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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-deп¬Ѓned product on various
spaces of functions, including Schwartz functions, smooth functions whose par-
tial derivatives of all orders are bounded, and О›-periodic smooth functions on E
152 20 WEYL ALGEBRAS

where О› is a lattice, i.e. smooth functions on a torus E/О›. This product does not
extend to continuous functions on E/О›, but it is possible to complete C в€ћ (E/О›)
to a noncommutative C в€— algebra called вЂњthe continuous functions on a quantum
в™¦
torusвЂќ .

20.3 Aп¬ѓne Invariance of the Weyl Product

The Weyl product on a Poisson vector space (E, О ) is invariant under aп¬ѓne Poisson
maps, i.e. if A : E в†’ E is an aп¬ѓne symplectic map, then the induced pull-back
map
Aв€— : C в€ћ (E)[[ ]] в€’в†’ C в€ћ (E)[[ ]]
is an algebra automorphism for the Weyl product.
By aп¬ѓne 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
aп¬ѓne coordinate changes. This condition on E amounts to the existence of a
п¬‚at connection without torsion, for which parallel transport preserves the Poisson
structure.
The inп¬Ѓnitesimal counterpart of aп¬ѓne 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, {В·, 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. DiracвЂ™s  quantum Poisson bracket

gв€’g
f f
[f, g] :=
i
satisп¬Ѓes the derivation law

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

just as a consequence of associativity. The similar property for {В·, k} is explained
by the fact that, for a polynomial k on E of degree в‰¤ 2, we have [В·, k] = {В·, k}.
In particular, for k1 and k2 polynomials of degree в‰¤ 2, we have [k1 , k2 ] =
{k1 , k2 }, which shows that polynomials of degree в‰¤ 2 form a Lie algebra. в™¦

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 в€ћ (E) modulo the functions which vanish
to inп¬Ѓnite order at 0.
20.5 Weyl Algebra Bundles 153

Theorem 20.1 Suppose that О  is a non-degenerate Poisson structure on E. Then
every derivation D of F (E)[[ ]] such that D = 0 is of the form [В·, f ] for some
f в€€ F (E)[[ ]].

Exercise 78
Prove this theorem. Hints:
A derivation D is determined by its eп¬Ђect on generators of the algebra
q1 , . . . , qn , p1 , . . . , pn . Notice that qi , pi have degree в‰¤ 2. Suppose that
D = [В·, f ] were a inner derivation. Then
в€‚f
{qi , f }
Dqi = [qi , f ] = = в€‚pi
в€‚f
{pi , f } в€’ в€‚q
Dpi = [pi , f ] = =
i

To п¬Ѓnd the element f , we must solve
df = (Dqi )dpi в€’ (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 в€ћ (E)[[ ]]. Let us check that the right-
hand side is closed:
в€‚ в€‚
в€’{pj , Dqi } + {qi , Dpj }
(Dqi ) + (Dpj ) =
в€‚qj в€‚pi
= [Dqi , pj ] + [qi , Dpj ]
= D[qi , pj ] = D(Оґi,j ) = 0 .
To п¬Ѓnish the proof that D = [В·, f ] , consider the п¬Ѓltration of F (E) by ideals
Ak generated by the homogeneous polynomials of degree k. Show that, if D is
a derivation, then DAk вЉ† Akв€’1 [[ ]].

Let (E, О ) be a Poisson vector space, and let П• be an automorphism of the Weyl
algebra C в€ћ (E)[[ ]] as a C[[ ]]-algebra..
The term in П• of 0-th order in shows that П• induces an automorphism of
в€ћ
C (E), hence a diп¬Ђeomorphism of E.
The term in П• of п¬Ѓrst order in shows that this diп¬Ђeomorphism is a Poisson
automorphism of (E, О ).
We hence obtain an exact sequence
E Aut(C в€ћ (E)[[ ]]) E P(E, О )
EI
1 E1

where P(E, О ) is the set of Poisson automorphisms of (E, О ). The kernel I of the
third arrow is the group of inner automorphisms of C в€ћ (E)[[ ]] corresponding to
invertible elements of C в€ћ (E)[[ ]] .

