EMOIRS

of the

American Mathematical Society

Number 972

The Moment Maps in Diffeology

Patrick Iglesias-Zemmour

September 2010

•

Volume 207

•

Number 972 (second of 5 numbers)

•

ISSN 0065-9266

American Mathematical Society

September 2010

• Volume 207 • Number 972 (second of 5 numbers)

• ISSN 0065-9266

The Moment Maps in Diffeology

Patrick Iglesias-Zemmour

Number 972

Library of Congress Cataloging-in-Publication Data

Iglesias-Zemmour, Patrick, 1953-

The moment maps in diﬀeology / Patrick Iglesias-Zemmour.

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Includes bibliographical references.
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1. Symplectic geometry.

I. Title.

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Contents

Introduction

Chapter 1.

Few words about diﬀeology

Chapter 2.

Diﬀeological groups and momenta

Chapter 3.

The paths moment map

Chapter 4.

The 2-points moment map

Chapter 5.

The moment maps

Chapter 6.

The moment maps for exact 2-forms

Chapter 7.

Functoriality of the moment maps

Chapter 8.

The universal moment maps

Chapter 9.

About symplectic manifolds

Chapter 10.

The homogeneous case

Chapter 11.

Examples of moment maps in diﬀeology

Bibliography

iii

Abstract

This memoir presents a generalization of the moment maps to the category

{Diﬀeology}. This construction applies to every smooth action of any diﬀeological
group G preserving a closed 2-form ω, deﬁned on some diﬀeological space X. In
particular, that reveals a universal construction, associated to the action of the
whole group of automorphisms Diff(X, ω). By considering directly the space of
momenta of any diﬀeological group G, that is the space

∗

of left-invariant 1-forms

on G, this construction avoids any reference to Lie algebra or any notion of vector
ﬁelds, or does not involve any functional analysis. These constructions of the various
moment maps are illustrated by many examples, some of them originals and others
suggested by the mathematical literature.

Received by the editor October 4, 2007.
Article electronically published on March 10, 2010; S 0065-9266(10)00582-X.
2000 Mathematics Subject Classiﬁcation. Primary 53C99, 53D30, 53D20.
Key words and phrases. Diﬀeology, Moment Map, Symplectic Geometry.

Thanks

. I am happy to thank the Hebrew University of Jerusalem Israel for its hospitality.

The friendly and studious atmosphere I found here helped me to complete this work. I am glad to
thank my friends with whom I discussed the matter developed in this memoir, Jean-Marie Souriau
of course, but also Paul Donato, Yael Karshon and Fran¸

cois Ziegler. Also I would like to thank

the referee who allowed me, by its remarks and questions, to enrich a part of this memoir.

2010 American Mathematical Society

Introduction

The moment map has been introduced in the 1970’s in Souriau’s work about

the structure of dynamical systems [Sou70]. It is the tool by excellence for dealing
with symmetries in symplectic, or pre-symplectic geometry. But, in recent decades,
the necessity appeared to extend the notion of symplectic formalism and moment
maps, outside the usual framework of manifolds, to include constructions in inﬁnite
dimension — spaces of connections of principal bundles, spaces of functions etc. —
or to include singular spaces — orbifolds, singular symplectic reduction spaces etc..

In this paper, we shall use the category

{Diﬀeology} as the framework for such a

generalization. We know already that diﬀeology is suitable to describe, in a unique
and satisfactory way, manifolds or inﬁnite dimensional spaces, as well as singular
quotients. But, if diﬀeology excels with covariant objects, as diﬀerential forms, it
is more subtle when it is question of contravariant objects like vector ﬁelds, Lie
algebra

, kernel etc.. Thus, in order to build a good diﬀeological theory of the

moment map, and to avoid useless debates, we need to get freed from everything
related to contravariant geometrical objects.

Actually, the notion of moment map is not really an object of the symplectic

world, but relates more generally to the category of space equipped with closed
2-forms. The non-degeneracy condition is secondary and can be skipped ﬁrst from
the data. This has been underlined explicitly by Souriau in his symplectic formula-
tion of Noether’s theorem, which involves pre-symplectic manifolds. On symplectic
manifolds, Noether’s theorem is empty. So, the moment map is just an object of
the world of diﬀerential closed form, and there is no reason a priori that it could
not be extended to diﬀeology which has a very well developed framework for De
Rham’s calculus.

Now, in order to generalize the moment map in diﬀeology, we need to under-

stand its meaning in the simplest possible case. Let M be a manifold equipped with
a closed 2-form ω. And, let G be a Lie group acting smoothly on M and preserving
ω. That is, g

∗

(ω) = ω for all elements g of G, where g

denotes the action of g on

M. Let us assume that ω is exact, ω = dλ, and moreover that λ is also invariant
by the action of G. So, for every point m of M, the pullback of λ, by the orbit map

m : g

→ g

(m) is a left-invariant 1-form of G. That is, an element of the dual of

the Lie algebra

∗

. The map, μ : m

→ ˆ

∗

(λ) is exactly the moment map of the

action of G on the pair (M, ω) (at least one of the moment maps, since they are
deﬁned up to constants). As we can see, this construction does not involve really
the Lie algebra of G but the space

∗

of left-invariant 1-forms on G. Since this space

is well deﬁned in diﬀeology, we have just to replace « manifold » by « diﬀeological

Several authors, beginning with Souriau, proposed some generalizations of Lie algebra in

diﬀeology. But, it does not seem to exist a unique good choice. Such generalizations rely actually
on the kind of problem treated.

PATRICK IGLESIAS-ZEMMOUR

space », and « Lie group » by « diﬀeological group », and everything works the same.
So, let us change the manifold M for a diﬀeological space

X, and let G be some

diﬀeological group. Let us continue to denote the space of left-invariant 1-forms on
G by

∗

, even if the star does not refer a priori to some duality, and let us call it

simply the space of momenta of the group G. Note that the group G continues to
act on

∗

by pullback of its adjoint action Ad : (g, k)

→ gkg

−1

, so we don’t lose

the notions of coadjoint action and coadjoint orbits.

So, if we got the good space of momenta, which is the space where the moment

maps are assumed to take their values, the problem remains that not every G-
invariant closed 2-form is exact. And moreover, even if such form is exact, there
is no reason, for some of its primitives to be G-invariant. We shall pass over this
diﬃculty by introducing an intermediary, on which we can realize the simple case
described above. This intermediary is the space Paths(X), of all the smooth paths
of X, where the group G acts naturally by composition. And since Paths(X) carries
a natural functional diﬀeology, it is legitimate to consider its diﬀerential forms, and
this is what we do. By integrating ω along the paths, we get a diﬀerential 1-form
deﬁned on Paths(X), and invariant by the action of G. The exact tool used here is
the chain-homotopy operator K [Piz05]. The 1-form Λ = Kω, deﬁned on Paths(X),
is a G-invariant primitive of the 2-form Ω = (ˆ

∗

− ˆ0

∗

)(ω), where ˆ

1 and ˆ

0 map every

path of X to its ends. Thus, thanks to the construction described above, we get a
moment map Ψ for the 2-form Ω = dΛ and the action of G on Paths(X). But, this
paths moment map Ψ is not the one we are waiting for. We need to push it down
on X, or moreover on X

× X. Now, if we get this way a 2-points moment map ψ

well deﬁned on X

× X, it doesn’t take anymore its value in G

∗

, as does Ψ, but in

the quotient

∗

/Γ, where Γ is the image by Ψ of all the loops of X. Fortunately,

Γ = Ψ(Loops(X)) is a subgroup of (

∗

, +) and depends on the loops only through

their free homotopy classes. In other words, Γ is an homomorphic image of the
fundamental group π

(X) of X, or more precisely of its abelianized. Well, it is not

a big deal to have the moment map taking its values in some quotient of the space
of momenta, we can live with that. Especially if the group Γ is invariant under
the coadjoint action of G, which is actually the case

. But, we are not completely

done. The usual moment map is not a 2-points function, but a 1-point function.
So, we have to extract our usual moment maps from this 2-points function ψ.
This is quite easy, thanks to its very deﬁnition, the moment map Ψ satisﬁes an
additive property for juxtaposition of paths. And, the moment map ψ inherits
this property as a cocycle condition: for any three point x, x

and x

of X we

have ψ(x, x

) + ψ(x

, x

) = ψ(x, x

). Hence, for X connected, there exists always

a map μ such that ψ(x, x

) = μ(x

)

− μ(x). And, any two such maps diﬀer just

by a constant. So, we get ﬁnally our wanted set of moment maps μ, deﬁned in
the diﬀeological framework. The only diﬀerence, with the simplest case described
above, is that the moment maps take their values in some quotient of the space
of momenta, instead of the space of momenta itself. But, this is in fact already
the case in the classical theory. It doesn’t appear explicitly because people focus
more on hamiltonian actions than just on symplectic actions. Actually, the group

The space X will be assumed to be connected, as many results need this hypothesis.

More precisely, the elements of Γ are not just elements of

∗

but are moreover closed, and

therefore invariant, each of them, by the coadjoint action of G.

INTRODUCTION

Γ represents the very obstruction, for the action of G on (X, ω), to be hamiltonian.
We shall call Γ, the holonomy of the action of G.

Now, let us come back to some properties of the various moment maps in-

troduced above. The paths moment maps Ψ and its projection ψ are equivariant
with respect to the action of G on X and the coadjoint action of G on

∗

, or the

projection of the coadjoint action on

∗

/Γ. But this is not anymore the case for

the moments maps μ. The variance of the maps μ reveals a family of cocycles θ
from G to

∗

/Γ diﬀering just by coboundaries, and generalizing Souriau’s cocycles

[Sou70]. This class of cocycles σ belongs to the cohomology group H

(G,

∗

/Γ),

and will be called Souriau’s class of the action of G of (X, ω). Souriau’s class σ is
precisely the obstruction for the 2-points moment map ψ to be exact, that is for
some moment map μ to be equivariant. Moreover, in parallel with the classical
situation, every Souriau’s cocycle θ deﬁnes a new action of G on

∗

/Γ, which we

still call the aﬃne coadjoint action (associated to θ). And, the image of a moment
maps μ is a collection of coadjoint orbits for this action. We call these orbits,
the (Γ, θ)-coadjoint orbits of G. Two diﬀerent cocycles give two families of orbits
translated by the same constant.

Let us remark that the holonomy group Γ and Souriau’s class σ appear clearly

on a diﬀerent level of meaning, the ﬁrst one is responsible of the non hamiltonian
character of the action of G, and the second characterizes the lack of equivariance
of the moment maps.

Well, until now we didn’t use all the facilities oﬀered by the diﬀeological frame-

work. Since we do not restrict ourselves to the category of Lie groups, nothing
prevents us to consider the group of all the automorphisms of the pair (X, ω). That
is, the group Diff(X, ω) of all the diﬀeomorphisms of X, preserving ω. This group
is a natural diﬀeological group, acting smoothly on X. Thus, everything built
above applies to Diff(X, ω), and every other action preserving ω, of any diﬀeo-
logical group, pass through Diff(X, ω), and through the associated object of the
theory developed here. Therefore, considering the whole group of automorphisms
of the closed 2-form ω of X, we get a natural notion of universal moment maps Ψ

and μ

, universal holonomy Γ

, universal Souriau’s cocycles θ

, and universal

Souriau’s class σ

. By the way, this universal construction suggests a simple and

new characterization, for any diﬀeological space X equipped with a closed 2-form ω,
of the group of hamiltonian diﬀeomorphisms Ham(X, ω), as the largest connected
subgroup of Diff(X, ω) whose holonomy vanishes.

It is interesting to notice that, contrary to the original constructions [Sou70]

and most of its generalizations, the theory described above is essentially global,
more or less algebraic, do not refer to any diﬀerential, or partial diﬀerential, equa-
tion and do not involve any notion of vector ﬁeld or functional analysis techniques.

I give, at the end of the memoir, several examples involving diﬀeological groups

which are not Lie groups, or involving diﬀeological spaces which are not manifolds.
We can see how the general theory applies to the singular « symplectic irrational
tori » for which topology is irrelevant.

These general constructions of moment

maps are also applied to a few examples in inﬁnite dimension, and an example
which mixes ﬁnite and inﬁnite dimensions. Finally, two examples of orbifolds are
also examined. These examples show without any doubt the ability of this theory
to treat correctly, in a unique framework, avoiding heuristic arguments, the large
variety of situations we can ﬁnd in the mathematical literature today. For inﬁnite

PATRICK IGLESIAS-ZEMMOUR

dimensional (heuristic) examples, see Donaldson’s paper [Dnl99]. By the way, I
developed on purpose some tedious computations, even if it is boring, just to show
diﬀeology at work. I mean, to show that diﬀeology is not just a formalism, but a
working calculus method too.

Considering the classical case of a closed 2-form ω deﬁned on a manifold M,

we show in particular that ω is non degenerate if and only if the group Diff(M, ω)
is transitive on M and if a universal moment maps μ

is injective. In other words,

symplectic manifolds are identiﬁed, by the universal moment maps, to some coad-
joint orbits (in our general sense) of their group of symplectomorphisms. This idea
that « every symplectic manifold is a coadjoint orbit » is not new, it is suggested
by a well known classiﬁcation theorem for symplectic homogeneous Lie group ac-
tions [Kir74], [Kos70], [Sou70], and has been stated already in a diﬀerent context
[Omo86]. What is new here is that diﬀeology make this statement rigorous without
the use of any functional analysis tools.

In conclusion, beside the point that the construction developed in this mem-

oir is a ﬁrst step in the elaboration of the symplectic diﬀeology program, I would
emphasize the fact that, since

{Manifolds} is a full and faithful subcategory of

{Diﬀeology}, all the constructions developed here apply to manifolds and give a
faithful description of the classical theory of moment maps. As we have seen, there
is no mention, and no use, of Lie algebra or vector ﬁelds in this exposition. This
reveal the fact that these objects are also superﬂuous in the traditional approach,
and can be avoided. And, I would add, they should be avoided. No just because
then, they can be extended to larger categories, but because the use of contravari-
ant object hide the deep fact that the theory of moment maps is a pure covariant
theory. For example, we know that since coadjoint orbits of Lie groups are sym-
plectic they are even dimensional. This is often regarded as a miracle, since it is not
necessarily the case for adjoint orbits. But if we think that Lie algebra have little to
do with the space of momenta of a Lie group, there is no more miracle, just diﬀerent
behaviors for diﬀerent objects, which is unsurprising. Moreover I would add, but
this can appear as more or less subjective, that avoiding all this va-et-vient between
Lie algebra and dual of Lie algebra, the diﬀeological approach of the moment maps
is much more simpler, and even deeper, than the classical approach. Compare for
example Souriau’s cocycle constructions in the original « Structure des syst`

emes

dynamiques » [Sou70] and in this memoir. The only crucial property used here is
connectedness, that is the existence of enough smooth paths connecting points in
spaces.

Now, this constructions, in particular the new diﬀeological symplectic frame-

work it suggests, come together with a lot of new questions which have not be
answered here. And I hope I’ll develop some of them in future works.

