes in Math. 836 (1980),

e

91-128.

[155] Sternberg, S., Celestial Mechanics Part II, W.A. Benjamin, New York, 1969.

[156] Str˜til˜, S., and Zsid´, L., Lectures on von Neumann Algebras, revision of

aa¸ o

the 1975 original, translated from the Romanian by S. Teleman, Editura

Academiei, Bucharest, Abacus Press, Tunbridge Wells, 1979.

[157] Takesaki, M., Theory of Operator Algebras, I, Springer-Verlag, New York-

Heidelberg, 1979.

[158] Thirring, W., A Course in Mathematical Physics, Vol. 3, Quantum Mechan-

ics of Atoms and Molecules, translated from the German by E. M. Harrell,

Lecture Notes in Physics 141, Springer-Verlag, New York-Vienna, 1981.

[159] Tr`ves, F., Topological Vector Spaces, Distributions and Kernels, Academic

e

Press, New York-London, 1967.

[160] Tsujishita, T., On variation bicomplexes associated to di¬erential equations,

Osaka J. Math. 19 (1982), 311-363.

[161] Vaintrob, A., Lie algebroids and homological vector ¬elds, Russ. Math. Surv.

52 (1997), 428-429.

[162] Vaisman, I., Lectures on the Geometry of Poisson Manifolds, Birkh¨user,

a

Basel, 1994.

[163] Weinstein, A., The local structure of Poisson manifolds, J. Di¬. Geom. 18

(1983), 523-557.

[164] Weinstein, A., Sophus Lie and symplectic geometry, Expos. Math. 1 (1983),

95-96.

[165] Weinstein, A., Poisson structures and Lie algebras, Ast´risque, hors s´rie

e e

(1985), 421-434.

[166] Weinstein, A., Poisson geometry of the principal series and nonlinearizable

structures, J. Di¬. Geom. 25 (1987), 55-73.

[167] Weinstein, A., Symplectic groupoids and Poisson manifolds, Bull. Amer.

Math. Soc. 16 (1987), 101-104.

[168] Weinstein, A., Coisotropic calculus and Poisson groupoids, J. Math. Soc.

Japan 40 (1988), 705-727.

[169] Weinstein, A., A¬ne Poisson structures, Intern. J. Math. 1 (1990), 343-360.

[170] Weinstein, A., Deformation quantization, S´minaire Bourbaki (1993-94),

e

Ast´risque 789 (1995), 389-409.

e

[171] Weinstein, A., Groupoids: unifying internal and external symmetry, a tour

through some examples, Notices Amer. Math. Soc. 43 (1996), 744-752.

[172] Weinstein, A., The modular automorphism group of a Poisson manifold, J.

Geom. Phys. 23 (1997), 379-394.

REFERENCES 173

[173] Weinstein, A., and Xu, P., Hochschild cohomology and characteristic classes

for star-products, Geometry of Di¬erential Equations, A. Khovanskii, A.

Varchenko, V. Vassiliev, eds., Amer. Math. Soc., Providence, 1997, pp. 177-

194.

[174] Weyl, H., The Theory of Groups and Quantum Mechanics, Dover, New York,

1931.

[175] Winkelnkemper, H., The graph of a foliation, Ann. Global Analysis and Ge-

ometry 1:3 (1983), 51-75.

[176] Xu, P., Morita Equivalence of Symplectic Groupoids and Poisson Manifolds,

Ph.D. thesis, University of California at Berkeley, 1990.

[177] Xu, P., Morita equivalence of Poisson manifolds, Comm. Math. Phys. 142

(1991), 493-509.

[178] Xu, P., Fedosov *-products and quantum momentum maps, Comm. Math.

Phys. 197 (1998), 167-197.

[179] Xu, P., Gerstenhaber algebras and BV-algebras in Poisson geometry, preprint

(1997), to appear in Comm. Math. Phys., dg-ga/9703001.

[180] Yakimov, M., Formal DGLA™s and deformation quantization, survey arti-

cle for the Berkeley Math 277 course taught by A. Weinstein, available at

http://math.berkeley.edu/∼alanw/ (1997).

[181] Zakrzewski, S., Quantum and classical pseudogroups, I and II, Comm. Math.

Phys. 134 (1990), 347-370, 371-395.

