P

g $$

$$ $

π ’1 (y)

$

$

W

π ’1 (x)

ry

xr

X

123

124 17 EXAMPLES OF LIE ALGEBROIDS

The projection π commutes with the H-action, and so there is a bundle map

Tπ

Tπ

E E TX

d

d

d

πd

‚

d ©

X

which is surjective. The induced map on sections is a Lie algebra homomorphism.

The kernel ker T π consists of the vertical part of T P/H. The sections of ker T π

are the H-invariant vector ¬elds on the ¬bers. Although each ¬ber of T P/H is

isomorphic to the Lie algebra h of H, there is no natural way to identify these two

Lie algebras. In fact, ker T π is the bundle associated to the principal bundle P by

the adjoint representation of H on h.

Exercise 63

Show that, when P is the bundle of frames for a vector bundle V ’ X, then

the gauge groupoid (P — P )/H of P is naturally isomorphic to the general

linear groupoid GL(V ) (see Section 14.4). Also show that the Lie algebroid

T P/H is naturally contained in gl(V ).

17.2 Connections on Transitive Lie Algebroids

We can use the Atiyah algebroid above to extend the notion of connection from

bundles to transitive Lie algebroids (see [110]).

A connection on the principal bundle P is a ¬eld of H-invariant direct comple-

ment subspaces to the ¬ber tangent spaces. Equivalently, a connection is simply a

splitting • of the exact sequence:

TπE

0 E ker T π EE' TX E0.

•

For any transitive Lie algebroid

ρE

0 E ker ρ EE TX E0,

we de¬ne a connection on E to be a linear splitting

E' TX

σ

of the sequence above, that is, a cross-section of ρ. The corresponding projection

ker ρ ' E

'

is called the connection form.

The curvature of a connection σ is its deviation from being a Lie algebra

homomorphism. Speci¬cally, for v, w ∈ “(T X), de¬ne the curvature form to be

„¦(v, w) = [σ(v), σ(w)]E ’ σ[v, w]T X ∈ “(ker ρ) .

17.3 The Lie Algebroid of a Poisson Manifold 125

An application of the Leibniz identity shows that „¦ is “tensorial,” i.e.

„¦(v, f w) = f „¦(v, w) .

One can verify that „¦ is a skew-symmetric bundle map T X — T X ’ ker ρ, i.e. „¦

is indeed a 2-form on X with values in ker ρ.

Exercise 64

Show that every (real-valued) 2-form on X is the curvature of a transitive Lie

algebroid

E X —R

0 EE E TX E 0.

(Hint: See Section 16.4.)

17.3 The Lie Algebroid of a Poisson Manifold

The symplectic structure on a symplectic manifold (X, ω) induces an isomorphism

Π = ω ’1

—

TX TX ,

E

where ω(v) = ω(v, ·). Pulling back the standard bracket on χ(X) by Π, we de¬ne

a bracket operation {·, ·} on di¬erential 1-forms „¦1 (X) = “(T — X). This makes

T — X into a Lie algebroid with anchor ρ = ’Π, called the Lie algebroid of the

symplectic manifold.

Furthermore, the bracket on 1-forms relates well to the Poisson bracket on

functions. Recall that the bracket of hamiltonian vector ¬elds Xf = Π(df ) and

Xg = Π(dg) satis¬es (see Section 3.5)

[Xf , Xg ] = ’X{f,g} .

We may pull the bracket back to “(T — X) by ’Π, and will denote by [·, ·] the bracket

on 1-forms. From the following computation

’Π[df, dg] = [’Π(df ), ’Π(dg)]

= [Xf , Xg ]

= ’X{f,g}

= ’Π(d{f, g}) .

we conclude that for exact 1-forms

[df, dg] = d{f, g} .

Now let (X, Π) be a Poisson manifold. The Poisson bivector ¬eld Π still induces

a map (see Section 4.2)

ΠE

T —X TX ,

though not necessarily an isomorphism. Nonetheless, there is a generalization of

the symplectic construction. This is the content of the following proposition, which

has been discovered many times, apparently ¬rst by Fuchssteiner [62].

126 17 EXAMPLES OF LIE ALGEBROIDS

Proposition 17.1 There is a natural Lie bracket [·, ·] on „¦1 (X) arising from a

Poisson structure on X, which satis¬es

• [df, dg] = d{f, g},

• Π : „¦1 (X) ’ χ(X) is a Lie algebra anti-homomorphism.

Proof. For general elements ±, β ∈ „¦1 (X), this bracket is de¬ned by

[±, β] := ’LΠ(±) β + LΠ(β) ± ’ dΠ(±, β) .

To check this de¬nition, we ¬rst note that the map Π was de¬ned by

Π(±) β = β(Π(±)) = Π(β, ±) .

If we then apply Cartan™s magic formula

LX · = X d· + d(X ·) ,

we can rewrite the bracket operation as

[±, β] = ’Π(±) dβ + Π(β) d± + dΠ(±, β) .

When ± = df and β = dg, it is easy to see that

[df, dg] = dΠ(df, dg) = d{f, g} .

Exercise 65

Show that this bracket on “(T — X) satis¬es the Leibniz identity

[±, f β] = f [±, β] + (’Π(±) · f )β .

It is also easy to show that this bracket satis¬es the Jacobi identity if we ¬rst

check it for df, dg, dh using [df, dg] = d{f, g}. Since any ± ∈ “(T — X) can be written

in a coordinate basis as

±= ui dfi ,

we may use the Leibniz identity to extend the Jacobi identity to arbitrary 1-forms.

Exercise 66

Check that Π de¬nes a Lie algebra anti-homomorphism from “(T — X) to

“(T X). Using the Leibniz identity, it su¬ces to check that Π is an anti-

homomorphism on exact 1-forms.

2

It was observed in [167] that the bracket on 1-forms makes T — X into a Lie

algebroid whose anchor is ’Π. This is called the Lie algebroid of the Poisson

manifold (X, Π). The orbits of this Lie algebroid are just the symplectic leaves

of X. The isotropy at a point x “ those cotangent vectors contained in ker Π

“ is the conormal space to the symplectic leaf Ox . The Lie algebra structure

which is inherited from the Lie algebroid T — X is exactly the transverse Lie algebra

structure from Section 5.2. Thus the Lie algebroid contains much of the information

associated with the Poisson structure! More on the Lie algebroid of a Poisson

manifold can be found in [162].

17.4 Vector Fields Tangent to a Hypersurface 127

Exercise 67

How canonical is this construction?

Speci¬cally, if • : X ’ Y is a Poisson map, is the induced map •— : „¦1 (Y ) ’

„¦1 (X) a Lie algebra homomorphism?

The Lie algebroid T — X is not always integrable to a Lie groupoid. However,

when it is integrable, at least one of its associated Lie groupoids carries a natural

symplectic structure compatible with the groupoid structure. Such an object is

called a symplectic groupoid (see [167]).