Note

— Diﬀeology is a maximal extension of the local category of smooth real

domains. It contains by the way, fully and faithfully, the category of manifolds.
Diﬀeology has been introduced by J.-M Souriau at the beginning of the 1980s
[Sou81], and it is a variant of the theory of K.-T. Chen’s diﬀerentiable spaces
introduced few years before [Che77]. Since then, the theory has been enhanced
by some authors. The reader is assumed to be familiar with diﬀeology even if we
remind some basics constructions in the ﬁrst Section. For an comprehensive report
on diﬀeology see [Piz05].

CHAPTER 1

Few words about diﬀeology

This is a reminder of the few diﬀeological notions we will use in the following.

More details about these constructions, and proofs, can be found in [Piz05].

1.1. Domains and parametrizations. We call numerical space any power

of the real numbers R, and we call numerical domain, or simply domain, any open
set of any numerical domain. If U is a domain of R

, we say that U is an n-domain.

Let X be a set, we call parametrization in X any map deﬁned on some numerical
domain with values in X. The set of all the parametrizations in X is denoted by
Param(X). For any parametrization P : U

→ X, the numerical domain U is called

the domain of P and is denoted by dom(P). If U is an n-domain we say that P is
a n-parametrization.

1.2. Diﬀeology and diﬀeological spaces. Let X be a set. A diﬀeology on

X is a set

D of parametrizations in X, that is D ⊂ Param(X), such that

D1. Covering

Every point of X is contained in the range of some P

∈ D.

D2. Locality

If P

∈ Param(X) and if for any r ∈ dom(P) there exists a

domain V such that r

∈ V ⊂ dom(P) and P V ∈ D, then P ∈ D.

D3. Smooth compatibility

If P

∈ D and F is a C

∞

mapping from some

domain V to dom(P), then P

◦ F ∈ D.

Equipped with a diﬀeology

D, X is a diﬀeological space. To make it short, the

elements of the diﬀeology are called the plots of the diﬀeological space. So, the plots
of a diﬀeological space are the elements of its diﬀeology. Note that the deﬁnition
of a diﬀeology does not assume any pre-existing structure on the underlying set.

1.3. Smooth maps and diﬀeomorphisms. Let X and X

be two sets e-

quipped with the diﬀeologies

D and D

respectively. A map F : X

→ Y is said to

be smooth if for each P

∈ D we have F◦P ∈ D

. The set of smooth maps from X to Y

is denoted by

∞

(X, Y). A bijective map F : X

→ Y is said to be a diﬀeomorphism

if both F and F

−1

are smooth. The set of diﬀeomorphisms of X is a group denoted

by Diﬀ(X). Diﬀeological spaces are the objects of the category

{Diﬀeology} whose

morphisms are smooth maps, and isomorphisms are diﬀeomorphisms.

1.4. Quotients and subspaces. The category

{Diﬀeology} is stable by set

theoretic operations. Products, sums of diﬀeological spaces are naturally diﬀeolog-
ical spaces, but also quotient and subsets. Let

∼ be any equivalence relation on

a diﬀeological space X, let Q = X/

∼ and π : X → Q be the projection. There

exists a natural quotient diﬀeology on Q, for which π is smooth, deﬁned by the
parametrizations which can be lifted locally along π by elements of

D. That is, a

parametrization P : U

→ Q is a plot if and only if for each r ∈ U there exists a

domain V containing r and a plot φ : V

→ X such that P V = π ◦ φ. On the other

1. FEW WORDS ABOUT DIFFEOLOGY

hand, there exists on every subset A

⊂ X a natural subset diﬀeology, for which the

inclusion is smooth, deﬁned by the elements of

D which take their values in A.

In the ﬁrst case, the map π : X

→ Q is a subduction, and in the second case the

injection j

: A

→ X is an induction.

1.5. Functional diﬀeology. Let X and X

be two diﬀeological spaces. There

exists on C

∞

(X, X

) a diﬀeology called the functional diﬀeology whose plots are

parametrizations P such that (r, x)

→ P(r)(x), deﬁned on dom(P) × X to X

smooth.

This diﬀeology is the coarsest (e.g.

largest) diﬀeology such that the

evaluation map (f, x)

→ f(x), from C

∞

(X, X

)

× X to X

, is smooth. In particular,

the set of paths C

∞

(R, X), denoted by Paths(X), is naturally a diﬀeological space,

equipped with the functional diﬀeology.

1.6. Diﬀerential forms. Let X be a diﬀeological space. A diﬀerential k-form

on X, for k

≥ 0, is a mapping α which associates to each plot P of X a smooth k-

form on dom(P). That is, if P is an n-plot, α(P) belongs to C

∞

(dom(P), Λ

)).

And satisfying the following compatibility condition: for any plot P of X and for
any smooth parametrization F : V

→ dom(P),

α(P

◦ F) = F

∗

(α(P)).

The space Ω

(X) of diﬀerential k-forms on X is naturally a vector space. It carries

also a natural diﬀeology called again functional diﬀeology for which the ordinary
vectorial operations are smooth. A parametrization r

→ α

of Ω

(X), deﬁned on

a domain U, is a plot for this functional diﬀeology if and only if for any n-plot
P : V

→ X, the parametrization (r, s) → α

(P)

, deﬁned on U

× V with values in

), is smooth.

Note that, if it is necessary for a diﬀerential form to check the compatibility

condition on all the plots of the space, two diﬀerential k-forms coincide if and only
if they coincide on the k-plots. In other words, the value of a diﬀerential k-form is
characterized by its values on the k-plots.

The exterior diﬀerential of a k-form α is the diﬀerential (k + 1)-form deﬁned

dα(P) = d(α(P)).

Let f : X

→ X

be a smooth map between diﬀeological spaces, let α

be a diﬀer-

ential k-form on X

, the pullback f

∗

(α

) is the diﬀerential k-form on X deﬁned by

∗

(α

)(P) = α

◦ P). The exterior diﬀerential and the pullback are linear and

smooth operations.

Let F :

I → Diff(X) be a 1-plot deﬁned on a open interval and centered at the

identity 1

, that is 0

∈ I and F(0) = 1

. Let α be a diﬀerential k-form on X, with

k > 0. The contraction i

(α) of α by F is the (k

− 1)-diﬀerential form deﬁned by

(α)(P)

, . . . , v

) = α

→ F(t)(P(r))

(

0
r

)

· · ·

· · · v

where P is any plot of X, r

∈ dom(P), and v

, . . . , v

are any k

− 1 vectors of R

n being the dimension of the plot P.

Let us continue with the 1-plot F :

I → Diff(X) deﬁned on I and centered at

. Let α be a diﬀerential k-form on X, with k

≥ 0. There exists a diﬀerential

1. FEW WORDS ABOUT DIFFEOLOGY

k-form on X, called the Lie derivative of α by F, deﬁned by

(α)(P)

∂α(F(t)

◦ P)

∂t

t=0

for every n-plot P and every r

∈ dom(P). Note that α(F(t)◦P) is just F(t)

∗

(α)(P),

and regarded as a function of t is smooth from

I to Λ

), so the derivative with

respect to t makes sense. Now, the so called classical Cartan formula extends to
diﬀeology and we have, for any diﬀerential k form α, with k > 0,

(α) = d[i

(α)] + i

(dα).

Let us ﬁx now some vocabulary we shall use in the later paragraphs. We call

automorphism of a diﬀerential k-form α on X any diﬀeomorphism ϕ of X which
preserves α, that is ϕ

∗

(α) = α. The set of all the automorphisms of the form α is

a group denoted by Diff(X, α),

Diff(X, α) =

{ϕ ∈ Diff(X) | ϕ

∗

(α) = α

The group Diff(X, α) will be called the group of automorphisms of α, and any of
its subgroups will be called a group of automorphisms of α.

1.7. Chain-Homotopy operator. Let X be a diﬀeological space. Let ˆ

0 and

1 be the maps deﬁned on Paths(X) to X by

0(p) = p(0)

and

1(p) = p(1).

There exists a smooth linear operator K, called Chain-Homotopy operator such that,
for any integer k > 0,

K : Ω

(X)

→ Ω

−1

(Paths(X))

and

◦ d + d ◦ K = ˆ1

∗

− ˆ0

∗

The value of the chain-homotopy operator K on a diﬀerential k-form α is given by
the following formulas. For k = 1, Kα is a real function

K(α)(p) =

α(p)

(1) dt

with

∈ Ω

(X)

and

∈ Paths(X).

For k > 1, let P : U

→ Paths(X) be a n-plot, let r ∈ U and let v

, . . . , v

be k

− 1

vectors of R

, so

(Kα)(P)

, . . . , v

) =

s
r

→ P(r)(s + t)

(

0
r

)

· · ·

· · · v

dt.

The chain-homotopy operator satisﬁes a natural equivariance relation. Let X

another diﬀeological space and f

∈ C

∞

(X, X

). Let f

∗

: Paths(X)

→ Paths(X

) be

the natural map f

∗

: p

→ f ◦ p. Let K

and K

be the chain-homotopy operators

associated to X and X

, so

◦ f

∗

= (f

∗

)

∗

◦ K

In particular, if X = X

and if f preserves a diﬀerential k-form α, that is f

∗

(α) = α,

then f

∗

preserves the diﬀerential (k

− 1)-form K(α), that is (f

∗

)

∗

(Kα) = Kα.

CHAPTER 2

Diﬀeological groups and momenta

Diﬀeological groups have been ﬁrst introduced as « groupes diﬀ´

erentiels » by

Souriau in [Sou81], [Sou84]. They are, with respect to diﬀeological spaces, what
Lie groups are to manifolds. We remind here their deﬁnition. Then, we propose a
diﬀeological equivalent of the « dual of the Lie algebra » as the space of invariant
1-forms on the group. We don’t consider any duality with a putative diﬀeological
Lie algebra. This is the simpler and the more natural way to work with coadjoint
action and coadjoint orbits in diﬀeology.

2.1. Diﬀeological groups. Let G be a group equipped with a diﬀeology

We say that G is a diﬀeological group, or

D is a group diﬀeology, if and only if the

multiplication as well as the inversion are smooth. That is,

[(g, g

)

→ gg

]

∈ C

∞

× G, G) and [g → g

−1

]

∈ C

∞

(G, G).

Note that if G is a standard manifold, this deﬁnition is nothing but the deﬁnition
of Lie groups. Note that any subgroup of a diﬀeological group, equipped with the
subset diﬀeology, is a diﬀeological group. As well, the quotient of any diﬀeological
group by a normal subgroup is a diﬀeological group for the quotient diﬀeology. We
denote by Hom

∞

(G, G

) the space of smooth homomorphisms from G to another

diﬀeological group G

An important example of diﬀeological group is the groups of all the diﬀeomor-

phisms of a diﬀeological space X, equipped with the functional diﬀeology of group
of diﬀeomorphisms. This diﬀeology is the coarsest group diﬀeology on Diff(X)
such that the evaluation map (f, x)

→ f(x) is smooth. A parametrization P : U →

Diff(X) is a plot if and only if the maps (r, x)

→ P(r)(x) and (r, x) → P(r)

−1

(x)

are smooth.

2.2. Covering diﬀeological groups. Let ˆ

G and G be two diﬀeological

groups. We say that a subduction pr : ˆ

→ G is a group covering if and only

if pr is an homomorphism and the ﬁber K = pr

−1

) is discrete

. Let G be a

connected diﬀeological group. Its universal covering ˜

G has a natural structure of

diﬀeological group such that the subduction π : ˜

→ G is an homomorphism. The

ﬁrst homotopy group π

(G) = ker(π) is a discrete invariant subgroup of ˜

G, so π is

a group covering. Any other connected covering pr : ˆ

→ G is the quotient of the

universal covering by a subgroup K of π

(G). If the subgroup K is normal then pr

is a group covering.

Let us remind that discrete means that the plots (here the plots for the subset diﬀeology)

are locally constant.

PATRICK IGLESIAS-ZEMMOUR

Proof.

This property has been stated originally in [Sou84], [Don84], but let

us remind the general construction given in [Igl85]. Let X be a connected diﬀeo-
logical space, let x

be a point of X, chosen at the base point. Let Paths(X, x

) be

the space of paths starting at x

. First of all, the end map ˆ

1 : p

→ p(1), deﬁned

on Paths(X, x

) is a subduction. The quotient of Paths(X, x

) by the ﬁxed ends

homotopy relation is exactly the universal covering pointed by the constant map

: t

→ x

, over the pointed space (X, x

). The ﬁber over x

is the homotopy

group π

(X, x

). Now if X = G we choose the identity 1

as base point. Thus,

the multiplication of paths (p, p

)

→ [t → p(t) · p

(t)] deﬁnes on ˜

G a group mul-

tiplication such that the projection π : ˜

→ G, deﬁned by π(class(p)) = ˆ1(p), is

an homomorphism. The kernel of this morphism is clearly the ﬁber over 1

, that

is π

(G). Now, the kernel of an homomorphism is always an invariant subgroup.

And, since π is a covering, π

−1

) is discrete. This last points are general results

of the diﬀeological theory of homotopy [Igl85].

2.3. Smooth actions of a diﬀeological group. Let G be a diﬀeological

group. Let X be a diﬀeological space. Let the group Diff(X), of all the diﬀeomor-
phisms of X, be equipped with the functional diﬀeology of group of diﬀeomorphisms.
A smooth action of G on X, or simply an action of G on X, is a smooth homomor-
phism ρ from G to Diff(X), that is ρ

∈ Hom

∞

(G, Diff(X)). Let us ﬁx or remind

some vocabulary used in the following.

(1) We says that the action is eﬀective if ker(ρ) =

(2) The orbits of G are the subsets ρ(G)(x) =

{ρ(g)(x) | g ∈ G}, where x ∈ X.

(3) We call orbit maps of a point x

∈ X, the smooth map ˆx : G → X, deﬁned

by ˆ

x : g

→ ρ(g)(x).

(4) The stabilizer St

(x) of a point x

∈ X is the subgroup of G deﬁned by the

equation ˆ

x(g) = x, g

∈ G.

(5) We say that X is homogeneous for the action ρ of G, or that X is an

homogeneous space of G, for ρ, if and only if the orbit map ˆ

x of some

point x

∈ X is a subduction, thus for every point. In this case, ˆx is

a principal ﬁbration [Igl85] with structure group the stabilizer St

(x).

That is X

G/St

(x), where g

∼ gh with h ∈ St

(x).

Let α be a diﬀerential k-form on X. We say that G acts by automorphisms on (X, α)
if ρ takes it values in Diff(X, α). That is, if ρ(G) is a group of automorphisms of
the diﬀerential form α.

2.4. Covering smooth actions. Let X be a connected diﬀeological space.

Let G be a connected diﬀeological group. Let ρ : G

→ Diff(X) be a smooth

action of G on X. Thus, ρ takes its values in the identity component Diff(X)

◦

comp(1

)

⊂ Diff(X). So, there exists a unique smooth action ˜ρ of the universal

covering ˜

G of G on the universal covering ˜

X of X, covering ρ.

Diff(X)

◦

Diff(X)

◦

Diff(X)

Proof.

