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P
g \$\$
\$\$ \$
ПЂ в€’1 (y)
\$
\$
W
ПЂ в€’1 (x)

ry

xr
X

123
124 17 EXAMPLES OF LIE ALGEBROIDS

The projection ПЂ commutes with the H-action, and so there is a bundle map
TПЂ
TПЂ
E E TX

d В
d В
d В
ПЂd В
В‚
В
X
which is surjective. The induced map on sections is a Lie algebra homomorphism.
The kernel ker T ПЂ consists of the vertical part of T P/H. The sections of ker T ПЂ
are the H-invariant vector п¬Ѓelds on the п¬Ѓbers. Although each п¬Ѓber 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 ПЂ 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 в†’ X, then
the gauge groupoid (P Г— 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 ).
A connection on the principal bundle P is a п¬Ѓeld of H-invariant direct comple-
ment subspaces to the п¬Ѓber tangent spaces. Equivalently, a connection is simply a
splitting П• of the exact sequence:

TПЂE
0 E ker T ПЂ EE' TX E0.
П•

For any transitive Lie algebroid

ПЃE
0 E ker ПЃ EE TX E0,

we deп¬Ѓne a connection on E to be a linear splitting

E' TX
Пѓ

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

ker ПЃ ' E
'

is called the connection form.
The curvature of a connection Пѓ is its deviation from being a Lie algebra
homomorphism. Speciп¬Ѓcally, for v, w в€€ О“(T X), deп¬Ѓne the curvature form to be

в„¦(v, w) = [Пѓ(v), Пѓ(w)]E в€’ Пѓ[v, w]T X в€€ О“(ker ПЃ) .
17.3 The Lie Algebroid of a Poisson Manifold 125

An application of the Leibniz identity shows that в„¦ is вЂњtensorial,вЂќ i.e.

в„¦(v, f w) = f в„¦(v, w) .

One can verify that в„¦ is a skew-symmetric bundle map T X Г— T X в†’ ker ПЃ, i.e. в„¦
is indeed a 2-form on X with values in ker ПЃ.

Exercise 64
Show that every (real-valued) 2-form on X is the curvature of a transitive Lie
algebroid
E X Г—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, П‰) induces an isomorphism

О  = П‰ в€’1
в€—
TX TX ,
E

where П‰(v) = П‰(v, В·). Pulling back the standard bracket on П‡(X) by О , we deп¬Ѓne
a bracket operation {В·, В·} on diп¬Ђerential 1-forms в„¦1 (X) = О“(T в€— X). This makes
T в€— X into a Lie algebroid with anchor ПЃ = в€’О , 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 п¬Ѓelds Xf = О (df ) and
Xg = О (dg) satisп¬Ѓes (see Section 3.5)

[Xf , Xg ] = в€’X{f,g} .

We may pull the bracket back to О“(T в€— X) by в€’О , and will denote by [В·, В·] the bracket
on 1-forms. From the following computation

в€’О [df, dg] = [в€’О (df ), в€’О (dg)]
= [Xf , Xg ]
= в€’X{f,g}
= в€’О (d{f, g}) .

we conclude that for exact 1-forms

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

Now let (X, О ) be a Poisson manifold. The Poisson bivector п¬Ѓeld О  still induces
a map (see Section 4.2)
О E
T в€—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 п¬Ѓrst by Fuchssteiner .
126 17 EXAMPLES OF LIE ALGEBROIDS

Proposition 17.1 There is a natural Lie bracket [В·, В·] on в„¦1 (X) arising from a
Poisson structure on X, which satisп¬Ѓes
вЂў [df, dg] = d{f, g},

вЂў О  : в„¦1 (X) в†’ П‡(X) is a Lie algebra anti-homomorphism.

Proof. For general elements О±, ОІ в€€ в„¦1 (X), this bracket is deп¬Ѓned by

[О±, ОІ] := в€’LО (О±) ОІ + LО (ОІ) О± в€’ dО (О±, ОІ) .

To check this deп¬Ѓnition, we п¬Ѓrst note that the map О  was deп¬Ѓned by

О (О±) ОІ = ОІ(О (О±)) = О (ОІ, О±) .

If we then apply CartanвЂ™s magic formula

LX О· = X dО· + d(X О·) ,

we can rewrite the bracket operation as

[О±, ОІ] = в€’О (О±) dОІ + О (ОІ) dО± + dО (О±, ОІ) .

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

[df, dg] = dО (df, dg) = d{f, g} .

Exercise 65
Show that this bracket on О“(T в€— X) satisп¬Ѓes the Leibniz identity

[О±, f ОІ] = f [О±, ОІ] + (в€’О (О±) В· f )ОІ .

It is also easy to show that this bracket satisп¬Ѓes the Jacobi identity if we п¬Ѓrst
check it for df, dg, dh using [df, dg] = d{f, g}. Since any О± в€€ О“(T в€— X) can be written
in a coordinate basis as
О±= ui dfi ,
we may use the Leibniz identity to extend the Jacobi identity to arbitrary 1-forms.

Exercise 66
Check that О  deп¬Ѓnes a Lie algebra anti-homomorphism from О“(T в€— X) to
О“(T X). Using the Leibniz identity, it suп¬ѓces to check that О  is an anti-
homomorphism on exact 1-forms.

2
It was observed in  that the bracket on 1-forms makes T в€— X into a Lie
algebroid whose anchor is в€’О . This is called the Lie algebroid of the Poisson
manifold (X, О ). The orbits of this Lie algebroid are just the symplectic leaves
of X. The isotropy at a point x вЂ“ those cotangent vectors contained in ker О
вЂ“ is the conormal space to the symplectic leaf Ox . The Lie algebra structure
which is inherited from the Lie algebroid T в€— 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 .
17.4 Vector Fields Tangent to a Hypersurface 127

Exercise 67
How canonical is this construction?
Speciп¬Ѓcally, if П• : X в†’ Y is a Poisson map, is the induced map П•в€— : в„¦1 (Y ) в†’
в„¦1 (X) a Lie algebra homomorphism?

The Lie algebroid T в€— 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 ).