20.5 Weyl Algebra Bundles

Let (E, ПЃ, [В·, В·]E ) be a Lie algebroid over a manifold M , with symplectic structure П‰ в€€
О“(в€§2 E в€— ). The symplectic E-2-form П‰ is non-degenerate and dE П‰ = 0; it determines
an E-Poisson structure О  (see Section 18.3) by О  = П‰ в€’1 , and an ordinary Poisson
structure ПЃ(О ) on T M .
Let W E be the Weyl algebra bundle over M whose п¬Ѓber at x в€€ 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 coeп¬ѓcient of each term is a smooth function
on M ; they form an algebra under п¬Ѓberwise multiplication. We think of О“(W E)
as вЂњfunctions on the quantized EвЂќ. Locally, we write a typical section as f (x, y, ),
where x в€€ 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 п¬Ѓber, just as PlanckвЂ™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 .
Interpret О“(W T M ) as the space of smooth functions on the вЂњquantized tangent
bundleвЂќ
О“(W T M ) = C в€ћ (Q T M ) .
The zero section is the map

C в€ћ (Q T M ) E C в€ћ (M )[[ ]]

given by evaluation at y = 0. We may think of Q T M as an inп¬Ѓnitesimal neighbor-
hood of the zero section.
In the next chapter, we will describe the quantization method of Fedosov, in
which C в€ћ (M )[[ ]] is identiп¬Ѓed with a subalgebra of О“(W T M ). The Weyl product
is then carried back to C в€ћ (M )[[ ]] to give a deformation quantization.
Geometrically, a subalgebra of О“(W T M ) annihilated by a Lie algebra of deriva-
tions corresponds to a вЂњfoliationвЂќ of Q T M . The foliation is transverse to the п¬Ѓbers
when the derivations are of the form X as X ranges over the vector п¬Ѓelds on M ,
deп¬Ѓning a п¬‚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 в€ћ (M )[[ ]]. Notice that
a п¬‚at linear connection on T M would not work: parallel sections of a п¬‚at connection
on C в€ћ (T M ) correspond to functions on a tangent п¬Ѓber, not C в€ћ (M )[[ ]] as we need.
Example. Let (M, П‰) be a symplectic vector space with coordinates x. Deп¬Ѓne
the connection by
в€‚ в€‚
в€’
= ,
в€‚
в€‚xi в€‚yi
в€‚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
в™¦
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 О  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 FedosovвЂ™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 aп¬ѓne automorphisms of V , we can construct a global
quantization of any Poisson manifold (M, О ) covered by canonical coordinate sys-
tems in the general case for which the transition maps are aп¬ѓne. Such a covering
exists when M admits a п¬‚at torsionless linear connection for which the covariant
derivative of О  is zero.
Fedosov overcomes the diп¬ѓculty of patching together local Weyl structures by
making the canonical coordinate neighborhoods вЂњinп¬Ѓnitely smallвЂќ. To understand
his idea, we should п¬Ѓrst think of elements of the deformed algebra C в€ћ (T M )[[ ]] as
sections of the bundle W T M over M whose п¬Ѓber at x в€€ M is W Tx M .
Of course we are most interested in dealing with the case where (M, О ) does
not admit a п¬‚at Poisson connection, and this is where the most interesting part of
FedosovвЂ™s proof comes in. He shows (in other terms) that the tangent bundle of
every symplectic (or regular Poisson) manifold does admit a п¬‚at Poisson connection,
if one gives the appropriate extended meaning to that concept, namely admitting
вЂњnonlinear quantum mapsвЂќ as the structure group.
FedosovвЂ™s connection is constructed on the bundle W T M of Weyl algebras.
The вЂњstructure Lie algebraвЂќ 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 ), it is not the full Weyl algebra of
R which serves as the typical п¬Ѓber, 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 inп¬Ѓnitesimal 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 п¬Ѓxed. In fact, the eп¬Ђect is to force us to forget the group and to work only
with the structure Lie algebra. A beneп¬Ѓcial, and somewhat surprising, result of this
eп¬Ђect is that a parallel section with respect to a п¬‚at connection is not determined by
its value at a single point. This situation is very close to that in formal diп¬Ђerential
geometry, where the bundle of inп¬Ѓnite jets of functions on a manifold M has a п¬‚at