The map ρ

◦π is smooth and

G is simply connected. So, thanks to the

monodromy theorem [Igl85], there exists a unique lifting ˜

ρ of ρ

◦ π mappings the

2. DIFFEOLOGICAL GROUPS AND MOMENTA

identity of ˜

G to the identity of

Diff(X)

◦

. Now, this lifting is an homomorphism

because its restriction on ker(π

) and its projection ρ are both homomorphisms.

2.5. Left, right and adjoint actions of a group onto itself. Let G be a

diﬀeological group. We denote by L(g) and R(g) the left and right actions of G onto
itself.

For all g

∈ G,

L(g) : g

→ gg

R(g) : g

→ g

Note that the « right action » is in fact an anti-action. That is, R(gg

) = R(g

)

◦R(g).

The adjoint action of G onto itself is denoted by Ad, and is deﬁned by:

For all g

∈ G, Ad(g) : k → gkg

−1

= L(g)

◦ R(g

−1

)(k).

The maps L and Ad are smooth homomorphisms from G to Diff(G), equipped
with the diﬀeology of group of diﬀeomorphisms. The map R is a smooth anti-
homomorphism from G to Diff(G).

2.6. Momenta of a diﬀeological group. We call left momentum — or sim-

ply momentum — of a diﬀeological group G, any 1-form of G, invariant by the left
action of G onto itself. We denote by

∗

the space of momenta of G. The space of

momenta of a diﬀeological group is naturally a diﬀeological vector space, equipped
with the functional diﬀeology. So,

∗

{α ∈ Ω

(G)

| For all g ∈ G, L(g)

∗

(α) = α

Note that, in spite of what the notation

∗

suggests, the space of momenta of a

diﬀeological group is not deﬁned by some duality. This notation is chosen here just
to remind us the connection with the dual of the Lie algebra in the case of Lie
groups.

2.7. Momenta and connectedness. Let G be a diﬀeological group. Let G

◦

be the identity component of G, that is G

◦

= comp(1

)

⊂ G. So, the pullback

∗

→ G

◦*

of the injection j : G

◦

→ G is an isomorphism. This property is

quite natural but needed to be checked up in our context of diﬀeological groups.

Note

— Said diﬀerently, the space of momenta of a connected diﬀeological

group, or any of its extensions by a discrete group, coincide. In particular, the only
momentum of a discrete group is the zero momentum.

Proof.

Let us check ﬁrst the injectivity. Let α

∈ G

∗

such that j

∗

(α) = 0, and

let P : U

→ G be a plot. Let r

∈ U and let B ⊂ U be a small open ball centered

at r

. Let g

= P(r

). Since B is connected, since L(g

−1

)

◦ P(r

) = 1

, and thanks

to the smoothness of group operations, the parametrization Q = [L(g

−1

)

◦ P] B

is a plot of G

◦

. So, α(Q) = 0. But, α(Q) = α(L(g

−1

)

◦ (P B)) = L(g

−1

)

∗

(α)(P

B) = α(P

B). Thus, α(P B) = 0. Since α vanishes locally at each point of U,

α = 0. And, j

∗

is injective. Now, let us prove the surjectivity. Let α

∈ G

◦*

. For

any component G

of G, let us choose an element g

∈ G

, and the identity for the

identity component. Let P : U

→ G be a plot, an let us assume that U is connected.

So, P(U) is contained in one connected component of G, let us say the component
G

. Let us deﬁne then, ¯

α(P) = α(R(g

−1

)

◦ P). Since R(g

−1

)

◦ P(r) ∈ G

◦

for all

∈ U, this is well deﬁned. Now, since any plot is the sum of its restrictions on the

components of its domain, the map ¯

α extends naturally to every plot of G. Now,

let P : U

→ G be a plot, let V be a domain, and let F ∈ C

∞

(V, U). Let s

∈ V, let

be the component of s

in V, let r

= F(s

), and let U

be the component of

PATRICK IGLESIAS-ZEMMOUR

in U. Let G

be the component of P

◦ F(s

) = P(r

) in G. We have, ¯

α((P

◦ F)

) = ¯

α((P

)

◦ (F V

)) = α(R(g

−1

)

◦ (P U

)

◦ (F V

)) = α([R(g

−1

)

◦ (P

)]

◦(F V

)) = (F

)

∗

[α(R(g

−1

)

◦(P U

)] = (F

)

∗

[ ¯

α(P

)]. So locally,

α(F

◦ P) =

loc

∗

( ¯

α(P)). And if it is satisﬁed locally, it is satisﬁed globally, thus

α(F

◦ P) = F

∗

( ¯

α(P)). The map ¯

α is a well deﬁned diﬀerential 1-form on G. Now,

let us check that ¯

α is invariant by left multiplication. Let g

∈ G, let P : U → G be

a plot, let r

∈ U, let U

be the component of r

in U, let G

be the component

of P(r

) in G, so P(U

)

⊂ G

. We have, L(g)

∗

( ¯

α(P

)) = ¯

α(L(g)

◦ (P U

)) =

α(R(g

−1

)

◦ L(g) ◦ (P U

)) = α(L(g)

◦ R(g

−1

)

◦ (P U

)) = [L(g)

∗

(α)](R(g

−1

)

◦ (P

)) = α(R(g

−1

)

◦ (P U

)) = ¯

α(P

). So locally, L(g)

∗

( ¯

α)(P) =

loc

α(P), and

therefore globally. So, L(g)

∗

( ¯

α) = ¯

α, thus ¯

α is an element of

∗

, which coincide

with α on G

◦

2.8. Momenta of coverings of diﬀeological groups. Let G be a diﬀeo-

logical group, let pr : ˆ

→ G be some group covering, see Subsection 2.2. Let G

∗

and ˆ

∗

be the spaces of momenta of G and ˆ

G. So, the pullback pr

∗

→ ˆG

∗

is a

smooth linear isomorphism.

Proof.

Thanks to Subsection 2.7, it is suﬃcient to assume that ˆ

G and G

are connected. And thanks to Subsection 2.2, it is suﬃcient to prove this for the
universal covering π : ˜

→ G. Now, π

∗

is obviously linear, let us show that π

∗

surjective. Let ˜

∈ G

∗

. The group G is isomorphic to

G/π

(G), with respect to

the left action of π

(G). That is ˜

∼ k˜g, for all k ∈ π

(G). Now, let ˜

∈ G

∗

, ˜

is left invariant by

G, thus by π

(G). That is, for all k

∈ π

(G), L(k)

∗

( ˜

α) = ˜

α.

But, since π

(G) = ker(π) is discrete, this is suﬃcient for the existence of a 1-

form α on G such that ˜

α = π

∗

(α). Now, let ˜

∈

G and g = π(˜

g). Since π is

an homomorphism, π

◦ L(˜g) = L(g) ◦ π. So, on one hand we have L(˜g)

∗

( ˜

α) =

L(˜

∗

(π

∗

(α)) = (π

◦ L(˜g))

∗

(α) = (L(g)

◦ π)

∗

(α) = π

∗

(L(g)

∗

(α)). And, on the other

hand, we have L(˜

∗

( ˜

α) = ˜

α = π

∗

(α). Hence, π

∗

(L(g)

∗

(α)) = π

∗

(α). But, since π is

a subduction, L(g)

∗

(α) = α. Thus, α

∈ G

∗

, and the map π

∗

is surjective. Now, let

α and ˜

β be such that π

∗

( ˜

α) = π

∗

( ˜

β). But, since π is a subduction, ˜

α = ˜

β. Finally,

∗

is injective. Finally, since the pullback is a smooth operation, π

∗

→ G

∗

is a

smooth linear isomorphism.

2.9. Linear coadjoint action and coadjoint orbits. Let G be a diﬀeolog-

ical group and let

∗

be the space of its momenta. The pushforward Ad(g)

∗

(α)

of a momentum α

∈ G

∗

, by the adjoint action of any element g of G, is again a

momentum of G, that is again a left-invariant 1-form. This deﬁnes a linear smooth
action of G on

∗

called coadjoint action, and denoted by Ad

∗

: (g, α)

→ Ad(g)

∗

(α) = Ad(g

−1

)

∗

(α).

We check immediately that for all g, g

in G, Ad

∗

(gg

) = Ad

∗

(g)

◦ Ad

∗

), and

that Ad

∗

(g) is linear. Note that, since α is left-invariant, Ad

∗

(g)(α) = R(g)

∗

(α).

The orbit of α by G is by deﬁnition a coadjoint orbit of G, and it will be denoted

or Ad

∗

(G)(α) =

{Ad

∗

(g)(α)

| g ∈ G}.

The orbit

can be regarded as a subset of

∗

, but also as the quotient of the

group G by the stabilizer of the moment α,

G/St

(α), with St

(α) =

{g ∈ G | Ad(g)

∗

(α) = α

2. DIFFEOLOGICAL GROUPS AND MOMENTA

Note

— The orbit

can be equipped with the subset diﬀeology of the functional

diﬀeology of

∗

, or with the quotient diﬀeology of G. There is no reason a priori that

these two diﬀeologies coincide. But it could be interesting however to understand
in which conditions they do.

2.10. Aﬃne coadjoint actions and (Γ, θ)-coadjoint orbits. Let G be a

diﬀeological group, and

∗

be the space of its momenta. Let Γ

⊂ G

∗

be a subgroup

of (

∗

, +), invariant by the coadjoint action Ad

∗

. That is, for all g

∈ G,

∗

(g)(Γ)

⊂ Γ.

So, the coadjoint action of G on

∗

project to the quotient

∗

/Γ, regarded as an

abelian group, on a smooth action. Let us denote this action by Ad

∗

. For every

∈ G and τ ∈ G

∗

/Γ,

∗

(g)(τ ) = class(Ad

∗

(g)(μ))

with

τ = class(μ)

∈ G

∗

/Γ.

Now, let θ be a smooth map from G to the space

∗

/Γ, such that for any pair g

and g

of elements of G,

θ(gg

) = Ad

∗

(g)(θ(g

)) + θ(g).

Such maps are formally known, in the literature as twisted 1-cocycles of G with
values in

∗

/Γ [Kir74]. We shall call them cocycles of G, with values in

∗

/Γ, or

simply (

∗

/Γ)-cocycles. A cocycle θ is a coboundary if and only if there exists a

constant c

∈ G

∗

/Γ, such that θ = Δc, with

Δc : g

→ Ad

∗

(g)(c)

− c.

Cocycles modulo coboundaries deﬁne a cohomology group denoted by H

(G,

∗

/Γ).

Every such cocycle θ deﬁnes a new action of G on

∗

/Γ by

Γ,θ

∗

: (g, τ )

→ Ad

∗

(g)(τ ) + θ(g).

The orbits for these actions will be called the (Γ, θ)-coadjoint orbits of G. If Γ =

{0}

we shall call them simply θ-coadjoint orbits. If θ = 0 we shall call them simply
Γ-coadjoint orbits. And, if Γ =

{0} and θ = 0 we ﬁnd again the ordinary coadjoint

orbits deﬁned in Subsection 2.9.

2.11. Closed momenta of a diﬀeological group. Let G be a diﬀeological

group, and let

∗

be its space of momenta. Let us denote by Z the subset of closed

momenta of G, and by B the subset of exact momenta of G. That is,

Z = Z

1
DR

(G)

∩ G

∗

and

B = B

1
DR

(G)

∩ G

∗

1) Let us assume that G is connected, and let ˜

G be its universal covering. By

factorization, the chain-homotopy operator deﬁnes a canonical De Rham isomor-
phism k, from the space of closed momenta Z to the vector space Hom

∞

( ˜

G, R).

That is, for all ζ

∈ Z,

k(ζ) = [˜

→ Kζ(p)], where Kζ(p) =

and

g = class(p).

Here, we have denoted by class(p) the ﬁxed ends homotopy class of the path
p

∈ Paths(G, 1

). The subspace of exact momenta B identiﬁes, through the iso-

morphism k, to the subspace Hom

∞

(G, R).

Hom

∞

( ˜

G, R)

and

Hom

∞

(G, R).

PATRICK IGLESIAS-ZEMMOUR

2) Let G be any diﬀeological group connected or not. Let ζ

∈ G

∗

, if ζ is closed

then ζ is Ad

∗

invariant.

For all ζ

∈ G

∗

, dζ = 0

⇒ Ad

∗

(g)(ζ) = ζ, for all g

∈ G.

Note

— Every homomorphism from a diﬀeological group G to an abelian

group factorizes through the abelianized group Ab(G) = G/[G, G], where [G, G] is
the normal subgroup of the commutators of G. So actually, Z

Hom

∞

(Ab( ˜

G), R)

and B

Hom

∞

(Ab(G), R).

Proof.

1) Let π : ˜

→ G be the universal covering deﬁned in Subsection

2.2. Since ˜

G is simply connected, every closed 1-form is exact [Piz05]. Thus, for

every ζ

∈ Z, the pullback π

∗

(ζ) is exact. So, let F be a primitive of π

∗

(α), that

is dF = π

∗

(α). We can even ﬁx uniquely F by choosing F(1

) = 0. Actually

F is deﬁned by integrating the form ζ along the paths starting at the identity,
that is F = k(ζ). Since α is left-invariant and since the projection π commutes
with the left actions, on G and ˜

G, π

∗

(α) is left invariant. So, for every ˜

∈ ˜G,

d[F

◦ L(˜g)] = dF. Since ˜G is connected, for every ˜g, ˜g

in ˜

G, F(˜

g˜

) = F(˜

) + f (˜

g).

Where f is a smooth real function. But since F(1

) = 0, f (˜

g) = F(˜

g), and F is

a smooth homomorphism from ˜

G to R. So, for every closed momentum ζ

∈ Z,

there exists a unique homomorphism F

∈ Hom

∞

( ˜

G, R) such that ζ = π

∗

(dF). The

homomorphism k is thus injective, and it is obviously surjective. Now, if ζ is exact,
that is if ζ = df , then F = π

∗

(f ). So, k(B) = π

∗

(Hom

∞

(G, R))

Hom

∞

(G, R).

2) Thanks to Subsection 2.7 we can assume that G is connected. Now, for every

g, ˜

in ˜

G, F(˜

g˜

−1

) = F(˜

). That is, F

◦ Ad(˜g) = Ad(˜g)

∗

(F) = F, for all ˜

∈ ˜G.

So, d[Ad(˜

∗

(F)] = dF, or Ad

∗

(˜

g)(π

∗

(ζ)) = π

∗

(ζ), or (π

◦ Ad(˜g))

∗

(ζ) = π

∗

(ζ). But

◦ Ad(˜g) = Ad(g) ◦ π, where g = π(˜g). So, π

∗

(Ad(g)

∗

(ζ)) = π

∗

(ζ). And since π

is a subduction, Ad(g)

∗

(ζ) = ζ. That is, Ad

∗

(g)(ζ) = ζ.

2.12. Equivalence between right and left momenta. Let G be a diﬀeo-

logical group, and let

denote the space of right momenta of the group G. That

is, the space of 1-forms of G, invariant by the right multiplication.

{α ∈ Ω

(G)

| For all g ∈ G, R(g)

∗

(α) = α

There exists a natural linear isomorphism flip :

∗

→ G

equivariant with respect

to the coadjoint action. That is, the following diagram commutes.