17.4 Vector Fields Tangent to a Hypersurface
Let Y be a hypersurface in a manifold X. Denote by П‡Y (X) the space of vector
п¬Ѓelds on X which are tangent to Y ; П‡Y (X) is closed under the bracket [В·, В·] of
vector п¬Ѓelds, it is a module over C в€ћ (X), and it acts on C в€ћ (X) by derivation. The
following theorem asserts that П‡Y (X) is the space of sections of some vector bundle.
This result was probably noticed earlier than the cited reference.

Theorem 17.2 (Melrose 8 ) There is a vector bundle whose space of sections
is isomorphic to П‡Y (X) as a C в€ћ (X)-module.

This is a consequence of П‡Y (X) being a locally free module over C в€ћ (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 вЉ† X of a point in Y , and adapted to the sub-
manifold Y in the sense that U в€© Y is deп¬Ѓned by x = 0.
A vector п¬Ѓeld
n
в€‚ в€‚
a, bi в€€ C в€ћ (U ) ,
v=a + bi ,
в€‚x i=2 в€‚yi

over U is the restriction of a vector п¬Ѓeld in П‡Y (X) if and only if the coeп¬ѓcient 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 п¬Ѓelds
в€‚в€‚ в€‚
x , ,...,
в€‚x в€‚y2 в€‚yn
form a local basis for П‡Y (X) as a module over C в€ћ (X). Call these local basis vectors
e1 , e2 , . . . , en . They satisfy [ei , ej ] = 0, just like a local basis for the tangent bundle.
The diп¬Ђerence between A and the tangent bundle lies in the anchor map ПЃ :
П‡Y (X) в†’ П‡(X), which is the inclusion
в€‚ в€‚
, jв‰Ґ2.
ПЃ(e1 ) = x , ПЃ(ej ) =
в€‚x в€‚yj
This induces an anchor map ПЃ at the level of vector bundles. Together, these data
form a Lie algebroid
(A, ПЃ, [В·, В·]) .
8 In
 Melrose handles the case Y = в€‚X, the boundary of X, but the idea works for any
hypersurface.
128 17 EXAMPLES OF LIE ALGEBROIDS

The orbits of A (orbits were deп¬Ѓned in Section 16.1) are the connected components
of Y and of X \ Y .
The isotropy of A, i.e. the kernel of ПЃ : A в†’ T X, is trivial over X \ Y . Over Y ,
the isotropy ker ПЃ|Y is the real line bundle spanned by e1 . This is clearly the trivial
line bundle Y Г— 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
в€‚
cooriented, x в€‚x still provides a trivialization of ker ПЃ|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
ПЃE
0 E ker ПЃ|Y E A|Y TY E0.

Therefore, a typical section of ker ПЃ|Y = Y Г— R has the form
в€‚
v = a(y) В· x В·
в€‚x
for some bundle morphism a : N Y в†’ N Y , expressing the rate at which v grows
as we move across Y . We conclude that sections of ker ПЃ|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 п¬Ѓrst 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 = в€‚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 П• : X в†’ [0, 1] be a deп¬Ѓning function for the boundary Y ; i.e.
П•в€’1 (0) = Y ,
dП• = 0 on Y , and
П• в‰Ў 1 oп¬Ђ a tubular neighborhood of Y .
With respect to the coordinates x, y2 , . . . , yn above, we deп¬Ѓne a map mП• вЉ• 1 on
vector п¬Ѓelds by
в€‚ в€‚ в€‚ в€‚
в€’в†’ П• В· a
a + bi + bi .
в€‚x в€‚yi в€‚x в€‚yi
We extend mП• вЉ• 1 as the identity map outside the tubular neighborhood of Y .
Then
mП• вЉ• 1 : П‡(X) в†’ П‡Y (X)
is an isomorphism of C в€ћ (X)-modules.
This isomorphism of the C в€ћ (X)-modules П‡(X) = О“(T X) and П‡Y (X) = О“(A)
induces an isomorphism between the underlying vector bundles
TX A
which we interpret over a tubular neighborhood of Y as T X ОЅ вЉ• П„ , where ОЅ and
Y Г—R
П„ 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 = в€‚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 MВЁbius 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).

в™¦
18 Diп¬Ђerential Geometry for Lie Algebroids
A useful way to view a Lie algebroid E over X is as an вЂњalternative tangent bundleвЂќ
for X, endowing X with a вЂњpeculiar diп¬Ђerentiable structureвЂќ. The Lie algebroid
axioms allow us to carry out virtually all of the usual diп¬Ђerential-geometric con-
structions, replacing T X by E. The reader may wish to keep the example E = T X
in mind during a п¬Ѓrst reading of this chapter.

18.1 The Exterior Diп¬Ђerential Algebra of a Lie Algebroid

Let (E, ПЃ, [В·, В·]E ) be a Lie algebroid over X, and let в€§вЂў E в€— be the exterior algebra
of its dual E в€— . Sections of в€§вЂў E в€— are called E-diп¬Ђerential forms on X, or simply
E-forms on X.
If Оё в€€ О“(в€§k E в€— ), we say that Оё is homogeneous, and furthermore that its
degree is |Оё| = k. In this case Оё is called an E-k-form.
We deп¬Ѓne a diп¬Ђerential operator taking an E-k-form Оё to an E-(k + 1)-form
dE Оё, which at E-vector п¬Ѓelds v1 , . . . , vk+1 в€€ О“(E) is

(в€’1)i+1 ПЃ(vi ) В· Оё(v1 , . . . , vi , . . . , vk+1 )
dE Оё(v1 , . . . , vk+1 ) = Л†
i
(в€’1)i+j Оё([vi , vj ]E , v1 , . . . , vi , . . . , vj , . . . , vk+1 ) .
+ Л† Л†
i<j

The Lie algebroid axioms for E imply the following properties for dE :
1. dE is C в€ћ (X)-multilinear,
2. d2 = 0, and
E

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

dE (Оё в€§ ОЅ) = dE Оё в€§ ОЅ + (в€’1)|Оё| Оё в€§ dE ОЅ .