Index

action Baer groupoid, 91

Baer, A., 91

coadjoint, 39

Berezin, F. A., 81, 82

e¬ective, 8

bisection, 106, 107

free, 8

bivector ¬eld, 12

groupoid, 90, 101

Borel groupoid, 93

groupoid algebra action, 103

Borel, E., 152

hamiltonian, 39

bracket

linear action, 102

cobracket, 132

of a groupoid bisection, 107

commutator, 150

of a Lie algebra, 8

Dirac™s quantum Poisson, 152, 157

right, 8

E-Gerstenhaber bracket, 133

adjoint

Gerstenhaber, 141

operation, 47

Gerstenhaber bracket on Hochschild

representation of C ∞ (M ), 14

cohomology, 143

admissible section, 106

Lie-Poisson, 11

Albert, C., 108

on alternating multilinear maps,

algebra

142

center, 50

Poisson, 149

dual pair, 50

properties of [·, ·]E , 133

factor, 50

Schouten-Nijenhuis, 12, 135, 144

von Neumann, 49

Brandt groupoid, 87

Weyl, 149

Brandt, W., 87, 89

Almeida, R., 118

bundle

almost

of groups, 93

commutativity of the universal en-

of Lie algebras, 114, 116

veloping algebra, 5

of Lie groups, 116

complex structure, 120

Lie algebra, 7 canonical

Poisson structure, 12 coordinate, 14

symplectic manifold, 20 1-form, 36

±-density, 77 Poisson relation, 14

anchor map symplectic structure on a cotan-

de¬nition, 113 gent bundle, 36, 119

image, 113 Cartan™s magic formula, 21, 126, 159

injective, 115 ´

Cartan, E., 9

kernel, 113 Casimir function, 14, 16, 136

surjective, 123 Cauchy-Riemann structure, 121

antipode, 69 center, 50

Arnold, V., 26 Chevalley cohomology, 132, 142

associativity Chevalley complex, 142

associative structure, 142 Chevalley-Eilenberg cohomology, 142

coassociativity, 70 classical observables, 151

of the cup product, 142 classical Yang-Baxter equation, 135

Atiyah algebra, 123 co-commutativity of the coproduct, 82

Atiyah sequence, 123 co-unit or coidentity, 69

Atiyah, M., 121, 123 coadjoint

175

176 INDEX

action, 39 form, 124

orbit, 39 iterative construction, 156, 160

coarse groupoid, 87 on a Lie algebroid, 124

coboundary, 43 on a transitive Lie algebroid, 124

cobracket, 132 torsionless ¬‚at Poisson, 155

cochain, 43 Connes, A., 89

cocycle, 43 conormal space, 25

cohomology convolution

Chevalley, 132, 142 of functions, 75

Chevalley-Eilenberg, 142 of measures, 73, 98

de Rham, 23 coproduct

E-cohomology, 132 co-commutativity, 82

E-Π-cohomology, 136 de¬nition, 69

Gel™fand-Fuks, 162 CR-functions, 121

Harrison, 142 CR-structure, 121

C — -algebra

Hochschild, 142

Lie algebra, 132, 142 de¬nition, 47

Lie algebroid, 132, 136 groupoid, 98

Poisson, 16 cup product

Poisson cohomology on a Lie al- associativity, 142

gebroid, 136 of multilinear maps, 142

squaring map, 139 on Hochschild cohomology, 143

coisotropic, 34 supercommutativity, 143

collective function, 66 curvature

commutant e¬ective, 160

de¬nition, 49 form, 124

double, 50 Lie algebroid, 124

double commutant theorem, 50 Weyl, 160

Poisson geometry, 51

Darboux™s theorem, 20, 21

commutative Hopf algebra, 72

Dazord, P., 108, 117

compact operator, 48

deformation

compatible equivalence relation, 34

Lie algebra, 2

complete

obstructions, 143

Poisson map, 31

of products, 144

symplectically complete foliation,

of products of functions, 144, 146

53

quantization, 6, 144, 146, 155

complex

quantization of R2n , 151

coordinates in symplectic geom-

theory, 141

etry, 62

degenerate Lie algebra, 26

Lie algebroid, 120

degree

structure, 120

multilinear map, 141

complexes

of [·, ·]E , 133

Chevalley, 142

of dE , 131

Hochschild, 142

of an E-di¬erential form, 131

complexi¬ed tangent bundle, 62