∗

flip

∗

flip

∗

(g)

∗

(g)

In other words, there is no reason to prefer left or right momenta of a diﬀeological
group. The particularization of left momenta comes because we are dealing with
actions of groups and not anti-actions.

Proof.

Let us denote by a dot the multiplication in G. Let α be any left

p-momentum of G. Let P : U

→ G be a n-plot. Let ¯α(P) be deﬁned by

α(P)(r) = α

→ P(s) · P(r)

−1

(s = r).

2. DIFFEOLOGICAL GROUPS AND MOMENTA

where r belongs to U. Let us show that ¯

α deﬁnes a p-form of G. First of all let us

remark that ¯

α(P) is the restriction of the 1-form α((s, r)

→ P(s) · P(r)

−1

) to the

diagonal s = r. Thus, ¯

α(P) is a smooth 1-form of U.

Now, let us prove that ¯

α is a well deﬁned 1-form on G, according to the

deﬁnition of diﬀerential forms in diﬀeology.

let F : V

→ U be a smooth m-

parametrization. Let v be a point of V, and δv be a vector of R

. We have:

α(P

◦ F)

(δv)

→ (P ◦ F)(s) · (P ◦ F)(v)

−1

(δv)

→ F(s) → (P ◦ F)(s) · (P ◦ F)(v)

−1

(δv)

→ r = F(s) → P(r) · P(F(v))

−1

(δv)

→ P(r) · P(F(v))

−1

◦ F

(δv)

∗

→ P(r) · P(F(v))

−1

(δv)

→ P(r) · P(F(v))

−1

F(v)

(D(F)(v)(δv))

α(P)

F(v)

(D(F)(v)(δv))

∗

[ ¯

α(P)]

(δv).

Then, let us check that ¯

α is right-invariant, that is ¯

∈ G

. For all g

∈ G, we have:

R(g)

∗

( ¯

α)(P)

(δr)

α(R(g)

◦ P)

(δr)

→ (R(g) ◦ P)(s) · (R(g) ◦ P)(r)

−1

(δr)

→ P(s) · g · (P(r) · g)

−1

(δr)

→ P(s) · g · g

−1

· P(r)

−1

(δr)

→ P(s) · P(r)

−1

(δr)

α(P)

(δr)

So, we have deﬁned a map flip : α

→ ¯α, from G

∗

. Let us prove now that flip

is bijective. Let β = ¯

α. Let P : U

→ G be a plot, and let us deﬁne ¯β by

β(P)(r) = β[s

→ P(r)

−1

· P(s)](s = r),

for all r

∈ U. So, we have:

β(P)(r)

→ P(r)

−1

· P(s)

(s = r)

→ P(r)

−1

· P(s)

(s = r)

→ P(r)

−1

· P(s) · P(r)

−1

· P(r)

(s = r)

→ P(r)

−1

· P(s)

(s = r)

L(P(r)

−1

)

∗

(α) [s

→ P(s)] (s = r)

α(P)(r).

Hence, ¯

β = α. Thus, flip is bijective. And, flip is clearly linear. Therefore, flip

is a linear isomorphism from

∗

. It is easy to check that it is a smooth

isomorphism.

Finally, let us check that flip is equivariant under the coadjoint action. Let

∈ G

∗

, let P : U

→ G be a plot and r ∈ U. On one hand we have,

flip[Ad(g)

∗

(α)](P)

flip[R(g)

∗

(α)](P)

R(g)

∗

(α)[s

→ P(s) · P(r)

−1

]

α(s

→ P(s) · P(r)

−1

· g)

PATRICK IGLESIAS-ZEMMOUR

And, on the other hand:

[Ad(g)

∗

(flip(α))](P)

[L(g)

∗

(flip(α))](P)

flip(α)(L(g

−1

)

◦ P)

α[s

→ (L(g

−1

)

◦ P)(s) · (L(g

−1

)

◦ P)(r))

−1

]

α[s

→ g

−1

· P(s) · P(r)

−1

· g]

L(g

−1

)

∗

(α)[s

→ P(s) · P(r)

−1

· g]

α[s

→ P(s) · P(r)

−1

· g]

Therefore, flip

◦ Ad(g)

∗

= Ad(g)

∗

◦ flip for all g ∈ G.

CHAPTER 3

The paths moment map

We shall now introduce the notion of moment map step by step. The ﬁrst step

consists to deﬁne the paths moment map.

3.1. Deﬁnition of the paths moment map. Let X be a diﬀeological space

and ω be a closed 2-form deﬁned on X. Let G be a diﬀeological group and ρ : G

→

Diff(X) be a smooth action. Let us denote by the same letter the natural action
of G on Paths(X), induced by the action ρ of G on X. That is, for all g

∈ G, for

all p

∈ Paths(X),

ρ(g)(p) = ρ(g)

◦ p = [t → ρ(g)(p(t))].

Let us assume now that the action ρ of G on X preserves ω. That is, for all g

∈ G,

ρ(g)

∗

(ω) = ω

∈ Hom

∞

(G, Diff(X, ω)).

Let K be the chain-homotopy operator, so Kω is a 1-form of Paths(X), and the
action of G on Paths(X) preserves the 1-form Kω. This is a consequence of the
variance of the chain-homotopy operator, see Subsection 1.7. Thus, for all g

∈ G,

ρ(g)

∗

(Kω) = Kω.

Now, let p be any paths of X, and let ˆ

p : G

→ Paths(X) be the orbit map. So, the

pullback ˆ

∗

(Kω) is a left-invariant 1-form of G, that is an element of

∗

. The map

Ψ : Paths(X)

→ G

∗

deﬁned by

Ψ(p) = ˆ

∗

(Kω),

is smooth with respect to the functional diﬀeology, Ψ

∈ C

∞

(Paths(X),

∗

). The

map Ψ will be called the paths moment map.

3.2. Evaluation of the paths moment map. Let X be a diﬀeological space

and ω be a closed 2-form deﬁned on X. Let G be a diﬀeological group and ρ be
a smooth action of G on X, preserving ω. Let p be a path in X. Thanks to the
explicit expression of the chain-homotopy operator given in Subsection 1.7, we get
the evaluation of the momentum Ψ(p) on any n-plot P of G,

(

♥)

Ψ(p)(P)

(δr) =

→ (ρ ◦ P)(u)(p(s + t))

(

s=0

u=r

)

1
0

δr

dt,

for all r in dom(P) and all δr in R

. Now, as a diﬀerential 1-form, Ψ(p) is char-

acterized by its values on the 1-plots [Piz05]. So, let f : t

→ f

be a 1-plot of G

centered at the identity 1

, that is f

∈ Paths(G) and f(0) = 1

. For any t

∈ R,

let F

be the path in Diff(X, ω) — centered at the identity 1

— deﬁned by

: s

→ ρ(f

−1

◦ f

t+s

PATRICK IGLESIAS-ZEMMOUR

So, we have

(

♣)

Ψ(p)(f )

(1) =

−

(ω) =

−

(ω)(p)

(1)ds,

where i

(ω) is the contraction of ω by F

, see Subsection 1.6.

But, as an invariant 1-form on G the moment Ψ(p) is characterized by its value

at the identity, that is for t = 0,

(

♦)

Ψ(p)(f )

(1) =

−

(ω) =

−

(ω)(p)

(1) dt

with

F = ρ

◦ f.

Note

— Let f

∈ Hom

∞

(R, G), so Ψ(p)(f ) is an invariant 1-form on R whose

coeﬃcient is just

(ω). That is,

Ψ(p)(f ) = h

(p)

× dt where h

(p) =

−

(ω).

The smooth map h

: Paths(X)

→ R is the hamiltonian of f, or the hamiltonian

of the 1-parameter group f (R). Note also that, the map h : Hom

∞

(R, G)

→

∞

(Paths(X), R), deﬁned above, is smooth.

Proof.

Let us prove

♥. Let us remind that for every p ∈ Paths(X) and every

∈ G, ˆp(g) = ρ(g)(p) = [t → ρ(g)(p(t))]. So, by deﬁnition

Ψ(p)(P)

(δr)

∗

(Kω)

(δr)

Kω(ˆ

◦ P)

(δr)

s
r

→ ˆp ◦ P(r)(s + t)

(

0
r

)

1
0

δr

s
r

→ (ρ ◦ P)(r)(p(s + t))

(

0
r

)

1
0

δr

dt.

Let us prove

♣. Let us apply the general formula ♥ for P = f. Introducing

= u

−t and s

= s + s

, using the compatibility property of ω(P

◦Q) = Q

∗

(ω(P))

3. THE PATHS MOMENT MAP

and the ρ(f

) invariance of ω, we get

Ψ(p)(f )

(1)

→ ρ(f

)(p(s + s

))

(

s=0
u=t

)

1
0

0
1

→ ρ(f

t+u

)(p(s

))

s=s

u =0

1
0

0
1

→ ρ(f

◦ f

−1

◦ f

t+u

)(p(s

))

s=s

u =0

1
0

0
1

→ ρ(f

)

)(p(s

))

s=s

u =0

1
0

0
1

→ F

)(p(s

))

s=s

u =0

1
0

0
1

→ F

)(p(s

))

u =0

s=s

0
1

1
0

−

→ F

)(p(s

))

u =0

s =s

1
0

0
1

−

(ω)(p)

(1)ds

−

(ω).

Let us prove the Note. Let f

∈ Hom

∞

(R, G). By deﬁnition of diﬀerential forms

and pullbacks, Ψ(p)(f ) = f

∗

(Ψ(p)), but since f is an homomorphism from R

to Diff(X, ω) and Ψ(p) is a left-invariant 1-form on Diff(X, ω), f

∗

(Ψ(p)) is an

invariant 1-form of R, so Ψ(p)(f ) = f

∗

(Ψ(p)) = a

× dt, for some real a. So,

Ψ(p)(f )

= Ψ(p)(f )

(1)

× dt = h

(p)

× dt, with h

(p) = Ψ(p)(f )

(1) =

−

(ω),

and dt is the canonical 1-form on R.

3.3. Variance of the paths moment map. Let X be a diﬀeological space

and ω be a closed 2-form deﬁned on X. Let G be a diﬀeological group and ρ be
a smooth action of G on X, preserving ω. The paths moment map Ψ, deﬁned in
Subsection 3.1, is equivariant under the action of G. That is, for all g

∈ G,

◦ ρ(g)

∗

= Ad(g)

∗

◦ Ψ.

Proof.

Let us denote here the orbit map ˆ

p of every path p

∈ Paths(X) by L(p).

That is, L(p)(g) = ρ(g)

∗

(p) = ρ(g)

◦ p. So, Ψ(ρ(g)

∗

(p)) = Ψ(ρ(g)

◦ p) = (L(ρ(g) ◦

∗

(Kω). But, L(ρ(g)

◦ p)(g

) = ρ(g

)(ρ(g)

◦ p) = ρ(g

◦ p = L(p)(g

g) = L(p)

◦

R(g)(g

). Thus, L(ρ(g)

◦ p) = L(p) ◦ R(g), and Ψ(ρ(g)

∗

(p)) = (L(p)

◦ R(g))

∗

(Kω) =

R(g)

∗

(L(p)

∗

(K(p)) = R(g)

∗

(Ψ(p)). But since Ψ(p) is left-invariant, R(g)

∗

(Ψ(p)) =

Ad(g)

∗

(Ψ(p)), and Ψ(ρ(g)

∗

(p)) = Ad(g)

∗

(Ψ(p)).

3.4. Additivity of the paths moment map. Let X be a diﬀeological space

and ω be a closed 2-form deﬁned on X. Let G be a diﬀeological group and ρ be
a smooth action of G on X, preserving ω. The paths moment map Ψ, deﬁned in

PATRICK IGLESIAS-ZEMMOUR

Subsection 3.1, satisﬁes the following additive property: for any two juxtaposable
paths p and p

in X,

Ψ(p

∨ p

) = Ψ(p) + Ψ(p

)

and

Ψ(¯

p) =

−Ψ(p), with ¯p(t) = p(1 − t).

Proof.

This is a direct application of the expression given in Subsection 3.2

♦, and of the additivity of the integral of diﬀerential form on paths.

3.5. Diﬀerential of the paths moment map. Let X be a diﬀeological space

and ω be a closed 2-form deﬁned on X. Let G be a diﬀeological group and ρ be
a smooth action of G on X, preserving ω. Let p be a path in X. So, the exterior
diﬀerential of the paths momentum Ψ(p) is given by

d(Ψ(p)) = ˆ

∗

(ω)

− ˆx

∗

(ω),

where x

= p(0) and x

= p(1), and the ˆ

denote the orbit maps.

Proof.

This is a direct application of the main property of the chain-homotopy

operator, d

◦ K + K ◦ d = ˆ1

∗

− ˆ0

∗

. Since dω = 0, we have d(Kω) = ˆ

∗

(ω)

− ˆ0

∗

(ω),

composed with ˆ

∗

, we get ˆ

∗

◦ d(Kω) = ˆp

∗

◦ ˆ1

∗

(ω)

− ˆp

∗

◦ ˆ0

∗

(ω). That is d(ˆ

∗

(Kω)) =

(ˆ

◦ ˆp)

∗

(ω)

− (ˆ0 ◦ ˆp)

∗

(ω). Thus, d(Ψ(p)) = ˆ

∗

(ω)

− ˆx

∗

(ω).

3.6. Homotopic invariance of the paths moment map. Let X be a dif-

feological space and ω be a closed 2-form deﬁned on X. Let G be a diﬀeological
group and ρ be a smooth action of G on X, preserving ω. Let p

and p

be any two

paths in X. If p

and p

are ﬁxed ends homotopic, then Ψ(p

) = Ψ(p

Proof.

Let s

→ p

be a ﬁxed ends homotopy connecting p

to p

, for example

let p

(0) = x

and p

(1) = x

, for all s. Let f be a 1-plot of G centered at the

identity 1

, that is f (0) = 1

, and let F = ρ

◦ f. We use the fact that the moment

of paths is characterized by its value at the identity, Ψ(p

)(f )

(1) =

−

(ω),

see Subsection 3.2

♦. Let us diﬀerentiate this equality with respect to s,

∂

∂s

Ψ(p

)(f )

(1)

−δ

(ω),

with

δ =

∂

∂s

The variation of the integral of diﬀerential forms on chains gives

(ω) =

d [i

(ω)](δp

) +

(ω)(δp

)

See [Piz05] for the deﬁnition of δp

and for the proof of this formula in diﬀeology.

Since the homotopy s

→ p

is a ﬁxed end homotopy, δp

(0) = 0 and δp

(1) = 0,

thus the second summand of the right term vanishes. Now, the Cartan formula
writes £

(ω) = d[i

(ω)] + i

(dω), see Subsection 1.6. But ω is invariant under the

action of G, so £

(ω) = 0, and since dω = 0 we get d[i

(ω)] = £

(ω) = 0. So,

(ω) = 0 and Ψ(p

) = Ψ(p

), for all s.