The triple (О“(в€§вЂў E в€— ), в€§, dE ) forms a diп¬Ђerential graded algebra, like the usual
algebra of diп¬Ђerential forms. We can recover the Lie algebroid structure on E from
(О“(в€§вЂў E в€— ), в€§, dE ):
вЂў the anchor map ПЃ is obtained from dE on functions by the formula:

for v в€€ О“(E) and f в€€ C в€ћ (X) ;
ПЃ(v) В· f = (dE f )(v) ,

вЂў the Lie bracket [В·, В·]E is determined by

ПЃ(v) В· Оё(w) в€’ ПЃ(w) В· Оё(v) в€’ dE Оё(v, w)
[v, w]E Оё =
v dE (w Оё) в€’ w dE (v Оё) в€’ (v в€§ w) dE Оё
=

for v, w в€€ О“(E) and Оё в€€ О“(E в€— ).
We conclude that there is a one-to-one correspondence between Lie algebroid
structures on E and diп¬Ђerential operators on О“(в€§вЂў E в€— ) satisfying properties 1-3.
Remark. The space of sections of в€§вЂў E в€— 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 п¬Ѓeld (since it is a
derivation), which is integrable because its superbracket with itself vanishes:

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

Hence, we may say that a Lie algebroid is a supermanifold with an odd integrable
supervector п¬Ѓeld. This idea permits one to apply to Lie algebroids some of the
в™¦
intuition attached to ordinary vector п¬Ѓelds. See .
The exterior diп¬Ђerential algebra (О“(в€§вЂў E в€— ), в€§, dE ) associated to a Lie algebroid
(E, ПЃ, [В·, В·]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 diп¬Ђerential complex

(в€§вЂў gв€— , в€§, dg ) : R в€’в†’ gв€— в€’в†’ gв€— в€§ gв€— в€’в†’ . . .

is the standard Lie algebra cohomology with trivial coeп¬ѓcients, also known
as Chevalley cohomology.
Notice that the п¬Ѓrst arrow is the zero map and the second arrow is the usual
cobracket with the opposite sign:

for Вµ в€€ gв€— , dg Вµ is the element of gв€— в€§ gв€—
which at v, w в€€ g gives dg Вµ(v, w) = в€’Вµ([v, w]) .

The higher diп¬Ђerentials are determined by the п¬Ѓrst two and the derivation
property.

2. When E = T X is a tangent bundle of a manifold X, the cohomology com-
puted by (О“(в€§вЂў E в€— ), в€§, dE ) = (в„¦вЂў (X), в€§, ddeRham ) is the usual de Rham coho-
mology.

в™¦

Exercise 69
Compute the Lie algebroid cohomology for the Y -tangent bundle of a manifold
X where Y = в€‚X is the boundary (see Sections 17.4 and 17.5 and ,
proposition 2.49).

Remark. There have been several theories of characteristic classes associated to
Lie algebroids. We refer to  for a recent study of these with ample references to
в™¦
earlier literature.

18.2 The Gerstenhaber Algebra of a Lie Algebroid

Sections of the exterior algebra в€§вЂў E of a Lie algebroid (E, ПЃ, [В·, В·]E ) are called Lie
algebroid multivector п¬Ѓelds or E-multivector п¬Ѓelds. If v в€€ О“(в€§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 [В·, В·]E to arbitrary E-multivector п¬Ѓelds by setting it, on
homogeneous E-multivector п¬Ѓelds v, w, to be

(в€’1)(|v|в€’1)(|w|в€’1) v dE (w Оё) в€’ w dE (v Оё)
[v, w]E Оё =
в€’(в€’1)|v|в€’1 (v в€§ w) dE Оё

where Оё в€€ О“(в€§вЂў E в€— ). If Оё в€€ О“(в€§k E в€— ), then [v, w]E Оё is homogeneous of degree
k в€’ (|v| + |w| в€’ 1). For [v, w]E Оё to be a function, the degree of Оё should be
k = |v| + |w| в€’ 1. Therefore, [v, w]E has degree |v| + |w| в€’ 1, and [В·, В·]E is a bracket
of degree в€’1.
Remark. In order to obtain a bracket of degree 0, we can redeп¬Ѓne the grading
on О“(в€§вЂў E), and let the new degree be the old degree minus 1:

for v в€€ О“(в€§k E) .
(v) := |v| в€’ 1 = k в€’ 1 ,

For the (В·) grading, we have

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

в™¦
The bracket [В·, В·]E on E-mutivector п¬Ѓelds has the following properties:

1. [В·, В·]E allows us to extend to arbitrary elements of v, w в€€ О“(в€§вЂў E) the E-Lie
derivative operation deп¬Ѓned for E-vector п¬Ѓelds in Section 16.1:

Lv w := [v, w]E .

2. [В·, В·]E is a super-Lie algebra (or вЂњgradedвЂќ Lie algebra) structure for the (В·)

[v, w]E = в€’(в€’1)(v)(w) [w, v]E = в€’(в€’1)(|v|в€’1)(|w|в€’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. [В·, В·]E satisп¬Ѓes a super-Jacobi identity:

(в€’1)(|y|в€’1)(|v|+|w|) [y, [v, w]E ]E
[v, [w, y]E ]E +
(в€’1)(|v|в€’1)(|w|+|y|) [w, [y, v]E ]E
+ = 0.

4. [v, В·]E satisп¬Ѓes a super-Leibniz identity (notice that both gradings appear
here):
[v, w в€§ y]E = [v, w]E в€§ y + (в€’1)(v)|w| w в€§ [v, y]E .

The triple (О“(в€§вЂў E), в€§, [В·, В·]E ) is called the Gerstenhaber algebra of the Lie
algebroid (E, ПЃ, [В·, В·]E ), or just the E-Gerstenhaber algebra. We will refer to
the bracket [В·, В·]E on О“(в€§вЂў E) as the E-Gerstenhaber bracket.
In general, a Gerstenhaber algebra (a, в€§, [В·, В·]) is the following structure:
a = a0 вЉ• a1 вЉ• . . .
together with
134 18 DIFFERENTIAL GEOMETRY FOR LIE ALGEBROIDS

2. a supercommutative associative multiplication of degree 0

ai в€§ aj вЉ† ai+j

and

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

[ai , aj ] вЉ† ai+jв€’1

satisfying the super-Leibniz identity

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

Historical Remark. Gerstenhaber found such a structure in 1963  in the
Hochschild cohomology of an associative algebra (see Sections 19.1 and 19.2). в™¦

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

ПЃв€—
О“(в€§вЂў E в€— ) ' О“(в€§вЂў T в€— X)

satisп¬Ѓes
ПЃв€— в—¦ d = dE в—¦ ПЃв€— ,
hence induces a map in cohomology. On the other hand, the wedge powers of ПЃ

в€§вЂў ПЃ
вЂў E О“(в€§вЂў T X)
О“(в€§ E)

в™¦
form a morphism of Gerstenhaber algebras.