155
156 21 DEFORMATION QUANTIZATION

connection whose sections are the lifts of functions on M . (See [160, Section 1] for
в™¦
a nice exposition with references.)
Fedosov uses an iterative method for вЂњп¬‚atteningвЂќ a connection which is similar
to that used in many diп¬Ђerential geometric problems. (See  for an example,
and  for a recent survey.) Over the domain of a local trivialization of a principal
G-bundle, a connection is given by a 1-form П† with values in the Lie algebra g; the
curvature of the connection is the Lie algebra valued 2-form
1
в„¦П† = dП† + [П†, П†] .
2
If the curvature is not zero, we may try to вЂњimproveвЂќ the connection by adding
another Lie algebra valued 1-form О±. The curvature zero condition for П† + О± is the
1
dО± + [П†, О±] = в€’в„¦П† в€’ [О±, О±] .
2
Rather than trying to solve this equation exactly, we linearize it by dropping the
term в€’ 1 [О±, О±]. The operator d + [П†, В·] is the covariant exterior derivative DП† , so
2
our linearized equation has the form

DП† О± = в€’в„¦П† .

From the Bianchi identity, DП† в„¦П† = 0, it appears that the obstruction to solving
the equation above for О± lies in a cohomology space. This is not quite correct, since
2
DП† = [в„¦П† , В·], which is not zero because the connection П† is not yet п¬‚at.
Up to now, we have essentially been following NewtonвЂ™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 diп¬Ђerential 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 в€’ 2 [О±, О±]. Such approximate solutions
are constructed by some version of the Hodge decomposition. In the diп¬Ђerential
geometric applications mentioned above, the full story involves elliptic diп¬Ђerential
operators, Sobolev spaces, and so on, but in the case at hand, it turns out that the
вЂњHodge theoryвЂќ is purely algebraic and quite trivial.

21.2 Preparing the Connection

We now start the construction of a п¬‚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 diп¬Ђerentiation operator on the dual
bundle, i.e. on the linear functions on п¬Ѓbers. In coordinates (x1 , . . . , xm )
on M :
в€‚ в€‚
= О“ijk П‰k .
в€‚
в€‚xi в€‚x в€‚x
j
21.2 Preparing the Connection 157

We introduce the coeп¬ѓcients (П‰k ) of the symplectic form to lower the
last index. For convenience, we assume that П‰k is constant (i.e. the xi вЂ™s
are Darboux coordinates).
If the connection has torsion, we can make it torsion-free by symmetriza-
tion 
О“ijk + О“jik
О“ijk .
2
Because this is a symplectic connection, symmetry in the last two indices

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 identiп¬Ѓed with
linear hamiltonian vector п¬Ѓelds on the manifold and hence with quadratic
functions. Thus the connection form can be written as
1
О“ijk yi yj вЉ— dxk ,
П†в€’1 =
2
where (y1 , . . . , ym ) is a basis of linear functions on the п¬Ѓbers correspond-
ing to the coordinates (x1 , . . . , xm ) on M .