3. THE PATHS MOMENT MAP

3.7. The holonomy group. Let X be a connected diﬀeological space, and

let ω be a closed 2-form deﬁned on X. Let G be a diﬀeological group and ρ be a
smooth action of G on X, preserving ω. Let Ψ be the paths moment map deﬁned
in Subsection 3.1. We deﬁne the holonomy Γ of the action ρ as

Γ =

{Ψ() | ∈ Loops(X)}.

(1) The holonomy Γ is an additive subgroup of the subspace of closed mo-

menta, Γ

⊂ Z (see Subsection 2.11). That is, for every elements γ and γ

of Γ,

dγ = 0

and

− γ

∈ Γ.

(2) The paths moment map Ψ, restricted to Loops(X), factorizes through an

homomorphism from π

(X) to

∗

. Thus, Γ is an homomorphic image of

(X), or its abelianized Ab(π

(X)).

(3) In particular, every element γ of Γ is invariant by the coadjoint action of

G on

∗

. For all g in G,

∗

(g)(γ) = γ.

The holonomy Γ is the obstruction for the action ρ to be « hamiltonian ». Precisely,
the action of G on X will be said to be hamiltonian if and only if Γ =

{0}. Note that,

if the group G has no Ad

∗

-invariant 1-forms except 0, the action ρ is necessarily

hamiltonian, see Subsection 2.11.

Proof.

We get immediately that γ

∈ Γ is closed, by application of the diﬀer-

ential of a path momentum: for all path p

∈ Paths(X), d(Ψ(p)) = ˆx

∗

(ω)

− ˆx

∗

(ω),

where x

= p(0) and x

= p(1), see Subsection 3.5. So, for any loop of X,

(0) = (1) and d(Ψ()) = 0. Now, let x

be any point of X. Thanks to Sub-

section 3.6, for every loop

∈ Loops(X, x

), the momentum Ψ() depends on

only through the its homotopy class. So Γ is the image of π

(X, x

). And, thanks

to the additive property of Ψ, see Subsection 3.4, the map class()

→ Ψ() is an

homomorphism. Now, since X is connected, for every other point x

of X, there

exists a path c connecting x

to x

, and let ¯

c = t

→ c(1 − t). Thanks to the

additive property, Ψ(¯

∨ ∨ c) = Ψ(¯c) + Ψ() + Ψ(c) = −Ψ(c) + Ψ() + Ψ(c) = Ψ().

And, since the map class()

→ class(¯c ∨ ∨ c) is a conjugation from π

(X, x

) to

(X, x

), Γ is the same homomorphic image of π

(X, x), for every point x

∈ X. So,

we proved the points 1 and 2, the third one is a direct consequence of Subsection
2.11.

CHAPTER 4

The 2-points moment map

The deﬁnition of the paths moment map leads immediately to the 2-points

moment map. The 2-points moment map satisﬁes a cocycle condition inherited
from the additive property of the paths moment map. This is the second step in
our general construction.

4.1. Deﬁnition of the 2-points moment map. Let X be a connected diﬀe-

ological space and ω be a closed 2-form deﬁned on X. Let G be a diﬀeological group
and ρ be a smooth action of G on X, preserving ω. Let Ψ be the paths moment
map and Γ be the holonomy of the action ρ, see Subsection 3.1 and Subsection 3.7.
So, there exists a smooth map ψ : X

× X → G

∗

/Γ such that the following diagram

commutes.

× X

∗

/Γ

Paths(X)

∗

ends

where pr is the canonical projection from

∗

onto its quotient, and ends = ˆ

× ˆ1,

that is ends(p) = (p(0), p(1)). The map ψ

∈ C

∞

× X, G

∗

/Γ) will be called the

2-points moment map.

(1) The 2-points moment map ψ satisﬁes the Chasles cocycle relation, for any

three points x, x

, x

of X,

(

♥)

ψ(x, x

) = ψ(x, x

) + ψ(x

, x

(2) The 2-points moment map ψ is equivariant under the action of G. That

is, for any g

∈ G, and any pair of points x and x

of X,

ψ(ρ(g)(x), ρ(g)(x

)) = Ad

∗

(g)(ψ(x, x

)).

Proof.

By construction ψ is deﬁned by ψ(x, x

) = class

(Ψ(p)), where p

∈

Paths(X), x = p(0), x

= p(1), and class

(α) denotes the class of α

∈ G

∗

/Γ.

The map ψ is smooth simply by general properties of subductions in diﬀeology.
Now, the ﬁrst point is a direct consequence of the additive property of the paths
moment map, see Subsection 3.4. The second point is a direct consequence of the
equivariance of the paths moment map of the Ad

∗

invariance of Γ, see Subsection

3.3, and of the deﬁnition of the Ad

∗

action, see Subsection 2.10.

Note

— T. Ratiu and A. Weinstein have kindly pointed out that Condevaux,

Dazord and Molino [CDM88] proposed a similar construction in the case where X
is a manifold, G is a Lie group, and Γ is closed in

∗

CHAPTER 5

The moment maps

From the construction of the paths moment map of Subsection 3.1 and the 2-

points moment map of Subsection 4.1 we get the notion of (1-point) moment map.
This is the third step of our general construction, and the generalization of the
notion of moment map coming from classical symplectic geometry.

5.1. Deﬁnition of the moment maps. Let X be a connected diﬀeological

space and let ω be a closed 2-form deﬁned on X. Let G be a diﬀeological group and
ρ be a smooth action of G on X, preserving ω. Let ψ be the 2-points moment map
deﬁned in Subsection 4.1. There exists always a smooth map μ : X

→ G

∗

/Γ, called

a primitive of ψ, such that, for any two points x and x

of X,

ψ(x, x

) = μ(x

)

− μ(x).

For every point x

∈ X, for every constant c ∈ G

∗

/Γ, the map μ deﬁned by

μ(x) = ψ(x

, x) + c.

is a primitive of ψ. Every primitive μ of ψ is of this kind, and any two primitive μ
and μ

of ψ diﬀer only by a constant.

The 2-points moment map ψ will be said to be exact if there exists a primitive

μ, equivariant by the action of G. That is, if there exists a primitive μ such that

◦ ρ(g) = Ad

∗

(g)

◦ μ,

for all g

∈ G. The primitives μ of ψ, equivariant or not, will be called the moment

maps

Note

— By the identity

♥ of Subsection 4.1, ψ is a 1-cocycle of the G-

equivariant cohomology of X with coeﬃcients in

∗

/Γ, twisted by the coadjoint

action. Two cocycles ψ and ψ

are cohomologous if and only if, there exists a

smooth equivariant map μ : X

→ G

∗

/Γ, such that ψ

(x, x

) = ψ(x, x

) + Δμ(x, x

)

where Δμ(x, x

) = μ(x

)

−μ(x), Δμ is a coboundary. So, the 2-points moment map

ψ deﬁnes a class belonging to H

1
G

(X,

∗

/Γ) which depends only on the form ω and

the action ρ of G on X. If the moment map ψ is exact, that is if class(ψ) = 0,
we shall say that the action ρ of G on X is exact, with respect to ω. In this case,
there exists a point x

of X and a constant c such that μ : x

→ ψ(x

, x) + c is an

equivariant primitive for ψ.

Proof.

Let x

be a chosen point of X. Since X is connected, for any x

∈ X

there exists always a path p

∈ X such that p(0) = x

and p(1) = x. Thus, deﬁning

μ(x) = ψ(x

, x) = class(Ψ(p)), and thanks to the cocycle properties of ψ, we have

ψ(x, x

) = ψ(x, x

) + ψ(x

, x

) = ψ(x

, x

)

− ψ(x

, x) = μ(x

)

− μ(x). Now, since ψ

These maps should have been called the 1-point moment maps. But to conform with the

usual denomination we chose to call them simply moment maps.

PATRICK IGLESIAS-ZEMMOUR

is smooth, μ is smooth. Therefore, the equation ψ(x

, x) = μ(x

)

− μ(x) has always

a solution in μ.

Now, let μ and μ

be two primitives of ψ. For each pair x, x

of points of X we

have μ

)

− μ

(x) = μ(x

)

− μ(x). That is, μ

)

− μ(x

) = μ

(x)

− μ(x). So, the

map x

→ μ

(x)

−μ(x) is constant. There exists c ∈ G

∗

/Γ such that μ

(x)

−μ(x) = c,

that is μ

(x) = μ(x) + c.

Since, the maps x

→ ψ(x

, x), where x

is a ﬁxed point of X, is a special solution

of the equation in μ, ψ(x

, x) = μ(x

)

−μ(x), any solution writes μ(x) = ψ(x

, x) +c

for some point x

∈ X and some constant c ∈ G

∗

/Γ.

5.2. Souriau’s cocycles. Let X be a connected diﬀeological space and ω be

a closed 2-form deﬁned on X. Let G be a diﬀeological group and ρ be a smooth
action of G on X, preserving ω. Let ψ be the 2-points moment map deﬁned in
Subsection 4.1 and let μ be a primitive of ψ as deﬁned in Subsection 5.1. So there
exists a map θ

∈ C

∞

(G,

∗

/Γ) such that

μ(ρ(g)(x)) = Ad

∗

(g)(μ(x)) + θ(g).

The map θ is a (

∗

/Γ)-cocycle, as deﬁned in Subsection 2.10. For all g, g

∈ G,

θ(gg

) = Ad

∗

(g)(θ(g

)) + θ(g).

We shall call the cocycle θ, Souriau’s cocycle of the moment μ.

(1) Two Souriau’s cocycles θ and θ

, associated to two moment maps μ and

are cohomologous. That is, they diﬀer by a coboundary

Δc : g

→ Ad

∗

(g)(c)

− c, where c ∈ G

∗

/Γ.

(2) For the aﬃne coadjoint action of G on

∗

/Γ deﬁned by θ, see Subsection

2.10, the moment map μ is equivariant. For all g

∈ G,

◦ ρ(g) = Ad

Γ,θ

∗

(g)

◦ μ.

(3) For every cocycle θ, associated to some moment μ, there exists always a

point x

∈ X and a constant c ∈ G

∗

/Γ such that, for all g in G

θ(g) = ψ(x

, ρ(g)(x

)) + Δc(g).

(4) The cohomology class σ of θ belongs to a cohomology group denoted by

(G,

∗

/Γ). And, it depends only on the cohomology class of the 2-points

moment map ψ. This class σ will be called Souriau’s cohomology class.

Note 1

— Let x

by some point of X. The 2-moment map (1-cocycle) ψ deﬁnes

a 1-cocycle f from G to

∗

/Γ by f (g, g

) = ψ(ρ(g)(x

), ρ(g

)(x

)). The cocycle f

associated to another point x

will diﬀer just by a coboundary. So, Souriau’s cocycle

σ represents just the class of this pullback f = ˆ

∗

(ψ) by the orbit map ˆ

, where

∗

: H

(X,

∗

/Γ)

→ H

(G,

∗

/Γ). And, by the way, depends only of the restriction

of ω on any one orbit of G on X. So, a good choice of the point x

can simplify

sometimes the computation of σ.

Note 2

— The nature of the action ρ has strong consequences on Souriau’s

class. For example, thanks to the third item, if the group G has a ﬁxed point x

that is ρ(g)(x

) = x

for all g in G, then Souriau’s class vanishes. So, the cocycle

ψ is exact, and there exists an equivariant primitive μ of ψ.

5. THE MOMENT MAPS

Proof.

Thanks to Subsection 5.1, every moment map μ writes μ(x) = ψ(x

, x)

+c, where x

is some ﬁxed point of X and c

∈ G

∗

/Γ. So, μ(ρ(g)(x))

−Ad

∗

(g)(μ(x)) =

ψ(x

, ρ(g)(x)) + c

− Ad

∗

(g)(ψ(x

, x) + c) = ψ(x

, ρ(g)(x)) + c

− Ad

∗

(g)(ψ(x

, x))

−

∗

(g)(c) = ψ(x

, ρ(g)(x))

− ψ(ρ(g)(x

), ρ(g)(x))

− Δc(g) = ψ(x

, ρ(g)(x)) +

ψ(ρ(g)(x), ρ(g)(x

))

− Δc(g) = ψ(x

, ρ(g)(x

))

− Δc(g). Therefore, μ(ρ(g)(x)) −

∗

(g)(μ(x)) is constant with respect to x. That proves the points 1) and 4).

Now, the variance of θ with respect to the multiplication of G is a classical result of
cohomology (see for example [Kir74]). It is then obvious that two moment maps
μ and μ

diﬀering just by a constant, the associated cocycles θ and θ

diﬀer by a

coboundary. The remaining items are just the results of elementary, or well known,
algebraic computations.

CHAPTER 6

The moment maps for exact 2-forms

The special case where the closed 2-form is the exterior diﬀerential of an in-

variant 1-form deserves a special care, since it justiﬁes the constructions above, by
analogy with the moment maps of classical symplectic geometry.

6.1. The exact case. Let X be a connected diﬀeological space and let ω be

a closed 2-form deﬁned on X. Let G be a diﬀeological group and ρ be a smooth
action of G on X, preserving ω. Let us assume that ω = dα and that α is also
invariant under the action of G, that is ρ(g)

∗

(α) = α for all g in G. Let Ψ be the

paths moment map deﬁned in Subsection 3.1, and ψ be the 2-points moment map
deﬁned in Subsection 4.1. So, for every p

∈ Paths(X)

Ψ(p) = ψ(x, x

) = ˆ

∗

(α)

− ˆx

∗

(α),

where x

= p(1) and x

= x

. Moreover, the 2-points moment map ψ is exact, and

every equivariant moment map is cohomologous to

μ : x

→ ˆx

∗

(α).

The action of G is hamiltonian, Γ =

{0} and exact σ = 0, see Subsection 3.7 and

Subsection 5.2. So, this shows in particular the coherence of the general construc-
tions developed until now.

Proof.

By deﬁnition of the paths moment map, Ψ(p) = ˆ

∗

(Kω). So, Ψ(p) =

∗

(K(dα)). But, K(dα) + d(Kα) = ˆ

∗

(α)

− ˆ0

∗

(α), thus K(dα) = ˆ

∗

[ˆ

∗

(α)

− ˆ0

∗

(α)

−

d(Kα)]. And, Ψ(p) = (ˆ

◦ ˆp)

∗

(α)

− (ˆ0 ◦ ˆp)

∗

(α)

− d[ˆp

∗

(K(α))]. But, ˆ

◦ ˆp = ˆx

, and

◦ ˆp = ˆx

. So Ψ(p) = ˆ

∗

(α)

− ˆx

∗

(α)

− d[ˆp

∗

(Kα)]. Now, Kα is the real function

Kα : p

→

α.

Since ˆ

∗

(Kα) = Kα

◦ ˆp, we have for all g ∈ G,

Kα(ˆ

p(g)) =

ρ(g)

◦p

α =

ρ(g)

∗

(α) =

α.

So, the function ˆ

∗

(Kα) : G

→ R is constant and equal to

α. So, d[ˆ

∗

(Kα)] = 0,

and Ψ(p) = ˆ

∗

(α)

− ˆx

∗

(α). Thus, Ψ(p) = ψ(x

, x

) and Γ =

{0}.