To summarize, from a Lie algebroid structure on E

(E, ПЃ, [В·, В·]E ) ,

we obtain a diп¬Ђerential algebra structure on О“(в€§вЂў E в€— )

(О“(в€§вЂў E в€— ), в€§, dE ) ,

and from that we get a Gerstenhaber algebra structure on О“(в€§вЂў E)

(О“(в€§вЂў E), в€§, [В·, В·]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, ПЃ, [В·, В·]E ) = (T X, id, [В·, В·]) ,
18.3 Poisson Structures on Lie Algebroids 135

dE is the de Rham diп¬Ђerential and the E-Gerstenhaber bracket is usually called the
Schouten-Nijenhuis bracket on multivector п¬Ѓelds (cf. Sections 3.2 and 3.3).9
A bivector п¬Ѓeld О  в€€ О“(в€§2 T X) is called a Poisson bivector п¬Ѓeld if and only
if [О , О ] = 0 (cf. Section 3.3). This condition is equivalent to the condition d2 = 0
О
for the diп¬Ђerential operator dО  := [О , В·].
If О  is a Poisson bivector п¬Ѓeld on X, then T в€— X is a Lie algebroid with anchor
в€’О  (as seen in Section 17.3), and dО  is the induced diп¬Ђerential on multivector
в™¦
п¬Ѓelds.
The notion of Poisson structure naturally generalizes to arbitrary Lie algebroids
as follows. Let (E, ПЃ, [В·, В·]E ) be a Lie algebroid over X. An element О  в€€ О“(в€§2 E)
is called an E-Poisson bivector п¬Ѓeld when [О , О ]E = 0, where [В·, В·]E is the E-
Gerstenhaber bracket.
Example. When E = g is a Lie algebra, a g-Poisson bivector п¬Ѓeld О  в€€ g в€§ g
corresponds to a left-invariant Poisson structure on the underlying Lie group G.
в™¦
The equation [О , О ]g = 0 is called the classical Yang-Baxter equation.

Remarks.

1. The push-forward ПЃв€— О  of an E-Poisson bivector п¬Ѓeld О  by the anchor ПЃ :
О“(в€§2 E) в†’ О“(в€§2 T X) deп¬Ѓnes an ordinary Poisson structure on the manifold
X.

2. By the Jacobi identity, an arbitrary (not necessarily Poisson) element О˜ в€€
О“(в€§2 E) satisп¬Ѓes
d2 + [ 1 [О˜, О˜]E , В·]E = 0 .
О˜ 2

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

в™¦
An E-Poisson bivector п¬Ѓeld О  в€€ О“(в€§2 E) is called an E-symplectic structure
when the induced bundle morphism

О  : Eв€— в†’ E

is an isomorphism. As in Section 17.3, О  satisп¬Ѓes

О±(О x (ОІ)) = О x (О±, ОІ)
в€—
for О±, ОІ в€€ Ex and x в€€ X.
An E-symplectic structure deп¬Ѓnes an element П‰О  в€€ О“(в€§2 E в€— ) by

П‰О  (v, w) = О (О в€’1 v, О в€’1 w)

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

dE П‰ О  = 0 .

Hence, П‰О  is called an E-symplectic form.
9 According to the deп¬Ѓnitions of Section 18.2, the signs here diп¬Ђer from the conventions of
Vaisman .
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 вЂњordinaryвЂќ Poisson
cohomology, introduced by Lichnerowicz , was studied from a general homo-
logical viewpoint by Huebschmann .
An E-Poisson structure О  on a Lie algebroid (E, ПЃ, [В·, В·]E ) over X induces an
operator
dО  = [О , В·]E
on О“(в€§вЂў E) (see Section 18.3). The super-Jacobi identity for [В·, В·]E , together with
the property [О , О ]E = 0, imply that

d2 = 0 ,
О

so (О“(в€§вЂў E), dО  ) forms a diп¬Ђerential complex. The cohomology of this complex is
called the Lie algebroid Poisson cohomology or E-О -cohomology. We will
вЂў
next interpret the corresponding cohomology groups HО  .
For f в€€ C в€ћ (X) and Оё в€€ О“(E в€— ),

[О , f ]E Оё = в€’О  (dE f в€§ Оё) = в€’О (dE f, Оё) = О (dE f ) Оё = Xf Оё ,

where the vector п¬Ѓeld
Xf := О (dE f )
is called the hamiltonian vector п¬Ѓeld of f with respect to О  (similar to Sec-
tion 4.5).
The computation above shows that

Xf = [О , f ]E = dО  f ,

so the image of dО  : C в€ћ (X) в†’ О“(E) is precisely the space of hamiltonian vector
п¬Ѓelds.

Exercise 70
Check that ПЃ maps the hamiltonian vector п¬Ѓeld of f with respect to О  to the
ordinary hamiltonian vector п¬Ѓeld of f with respect to ПЃв€— О .

The Poisson bracket of functions f, g в€€ C в€ћ (X) with respect to an E-Poisson
structure О
{f, g} = О (dE f, dE g) ,
coincides with the ordinary Poisson bracket with respect to ПЃв€— О

{f, g} = (ПЃв€— О )(df, dg) .

Exercise 71
Check this assertion.

Hence the kernel of dО  : C в€ћ (X) в†’ О“(E) is the set of usual Casimir functions.
For an E-vector п¬Ѓeld v, we have

[О , v]E = в€’[v, О ]E = в€’Lv О
18.5 Inп¬Ѓnitesimal Deformations of Poisson Structures 137

where Lv is the E-Lie derivative (deп¬Ѓned in Sections 16.1 and 18.2). We naturally
call Poisson vector п¬Ѓelds those v в€€ О“(E) satisfying Lv О  = 0; these form the
kernel of dО  : О“(E) в†’ О“(в€§2 E).
A synopsis of these observations is
0
HО  = Casimir functions

Poisson vector п¬Ѓelds
1
HО  =
hamiltonian vector п¬Ѓelds
2 3
The next two sections demonstrate how HО  and HО  are related to deformations of
the Poisson structure О .