17.4 Vector Fields Tangent to a Hypersurface

Let Y be a hypersurface in a manifold X. Denote by χY (X) the space of vector

¬elds on X which are tangent to Y ; χY (X) is closed under the bracket [·, ·] of

vector ¬elds, it is a module over C ∞ (X), and it acts on C ∞ (X) by derivation. The

following theorem asserts that χY (X) is the space of sections of some vector bundle.

This result was probably noticed earlier than the cited reference.

Theorem 17.2 (Melrose [117]8 ) There is a vector bundle whose space of sections

is isomorphic to χY (X) as a C ∞ (X)-module.

This is a consequence of χY (X) being a locally free module over C ∞ (X). The

corresponding vector bundle A can be constructed from its space of sections and it

is called the Y -tangent bundle of X.

The Y -tangent bundle A comes equipped with a Lie algebroid structure over

X. To see the anchor map at the level of sections, introduce local coordinates

x, y2 , . . . , yn in a neighborhood U ⊆ X of a point in Y , and adapted to the sub-

manifold Y in the sense that U © Y is de¬ned by x = 0.

A vector ¬eld

n

‚ ‚

a, bi ∈ C ∞ (U ) ,

v=a + bi ,

‚x i=2 ‚yi

over U is the restriction of a vector ¬eld in χY (X) if and only if the coe¬cient a

vanishes when x = 0, that is, if and only if the smooth function a is divisible by x.

Hence, with respect to these coordinates, the vector ¬elds

‚‚ ‚

x , ,...,

‚x ‚y2 ‚yn

form a local basis for χY (X) as a module over C ∞ (X). Call these local basis vectors

e1 , e2 , . . . , en . They satisfy [ei , ej ] = 0, just like a local basis for the tangent bundle.

The di¬erence between A and the tangent bundle lies in the anchor map ρ :

χY (X) ’ χ(X), which is the inclusion

‚ ‚

, j≥2.

ρ(e1 ) = x , ρ(ej ) =

‚x ‚yj

This induces an anchor map ρ at the level of vector bundles. Together, these data

form a Lie algebroid

(A, ρ, [·, ·]) .

8 In

[117] Melrose handles the case Y = ‚X, the boundary of X, but the idea works for any

hypersurface.

128 17 EXAMPLES OF LIE ALGEBROIDS

The orbits of A (orbits were de¬ned in Section 16.1) are the connected components

of Y and of X \ Y .

The isotropy of A, i.e. the kernel of ρ : A ’ T X, is trivial over X \ Y . Over Y ,

the isotropy ker ρ|Y is the real line bundle spanned by e1 . This is clearly the trivial

line bundle Y — R when Y is cooriented (meaning that the normal bundle N Y is

trivial, or equivalently that Y is a two-sided hypersurface). But even if Y is not

‚

cooriented, x ‚x still provides a trivialization of ker ρ|Y , as this section is invariant

under change of orientation of N Y over U .

Restricting the vector bundle A to Y , we obtain the exact sequence

ρE

0 E ker ρ|Y E A|Y TY E0.

Therefore, a typical section of ker ρ|Y = Y — R has the form

‚

v = a(y) · x ·

‚x

for some bundle morphism a : N Y ’ N Y , expressing the rate at which v grows

as we move across Y . We conclude that sections of ker ρ|Y coincide with endomor-

phisms of the normal bundle of Y . Note that A|Y is the gauge algebroid (or Atiyah

algebroid) of N Y ; see also the ¬rst remark at the end of Section 17.5.

17.5 Vector Fields Tangent to the Boundary

The construction of the previous section extends to the case where X is a manifold

with boundary Y = ‚X.

Recall that the tangent space to X at a point in the boundary is just the usual

tangent space as if the manifold was enlarged by a collar extension so that the point

became interior.

Let • : X ’ [0, 1] be a de¬ning function for the boundary Y ; i.e.

•’1 (0) = Y ,

d• = 0 on Y , and

• ≡ 1 o¬ a tubular neighborhood of Y .

With respect to the coordinates x, y2 , . . . , yn above, we de¬ne a map m• • 1 on

vector ¬elds by

‚ ‚ ‚ ‚

’’ • · a

a + bi + bi .

‚x ‚yi ‚x ‚yi

We extend m• • 1 as the identity map outside the tubular neighborhood of Y .

Then

m• • 1 : χ(X) ’ χY (X)

is an isomorphism of C ∞ (X)-modules.

This isomorphism of the C ∞ (X)-modules χ(X) = “(T X) and χY (X) = “(A)

induces an isomorphism between the underlying vector bundles

TX A

which we interpret over a tubular neighborhood of Y as T X ν • „ , where ν and

Y —R

„ are the pull-back to the tubular neighborhood of the normal bundle N Y

and of the tangent bundle T Y , respectively.

17.5 Vector Fields Tangent to the Boundary 129

Remarks.

1. When Y = ‚X, A is the Lie algebroid of a groupoid over X, namely the

groupoid built from the pair groupoid of X \ Y together with the gauge

groupoid of the normal bundle of Y in X.

Exercise 68

What if Y is not the boundary of X?

2. In general, if the hypersurface Y is not the boundary of X, then the Y -tangent

bundle A might be not isomorphic to the tangent bundle T X.

For example, let X be a circle and let Y be one point. Then the Y -tangent

bundle is a M¨bius band rather than the trivial bundle. A similar construction

o

works when X is a 2-torus and Y is a single homologically nontrivial closed

curve.

Notice that, if Y is two points on a circle X, then the Y -tangent bundle is

again the trivial bundle.

It would be interesting to understand how much of the structure of the Y -

tangent bundle is determined by the cohomology class dual to Y (and the

original tangent bundle).

™¦

18 Di¬erential Geometry for Lie Algebroids

A useful way to view a Lie algebroid E over X is as an “alternative tangent bundle”

for X, endowing X with a “peculiar di¬erentiable structure”. The Lie algebroid

axioms allow us to carry out virtually all of the usual di¬erential-geometric con-

structions, replacing T X by E. The reader may wish to keep the example E = T X

in mind during a ¬rst reading of this chapter.

18.1 The Exterior Di¬erential Algebra of a Lie Algebroid

Let (E, ρ, [·, ·]E ) be a Lie algebroid over X, and let §• E — be the exterior algebra

of its dual E — . Sections of §• E — are called E-di¬erential forms on X, or simply

E-forms on X.

If θ ∈ “(§k E — ), we say that θ is homogeneous, and furthermore that its

degree is |θ| = k. In this case θ is called an E-k-form.