Now, the function μ : x

→ ˆx

∗

(α) is clearly a primitive of ψ. That is, ψ(x

, x

) =

μ(x

)

− μ(x

). But R(ρ(g)(x)) = ˆ

◦ R(g), where R(ρ(g)(x)) denotes the orbit

map of ρ(g)(x), with g

∈ G. So, μ(ρ(g)(x)) = (ˆx ◦ R(g))

∗

(α) = R(g)

∗

(ˆ

∗

(α)) =

R(g)

∗

(μ(x)) = Ad

∗

(g)(μ(x)).

Thus, μ is an equivariant primitive of ψ.

And,

Souriau’s class σ vanishes.

CHAPTER 7

Functoriality of the moment maps

We inspect now, the behavior of the moment maps and the various associated

objects under natural transformations.

7.1. Images of the moment maps by morphisms. Let X be a connected

diﬀeological space and ω be a closed 2-form deﬁned on X. Let G be a diﬀeological
group and ρ be a smooth action of G on X, preserving ω. Let G

be another

diﬀeological group, and let h : G

→ G be a smooth homomorphism. Let ρ

= ρ

◦ h

be the induced action of G

on X. Let us remind that the pullback h

∗

→ G

∗

a linear smooth map.

(1) Let Ψ : Paths(X)

→ G, and Ψ

: Paths(X)

→ G

be the paths moment

map with respect to the actions of G and G

on X. So, Ψ

= h

∗

◦ Ψ.

(2) Let Γ and Γ

be the holonomy groups with respect to the actions of G and

on X. So, Γ

= h

∗

(Γ).

(3) The linear map h

∗

projects on a smooth homomorphism h

∗

G/Γ →

∗

/Γ

, such that the following diagram commutes.

∗

/Γ

∗

/Γ

∗

(4) Let ψ and ψ

be the 2-points moment maps with respect to the actions of

G and G

. So, ψ

= h

∗

◦ ψ.

(5) Let μ be a moment map relative to the action ρ of G. So μ

= h

∗

◦ μ is a

moment map relative to the action ρ

of G

(6) Let μ be a moment map relative to the action ρ of G, and let μ

= μ

◦ h

∗

be the associated moment map relative to the action ρ

of G

. So, the

associated Souriau’s cocycles satisfy θ

= h

∗

◦ θ ◦ h, summarized by the

following commutative diagram.

∗

/Γ

∗

/Γ

∗

Said diﬀerently, if θ is Souriau’s cocycle associated to a moment μ of the
action ρ of G, and μ

is a moment of the action ρ

of G

, so θ

and h

∗

◦θ ◦h

are cohomologous.

Note

— Thanks to the identiﬁcation between the space of momenta of a

diﬀeological group and any of its extensions by a discrete group, stated in Subsection

PATRICK IGLESIAS-ZEMMOUR

2.7, the moment maps of the action of a group or the moment map of the restriction
of this action to its identity component coincide. Said diﬀerently, the moment maps
doesn’t say anything about actions of discrete groups.

Proof.

To avoid confusion, let us denote by R(p) and R

(p) the orbit maps of

G and G

of p

∈ Paths(X). That is, R(p)(g) = ρ(g) ◦ p and R

(p)(g) = ρ

(g)

◦ p. So,

we have, R

(p)(g) = ρ

(g)

◦ p = ρ(h(g)) ◦ p = (R(p) ◦ h)(g)). Thus, R

(p) = R(p)

◦ h.

1. By deﬁnition of the paths moment map, we have Ψ

(p) = R

(p)

∗

(Kω) =

(R(p)

◦ h)

∗

(Kω) = h

∗

(R(p)

∗

(Kω)) = h

∗

(Ψ(p)). Thus, Ψ

= h

∗

◦ Ψ.

2. Since Γ

= Ψ

(Loops(X)), and thanks to item 1, we have Γ

= h

∗

(Γ).

3. The map h

∗

is deﬁned by class

(α)

→ class

∗

(α)), for all α

∈ G

∗

. If

β = α + γ, with γ

∈ Γ, then h

∗

(β) = h

∗

(α) + γ

, with γ

= h

∗

(γ)

∈ Γ

(item 2). So,

class

∗

(β)) = class

∗

(α)). And, h

∗

is well deﬁned. Thanks to the linearity

of h

∗

, h

∗

is clearly an homomorphism. And, for

∗

/Γ and

∗

/Γ

equipped with the

quotient diﬀeologies, h

∗

is naturally smooth.

4. With to the notations above, ψ and ψ

are deﬁned by, pr

◦ Ψ = ψ ◦ ends and

◦ Ψ

= ψ

◦ ends, where ends(p) = ˆ0 × ˆ1(p) = (p(0), p(1)), with p ∈ Paths(X).

So, by item 1 and 3, we have pr

◦h

∗

◦Ψ = h

∗

◦ψ◦pr. That is, pr

◦Ψ

= (h

∗

◦ψ)◦pr.

So, h

∗

◦ ψ = ψ

5. Let μ

= h

∗

◦ μ, and let x, y ∈ X. So, μ

(y)

− μ

(x) = h

∗

◦ μ(y) − h

∗

◦ μ(y) =

∗

(μ(y)

− μ(x)) = h

∗

◦ ψ(y, x) = ψ

(y, x). So, μ

is a moment map for the action

of G.

6. According to Subsection 5.2, there exists a point x

∈ X such that, for

all g

∈ G

, θ

) = ψ

, ρ

)(x

)). So, thanks to the previous items we have,

) = (h

∗

◦ψ)(x

, ρ(h(g

))(x

)) = h

∗

(ψ(x

, ρ(h(g

))(x

))) = h

∗

(θ(h(g

))) = (h

∗

◦

◦ h)(g

). Thus, we get θ

= h

∗

◦ θ ◦ h

7.2. Pushing forward moment maps. Let X and X

be two connected

diﬀeological spaces. Let ω and ω

be two closed 2-forms deﬁned respectively on

X and X

. Let G be a diﬀeological group, let ρ be a smooth action of G on X,

preserving ω, and let ρ

be a smooth action of the same group G on X

, preserving

. Let f : X

→ X

be a smooth map such that ω = f

∗

(ω

), and f

◦ ρ(g) = ρ

(g)

◦ f,

for all g

∈ G.

(1) Let f

∗

: Paths(X)

→ Paths(X

) deﬁned by f

∗

(p) = f

◦ p. So, the paths

moment maps Ψ and Ψ

relative to the action ρ and ρ

are related by

Ψ = Ψ

◦ f

∗

and the associated holonomy groups Γ and Γ

satisfy

Γ =

{Ψ

◦ ) | ∈ Loops(X)} ⊂ Γ

(2) Let φ :

∗

/Γ

→ G

∗

/Γ

be the projection induced by the inclusion Γ

⊂ Γ

Let ψ and ψ

be the 2-points moment maps relative to the actions ρ and

. So, for all pairs of points x

, x

of X,

(f (x

), f (x

)) = φ(ψ(x

, x

)).

(3) For every moment map μ relative to the action ρ, there exists a moment

map μ

relative to the action ρ

, such that

◦ f = φ ◦ μ.

7. FUNCTORIALITY OF THE MOMENT MAPS

(4) Let θ and θ

be two Souriau’s cocycles relative to the actions ρ and ρ

So, the map φ

◦ θ is a Souriau cocycle, cohomologous to θ

. Thus, the

two Souriau’s classes σ and σ

satisfy σ

= φ

∗

(σ). Where φ

∗

denotes the

action of φ on cohomology, φ

∗

(class(θ)) = class(φ

◦ θ).

Proof.

1. By deﬁnition Ψ(p) = ˆ

∗

(Kω), that is Ψ(p) = ˆ

∗

(K(f

∗

(ω

))). And

thanks to the variance of the chain-homotopy operator K

◦ f

∗

= (f

∗

)

∗

◦ K

, see

Subsection 1.7, we have Ψ(p) = ˆ

∗

◦ (f

∗

)

∗

) = (f

∗

◦ ˆp)

∗

). But, for all

∈ G, f

∗

◦ ˆp(g) = f ◦ ρ(g) ◦ p = ρ

(g)

◦ f ◦ p = ˆp

(g), where p

= f

◦ p. So,

Ψ(p) = ˆ

∗

) = Ψ

∗

(p)). Therefore, Ψ = Ψ

◦ f

∗

. Now, by deﬁ-

nition of the holonomy groups, Γ = Ψ(Loops(X)) = Ψ

∗

(Loops(X))), and since

∗

(Loops(X))

⊂ Loops(X

), we get Γ

⊂ Γ

2. Since Γ

⊂ Γ

, the map φ : class

(α)

→ class

(α), from

∗

/Γ

→ G

∗

/Γ

is well deﬁned. Now, let x

= f (x

) and x

= f (x

), there exists p

∈ Paths(X)

connecting x

to x

. So the path f

∗

(p) connects x

to x

. Thus, by deﬁnition

of ψ

, ψ

, x

) = class

(Ψ

)) = class

(Ψ

◦ f

∗

(p)), and thanks to the ﬁrst

item, class

(Ψ

)) = class

(Ψ(p)) = φ(class

(Ψ(p))). But class

(Ψ(p)) =

ψ(x

, x

). So, ψ

, x

) = φ(ψ(x

, x

)), that is ψ

(f (x

), f (x

)) = ψ(x

, x

3. According to Subsection 5.1, for every moment map μ there exists a point

∈ X and a constant c ∈ G

∗

/Γ such that μ(x) = ψ(x

, x) + c . Let us deﬁne μ

) = ψ

, x

) + c

, where x

= f (x

) and c

= φ(c). So, thanks to the item 2,

(f (x

), f (x)) = φ(ψ(x

, x)), so μ

(f (x)) = φ(ψ(x

, x)) + φ(c) = φ(ψ(x

, x) + c) =

φ(μ(x)). Thus, μ

satisﬁes μ

◦ f = φ ◦ μ.

4. Let θ be a Souriau cocycle for the action ρ. According to Subsection 5.2,

θ is cohomologous to ϑ : g

→ ψ(x

, ρ(g)(x)), where x

is some point of X. So, let

= f (x

), and ϑ

: g

→ ψ

, ρ

(g)(x

)). Thus, ϑ

(g) = ψ

(f (x

), ρ

(g)(f (x

))) =

(f (x

), f (ρ(g)(x

))) = φ(ψ(x

, ρ(g)(x

))) = φ

◦ ϑ(g). Now since all Souriau’s

cocycles, with respect to a given action of G, are cohomologous, the cocycle θ

cohomologous to ϑ

, and then cohomologous to φ

◦ ϑ, and thus to φ ◦ θ. Therefore,

= class(θ

) = class(φ

◦ θ) = φ

∗

(class(θ)) = φ

∗

(σ).

CHAPTER 8

The universal moment maps

The theory of moment maps developed in the previous paragraph applies in

particular to the whole group of automorphisms Diff(X, ω) of a closed 2-form ω
deﬁned on a diﬀeological space X. We will describe, in this paragraph, the relation-
ships between the « universal » moment maps and associated objects obtained by
considering the whole group Diff(X, ω) and the equivalent objects associated to a
smooth action of some other group G on X, preserving ω.

8.1. Universal moment maps. Let X be a connected diﬀeological space and

let ω be a closed 2-form deﬁned on X. Let us remind that the group Diff(X, ω) of
all the automorphisms of (X, ω) is equipped with the functional diﬀeology of group
of diﬀeomorphisms. Let us denote also this group by G

. Every constructions

deﬁned above, the moment space, the paths moment map, the holonomy group,
the 2-points moment map, the moment maps, Souriau’s cocycle and Souriau’s class,
apply for G

. We shall distinguish these objects by the index ω. So, we denote by

∗

the momenta space of G

, by Ψ

: Paths(X)

→ G

∗

the paths moment map, by

= Ψ

(Loops(X)) the holonomy group, by ψ

the 2-points moment map, by μ

the moment maps, by θ

Souriau’s cocycles, and by σ

Souriau’s class. Since G

and its action on X are uniquely deﬁned by ω, these objects depend only on the
2-form ω.

Now, let G be a diﬀeological group and ρ be a smooth action of G on X,

preserving ω. That is, a smooth homomorphism ρ from G to G

. The values of

the various objects Ψ, Γ, ψ, μ, θ, with respect to the action ρ of G on X, depend
only on ρ

∗

, Ψ

, Γ

, ψ

, μ

, and θ

, as described in Subsection 7.1. And, we have:

⎧

⎨
⎩

∗

◦ Ψ

∗

(Γ

)

∗

◦ ψ

∗

◦ μ

∗

◦ θ

◦ ρ.

In this sense the objects G

, Γ

, Ψ

, Γ

, ψ

, μ

, θ

and σ

are universal. So,

we shall call Ψ

the universal paths moment map, Γ

the universal holonomy, ψ

the universal 2-points moment map, μ

the universal moment maps, θ

universal

Souriau’s cocycles, and σ

universal Souriau’s class of ω.

Note that in particular, this gives us a notion of hamiltonian spaces, those for

which, for one reason or another, the universal holonomy is trivial Γ

{0}.

8.2. The group of hamiltonian diﬀeomorphisms. Let X be a connected

diﬀeological space equipped with a closed 2-form ω. There exists a largest con-
nected subgroup Ham(X, ω)

⊂ Diff(X, ω) whose action is hamiltonian, that is

whose holonomy vanishes. The elements of Ham(X, ω) are called hamiltonian dif-
feomorphisms. An action ρ of a diﬀeological group G on X is hamiltonian if and
only if, restricted to the identity component of G, ρ takes its values in Ham(X, ω).

PATRICK IGLESIAS-ZEMMOUR

The construction of Ham(X, ω) is actually given as follows. Let us denote by G

the group Diff(X, ω) and by G

◦

its identity component. Let π :

◦

→ G

◦

be the

universal covering. Since the universal holonomy Γ

is made up of closed momenta,

according to Subsection 2.11 every γ

∈ Γ

deﬁnes a unique homomorphism k(γ)

from

◦

to R such that π

∗

(γ) = d[k(γ)]. Let

∈Γ

ker(k(γ)),

and let

◦

be its identity component. So,

Ham(X, ω) = π(

◦

Note 1

— The map f :

◦

→ Hom(π

(X), R) deﬁned by f(˜

g) = [τ

→ k(γ)(˜g)],

with τ = class() and γ = Ψ(), is an homomorphism. And,

= ker(f). In

classical symplectic geometry, the image F = val(f) is called, by some authors, the
group of ﬂux of ω.

Note 2

— Since to be hamiltonian for a group of automorphisms depends only

on its connected component, see Subsection 2.7 and Subsection 2.8, any extension
H

⊂ Diff(X, ω) of Ham(X, ω), such that H/Ham(X, ω) is discrete

, is hamiltonian.

In particular π(

) is hamiltonian, or if Γ

{0} then Diff(X, ω) is hamiltonian,

and Ham(X, ω) is the identity component of Diff(X, ω).

Note 3

— Let us choose a point x

in X and let μ be the moment map with

respect to the group Ham(X, ω), deﬁned by μ(x

) = 0. Let f be a 1-parameter

subgroup of Ham(X, ω). Applying the note of Subsection 3.2, we get for all x

∈ X

the expression of μ(x), evaluated on f

μ(x)(f ) = h

(x)

× dt with h

(x) =

−

(ω).