Exercise 72
Compute the О -cohomology for the following Poisson manifolds:
(a) gв€— with its Lie-Poisson structure,
(b) the 3-torus T3 with a translation-invariant regular Poisson structure
(see ),
(c) R2 with {x, y} = x2 + y 2 (see [70, 123]).

Remark. Let О  be a Poisson structure on a Lie algebroid E. The operator dО
induces a Lie algebroid structure on E в€— , hence a bracket on О“(в€§вЂў E в€— ). The E в€— -de
Rham complex (О“(в€§вЂў E), dEв€— ) coincides with the О -complex for E, (О“(в€§вЂў E), dО  ).
Therefore, the E-О -cohomology equals the E в€— -cohomology.
2
The canonical cohomology class [О ] в€€ HО  is zero if and only if there exists
X в€€ О“(E) such that LX О  = О . An element О  в€€ О“(E в€§ E) satisfying LX О  = О  for
some X в€€ О“(E) is called exact; X is called a Liouville vector п¬Ѓeld for О  (as in
в™¦
the symplectic case).

Exercise 73
Find an example of an exact Poisson structure on a compact manifold (see ).

18.5 Inп¬Ѓnitesimal Deformations of Poisson Structures

Let О (Оµ) be a smooth family of sections of в€§2 E for a Lie algebroid (E, ПЃ, [В·, В·]E ).
Write
О (Оµ) = О 0 + ОµО 1 + Оµ2 О 2 + . . .
as a formal power series expansion.
The equation for each О (Оµ) to be a Poisson structure is

0 = [О (Оµ), О (Оµ)]E
[О 0 , О 0 ]E + 2Оµ[О 0 , О 1 ]E + Оµ2 (2[О 0 , О 2 ]E + [О 1 , О 1 ]E ) + . . .
= ()

Assume that О (0) = О 0 is a Poisson structure, so that [О 0 , О 0 ]E vanishes.
The coeп¬ѓcient О 1 is called an inп¬Ѓnitesimal deformation of О 0 when

dО 0 О 1 = [О 0 , О 1 ]E = 0 .

This is a cocycle condition in the complex (О“(в€§вЂў E), dО 0 ).
138 18 DIFFERENTIAL GEOMETRY FOR LIE ALGEBROIDS

Suppose that
О 1 = dО 0 v = [О 0 , v]E = в€’Lv О 0
for some v в€€ О“(E). Then О 1 is considered a trivial inп¬Ѓnitesimal deformation
of О 0 .
Remark. The term вЂњtrivialвЂќ is suggested by the tangent bundle E = T X case
with the (local) п¬‚ow П•t of в€’v. For each t = Оµ, the pull-back П•в€— О 0 is again a Poisson
Оµ
structure. Furthermore,
dв€—
= Lв€’v О 0 = О 1 .
П• О 0
dОµ Оµ Оµ=0

The inп¬Ѓnitesimal deformation О 1 is trivial in the sense that all Poisson structures
О (Оµ) = П•в€— О 0 are essentially the same expressed in diп¬Ђerent coordinates. The inter-
Оµ
pretation of this inп¬Ѓnitesimal triviality for general Lie algebroids (with or without
в™¦
using an associated groupoid) is not so clear.

We conclude that
inп¬Ѓnitesimal deformations of О
2
HО  =
trivial inп¬Ѓnitesimal deformations of О
2
The group HО  is a candidate for the tangent space at О  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 [О 0 , О 0 ]E =
[О 0 , О 1 ]E = 0. To eliminate the Оµ2 term, we need the vanishing of
[О 0 , О 2 ]E + 1 [О 1 , О 1 ]E ,
2

i.e. having found О 1 , we need to solve for О 2 in the non-homogeneous diп¬Ђerential
equation
dО 0 О 2 = в€’ 1 [О 1 , О 1 ]E .
2
By the super-Jacobi identity,
dО 0 ([О 1 , О 1 ]E ) = 0 ,
3
so [О 1 , О 1 ]E determines an element of HО 0 . This element is zero if and only if
the solution О 2 of dО 0 О 2 = в€’ 1 [О 1 , О 1 ]E exists. Therefore, HО 0 is the home of
3
2
obstructions to continuing inп¬Ѓnitesimal deformations.
In general, the recursive solution of equation ( ) involves at each step working
out an equation of type
dО 0 О n = quadratic expression in the О i вЂ™s with i < n .

Exercise 74
Let О  be a Poisson structure on E.
Show that О  induces, via О  : E в€— в†’ E, a chain map
(О“(в€§вЂў E в€— ), dE ) в€’в†’ (О“(в€§вЂў E), dО  ) .
Hence, О  induces a map from E-О -cohomology to E-cohomology.
Show that, if О  is symplectic, then all the maps above are isomorphisms, so
E-О -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 [В·, В·]E on О“(в€§вЂў E) passes to E-О -cohomology. In particular, it gives
rise to a squaring map
1 2 3
2 [В·, В·]E : HО  в€’в†’ HО  .

This is a quadratic map whose zeros are the inп¬Ѓnitesimal deformations which can
be extended to second order in Оµ.

Exercise 75
Show that the squaring map is zero when О  is symplectic.

The exercise implies that, in the symplectic case, any inп¬Ѓnitesimal 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-
п¬Ѓnitesimal deformation: one may invert the Poisson structure, extend the resulting
deformation of symplectic structure, and invert back.

Remark. If a formal power series О (Оµ) satisп¬Ѓes all the stepwise equations for
[О (Оµ), О (Оµ)]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
в™¦
problem can be answered in terms of the E-О -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 Г— . . . Г— V | m is linear in each argument } .
k

Let Ak (V ) вЉ† 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 в€€ M k (V ) and b в€€ M (V ), let

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

в€€ V . Then let
where x1 , x2 , . . . , xk+ в€’1

в€’1)
(в€’1)(iв€’1)(
a b := N В· a в—¦i b
i

where N is a combinatorial factor not relevant to our study. The Gerstenhaber
bracket [В·, В·]G (see ) is deп¬Ѓned to be
в€’1)
[a, b]G := a b в€’ (в€’1)(kв€’1)( b a.