We de¬ne a di¬erential operator taking an E-k-form θ to an E-(k + 1)-form

dE θ, which at E-vector ¬elds v1 , . . . , vk+1 ∈ “(E) is

(’1)i+1 ρ(vi ) · θ(v1 , . . . , vi , . . . , vk+1 )

dE θ(v1 , . . . , vk+1 ) = ˆ

i

(’1)i+j θ([vi , vj ]E , v1 , . . . , vi , . . . , vj , . . . , vk+1 ) .

+ ˆ ˆ

i<j

The Lie algebroid axioms for E imply the following properties for dE :

1. dE is C ∞ (X)-multilinear,

2. d2 = 0, and

E

3. dE is a superderivation of degree 1, i.e.

dE (θ § ν) = dE θ § ν + (’1)|θ| θ § dE ν .

The triple (“(§• E — ), §, dE ) forms a di¬erential graded algebra, like the usual

algebra of di¬erential forms. We can recover the Lie algebroid structure on E from

(“(§• E — ), §, dE ):

• the anchor map ρ is obtained from dE on functions by the formula:

for v ∈ “(E) and f ∈ C ∞ (X) ;

ρ(v) · f = (dE f )(v) ,

• the Lie bracket [·, ·]E is determined by

ρ(v) · θ(w) ’ ρ(w) · θ(v) ’ dE θ(v, w)

[v, w]E θ =

v dE (w θ) ’ w dE (v θ) ’ (v § w) dE θ

=

for v, w ∈ “(E) and θ ∈ “(E — ).

We conclude that there is a one-to-one correspondence between Lie algebroid

structures on E and di¬erential operators on “(§• E — ) satisfying properties 1-3.

Remark. The space of sections of §• E — can be regarded as the space of functions

on a supermanifold.

131

132 18 DIFFERENTIAL GEOMETRY FOR LIE ALGEBROIDS

In this language, dE is an odd (since the degree is 1) vector ¬eld (since it is a

derivation), which is integrable because its superbracket with itself vanishes:

[dE , dE ] = dE dE ’ (’1)1 dE dE = 2d2 = 0 .

E

Hence, we may say that a Lie algebroid is a supermanifold with an odd integrable

supervector ¬eld. This idea permits one to apply to Lie algebroids some of the

™¦

intuition attached to ordinary vector ¬elds. See [161].

The exterior di¬erential algebra (“(§• E — ), §, dE ) associated to a Lie algebroid

(E, ρ, [·, ·]E ) determines de Rham cohomology groups, called the Lie algebroid

cohomology of E or E-cohomology.

Examples.

1. When E = g is a Lie algebra (i.e. a Lie algebroid over a one-point space), the

cohomology of the di¬erential complex

(§• g— , §, dg ) : R ’’ g— ’’ g— § g— ’’ . . .

is the standard Lie algebra cohomology with trivial coe¬cients, also known

as Chevalley cohomology.

Notice that the ¬rst arrow is the zero map and the second arrow is the usual

cobracket with the opposite sign:

for µ ∈ g— , dg µ is the element of g— § g—

which at v, w ∈ g gives dg µ(v, w) = ’µ([v, w]) .

The higher di¬erentials are determined by the ¬rst two and the derivation

property.

2. When E = T X is a tangent bundle of a manifold X, the cohomology com-

puted by (“(§• E — ), §, dE ) = („¦• (X), §, ddeRham ) is the usual de Rham coho-

mology.

™¦

Exercise 69

Compute the Lie algebroid cohomology for the Y -tangent bundle of a manifold

X where Y = ‚X is the boundary (see Sections 17.4 and 17.5 and [117],

proposition 2.49).

Remark. There have been several theories of characteristic classes associated to

Lie algebroids. We refer to [85] for a recent study of these with ample references to

™¦

earlier literature.

18.2 The Gerstenhaber Algebra of a Lie Algebroid

Sections of the exterior algebra §• E of a Lie algebroid (E, ρ, [·, ·]E ) are called Lie

algebroid multivector ¬elds or E-multivector ¬elds. If v ∈ “(§k E), then v

is called homogeneous with degree |v| = k.

18.2 The Gerstenhaber Algebra of a Lie Algebroid 133

We extend the bracket [·, ·]E to arbitrary E-multivector ¬elds by setting it, on

homogeneous E-multivector ¬elds v, w, to be

(’1)(|v|’1)(|w|’1) v dE (w θ) ’ w dE (v θ)

[v, w]E θ =

’(’1)|v|’1 (v § w) dE θ

where θ ∈ “(§• E — ). If θ ∈ “(§k E — ), then [v, w]E θ is homogeneous of degree

k ’ (|v| + |w| ’ 1). For [v, w]E θ to be a function, the degree of θ should be

k = |v| + |w| ’ 1. Therefore, [v, w]E has degree |v| + |w| ’ 1, and [·, ·]E is a bracket

of degree ’1.

Remark. In order to obtain a bracket of degree 0, we can rede¬ne the grading

on “(§• E), and let the new degree be the old degree minus 1:

for v ∈ “(§k E) .

(v) := |v| ’ 1 = k ’ 1 ,

For the (·) grading, we have

= [v, w]E ’ 1 = |v| + |w| ’ 2 = (v) + (w) .

[v, w]E

™¦

The bracket [·, ·]E on E-mutivector ¬elds has the following properties:

1. [·, ·]E allows us to extend to arbitrary elements of v, w ∈ “(§• E) the E-Lie

derivative operation de¬ned for E-vector ¬elds in Section 16.1:

Lv w := [v, w]E .

2. [·, ·]E is a super-Lie algebra (or “graded” Lie algebra) structure for the (·)

grading:

[v, w]E = ’(’1)(v)(w) [w, v]E = ’(’1)(|v|’1)(|w|’1) [w, v]E .

In words, [v, w]E is symmetric in v and w when both |v| and |w| are even and

is antisymmetric otherwise.

3. [·, ·]E satis¬es a super-Jacobi identity:

(’1)(|y|’1)(|v|+|w|) [y, [v, w]E ]E

[v, [w, y]E ]E +

(’1)(|v|’1)(|w|+|y|) [w, [y, v]E ]E

+ = 0.

4. [v, ·]E satis¬es a super-Leibniz identity (notice that both gradings appear

here):

[v, w § y]E = [v, w]E § y + (’1)(v)|w| w § [v, y]E .

The triple (“(§• E), §, [·, ·]E ) is called the Gerstenhaber algebra of the Lie

algebroid (E, ρ, [·, ·]E ), or just the E-Gerstenhaber algebra. We will refer to

the bracket [·, ·]E on “(§• E) as the E-Gerstenhaber bracket.

In general, a Gerstenhaber algebra (a, §, [·, ·]) is the following structure:

1. a graded vector space

a = a0 • a1 • . . .

together with

134 18 DIFFERENTIAL GEOMETRY FOR LIE ALGEBROIDS

2. a supercommutative associative multiplication of degree 0

ai § aj ⊆ ai+j

and

3. a super-Lie algebra structure of degree ’1

[ai , aj ] ⊆ ai+j’1

satisfying the super-Leibniz identity

[a, b § c] = [a, b] § c + (’1)(|a|’1)|b| b § [a, c] .