Step 3 The symplectic connection lifts to the Weyl algebra bundle. A co-
variant diп¬Ђerentiation D on the Weyl algebra bundle is described with
respect to a local trivialization by

Du = du + П€u

for a local section u, where П€ is a 1-form with values in Der(W T M ). We
can rewrite this local expression in the form

Du = du + [П†, u] ,

where now П† is a 1-form with values in W T M itself, and [В·, В·] is (1/i )
times the commutator bracket; the bracket [В·, В·] is the quantum Poisson
bracket of Dirac deп¬Ѓned in Section 20.3. The generator П† of this
вЂњinner derivationвЂќ 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 П†в€’1 (with the yi вЂ™s now interpreted as formal
variables) as taking values in the bundle F W (T M ),

П†в€’1 в€€ О“(T в€— M вЉ— F W (T M ))

becomes the connection form for the associated connection on that bun-
dle. Even if this connection were п¬‚at, it would not be the correct one to
use for quantization, since its parallel sections would not be identiп¬Ѓable
in any reasonable way with functions on M . Instead we must use for our
п¬Ѓrst approximation

1
О“ijk yi yj ) вЉ— dxk .
П†0 = ( П‰kj 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 ), that its curvature is

dП†0 + 1 [П†0 , П†0 ]
в„¦0 = 2
в€’1 1
П‰ir вЉ— dxi в€§ dxr + Rijkl yi yj вЉ— dxk в€§ dxl
= 2 4
в€’1 вЉ— П‰ + 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 в€’1вЉ—П‰ appears even when the linear connection is п¬‚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 coeп¬ѓcients 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 п¬Ѓltration 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  deп¬Ѓned in Section 20.3. The factor (1/i )
makes the quantum bracket reduce to the classical one (rather than to zero) when
в†’ 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 )] вЉ† F Wr+sв€’2 (V ) ,
so that the adjoint action of any element of F W2 (V ) preserves the п¬Ѓltration.
We introduce the algebra

W(V ) = F W (V ) вЉ— в€§в€— (V )

whose elements may be regarded as diп¬Ђerential forms on the вЂњquantum space whose
algebra of functions is F W (V )вЂќ. The algebra W(V ) inherits a п¬Ѓltration 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 inп¬Ѓnite jets at the origin of
diп¬Ђerential 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 Оґ. Remarkably, Оґ is also a derivation for the quantized algebra structure
on W(V ).
We may describe the operator Оґ 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

в€§в€— (V ) by (dx1 , . . . , dxm ). Then W(V ) is formally generated by the elements yi вЉ— 1,
вЉ— 1, and 1 вЉ— dxi , and we have

Оґ(yi вЉ— 1) = 1 вЉ— dxi , Оґ( вЉ— 1) = 0 , and Оґ(1 вЉ— dxi ) = 0 .

Notice that Оґ decreases the Weyl algebra п¬Ѓltration degree by 1 while it increases
the exterior algebra grading by 1.
Since Оґ is essentially the de Rham operator on a contractible space, we expect
the cohomology of the complex which it deп¬Ѓnes to be trivial. Fedosov makes this
explicit by introducing the dual operator Оґ в€— of contraction with the Euler vector
п¬Ѓeld i yi вЉ— в€‚xi . More precisely, Оґ в€— maps the monomial yi1 В· В· В· yip вЉ— dxj1 в€§ В· В· В· в€§ dxjq
в€‚

to
(в€’1)kв€’1 yi1 В· В· В· yip yjk вЉ— dxj1 в€§ В· В· В· в€§ dxjk в€§ В· В· В· в€§ 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 п¬Ѓeld)
shows that, on the monomial above, we have

ОґОґ в€— + Оґ в€— Оґ = (p + q)id ,

so that if we deп¬Ѓne the operator Оґ в€’1 to be p+q Оґ в€— on the monomial above, and 0 on
1

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

u = ОґОґ в€’1 u + Оґ в€’1 Оґu + Hu ,

where the вЂњharmonicвЂќ part Hu of u is the part involving only powers of and
no yi вЂ™s or dxi вЂ™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 PoincarВґ lemma via a
e
в€’1
homotopy operator Оґ from H to the identity.
When the Poisson vector space V is symplectic, the operator Оґ has another
description. For any a в€€ F W (V ), [yi , a] = {yi , a} = j ПЂij (в€‚a/в€‚yj ). If (П‰ij ) is the
matrix of the symplectic structure, inverse to (ПЂij ), we get в€‚a/в€‚yi = [ j П‰ij yj , a],
and hence

Оґ(a вЉ— 1) = (в€‚a/в€‚yi ) вЉ— dxi = [ П‰ij yj вЉ— dxi , a вЉ— 1] .
i ij

It follows from the derivation property that a similar equation holds for any element
of W(V ); i.e. the operator Оґ is equal to the adjoint action of the element ij П‰ij yj вЉ—
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) вЉ— в€§в€— (E) .