Theorem 19.1 (Gerstenhaber ) The bracket [В·, В·]G satisп¬Ѓes the super-Jacobi
identity if we declare elements of M k (V ) to have degree k в€’ 1.

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

a(b(x, y), z) в€’ a(x, b(y, z))
(a b)(x, y, z) =

a(b(x, y), z) в€’ a(x, b(y, z))
[a, b]G (x, y, z) =
+b(a(x, y), z) в€’ b(x, a(y, z)

1
a(a(x, y), z) в€’ a(x, a(y, z))
2 [a, a]G (x, y, z) =

Writing x В· y for a(x, y), we obtain
1
= (x В· y) В· z в€’ x В· (y В· 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 в€€ M 2 (V ) .

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

: M 2 (V ) M 3 (V )
в€’в†’
sq
1
в€’в†’
a 2 [a, a]G

the associative algebra structures on V are the elements of ker(sq).
Given an associative multiplication m в€€ M 2 (V ), [m, m]G = 0, we denote the
multiplication by
x В· y := m(x, y) .

We may then deп¬Ѓne a cup product on M вЂў (V ) by the formula

(a в€Є b)(x1 , x2 , . . . , xk+ ) = a(x1 , . . . , xk ) В· b(xk+1 , . . . , xk+ )

where a в€€ M k (V ), b в€€ M (V ) and x1 , . . . , xk+ в€€ V .
The associativity of the cup product follows from the associativity of m. Notice
that, while the Gerstenhaber bracket is deп¬Ѓned on any vector space V , the cup
product structure depends on the choice of a multiplication on V .

Remark. AвЂў (V ) is not closed under [В·, В·]G . However, using anti-symmetrization,
we п¬Ѓnd a similar bracket on AвЂў (V ) for which the equation [a, a]G = 0 amounts to
the Jacobi identity for a в€€ A2 (V ). In the case of symmetric multilinear maps on
V , S вЂў (V ) вЉ† M вЂў (V ), we may use symmetrization to obtain a bracket. в™¦

19.2 Hochschild Cohomology

Suppose that m is an associative multiplication on V , i.e. m в€€ M 2 (V ) and [m, m]G =
0. Deп¬Ѓne the map
Оґm := [m, В·]G : M вЂў (V ) в†’ M вЂў+1 (V ) .

By the super-Jacobi identity, we have
2
Оґm = 0 .

We hence obtain a complex (M вЂў (V ), Оґm ), called the Hochschild complex of
(V, m).
The cohomology of (M вЂў (V ), Оґm ) is known as Hochschild cohomology. The
вЂў
cohomology groups are denoted by HHm .

Remark. For the alternating version of the bracket [В·, В·]G , consider Оґa := [a, В·]G :
AвЂў (V ) в†’ AвЂў+1 (V ) where a в€€ A2 (V ), [a, a]G = 0. The corresponding complex
(AвЂў (V ), Оґa ) is the Chevalley complex of (V, a) and its cohomology is known as
Chevalley cohomology, or Lie algebra cohomology or Chevalley-Eilenberg
cohomology . For the case of symmetric multilinear maps S вЂў (V ), we obtain
в™¦
Harrison cohomology .
19.2 Hochschild Cohomology 143

Repeating the computations and deп¬Ѓnitions of Sections 18.4 and 18.5, we п¬Ѓnd
that:
0
HHm = center of the algebra (V, m)

derivations of the algebra (V, m)
1
HHm =
inner derivations of the algebra (V, m)

inп¬Ѓnitesimal deformations of m
2
HHm =
trivial inп¬Ѓnitesimal deformations of m

Exercise 76
Check the assertions above.

вЂў
The groups HHm have the following algebraic structures:
вЂў
1. The Gerstenhaber bracket [В·, В·]G passes to HHm , since it commutes with
Оґm . Notice that [В·, В·]G is independent of the algebra structure on V , while
вЂў
HHm is deп¬Ѓned for a particular choice of m в€€ 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 [В·, В·]G : HHm в†’ HHm .
This map describes the obstructions to extending inп¬Ѓnitesimal deformations
of m as we will see in Section 19.4.
3. The cup product operation on M вЂў (V ), for a п¬Ѓxed associative multiplication
вЂў
m, satisп¬Ѓes a derivation law with respect to [В·, В·]G which passes to HHm :

[a, b в€Є c]G = [a, b]G в€Є c + (в€’1)(|a|в€’1)|b| b в€Є [a, c]G

where a, b, c are Hochschild cohomology classes.
Since, for a, b, c в€€ M вЂў (V ), we have

a Оґm b в€’ Оґm (a b) + (в€’1)|b|в€’1 Оґm a b = (в€’1)|b|в€’1 (b в€Є a в€’ (в€’1)|a||b| a в€Є b) ,

on cohomology we have supercommutativity

a в€Є b = (в€’1)|a||b| b в€Є a .

Remark. Notice that the cup product is supercommutative only in co-
homology, whereas the Gerstenhaber bracket [В·, В·]G was supercommutative
в™¦

4. The action of the permutation (or symmetric) groups on the spaces
M k (V ) gives rise to a п¬Ѓner structure in Hochschild cohomology, analogous to
the Hodge decomposition .

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 , П•, m2 ), where m1 , m2 are objects and П•
is a linear isomorphism with m1 = П•в€— m2 . в™¦
144 19 ALGEBRAIC DEFORMATION THEORY

19.3 Case of Functions on a Manifold

In the case where V = C в€ћ (M ) for some manifold M , HH 0 is the center C в€ћ (M ),
while HH 1 = П‡1 (M ), since every derivation comes from a vector п¬Ѓeld, and the
only inner derivation is 0. More generally, we have the following result, after an
algebraic version by Hochschild, Kostant and Rosenberg .