Historical Remark. Gerstenhaber found such a structure in 1963 [66] in the

Hochschild cohomology of an associative algebra (see Sections 19.1 and 19.2). ™¦

Remark. For a Lie algebroid (E, ρ, [·, ·]E ), the pull-back by ρ

ρ—

“(§• E — ) ' “(§• T — X)

satis¬es

ρ— —¦ d = dE —¦ ρ— ,

hence induces a map in cohomology. On the other hand, the wedge powers of ρ

§• ρ

• E “(§• T X)

“(§ E)

™¦

form a morphism of Gerstenhaber algebras.

To summarize, from a Lie algebroid structure on E

(E, ρ, [·, ·]E ) ,

we obtain a di¬erential algebra structure on “(§• E — )

(“(§• E — ), §, dE ) ,

and from that we get a Gerstenhaber algebra structure on “(§• E)

(“(§• E), §, [·, ·]E ) .

This process can be reversed, so these structures are equivalent.

For more on this material, see [84, 98, 162, 179].

18.3 Poisson Structures on Lie Algebroids

Example. For the tangent bundle Lie algebroid

(E, ρ, [·, ·]E ) = (T X, id, [·, ·]) ,

18.3 Poisson Structures on Lie Algebroids 135

dE is the de Rham di¬erential and the E-Gerstenhaber bracket is usually called the

Schouten-Nijenhuis bracket on multivector ¬elds (cf. Sections 3.2 and 3.3).9

A bivector ¬eld Π ∈ “(§2 T X) is called a Poisson bivector ¬eld if and only

if [Π, Π] = 0 (cf. Section 3.3). This condition is equivalent to the condition d2 = 0

Π

for the di¬erential operator dΠ := [Π, ·].

If Π is a Poisson bivector ¬eld on X, then T — X is a Lie algebroid with anchor

’Π (as seen in Section 17.3), and dΠ is the induced di¬erential on multivector

™¦

¬elds.

The notion of Poisson structure naturally generalizes to arbitrary Lie algebroids

as follows. Let (E, ρ, [·, ·]E ) be a Lie algebroid over X. An element Π ∈ “(§2 E)

is called an E-Poisson bivector ¬eld when [Π, Π]E = 0, where [·, ·]E is the E-

Gerstenhaber bracket.

Example. When E = g is a Lie algebra, a g-Poisson bivector ¬eld Π ∈ g § g

corresponds to a left-invariant Poisson structure on the underlying Lie group G.

™¦

The equation [Π, Π]g = 0 is called the classical Yang-Baxter equation.

Remarks.

1. The push-forward ρ— Π of an E-Poisson bivector ¬eld Π by the anchor ρ :

“(§2 E) ’ “(§2 T X) de¬nes an ordinary Poisson structure on the manifold

X.

2. By the Jacobi identity, an arbitrary (not necessarily Poisson) element ˜ ∈

“(§2 E) satis¬es

d2 + [ 1 [˜, ˜]E , ·]E = 0 .

˜ 2

Notice the resemblance to the equation for a ¬‚at connection.

™¦

An E-Poisson bivector ¬eld Π ∈ “(§2 E) is called an E-symplectic structure

when the induced bundle morphism

Π : E— ’ E

is an isomorphism. As in Section 17.3, Π satis¬es

±(Πx (β)) = Πx (±, β)

—

for ±, β ∈ Ex and x ∈ X.

An E-symplectic structure de¬nes an element ωΠ ∈ “(§2 E — ) by

ωΠ (v, w) = Π(Π’1 v, Π’1 w)

for v, w ∈ “(E). This E-2-form on X is non-degenerate and E-closed:

dE ω Π = 0 .

Hence, ωΠ is called an E-symplectic form.

9 According to the de¬nitions of Section 18.2, the signs here di¬er from the conventions of

Vaisman [162].

136 18 DIFFERENTIAL GEOMETRY FOR LIE ALGEBROIDS

18.4 Poisson Cohomology on Lie Algebroids

In this section, we study Poisson cohomology on general Lie algebroids, but the

most interesting case is of course that where E = T M . This “ordinary” Poisson

cohomology, introduced by Lichnerowicz [105], was studied from a general homo-

logical viewpoint by Huebschmann [83].

An E-Poisson structure Π on a Lie algebroid (E, ρ, [·, ·]E ) over X induces an

operator

dΠ = [Π, ·]E

on “(§• E) (see Section 18.3). The super-Jacobi identity for [·, ·]E , together with

the property [Π, Π]E = 0, imply that

d2 = 0 ,

Π

so (“(§• E), dΠ ) forms a di¬erential complex. The cohomology of this complex is

called the Lie algebroid Poisson cohomology or E-Π-cohomology. We will

•

next interpret the corresponding cohomology groups HΠ .

For f ∈ C ∞ (X) and θ ∈ “(E — ),

[Π, f ]E θ = ’Π (dE f § θ) = ’Π(dE f, θ) = Π(dE f ) θ = Xf θ ,

where the vector ¬eld

Xf := Π(dE f )

is called the hamiltonian vector ¬eld of f with respect to Π (similar to Sec-

tion 4.5).

The computation above shows that

Xf = [Π, f ]E = dΠ f ,

so the image of dΠ : C ∞ (X) ’ “(E) is precisely the space of hamiltonian vector

¬elds.

Exercise 70

Check that ρ maps the hamiltonian vector ¬eld of f with respect to Π to the

ordinary hamiltonian vector ¬eld of f with respect to ρ— Π.

The Poisson bracket of functions f, g ∈ C ∞ (X) with respect to an E-Poisson

structure Π

{f, g} = Π(dE f, dE g) ,

coincides with the ordinary Poisson bracket with respect to ρ— Π

{f, g} = (ρ— Π)(df, dg) .

Exercise 71

Check this assertion.

Hence the kernel of dΠ : C ∞ (X) ’ “(E) is the set of usual Casimir functions.

For an E-vector ¬eld v, we have

[Π, v]E = ’[v, Π]E = ’Lv Π

18.5 In¬nitesimal Deformations of Poisson Structures 137

where Lv is the E-Lie derivative (de¬ned in Sections 16.1 and 18.2). We naturally

call Poisson vector ¬elds those v ∈ “(E) satisfying Lv Π = 0; these form the

kernel of dΠ : “(E) ’ “(§2 E).

A synopsis of these observations is

0

HΠ = Casimir functions

Poisson vector ¬elds

1

HΠ =

hamiltonian vector ¬elds

2 3

The next two sections demonstrate how HΠ and HΠ are related to deformations of

the Poisson structure Π.