The smooth function h

: X

→ R is the hamiltonian (vanishing at x

) of the

1-parameter subgroup f .

Proof.

Let us remark, ﬁrst of all, that for every γ

∈ Γ

, π

∗

(γ)

= 0.

Indeed, π

∗

(γ)

= d[k(γ)]

= d[k(γ)

]. But, by the very deﬁnition of

, k(γ)

= 0, so π

∗

(γ)

= 0.

a) Let us prove that the holonomy of Ham(X, ω) is trivial. Let H

= π(

)

and let us denote by j

the inclusion H

⊂ G

, by j

the inclusion

⊂

◦

and by π

→ H

the projection. So, j

◦ π

= π

◦ j

. Let Γ

be the

holonomy of H

, so according to Subsection 7.1, Γ

= j

∗

(Γ

). Thus, for every

∈ Γ

there exists γ

∈ Γ

such that ¯

γ = γ

= j

∗

(γ). So, for all ¯

∈ Γ

∗

(¯

γ) = π

∗

(γ)) = (j

◦ π

)

∗

(γ) = (π

◦ j

)

∗

(γ) = j

∗

(π

∗

(γ)) = π

∗

(γ)

. But, π

∗

(γ)

= 0, so π

∗

(¯

γ) = 0. And since π

is a subduction, ¯

γ = 0.

Therefore, the holonomy of H

vanishes, Γ

{0}.

b) Let us prove now that every connected subgroup H

⊂ G

whose action is

hamiltonian is a subgroup of Ham(X, ω). Let

H = π

−1

(H) and

◦

be its identity

component. Let j

be the inclusion H

⊂ G

, and j

◦

be the inclusion

◦

⊂

◦

Let π

= π

◦

. So, j

◦ π

= π

◦ j

◦

. Let Γ

be the holonomy of H. Since

Where H and Ham(X, ω) are equipped with the subset diﬀeology of the functional diﬀeology

of Diff(X, ω).

8. THE UNIVERSAL MOMENT MAPS

= j

∗

(Γ

) and Γ

{0}, for all γ ∈ Γ

, j

∗

(γ) = 0. Thus, for all γ

∈ Γ

∗

(γ)) = 0. But, π

∗

(γ)) = (j

◦ π

)

∗

(γ) = (π

◦ j

◦

)

∗

(γ) = j

∗

◦

(π

∗

(γ)) =

∗

(γ)

◦

. So, for all γ

∈ Γ

, π

∗

(γ)

◦

= 0. But π

∗

(γ) = d[k(γ)], hence

d[k(γ)

◦

] = 0. So, since H

◦

is connected, k(γ) is constant on

◦

, and since k(γ)

is an homomorphism to R, this constant is necessarily 0. Thus,

◦

⊂ ker(k(γ)),

for all γ

∈ Γ

, that is

◦

⊂

. But, since H

◦

is connected

◦

⊂

◦

⊂ H

and

thus H = π(

◦

)

⊂ Ham(X, ω) = π(

◦

8.3. Time-dependent hamiltonian. Let X be a connected diﬀeological

space and ω be a closed 2-form deﬁned on X. A diﬀeomorphism f of X belongs to
Ham(X, ω) if and only if:

(1) There exists a smooth path t

→ f

in Diff(X, ω) connecting the identity

= f

to f = f

(2) There exists a smooth path t

→ Φ

in C

∞

(X, R) such that for all t,

(ω) =

−dΦ

with

: s

→ f

−1

◦ f

t+s

According to the tradition of classical symplectic geometry, the path t

→ Φ

can

be called a time-dependent hamiltonian of the 1-parameter family of hamiltonian
diﬀeomorphisms t

→ f

Proof.

Let us assume ﬁrst that f satisﬁes the condition above. That is, there

exists a smooth path t

→ f

in Diff(X, ω) such that f

= 1

, f

= f , and there

exists a smooth path t

→ Φ

in C

∞

(X, R) such that i

(ω) =

−dΦ

for all t where

: s

→ f

−1

◦ f

t+s

. Let us remind that Ham(X, ω) = π(

◦

), with

◦

the identity

component of

∩

∈Γ

ker(k(γ)), and let ˜

∈ G

◦

be the homotopy class of the

path t

→ f

, notations of Subsection 8.2. So, let γ

∈ Γ

, that is γ = Ψ

() where

is some loop in M. By deﬁnition, we have

k(γ)( ˜

f ) =

→f

]

γ =

→f

]

() =

()([t

→ f

])

(1)dt

Now, thanks to Subsection 3.2

♣, we have

()([t

→ f

])

(1) =

−

(ω) =

dΦ

∂

= 0.

So, k(γ)( ˜

f ) = 0 for all γ

∈ Γ

and ˜

f belongs to

and more precisely in the

identity component of

. Therefore f

∈ Ham(X, ω).

Conversely, let f

∈ Ham(M, ω). Since Ham(M, ω) is connected there exists

a path t

→ f

in Ham(M, ω) connecting 1

to f .

And, since the projection

◦

→ Ham(M, ω) is a covering, there exists a (unique) lifting t → ˜

→ f in

◦

, along π

◦

, such that ˜

= 1

. This lifting is actually given by

= class(p

), with p

: s

→ f

. So, for all t, ˜

∈

◦

⊂

∩

∈Γ

ker(k(γ)).

That is, for all γ

∈ Γ

, k(γ)( ˜

) = 0, or in other words, for all

∈ Loops(M),

PATRICK IGLESIAS-ZEMMOUR

k(Ψ

())( ˜

) = 0. But,

k(Ψ

())( ˜

)

()

()(s

→ f

)

(1)ds

()(s

→ st → f

)

(1)ds

[Ψ

()(u

→ f

)]

u=st

dst

()(u

→ f

)

(1)du.

So,

k(Ψ

())( ˜

) = 0

⇒

()(u

→ f

)

(1)du = 0,

and taking the limit for t

→ 0 we get,

k(Ψ

())( ˜

) = 0

⇒ Ψ

()(t

→ f

)

(1) = 0.

But, Ψ

()([t

→ f

])

(1) =

−

(ω), see Subsection 3.2

♣. So, for all t and all

∈ Loops(X)

(ω) = 0.

But F

is a path in Diff(X, ω) centered at the identity, so the Lie derivative of ω

by F

vanishes, and applying the Cartan formula given in Subsection 1.6, we get

ω = 0

⇒ d[i

(ω)] + i

(dω) = d[i

(ω)] = 0.

So, the 1-form i

(ω) is closed and its integral on any loop of X vanishes, therefore

(ω) is exact [Piz05].

Thus, for all real t there exists a real function Φ

∈

∞

(X, R) such that i

(ω) =

−dΦ

. The fact that t

→ Φ

is a smooth map

from R to C

∞

(X, R), for the functional diﬀeology, is a consequence of the explicit

construction of the function Φ

by integration along the paths, see [Piz05].

CHAPTER 9

About symplectic manifolds

The case of symplectic manifolds (M, ω) deserves a special care: any universal

moment map μ

is injective and therefore identiﬁes M with a coadjoint orbit — in

the general sense given in Subsection 2.10 — of Diff(M, ω).

9.1. Value of the moment maps for manifolds. Let M be a connected

manifold equipped with a closed 2-form ω. In this context, the paths moment map
Ψ

takes a special expression. Let p be a path in M, let F : U

→ Diff(M, ω) be a

n-plot, we have

(

♦)

(p)(F)

(δr) =

p(t)

( ˙

p(t), δp(t)) dt

for all r

∈ U and δr ∈ R

, where δp is the lifting in the tangent space TM of the

path p, deﬁned by

(

♥)

δp(t) = [D(F(r))(p(t))]

−1

∂F(r)(p(t))

∂r

(δr).

Proof.

By deﬁnition, Ψ(p)(F) = ˆ

∗

(Kω)(F) = Kω(ˆ

◦ F). The explicit expres-

sion of the operator K given in Subsection 1.7, applied to the plot ˆ

◦F : r → F(r)◦p

of Paths(X), gives

(Kω)(ˆ

◦ F)

(δr) =

→ (ˆp ◦ F)(u)(s + t)

(

s=0

u=r

)

1
0

δr

dt.

But (ˆ

◦ F)(u)(s + t) = F(u)(p(s + t)), let us denote temporarily by Φ

the plot

(s, u)

→ F(u)(p(s + t)), so F(u)(p(s + t)) writes Φ

(s, u). Now, let us denote by

the integrand of the right term of this expression. We have,

I = ω

→ Φ

(s, u)

(

s=0

u=r

)

1
0

δr

∗

(ω)(

0
r

)

1
0

δr

(

0
r

)

D(Φ

)(

0
r

)

1
0

, D(Φ

)(

0
r

)

δv

F(r)(p(t))

∂

∂s

F(r)(p(s + t))

s=0

∂

∂r

F(r)(p(t))

(δr)

But,

∂

∂s

F(r)(p(s + t))

s=0

= D(F(r))(p(t))

∂p(s + t)

∂s

s=0

= D(F(r))(p(t))( ˙p(t)).

PATRICK IGLESIAS-ZEMMOUR

So, using this last expression and the fact that F is a plot of Diff(M, ω), that is for
all r in U, F(r)

∗

ω = ω, we have

→ Φ

(s, u)

(

s=0

u=r

)

F(r)(p(t))

D(F(r))(p(t))( ˙p(t)),

∂F(r)(p(t))

∂r

(δr)

p(t)

p(t), [D(F(r))(p(t))]

−1

∂F(r)(p(t))

∂r

(δr)

p(t)

( ˙

p(t), δp(t)).

Therefore, Ψ

(p)(F)

(δr) = Kω(ˆ

◦ F)

(δr) =

p(t)

( ˙

p(t), δp(t)) dt.

9.2. The paths moment maps for symplectic manifolds. Let M be a

Hausdorﬀ manifold and ω be a non degenerate closed 2-form deﬁned on M. Let
m

and m

be two points of M connected by a path p. Let f

∈ C

∞

(M, R) with

compact support. Let F be the exponential of the symplectic gradient

grad

(f ),

F is a 1-plot of Diff(M, ω), and precisely a 1-parameter homomorphism. So, the
universal paths moment map Ψ

, computed at the path p, evaluated to the 1-plot

F, is the constant 1-form of R,

(p)(F) = [f (m

)

− f(m

)]

× dt with F : t → e

t grad

(f )

and dt the standard 1-form of R. Note that we are in the special case where F
is actually a 1-parameter homomorphism of Ham(M, ω)

⊂ Diff(M, ω), and the

function f is one hamiltonian of F.

Proof.

Let us remark that, in our case, the lift δp deﬁned by

♥ of Subsection

9.1 writes simply

δp(t) = [D(e

rξ

)(p(t))]

−1

∂e

rξ

(p(t))

∂r

(δr) = ξ(p(t))

× δr with ξ = grad

(f ),

where r and δr are reals. So, the expression

♦ of Subsection 9.1 becomes

(p)(F)

(δr)

p(t)

( ˙

p(t), ξ(p(t)) dt

× δr

p(t)

( ˙

p(t), grad

(f )(p(t)) dt

× δr

dp(t)

× δr

[f (p(1))

− f(p(0))] × δr

That is, Ψ

(p)(F) = [f (m

)

− f(m

)]

× dt.

9.3. Moment maps for symplectic manifolds. Let M be a connected

Hausdorﬀ manifold and ω be a closed 2-form deﬁned on M. The form ω is non-
degenerated, that is symplectic, if and only if

(1) The manifold M is an homogeneous space of Diff(M, ω).
(2) Any one of its universal moment maps μ

: M

→ G

∗

/Γ

is injective.

Let us remind that the symplectic gradient is deﬁned by ω(grad

(f ),

·) = −df.

9. ABOUT SYMPLECTIC MANIFOLDS

Note that, if one of the universal moment maps μ

is injective so are every ones.

Note also that, if ω is symplectic, then the image of the moment map,

(M)

∈ G

∗

/Γ

, is a (Γ

, θ

)-coadjoint orbit of Diff(M, ω). And, μ

identiﬁes M

, where

is equipped with the quotient diﬀeology of Diff(M, ω). In other

words, every symplectic manifold is a coadjoint orbit.

Remark

— Let us consider the example M = R

and ω = (x

+ y

) dx

∧ dy.

This form is non degenerate on R

−{0}, but degenerates at the point (0, 0). Thus,

(0, 0) is an orbit of the group Diff(X, ω), and actually R

− {0} is the other orbit.

Since R

is contractible the holonomy Γ

is trivial and the universal moment map

deﬁned by μ

(0, 0) = 0

∗

is equivariant. Now, μ

is injective, and ω is not

symplectic. So, the hypothesis of transitivity of Diff(M, ω) on M is not superﬂuous
is this proposition.

Proof.

Let us assume ﬁrst that ω is nondegenerate, that is symplectic. So,

the group Diff(M, ω) is transitive on M [Boo69]. Moreover, for every m

∈ M,

the orbit map ˆ

m : ϕ

→ ϕ(m) is a subduction [Don84]. So, the image of moment

moment map μ

is one orbit

of the aﬃne coadjoint action of G

∗

/Γ

associated to the cocycle θ

. Thus, for the orbit

equipped with the quotient

diﬀeology of G

, the moment map μ

is a subduction.

Now, let m

and m

two points of M such that μ

) = μ

), that is

, m

) = μ

)

− μ

) = 0. Let p

∈ Paths(M) such that p(0) = m

and p(1) = m

. Thus, ψ

, m

) = 0 is equivalent to Ψ

(p) = Ψ

(), where

is some loop of M, we can choose (0) = (1) = m

. Now, let us assume that

= m

. Since M is Hausdorﬀ there exists a smooth real function f

∈ C

∞

(M, R),

with compact support, such that f (m

) = 0 and f (m

) = 1. Let us denote by ξ the

symplectic gradient ﬁeld associated to f and by F the exponential of ξ. Thanks to
Subsection 9.2, on one hand we have Ψ(p)(F) = [f (m

)

−f(m

)]dt = dt, and on the

other hand Ψ

()(F) = [f (m

)

− f(m

)]dt = 0. But dt

= 0, thus ψ

, m

)

= 0,

and the moment map μ

is injective. Therefore, μ

is an injective subduction on

, that is a diﬀeomorphism.

Conversely, let us assume that M is an homogeneous space of Diff(M, ω) and

is injective. Let us notice ﬁrst that, since Diff(M, ω) is transitive, the rank of

ω is constant. In other words, dim ker ω = const. Now, let us assume that ω is
degenerated, that is dim(ker ω)

≥ 1. Since m → ker ω

is a smooth foliation, for

any point m of M there exists a smooth path p of M such that p(0) = m and for
t belonging to a small interval around 0

∈ R, ˙p(t) = 0 and ˙p(t) ∈ ker ω

p(t)

for all

t in this interval. So, we can re-parametrize the path p and assume now that p
is deﬁned on the whole R and satisﬁes p(0) = m, p(1) = m

with m

= m

, and

p(t)

∈ ker ω

p(t)

for all t. Now, since ˙

p(t)

∈ ker ω

p(t)

for all t, using the expression

♦ given in Subsection 9.1, we get Ψ

(p) = 0

∗

and thus μ

(m) = μ

). But

= m

and we have assumed that μ

is injective. So the kernel of ω is reduced to

{0}, ω is nondegenerate, that is symplectic.