Theorem 19.2 (Cahen-Gutt-De Wilde ) The subcomplex of M вЂў (C в€ћ (M ))
consisting of those multilinear maps which are diп¬Ђerential operators in each argu-
ment, has cohomology

HHdiп¬Ђ (C в€ћ (M ))
k
П‡k (M ) = О“(в€§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 п¬Ѓeld.
3. A k-vector п¬Ѓeld is a coboundary only if it is zero.
The inclusion
(П‡вЂў (M ), 0) E (Mloc (C в€ћ (M )), Оґ)
вЂў

is a linear isomorphism on the level of cohomology, but it is not a morphism for
the Gerstenhaber bracket. Kontsevich has recently  proven his formality con-
jecture, which states that the inclusion can be deformed to a morphism of diп¬Ђer-
ential graded Lie algebras which still induces an isomorphism on cohomology. As
a consequence of this theorem, Kontsevich establishes an equivalence between the
classiп¬Ѓcation of formal deformations of the standard associative multiplication on
C в€ћ (M ) and formal deformations of the zero Poisson structure on M . We discuss
these issues from a вЂњpre-KontsevichвЂќ viewpoint in the remainder of these notes.

19.4 Deformations of Associative Products

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

m(Оµ) = m0 + Оµm1 + Оµ2 m2 + . . .

to be associative, identically in Оµ, is

0 = [m(Оµ), m(Оµ)]G
[m0 , m0 ]G + 2Оµ[m0 , m1 ]G + Оµ2 (2[m0 , m2 ]G + [m1 , m1 ]G ) + . . .
= ()

cf. Section 18.5. We will try to solve this equation stepwise:
We п¬Ѓrst need the term m0 to be associative, i.e. [m0 , m0 ]G = 0. Next, for the
coeп¬ѓcient of Оµ in ( ) to vanish, we need

0 = [m0 , m1 ]G = Оґm0 m1 .

Writing
x В· y := m0 (x, y) ,
19.4 Deformations of Associative Products 145

Оґm0 m1 is:
Оґm0 m1 (x, y, z) = x В· m1 (y, z) в€’ m1 (x В· y, z) + m1 (x, y В· z) в€’ m1 (x, y) В· z .
If m1 were a biderivation (i.e. a derivation in each argument), this would become
= x В· m1 (y, z) в€’ x В· m1 (y, z) в€’ m1 (x, z) В· y
Оґm0 m1 (x, y, z)
+y В· m1 (x, z) + m1 (x, y) В· z в€’ m1 (x, y) В· z
= в€’m1 (x, z) В· y + y В· m1 (x, z) .
If m0 is symmetric (i.e. commutative), then every biderivation m1 is a cocycle with
respect to Оґm0 .
Suppose that m1 is antisymmetric.10 We then have
= x В· m1 (y, z) в€’ m1 (x В· y, z) + m1 (x, y В· z) в€’ m1 (x, y) В· z
Оґm0 m1 (x, y, z)
= x В· m1 (z, y) в€’ m1 (x В· z, y) + m1 (x, z В· y) в€’ m1 (x, z) В· y
Оґm0 m1 (x, z, y)
= z В· m1 (x, y) в€’ m1 (z В· x, y) + m1 (z, x В· y) в€’ m1 (z, x) В· y
Оґm0 m1 (z, x, y)
Writing
{x, y} := m1 (x, y) ,
and assuming that m0 is symmetric, we obtain
1
в€’ Оґm0 m1 (x, z, y) + Оґm0 m1 (z, x, y)]
2 [Оґm0 m1 (x, y, z)
= x В· {y, z} + {x, z} В· y в€’ {x В· 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
Оґm0 -cocycle, then m1 is a biderivation.
Similarly, we п¬Ѓnd
1
= {{x, y}, z} в€’ {x, {y, z}} .
2 [m1 , m1 ]G (x, y, z)

The equation for eliminating the Оµ2 coeп¬ѓcient in ( ) is
1
Оґm0 m2 + 2 [m1 , m1 ]G = 0 , i.e.
{{x, y}, z} в€’ {x, {y, z}} + x В· m2 (y, z) в€’ m2 (x В· y, z) + m2 (x, y В· z) в€’ m2 (x, y) В· z = 0 .
Assume that m0 is symmetric, m1 is antisymmetric and m2 is symmetric:
xВ·y = yВ·x
{x, y} = в€’{y, x}
m2 (x, y) = m2 (y, x)
The equation for the vanishing of the coeп¬ѓcient of Оµ2 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 {В·, В·}.
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
в‡ђв‡’ jacobiator for m1 is zero
в‡ђв‡’ Jacobi identity for m1 .
local cochains on C в€ћ (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 в€ћ (M ) is the space of smooth functions on a Poisson manifold (M, О ) (see also
Section 19.3).
Let m0 be pointwise multiplication of functions, and let m1 be the Poisson
bracket {В·, В·}.
Take a formal deformation of m0 with linear term m1 . The formal variable
Оµ is traditionally replaced by i2 , where the symbol plays the role of PlanckвЂ™s
i
constant from physics. We redeп¬Ѓne m1 = 2 {В·, В·}, and take Оµ = instead. The
formal deformation is then
2
m( ) = m0 + m1 + m2 + . . .

The equation for m( ) to be an associative product for each вЂњvalueвЂќ of is

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

cf. Sections 18.5 and 19.4.
For these particular m0 and m1 , we have

в‡ђв‡’
[m0 , m0 ]G = 0 m0 is associative
в‡ђв‡’
[m0 , m1 ]G = 0 m1 satisп¬Ѓes the Leibniz identity
в€ѓm2 : 2[m0 , m2 ]G + [m1 , m1 ]G = 0 в‡ђв‡’ m1 satisп¬Ѓes the Jacobi identity .
0 1 2
Hence, the coeп¬ѓcients of , and in [m( ), m( )]G vanish. To eliminate
the coeп¬ѓcient of 3 , we need

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

This is equivalent to requiring the Оґm0 -cocycle [m1 , m2 ]G to be a Оґm0 -coboundary:

Оґm0 m3 = в€’[m1 , m2 ]G .

The obstruction to solving the equation lies hence in HHm0 (C в€ћ (M )).
3

Exercise 77
Check that Оґm0 [m1 , m2 ]G = 0.