In particular, when E is the tangent bundle of a symplectic manifold M , the op-
erator Оґ and its relatives act on the algebra of diп¬Ђerential forms on M with values
in F W (T M ). These operators are purely algebraic with respect to the variable in
M , with Оґ 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 п¬Ѓltration) sequence П†n of connections whose curvatures в„¦n tend to the central
element в€’1 вЉ— П‰. Fedosov calls this central element the Weyl curvature of the
limit connection; to simplify notation, we will write в„¦ = в„¦ + 1 вЉ— П‰ for the form
which should be zero, and we call this the eп¬Ђective curvature.

Step 6 As suggested in Section 21.1, we let

П†n+1 = П†n + О±n+1 ,

where О±n+1 is a section of W(T M ).
The corresponding curvature is

dП†n+1 + 1 [П†n+1 , П†n+1 ]
в„¦n+1 = 2
в„¦n + dО±n+1 + [П†n , О±n+1 ] + 1 [О±n+1 , О±n+1 ]
= 2
Dn О±n+1

where Dn = DП†n = d + [П†n , В·]. Instead of solving
1
Dn О±n+1 = в€’в„¦n в€’ 2 [О±n+1 , О±n+1 ] в€’ 1 вЉ— П‰ ,

we drop the quadratic term and look at the simpler equation

Dn О±n+1 = в€’в„¦n в€’ 1 вЉ— П‰ .

This would solve approximately the linearized equation for zero eп¬Ђective
curvature
Dn О±n+1 + в„¦n = 0 .

The operator Dn = DП†n will have the form d + Оґ + [cn , В·], 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 , В·], like d, is п¬Ѓltration preserving.
Since Оґ lowers the п¬Ѓltration degree by 1, the principal part of the diп¬Ђer-
ential operator Dn will actually be the algebraic operator Оґ (and not d
as it would be if we measured forms by the size of their derivatives.)
We cannot even solve
ОґО±n+1 + в„¦n = 0
exactly, because the Bianchi identity gives Dn в„¦n = 0 instead of Оґ в„¦n = 0.
(The term 1 вЉ— П‰ is killed by both operators.) Nevertheless, we deп¬Ѓne

О±n+1 = в€’Оґ в€’1 (в„¦n ) ,

and take care of the errors later.
Step 8 From the recursion relation
1
в„¦n+1 = в„¦n + Dn О±n+1 + 2 [О±n+1 , О±n+1 ] ,

we п¬Ѓnd after a straightforward calculation using the decompositions

u = ОґОґ в€’1 u + Оґ в€’1 Оґu + Hu
Dn = d + Оґ + [cn , В·] and
21.5 Classiп¬Ѓcation of Deformation Quantizations 161

that

Оґ в€’1 Оґ в„¦n
в„¦n+1 =
+Hв„¦n + dО±n+1 + [cn , О±n+1 ] + 1 [О±n+1 , О±n+1 ] .
2

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

Оґ в€’1 Dn в„¦n в€’ Оґ в€’1 dв„¦n в€’ Оґ в€’1 [cn , в„¦n ]
в„¦n+1 =
+Hв„¦n + dО±n+1 + [cn , О±n+1 ] + 1 [О±n+1 , О±n+1 ] .
2

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

в„¦n+1 = Hв„¦n в€’Оґ в€’1 dв„¦n в€’Оґ в€’1 [cn , в„¦n ]+dО±n+1 +[cn , О±n+1 ]+ 1 [О±n+1 , О±n+1 ] .
2