Exercise 72

Compute the Π-cohomology for the following Poisson manifolds:

(a) g— with its Lie-Poisson structure,

(b) the 3-torus T3 with a translation-invariant regular Poisson structure

(see [81]),

(c) R2 with {x, y} = x2 + y 2 (see [70, 123]).

Remark. Let Π be a Poisson structure on a Lie algebroid E. The operator dΠ

induces a Lie algebroid structure on E — , hence a bracket on “(§• E — ). The E — -de

Rham complex (“(§• E), dE— ) coincides with the Π-complex for E, (“(§• E), dΠ ).

Therefore, the E-Π-cohomology equals the E — -cohomology.

2

The canonical cohomology class [Π] ∈ HΠ is zero if and only if there exists

X ∈ “(E) such that LX Π = Π. An element Π ∈ “(E § E) satisfying LX Π = Π for

some X ∈ “(E) is called exact; X is called a Liouville vector ¬eld for Π (as in

™¦

the symplectic case).

Exercise 73

Find an example of an exact Poisson structure on a compact manifold (see [81]).

18.5 In¬nitesimal Deformations of Poisson Structures

Let Π(µ) be a smooth family of sections of §2 E for a Lie algebroid (E, ρ, [·, ·]E ).

Write

Π(µ) = Π0 + µΠ1 + µ2 Π2 + . . .

as a formal power series expansion.

The equation for each Π(µ) to be a Poisson structure is

0 = [Π(µ), Π(µ)]E

[Π0 , Π0 ]E + 2µ[Π0 , Π1 ]E + µ2 (2[Π0 , Π2 ]E + [Π1 , Π1 ]E ) + . . .

= ()

Assume that Π(0) = Π0 is a Poisson structure, so that [Π0 , Π0 ]E vanishes.

The coe¬cient Π1 is called an in¬nitesimal deformation of Π0 when

dΠ0 Π1 = [Π0 , Π1 ]E = 0 .

This is a cocycle condition in the complex (“(§• E), dΠ0 ).

138 18 DIFFERENTIAL GEOMETRY FOR LIE ALGEBROIDS

Suppose that

Π1 = dΠ0 v = [Π0 , v]E = ’Lv Π0

for some v ∈ “(E). Then Π1 is considered a trivial in¬nitesimal deformation

of Π0 .

Remark. The term “trivial” is suggested by the tangent bundle E = T X case

with the (local) ¬‚ow •t of ’v. For each t = µ, the pull-back •— Π0 is again a Poisson

µ

structure. Furthermore,

d—

= L’v Π0 = Π1 .

• Π0

dµ µ µ=0

The in¬nitesimal deformation Π1 is trivial in the sense that all Poisson structures

Π(µ) = •— Π0 are essentially the same expressed in di¬erent coordinates. The inter-

µ

pretation of this in¬nitesimal triviality for general Lie algebroids (with or without

™¦

using an associated groupoid) is not so clear.

We conclude that

in¬nitesimal deformations of Π

2

HΠ =

trivial in¬nitesimal deformations of Π

2

The group HΠ is a candidate for the tangent space at Π of the moduli space of

Poisson structures on E modulo isomorphism.

18.6 Obstructions to Formal Deformations

Returning to the equation ( ) of the previous section, suppose that [Π0 , Π0 ]E =

[Π0 , Π1 ]E = 0. To eliminate the µ2 term, we need the vanishing of

[Π0 , Π2 ]E + 1 [Π1 , Π1 ]E ,

2

i.e. having found Π1 , we need to solve for Π2 in the non-homogeneous di¬erential

equation

dΠ0 Π2 = ’ 1 [Π1 , Π1 ]E .

2

By the super-Jacobi identity,

dΠ0 ([Π1 , Π1 ]E ) = 0 ,

3

so [Π1 , Π1 ]E determines an element of HΠ0 . This element is zero if and only if

the solution Π2 of dΠ0 Π2 = ’ 1 [Π1 , Π1 ]E exists. Therefore, HΠ0 is the home of

3

2

obstructions to continuing in¬nitesimal deformations.

In general, the recursive solution of equation ( ) involves at each step working

out an equation of type

dΠ0 Πn = quadratic expression in the Πi ™s with i < n .

Exercise 74

Let Π be a Poisson structure on E.

Show that Π induces, via Π : E — ’ E, a chain map

(“(§• E — ), dE ) ’’ (“(§• E), dΠ ) .

Hence, Π induces a map from E-Π-cohomology to E-cohomology.

Show that, if Π is symplectic, then all the maps above are isomorphisms, so

E-Π-cohomology and E-cohomology are the same.

18.6 Obstructions to Formal Deformations 139

In view of the exercise, we conclude that, in the symplectic case, the obstructions

2 3

to formal deformations of a Poisson structure lie in HdeRham and HdeRham (see

below).

The bracket [·, ·]E on “(§• E) passes to E-Π-cohomology. In particular, it gives

rise to a squaring map

1 2 3

2 [·, ·]E : HΠ ’’ HΠ .

This is a quadratic map whose zeros are the in¬nitesimal deformations which can

be extended to second order in µ.

Exercise 75

Show that the squaring map is zero when Π is symplectic.

The exercise implies that, in the symplectic case, any in¬nitesimal deformation

can be extended to second order. In fact, since symplectic structures are open in

the vector space of closed 2-forms, there are no obstructions to extending an in-

¬nitesimal deformation: one may invert the Poisson structure, extend the resulting

deformation of symplectic structure, and invert back.

Remark. If a formal power series Π(µ) satis¬es all the stepwise equations for

[Π(µ), Π(µ)]E = 0, there remains the question of whether there exists a smooth

deformation corresponding to that power series. It is not known how or if this

™¦

problem can be answered in terms of the E-Π-cohomology groups.

Part VIII

Deformations of Algebras of

Functions

19 Algebraic Deformation Theory

Let V be a vector space (or just a module over a ring). We will study product-type

structures associated to V .

19.1 The Gerstenhaber Bracket

For k = 0, 1, 2, . . ., consider the set of all k-multilinear maps on V :

M k (V ) = {m : V — . . . — V | m is linear in each argument } .

k

Let Ak (V ) ⊆ M k (V ) be the subset of alternating k-multilinear maps on V .

Candidates for an associative product structure on V lie in M 2 (V ).

Candidates for a Lie bracket structure on V lie in A2 (V ).

For a ∈ M k (V ) and b ∈ M (V ), let

(a —¦i b)(x1 , x2 , . . . , xk+ ’1 ) := a(x1 , . . . , xi’1 , b(xi , . . . , xi+ ’1 ), xi+ , . . . , xk+ ’1 )

∈ V . Then let

where x1 , x2 , . . . , xk+ ’1

’1)

(’1)(i’1)(

a b := N · a —¦i b

i

where N is a combinatorial factor not relevant to our study. The Gerstenhaber

bracket [·, ·]G (see [66]) is de¬ned to be

’1)

[a, b]G := a b ’ (’1)(k’1)( b a.