Historical Remarks. The program of quantizing a symplectic manifold M with a
в€—-product, that is an associative multiplication on formal power series C в€ћ (M )[[ ]],
was п¬Ѓrst set out by Bayen, Flato, Fronsdal, Lichnerowicz and Sternheimer in the
70вЂ™s .
In 1983 , 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 , Karasev and Maslov gave
further details of a proof, whose п¬Ѓrst outline was sketched in , 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 . 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 aп¬ѓne 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 п¬Ѓbrewise
Poisson bracket, and with a п¬Ѓbrewise quantization. To quantize M itself, we may
try to identify a subalgebra of the quantized algebra C в€ћ (T M )[[ ]] with the vector
space C в€ћ (M )[[ ]] in such a way that the induced multiplication on C в€ћ (M )[[ ]]
gives a deformation quantization of M . Such an identiп¬Ѓcation is called a Weyl
structure in .
In Chapter 21 we will discuss FedosovвЂ™s proof of existence of deformation quan-
tization on symplectic manifolds.
в™¦
For the history of these developments, see [14, 60].
20 Weyl Algebras
Let (E, О ) be a Poisson vector space. We will regard the Poisson structure О  в€€ Eв€§E
as a bivector п¬Ѓeld on E with constant coeп¬ѓcients.

20.1 The Moyal-Weyl Product

For local canonical coordinates (q1 , . . . , qk , p1 , . . . , pk , c1 , . . . , cl ) (deп¬Ѓned in Sec-
tions 3.4 and 4.2), we use the symbols
в†ђв€’
в€’ в†ђв€’
в€’
в€‚ в€‚
and
в€‚qj в€‚pj

for diп¬Ђerential operators acting on functions to their left, and
в€’в†’
в€’ в€’в†’
в€’
в€‚ в€‚
and
в€‚qj в€‚pj

for diп¬Ђerential operators acting on functions to their right, so that
в†ђв€’в€’в†’ в†ђв€’в€’в†’
в€’в€’ в€’в€’
в€‚в€‚ в€‚в€‚
{f, g} = f в€’ g.
в€‚qj в€‚pj в€‚pj в€‚qj
j

P

Let m1 be the following bidiп¬Ђerential operator on C в€ћ (E):
в†ђв€’в€’в†’ в†ђв€’в€’в†’
в€’в€’ в€’в€’
i i в€‚в€‚ в€‚в€‚
в€’
m1 = P = .
2 2 в€‚qj в€‚pj в€‚pj в€‚qj
j

The operator P is closely related to an operator on functions on the product
space
P : C в€ћ (E Г— E) в€’в†’ C в€ћ (E Г— E)
deп¬Ѓned in coordinates (q , p , c , q , p , c ) on E Г— E as

в€‚в€‚ в€‚в€‚
в€’
P= .
в€‚qj в€‚pj в€‚pj в€‚qj
j

Consider the maps

C в€ћ (E) вЉ— C в€ћ (E) E C в€ћ (E Г— E)

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

and
в€†E в€ћ
C в€ћ (E Г— E) C (E)

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

149
150 20 WEYL ALGEBRAS

The bidiп¬Ђerential operator P is the composition

P в€†
C в€ћ (E) вЉ— C в€ћ (E) E C в€ћ (E Г— E) E C в€ћ (E Г— E) E C в€ћ (E) .

Powers P k are deп¬Ѓned by taking P k in this composition. Adding all the powers
(with the usual factorial coeп¬ѓcients), we deп¬Ѓne the formal power series of operators

C в€ћ (E) вЉ— C в€ћ (E) E C в€ћ (E)

f вЉ—g Ef g

by the formula
в€ћ
1i j j
f g := j! ( 2 ) f P g
j=0
i
f В·e2P В·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 + Оµ) = (eОµ dx f )(x0 ) ,

в™¦
which converges for small Оµ 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 deп¬Ѓned as above.
Note that, in the formal Weyl algebra:

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

вЂў the variables c and commute with everything, and

i
вЂў qj pj в€’ pj qj = ,
2

whence the following relations:

i
[ci , В·] = 0
[qj , pj ] = 2
[ , В·] = 0
[qi , qj ] = [pi , pj ] = 0

where [В·, В·] is the usual commutator bracket.
The aп¬ѓne functions on E

h := E в€— вЉ• Rв€— = (E вЉ• R)в€—

form a Lie algebra. When О  is non-degenerate, h is the Heisenberg algebra,
with central element . The universal enveloping algebra U(h) may be identiп¬Ѓed
by symmetrization with the polynomial algebra Pol(E вЉ• R).
20.2 The Moyal-Weyl Product as an Operator Product 151

20.2 The Moyal-Weyl Product as an Operator Product

Let (E, О ) be a symplectic Poisson vector space with canonical coordinates (q1 , . . . , qn , p1 , . . . , pn ).
The Moyal-Weyl product on (E, О ) (deп¬Ѓned in the previous section) can be in-
terpreted as an operator product for operators on R2n . (This is in fact how it
originated .)
The following map Op(В·) from the coordinate functions (q1 , . . . , qn , p1 , . . . , pn )
on R2n to operators on Rn equipped with coordinates (x1 , . . . , xn ):

в€’в†’
qj Op(qj ) = qj := multiplication by xj
в€‚
в€’в†’
pj Op(pj ) = pj := i в€‚xj
в€’в†’
1 Op(1) := multiplication by 1

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

Remark. In the language of Dirac and SchrВЁdinger, we are mapping the classical
o
observables q and p to the corresponding quantum operators q and p. The Poisson
в™¦
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 )в€—

where Q and P are variables on (R2n )в€— 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 )в€—

since the exponential factor is a unitary operator. The function f is called the
Weyl symbol  of the operator Op(f ).
We then deп¬Ѓne
g := Opв€’1 (Op(f ), Op(g)) .
f
Here f and g are Schwartz functions, and Opв€’1 is the map taking an operator to
its Weyl symbol. For this new (noncommutative) product of functions, the map
f в†’ Op(f ) is an algebra homomorphism.
Remark. An integral formula for in the symplectic case was found by von Neu-
mann  (well before Moyal):
2n
1 i
f (y) g(z) e S(x, y, z) dy dz ,
(f g)(x) =
ПЂ
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