Suppose now that в„¦n в€€ Wr (T M ) with r в‰Ґ 1. Then Hв„¦n = 0 and
О±n+1 в€€ Wr+1 (T M ), so that cn в€€ 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 в„¦0 = R has п¬Ѓltration-degree 2, we conclude that в„¦n has degree
at least n + 2, and О±n+1 has degree at least n + 3, so the sequence П†n
converges to a connection form

П† = П†0 + О±1 + О±2 + . . .

for which the curvature is в„¦ = в€’1 вЉ— П‰. This curvature is a central
section, so the connection on F W (T M ) associated to П† by the adjoint
representation F W (T M ) is п¬‚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 FedosovвЂ™s construction is to show by a recursive con-
struction, similar to the one above, that each element of C в€ћ (M )[[ ]]
is the harmonic part of a unique parallel section of F W (T M ), so that
C в€ћ (M )[[ ]] is identiп¬Ѓed 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 П‰.

21.5 Classiп¬Ѓcation of Deformation Quantizations

Fedosov  showed that his iterative construction of a connection on F W (T M )
j
вЉ—П‰j , for any sequence of closed
can be modiп¬Ѓed so that the curvature becomes
2-forms П‰j such that П‰0 is the original symplectic structure П‰. He also showed that
the isomorphism class of the resulting -product depends precisely on the sequence
of de Rham cohomology classes [П‰j ] в€€ H 2 (M, R) and in particular is independent
of the initial choice of connection П†0 .
In summary, the relevant data for an equivalence class of deformation quanti-
zations on a manifold M is
П‰ , [П‰1 ] , [П‰2 ] , . . .
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 FedosovвЂ™s construction. A positive answer to this question has been given
by Nest and Tsygan. Using a noncommutative version of GelвЂ™fand-Fuks cohomol-
ogy, they construct in  for each deformation quantization a characteristic class
in H 2 (M, R)[[ ]] with constant term П‰. In , they show that this class deter-
mines the -product up to isomorphism and that it agrees with FedosovвЂ™s curvature
for the -products constructed by his method. By MoserвЂ™s classiп¬Ѓcation  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 classiп¬Ѓcation are Bertelson-Cahen-Gutt  Kontsevich ,
and Weinstein-Xu .
One consequence of this classiп¬Ѓcation 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,  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 .
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Index
action Baer groupoid, 91
Baer, A., 91
Berezin, F. A., 81, 82
eп¬Ђective, 8
bisection, 106, 107
free, 8
bivector п¬Ѓeld, 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
DiracвЂ™s quantum Poisson, 152, 157
right, 8
E-Gerstenhaber bracket, 133
Gerstenhaber, 141
operation, 47
Gerstenhaber bracket on Hochschild
representation of C в€ћ (M ), 14
cohomology, 143
Lie-Poisson, 11
Albert, C., 108
on alternating multilinear maps,
algebra
142
center, 50
Poisson, 149
dual pair, 50
properties of [В·, В·]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
О±-density, 77 Poisson relation, 14
anchor map symplectic structure on a cotan-
deп¬Ѓnition, 113 gent bundle, 36, 119
image, 113 CartanвЂ™s magic formula, 21, 126, 159
injective, 115 Вґ
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

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 п¬‚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-О -cohomology, 136 deп¬Ѓnition, 69
GelвЂ™fand-Fuks, 162 CR-functions, 121
Harrison, 142 CR-structure, 121
C в€— -algebra
Hochschild, 142
Lie algebra, 132, 142 deп¬Ѓnition, 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 eп¬Ђective, 160
deп¬Ѓnition, 49 form, 124
double, 50 Lie algebroid, 124
double commutant theorem, 50 Weyl, 160
Poisson geometry, 51
DarbouxвЂ™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 [В·, В·]E , 133
Chevalley, 142
of dE , 131
Hochschild, 142
of an E-diп¬Ђerential form, 131
complexiп¬Ѓed tangent bundle, 62
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