Theorem 19.1 (Gerstenhaber [66]) The bracket [·, ·]G satis¬es the super-Jacobi

identity if we declare elements of M k (V ) to have degree k ’ 1.

When a, b ∈ M 2 (V ) are bilinear maps,

a(b(x, y), z) ’ a(x, b(y, z))

(a b)(x, y, z) =

a(b(x, y), z) ’ a(x, b(y, z))

[a, b]G (x, y, z) =

+b(a(x, y), z) ’ b(x, a(y, z)

1

a(a(x, y), z) ’ a(x, a(y, z))

2 [a, a]G (x, y, z) =

Writing x · y for a(x, y), we obtain

1

= (x · y) · z ’ x · (y · z) .

2 [a, a]G (x, y, z)

141

142 19 ALGEBRAIC DEFORMATION THEORY

Therefore, associative algebra structures on V are the solutions of the quadratic

equation

[a, a]G = 0 , a ∈ M 2 (V ) .

In terms of the squaring map (similar to the one mentioned in Section 18.6)

: M 2 (V ) M 3 (V )

’’

sq

1

’’

a 2 [a, a]G

the associative algebra structures on V are the elements of ker(sq).

Given an associative multiplication m ∈ M 2 (V ), [m, m]G = 0, we denote the

multiplication by

x · y := m(x, y) .

We may then de¬ne a cup product on M • (V ) by the formula

(a ∪ b)(x1 , x2 , . . . , xk+ ) = a(x1 , . . . , xk ) · b(xk+1 , . . . , xk+ )

where a ∈ M k (V ), b ∈ M (V ) and x1 , . . . , xk+ ∈ V .

The associativity of the cup product follows from the associativity of m. Notice

that, while the Gerstenhaber bracket is de¬ned on any vector space V , the cup

product structure depends on the choice of a multiplication on V .

Remark. A• (V ) is not closed under [·, ·]G . However, using anti-symmetrization,

we ¬nd a similar bracket on A• (V ) for which the equation [a, a]G = 0 amounts to

the Jacobi identity for a ∈ A2 (V ). In the case of symmetric multilinear maps on

V , S • (V ) ⊆ M • (V ), we may use symmetrization to obtain a bracket. ™¦

19.2 Hochschild Cohomology

Suppose that m is an associative multiplication on V , i.e. m ∈ M 2 (V ) and [m, m]G =

0. De¬ne the map

δm := [m, ·]G : M • (V ) ’ M •+1 (V ) .

By the super-Jacobi identity, we have

2

δm = 0 .

We hence obtain a complex (M • (V ), δm ), called the Hochschild complex of

(V, m).

The cohomology of (M • (V ), δm ) is known as Hochschild cohomology. The

•

cohomology groups are denoted by HHm .

Remark. For the alternating version of the bracket [·, ·]G , consider δa := [a, ·]G :

A• (V ) ’ A•+1 (V ) where a ∈ A2 (V ), [a, a]G = 0. The corresponding complex

(A• (V ), δa ) is the Chevalley complex of (V, a) and its cohomology is known as

Chevalley cohomology, or Lie algebra cohomology or Chevalley-Eilenberg

cohomology [69]. For the case of symmetric multilinear maps S • (V ), we obtain

™¦

Harrison cohomology [69].

19.2 Hochschild Cohomology 143

Repeating the computations and de¬nitions of Sections 18.4 and 18.5, we ¬nd

that:

0

HHm = center of the algebra (V, m)

derivations of the algebra (V, m)

1

HHm =

inner derivations of the algebra (V, m)

in¬nitesimal deformations of m

2

HHm =

trivial in¬nitesimal deformations of m

Exercise 76

Check the assertions above.

•

The groups HHm have the following algebraic structures:

•

1. The Gerstenhaber bracket [·, ·]G passes to HHm , since it commutes with

δm . Notice that [·, ·]G is independent of the algebra structure on V , while

•

HHm is de¬ned for a particular choice of m ∈ M 2 (V ) with [m, m]G = 0.

2. In particular, the Gerstenhaber bracket on Hochschild cohomology induces a

squaring map

1 2 3

2 [·, ·]G : HHm ’ HHm .

This map describes the obstructions to extending in¬nitesimal deformations

of m as we will see in Section 19.4.

3. The cup product operation on M • (V ), for a ¬xed associative multiplication

•

m, satis¬es a derivation law with respect to [·, ·]G which passes to HHm :

[a, b ∪ c]G = [a, b]G ∪ c + (’1)(|a|’1)|b| b ∪ [a, c]G

where a, b, c are Hochschild cohomology classes.

Since, for a, b, c ∈ M • (V ), we have

a δm b ’ δm (a b) + (’1)|b|’1 δm a b = (’1)|b|’1 (b ∪ a ’ (’1)|a||b| a ∪ b) ,

on cohomology we have supercommutativity

a ∪ b = (’1)|a||b| b ∪ a .

Remark. Notice that the cup product is supercommutative only in co-

homology, whereas the Gerstenhaber bracket [·, ·]G was supercommutative

™¦

already before passing to cohomology.

4. The action of the permutation (or symmetric) groups on the spaces

M k (V ) gives rise to a ¬ner structure in Hochschild cohomology, analogous to

the Hodge decomposition [69].

Remark. There is a groupoid related to HH 1 and HH 2 . It is the transformation

groupoid of the category whose objects are the associative multiplications on V ,

and whose morphisms are the triples (m1 , •, m2 ), where m1 , m2 are objects and •

is a linear isomorphism with m1 = •— m2 . ™¦

144 19 ALGEBRAIC DEFORMATION THEORY

19.3 Case of Functions on a Manifold

In the case where V = C ∞ (M ) for some manifold M , HH 0 is the center C ∞ (M ),

while HH 1 = χ1 (M ), since every derivation comes from a vector ¬eld, and the

only inner derivation is 0. More generally, we have the following result, after an

algebraic version by Hochschild, Kostant and Rosenberg [82].

Theorem 19.2 (Cahen-Gutt-De Wilde [21]) The subcomplex of M • (C ∞ (M ))

consisting of those multilinear maps which are di¬erential operators in each argu-

ment, has cohomology

HHdi¬ (C ∞ (M ))

k

χk (M ) = “(§k T M ) ,

and the Gerstenhaber bracket becomes the Schouten-Nijenhuis bracket.

The theorem is saying that:

1. Every k-cocycle is cohomologous to a skew-symmetric cocycle.

2. Every skew-symmetric cocycle is given by a k-vector ¬eld.

3. A k-vector ¬eld is a coboundary only if it is zero.

The inclusion

(χ• (M ), 0) E (Mloc (C ∞ (M )), δ)

•

is a linear isomorphism on the level of cohomology, but it is not a morphism for

the Gerstenhaber bracket. Kontsevich has recently [97] proven his formality con-

jecture, which states that the inclusion can be deformed to a morphism of di¬er-

ential graded Lie algebras which still induces an isomorphism on cohomology. As

a consequence of this theorem, Kontsevich establishes an equivalence between the

classi¬cation of formal deformations of the standard associative multiplication on

C ∞ (M ) and formal deformations of the zero Poisson structure on M . We discuss

these issues from a “pre-Kontsevich” viewpoint in the remainder of these notes.

19.4 Deformations of Associative Products

The equation for a formal series in M 2 (V )

m(µ) = m0 + µm1 + µ2 m2 + . . .

to be associative, identically in µ, is

0 = [m(µ), m(µ)]G

[m0 , m0 ]G + 2µ[m0 , m1 ]G + µ2 (2[m0 , m2 ]G + [m1 , m1 ]G ) + . . .

= ()

cf. Section 18.5. We will try to solve this equation stepwise:

We ¬rst need the term m0 to be associative, i.e. [m0 , m0 ]G = 0. Next, for the

coe¬cient of µ in ( ) to vanish, we need

0 = [m0 , m1 ]G = δm0 m1 .

Writing

x · y := m0 (x, y) ,

19.4 Deformations of Associative Products 145

δm0 m1 is:

δm0 m1 (x, y, z) = x · m1 (y, z) ’ m1 (x · y, z) + m1 (x, y · z) ’ m1 (x, y) · z .

If m1 were a biderivation (i.e. a derivation in each argument), this would become

= x · m1 (y, z) ’ x · m1 (y, z) ’ m1 (x, z) · y

δm0 m1 (x, y, z)

+y · m1 (x, z) + m1 (x, y) · z ’ m1 (x, y) · z

= ’m1 (x, z) · y + y · m1 (x, z) .

If m0 is symmetric (i.e. commutative), then every biderivation m1 is a cocycle with

respect to δm0 .

Suppose that m1 is antisymmetric.10 We then have

= x · m1 (y, z) ’ m1 (x · y, z) + m1 (x, y · z) ’ m1 (x, y) · z

δm0 m1 (x, y, z)

= x · m1 (z, y) ’ m1 (x · z, y) + m1 (x, z · y) ’ m1 (x, z) · y

δm0 m1 (x, z, y)

= z · m1 (x, y) ’ m1 (z · x, y) + m1 (z, x · y) ’ m1 (z, x) · y

δm0 m1 (z, x, y)

Writing

{x, y} := m1 (x, y) ,

and assuming that m0 is symmetric, we obtain

1

’ δm0 m1 (x, z, y) + δm0 m1 (z, x, y)]

2 [δm0 m1 (x, y, z)

= x · {y, z} + {x, z} · y ’ {x · y, z} .

The vanishing of this expression is the Leibniz identity for m1 with respect to

m0 .

Hence, assuming that m0 is symmetric and m1 is antisymmetric, if m1 is a

δm0 -cocycle, then m1 is a biderivation.

Similarly, we ¬nd

1

= {{x, y}, z} ’ {x, {y, z}} .

2 [m1 , m1 ]G (x, y, z)

The equation for eliminating the µ2 coe¬cient in ( ) is

1

δm0 m2 + 2 [m1 , m1 ]G = 0 , i.e.

{{x, y}, z} ’ {x, {y, z}} + x · m2 (y, z) ’ m2 (x · y, z) + m2 (x, y · z) ’ m2 (x, y) · z = 0 .

Assume that m0 is symmetric, m1 is antisymmetric and m2 is symmetric:

x·y = y·x

{x, y} = ’{y, x}

m2 (x, y) = m2 (y, x)

The equation for the vanishing of the coe¬cient of µ2 of in ( ) added to itself

under cyclic permutations (x, y, z) yields:

{{x, y}, z} + {{y, z}, x} + {{z, x}, y} = 0 ,

that is, the Jacobi identity for {·, ·}.

We conclude that the extendibility of the deformation to second order, with the

(anti)symmetry conditions imposed above, is equivalent to

[m1 , m1 ]G is a coboundary

⇐’ jacobiator for m1 is zero

⇐’ Jacobi identity for m1 .

local cochains on C ∞ (M ), this can always be arranged by subtracting a coboundary from

10 For

m1 .

146 19 ALGEBRAIC DEFORMATION THEORY

19.5 Deformations of the Product of Functions

We now apply the observations of the previous section to the case where V =

C ∞ (M ) is the space of smooth functions on a Poisson manifold (M, Π) (see also

Section 19.3).

Let m0 be pointwise multiplication of functions, and let m1 be the Poisson

bracket {·, ·}.

Take a formal deformation of m0 with linear term m1 . The formal variable

µ is traditionally replaced by i2 , where the symbol plays the role of Planck™s

i

constant from physics. We rede¬ne m1 = 2 {·, ·}, and take µ = instead. The

formal deformation is then

2

m( ) = m0 + m1 + m2 + . . .

The equation for m( ) to be an associative product for each “value” of is

[m( ), m( )]G = 0 ,

cf. Sections 18.5 and 19.4.

For these particular m0 and m1 , we have

⇐’

[m0 , m0 ]G = 0 m0 is associative

⇐’

[m0 , m1 ]G = 0 m1 satis¬es the Leibniz identity

∃m2 : 2[m0 , m2 ]G + [m1 , m1 ]G = 0 ⇐’ m1 satis¬es the Jacobi identity .

0 1 2

Hence, the coe¬cients of , and in [m( ), m( )]G vanish. To eliminate

the coe¬cient of 3 , we need

[m0 , m3 ]G + [m1 , m2 ]G = 0 .

This is equivalent to requiring the δm0 -cocycle [m1 , m2 ]G to be a δm0 -coboundary:

δm0 m3 = ’[m1 , m2 ]G .

The obstruction to solving the equation lies hence in HHm0 (C ∞ (M )).

3

Exercise 77

Check that δm0 [m1 , m2 ]G = 0.

Historical Remarks. The program of quantizing a symplectic manifold M with a

—-product, that is an associative multiplication on formal power series C ∞ (M )[[ ]],

was ¬rst set out by Bayen, Flato, Fronsdal, Lichnerowicz and Sternheimer in the

70™s [12].

In 1983 [43], De Wilde and Lecomte showed that every symplectic manifold

admits a formal deformation quantization. Their proof involved rather complicated

calculations which made the result look quite technical.

Some later versions of the existence proof relied on patching together local Weyl

algebras with nonlinear coordinate changes. In [92], Karasev and Maslov gave

further details of a proof, whose ¬rst outline was sketched in [91], which reduces

the patching to standard sheaf-theoretic ideas.

Another proof of the existence of deformation quantization which uses patching

ideas was given by Omori, Maeda and Yoshioka [133]. Although their proof still

19.5 Deformations of the Product of Functions 147

involved substantial computations, it used a fundamental idea which is also basic

in the proof of Fedosov (who discovered it independently). Each tangent space of

a Poisson manifold M can be viewed as an a¬ne space with a constant Poisson

structure, so it carries a natural Moyal-Weyl quantization (see Section 20.1). In

this way, the tangent bundle T M becomes a Poisson manifold with the ¬brewise

Poisson bracket, and with a ¬brewise quantization. To quantize M itself, we may

try to identify a subalgebra of the quantized algebra C ∞ (T M )[[ ]] with the vector

space C ∞ (M )[[ ]] in such a way that the induced multiplication on C ∞ (M )[[ ]]

gives a deformation quantization of M . Such an identi¬cation is called a Weyl

structure in [133].

In Chapter 21 we will discuss Fedosov™s proof of existence of deformation quan-

tization on symplectic manifolds.

™¦

For the history of these developments, see [14, 60].

20 Weyl Algebras

Let (E, Π) be a Poisson vector space. We will regard the Poisson structure Π ∈ E§E

as a bivector ¬eld on E with constant coe¬cients.

20.1 The Moyal-Weyl Product

For local canonical coordinates (q1 , . . . , qk , p1 , . . . , pk , c1 , . . . , cl ) (de¬ned in Sec-

tions 3.4 and 4.2), we use the symbols

←’

’ ←’

’

‚ ‚

and

‚qj ‚pj

for di¬erential operators acting on functions to their left, and

’’

’ ’’

’

‚ ‚

and

‚qj ‚pj

for di¬erential operators acting on functions to their right, so that

←’’’ ←’’’

’’ ’’

‚‚ ‚‚

{f, g} = f ’ g.

‚qj ‚pj ‚pj ‚qj

j

P

Let m1 be the following bidi¬erential operator on C ∞ (E):

←’’’ ←’’’

’’ ’’

i i ‚‚ ‚‚

’

m1 = P = .

2 2 ‚qj ‚pj ‚pj ‚qj

j

The operator P is closely related to an operator on functions on the product

space

P : C ∞ (E — E) ’’ C ∞ (E — E)

de¬ned in coordinates (q , p , c , q , p , c ) on E — E as

‚‚ ‚‚

’

P= .

‚qj ‚pj ‚pj ‚qj

j

Consider the maps

C ∞ (E) — C ∞ (E) E C ∞ (E — E)

f (q, p, c) — g(q, p, c) E f (q , p , c ) g(q , p , c )

and

∆E ∞

C ∞ (E — E) C (E)

f (q , p , c , q , p , c ) E f (q, p, c, q, p, c)

149

150 20 WEYL ALGEBRAS

The bidi¬erential operator P is the composition

P ∆

C ∞ (E) — C ∞ (E) E C ∞ (E — E) E C ∞ (E — E) E C ∞ (E) .

Powers P k are de¬ned by taking P k in this composition. Adding all the powers

(with the usual factorial coe¬cients), we de¬ne the formal power series of operators

C ∞ (E) — C ∞ (E) E C ∞ (E)

f —g Ef g

by the formula

∞

1i j j

f g := j! ( 2 ) f P g

j=0

i

f ·e2P ·g .

=:

This is called the Moyal-Weyl product [122, 174] or simply the Weyl product.

Remark. This exponential series is analogous to the Taylor expansion

d

f (x0 + µ) = (eµ dx f )(x0 ) ,

™¦

which converges for small µ only when f is real analytic.

Similarly, the Moyal-Weyl product will not converge in general, so we must

regard it as a formal power series in . The formal Weyl algebra is the algebra

of formal series in q, p, c, equipped with the Moyal-Weyl product de¬ned as above.

Note that, in the formal Weyl algebra:

• the polynomials in q, p, c, form a subalgebra,

• the variables c and commute with everything, and

i

• qj pj ’ pj qj = ,

2

whence the following relations:

i

[ci , ·] = 0

[qj , pj ] = 2

[ , ·] = 0

[qi , qj ] = [pi , pj ] = 0

where [·, ·] is the usual commutator bracket.

The a¬ne functions on E

h := E — • R— = (E • R)—

form a Lie algebra. When Π is non-degenerate, h is the Heisenberg algebra,

with central element . The universal enveloping algebra U(h) may be identi¬ed

by symmetrization with the polynomial algebra Pol(E • R).

20.2 The Moyal-Weyl Product as an Operator Product 151

20.2 The Moyal-Weyl Product as an Operator Product

Let (E, Π) be a symplectic Poisson vector space with canonical coordinates (q1 , . . . , qn , p1 , . . . , pn ).

The Moyal-Weyl product on (E, Π) (de¬ned in the previous section) can be in-

terpreted as an operator product for operators on R2n . (This is in fact how it

originated [174].)

The following map Op(·) from the coordinate functions (q1 , . . . , qn , p1 , . . . , pn )

on R2n to operators on Rn equipped with coordinates (x1 , . . . , xn ):

’’

qj Op(qj ) = qj := multiplication by xj

‚

’’

pj Op(pj ) = pj := i ‚xj

’’

1 Op(1) := multiplication by 1

satis¬es

[Op(qj ), Op(pj )] = i Op(1) = i Op({qj , pj }) .

Remark. In the language of Dirac and Schr¨dinger, we are mapping the classical

o

observables q and p to the corresponding quantum operators q and p. The Poisson

™¦

bracket of classical observables maps to the commutator of operators.

In order to avoid ordering ambiguity, products of observables qj pj = pj qj may

1

be mapped to 2 (qj pj + pj qj ). For arbitrary functions f (q, p), a device of Weyl

extends this symmetric ordering. Write f (q, p) in terms of its Fourier transform as

ei (qj Qj +pj Pj )

f (q, p) = (Ff ) (Q, P ) dQ dP

(R2n )—

where Q and P are variables on (R2n )— dual to q and p on R2n . Restricting to

Schwartz functions on R2n , we may set

ei (Qj Op(qj )+Pj Op(pj ))

Op(f ) := (Ff ) (Q, P ) dQ dP

(R2n )—

since the exponential factor is a unitary operator. The function f is called the

Weyl symbol [174] of the operator Op(f ).

We then de¬ne

g := Op’1 (Op(f ), Op(g)) .

f

Here f and g are Schwartz functions, and Op’1 is the map taking an operator to

its Weyl symbol. For this new (noncommutative) product of functions, the map

f ’ Op(f ) is an algebra homomorphism.

Remark. An integral formula for in the symplectic case was found by von Neu-

mann [128] (well before Moyal):

2n

1 i

f (y) g(z) e S(x, y, z) dy dz ,

(f g)(x) =

π