Let us ﬁnish by proving the remark. That is, the universal moment map μ

ω = (x

+ y

) dx

∧ dy is injective. First of all μ

(0, 0) = 0

∗

. Now if z = (x, y) and

= (x

, y

) are two diﬀerent points of R

and diﬀerent from (0, 0), there is a smooth

function with compact support contained in a small ball not containing (0, 0) nor z
and such that f (z

) = 1. So the 1-parameter group generated by grad

(f ) belongs

to Diff(R

, ω), and then a similar argument as the one of the proof above shows

that μ

(z)

= μ

). Now it remains to prove that if z

= (0, 0), μ

(z)

= 0

∗

. Let

PATRICK IGLESIAS-ZEMMOUR

us consider p(t) = tz and F(r) be the positive rotation of angle 2πr, where r

∈ R.

The application of the formula

♦ of Subsection 9.1, computed at the point r = 0

and applied to the vector δr = 1 gives (2π/3)(x

+ y

)

which is not zero. So, the

moment map μ

is injective.

9.4. Restriction to hamiltonian diﬀeomorphisms. Let (M, ω) be a con-

nected Hausdorﬀ symplectic manifold. Let Ham(M, ω) be the group of hamiltonian
diﬀeomorphisms, and let

∗

be the space of its momenta. Let μ

: M

→ H

∗

any moment map associated to the action of Ham(M, ω), and let θ

be the associ-

ated Souriau cocycle. So, μ

is injective, and identiﬁes M to a θ

-coadjoint orbit

of Ham(M, ω) in

Proof.

It is known also that the group Ham(M, ω) acts transitively on M

[Boo69]. With respect to that group, and by construction, the holonomy is trivial:
the associated paths moment map Ψ

and the moment maps μ

take their values in

the space

∗

. Let j : Ham(M, ω)

→ Diff(M, ω) be the inclusion, so the universal

holonomy Γ

is in the kernel of j

∗

, and we get a natural mapping j

∗

/Γ

→

∗

. Now, the paths moment maps satisfy Ψ

= j

∗

◦ Ψ

, and μ

= j

∗

◦ μ

, see

Subsection 8.1. Then, since the Subsection 9.2 involves only plots of Ham(X, ω), the
ﬁrst part of the proof of Subsection 9.3 applies mutatis mutandis to the hamiltonian
case and we deduce that the moment maps μ

are injective and identify M with

some θ

-coadjoint orbits of Ham(M, ω).

9.5. Hamiltonian diﬀeomorphisms of symplectic manifolds. Let (M, ω)

be a connected Hausdorﬀ symplectic manifold. According to Banayaga, a diﬀeo-
morphism f is said to be hamiltonian if it can be connected to the identity 1

a smooth path t

→ f

in Diff(M, ω) such that

ω( ˙

·) = dφ

with

(x) =

◦ f

−1

(x)

s=t

where (t, x)

→ φ

(x) is a smooth real function, see [Ban78]. If, according to

this deﬁnition, f is hamiltonian then it is an element of Ham(M, ω), as deﬁned
in Subsection 8.2. Conversely, any element f of Ham(M, ω) satisﬁes the condition
above. So, the deﬁnition of hamiltonian diﬀeomorphisms given in Subsection 8.2 is a
faithful generalization of the classical deﬁnition for symplectic manifolds. Note that
the technical requirement of compacity of the original deﬁnition op. cit. doesn’t
play any role in this characterization of hamiltonian diﬀeomorphisms.

Proof.

This proposition is a direct consequence of the general statement given

in Subsection 8.3 and the following comparison between the above 1-parameter
family of vector ﬁelds ˙

and the family F

of the Subsection 8.3.

Since f

◦f

−1

= f

◦(f

−1

◦f

)

◦f

−1

, the vector ﬁelds ˙

and F

are conjugated

by f

, precisely:

= (f

)

∗

)

(x) = D(f

)(f

−1

(x))(F

−1

(x))).

This implies in particular that if the vector ﬁeld ˙

satisﬁes Banyaga’s condition

for the function φ

then the vector ﬁeld F

satisﬁes Banyaga’s condition for the

function Φ

−φ

◦ f

, and conversely. That is:

ω( ˙

·) = dφ

⇔ ω(F

·) = −dΦ

with

−φ

◦ f

9. ABOUT SYMPLECTIC MANIFOLDS

Indeed, let x

∈ M, x

= f

(x), δx

∈ T

M, and δx

= D(f

)(x)(δx), we have:

( ˙

), δx

)

[dφ

]

(δx

)

(x)

( ˙

(x)), D(f

)(x)(δx))

[dφ

]

(x)

(D(f

)(x)(δx))

(x)

(D(f

)(x)(F

(x)), D(f

)(x)(δx))

∗

(dφ

)]

(δx)

∗

(ω)]

(x), δx)

d[f

∗

(φ

)]

(δx)

(x), δx)

d[φ

◦ f

]

(δx).

Thus, we get Φ

−φ

◦ f

CHAPTER 10

The homogeneous case

As it is suggested by Subsection 9.3, the case of an homogeneous action of a

diﬀeological group G on a space X, preserving a closed 2-form ω, deserves a special
attention.

10.1. The homogeneous case. Let X be a connected diﬀeological space

equipped with a closed 2-form ω. Let ρ be a smooth action of a diﬀeological group
G on X, preserving ω. Let us assume that X is homogeneous for this action, see
Subsection 2.3. Let Γ be the holonomy of the action ρ, let μ be a moment, and let
θ be the cocycle associated to μ. Let x

be any point of X, and let μ

= μ(x

Let St

Γ,θ

∗

(μ

) be the stabilizer of μ

for the aﬃne coadjoint action of G on

∗

/Γ.

Thanks to the equivariance of the moment map μ, with respect to the aﬃne coad-
joint action of G on

∗

/Γ, μ

◦ ρ(g) = Ad

Γ,θ

∗

(g)

◦ μ, the image O = μ(X) is a

(Γ, θ)-orbit of G. Let us equip

O with the quotient diﬀeology of G, such that

O G/St

Γ,θ

∗

(μ

). So, the orbit map ˆ

: G

→ X is a principal ﬁbration with

structure group St

), the orbit map ˆ

: G

→ O is a principal ﬁbration with

structure group St

Γ,θ

∗

(μ

), and St

)

⊂ St

Γ,θ

∗

(μ

). So, the moment map

μ : X

→ O is a ﬁbration with ﬁber, the homogeneous space St

Γ,θ

∗

(μ

)/St

Γ,θ

∗

(μ

)/St

)

Γ,θ

∗

(μ

)

Note

— The moment maps μ are deﬁned up to a constant, but the character-

istics of μ, that is the subspaces deﬁned by μ(x) = const, are not. They are the
solutions of the equation ψ(x

, x) = 0, where const = μ(x

) and ψ is the 2-points

moment map.

Proof.

This is just an application of standard diﬀeological relations.

10.2. Symplectic homogeneous diﬀeological spaces. Let X be a con-

nected diﬀeological space and ω be a closed 2-form deﬁned on X.

Definition.

We say that (X, ω) is an homogeneous symplectic space if it is

homogeneous under the action of Diff(X, ω) and if a universal moment map μ

a covering onto its image.

The homogeneous situation where the moment maps μ

are not coverings onto

their images can be regarded as the homogeneous pre-symplectic case.

Now, let G be some diﬀeological group, and let ρ be a smooth action of G

on X, preserving ω. So, if the action ρ of G on X is homogeneous, then X is

PATRICK IGLESIAS-ZEMMOUR

an homogeneous space of Diff(X, ω). And, if a moment map μ : X

→ G

∗

/Γ is a

covering onto its image, then any universal moment map μ

: X

→ G

∗

/Γ

is a

covering onto its image.

Thus, to check that an homogeneous pair (X, ω) is symplectic it is suﬃcient to

ﬁnd a smooth homogeneous smooth action of some diﬀeological group G for which
one moment map is a covering onto its image.

Proof.

To be homogeneous under the action of G means that, for some point

(and thus for any point) x

∈ X, the orbit map ˆx : G → X, deﬁned by ˆx(g) = ρ(g)(x),

is a subduction. So, ˆ

x is surjective and, for any plot P : U

→ X, for any r

∈ U,

there exists a superset V of r

and a plot Q : V

→ G such that P V = ˆx ◦ Q.

That is, P(r) = ρ(Q(r))(x) for all r

∈ V. Since ρ is smooth, ¯Q = ρ ◦ Q is a plot

of Diff(X, ω), and P

V = ˆx ◦ ¯Q. Since, ˆx : Diff(X, ω) → X is surjective, it is a

subduction and X is an homogeneous space of Diff(X, ω).

Now, let us remark that, since the moment maps diﬀer just by a constant,

if a moment map μ is a covering onto its image

O equipped with the quotient

diﬀeology of G, then every other moment map μ

= μ + const is a covering onto

its image

O + const. So, let x

be a point of X, and let μ(x) = ψ(x

, x),

where ψ is the 2-points moment map. Let μ

= ψ

, x). According to Subsection

8.1, μ = ρ

∗

◦ μ

. Let

O = μ(X) and O

= μ

(X), equipped with the quotient

diﬀeologies of G and G

= Diff(X, ω). So,

O = ρ

∗

(

). Let m

∈ O and m

∈ O

such that ρ

∗

) = m. So, μ

−1

) =

{x ∈ X | μ

(x) = m

} ⊂ μ

−1

(m) =

{x ∈

| μ(x) = ρ

∗

(μ

(x)) = m

}. Thus, if μ

−1

(m) is discrete, a fortiori μ

−1

)

⊂

−1

(m). Thus, if μ is a ﬁbration onto its image, then μ

is a ﬁbration onto its

image too. And of course if μ is injective, a fortiori μ

CHAPTER 11

Examples of moment maps in diﬀeology

This short list of examples shows how the theory of moment map in diﬀeology

can be applied to the folklore of inﬁnite dimensional situations, but also to the less
familiar cases of singular spaces.

11.1. The moments of imprimitivity. Let X be a diﬀeological space. Let

us remind, and make some preliminary remarks on, the construction of the cotan-
gent bundle and the deﬁnition of the Liouville form [Piz05]. Let Ω

(X) denotes the

vector space of 1-form of X, equipped with the functional diﬀeology. The mapping
Taut, which associates to each n-plot Q

× P of the product X × Ω

(X) the 1-form

Taut(P

× Q) : r → P(r)(Q)

of dom(Q

× P), is a 1-form of X × Ω

(X). We call it the tautological form.

Now, let us consider the value equivalence relation. Let α and α

be two 1-

forms of X, let x be a point of X. We say that α and α

have the same value at

the point x, and we denote value(α)(x) = value(α

)(x), if and only if, for every

plot Q of X centered

at x , α(Q)

= α

(Q)

. Then, the cotangent bundle of X is

deﬁned as the quotient X

× Ω

(X) by the relation value, and denoted

by T

∗

X = X

× Ω

(X)/value.

This notion of value, for smooth forms on numerical domains, coincides with the
ordinary deﬁnition. So, when there will be no risk of confusion

, we shall denote

simply by α(x) the value of α at the point x, that is α(x) = value(α)(x).

Let pr : X

× Ω

(X)

→ T

∗

X be the canonical projection. So, there exists a

1-form on T

∗

X, denoted by Liouv and called the Liouville form such that

Taut = pr

∗

(Liouv)

Liouv = pr

∗

(Taut),

Liouv

∈ Ω

∗

X).

The characteristic property of the Liouville form is the following. Let α be a 1-
form of X, let ¯

α be the section of the canonical projection π : T

∗

→ X deﬁned

by ¯

α : x

→ value(α)(x), so α = ¯α

∗

(Liouv). Note also that, the group Diff(X)

acts naturally on the product X

× Ω

(X) by ¯

ϕ(x, α) = (ϕ(x), ϕ

∗

(α)), where ϕ

is a diﬀeomorphism of X. So, the tautological form is invariant by this action.
Moreover, this action is compatible with the relation value, and the group Diff(X)
has a natural projected action on T

∗

X. By equivariance, the Liouville form is

invariant by this action. Note that, the moment map for the action of Diff(X) on
(T

∗

X, dLiouv) is given by the general construction of Subsection 6.1. This can be

compared to Donato’s construction for manifolds in [Don88].

We say that a plot Q is centered at x if and only if 0

∈ dom(Q) and Q(0) = x.

Note that, as well as for the notation

∗

of the space of momenta of a diﬀeological group,

the star in T

∗

X do not rely to any kind of duality a priori.

This notation α(x) has not to be mixed up with the notation α(Q) for the value of α in the

plot Q. But the diﬀerent nature of x: a point of X, and Q: a plot of X, makes the diﬀerence.

PATRICK IGLESIAS-ZEMMOUR

Now, let us introduce the additive diﬀeological group of smooth functions

∞

(X, R), acting smoothly on X

× Ω

(X) by,

f : (x, α)

→ (x, α + df),

for all f

∈ C

∞

(X, R). This action projects naturally on the cotangent T

∗

X into an

action, denoted by the same way,

f : (x, a)

→ (x, a + df(x)),

for all (x, a)

∈ T

∗

X. So,

(1) For all f

∈ C

∞

(X, R), the variance of the tautological form and the Liou-

ville form are given by,

∗

(Taut) = Taut + pr

∗

(df )

and

∗

(Liouv) = Liouv + π

∗

(df ).

So, the exterior diﬀerentials dTaut and ω = dLiouv are invariant by the
action of C

∞

(X, R).

(2) Let p be a path of T

∗

X, connecting (x

, a

) = p(0) to (x

, a

) = p(1). So,

the paths moment map Ψ and the 2-points moment map ψ, with respect
to the 2-form ω = dLiouv, are given by

Ψ(p) = ψ((x

, a

), (x

, a

)) = d[f

→ f(x

)]

− d[f → f(x

)].

(3) For every x

∈ X, the real function [f → f(x)] is smooth. We call it the

Dirac function of the point x, and we denote it by δ

= [f

→ f(x)] ∈ C

∞

(X, R), R).

The diﬀerential dδ

= d[f

→ f(x)] is an invariant 1-form

of the additive

group C

∞

(X, R). Every moment map of the action of C

∞

(X, R) on T

∗

is cohomologous to the invariant moment map

μ : (x, a)

→ −dδ

Note that, the moment μ is constant on the ﬁbers T

∗

X = π

−1

(x). And, if

the real smooth functions separate

the points of X, the image of the mo-

ment map μ is the space X, identiﬁed with the space of Dirac’s functions.

(4) The action of C

∞

(X, R) on (T

∗

X, ω) is hamiltonian and exact. That is,

Γ =

{0} and σ = 0.

This example has been drawn to my attention by Fran¸cois Ziegler. This moment

appears informally in Ziegler’s construction of a symplectic analogue for « systems
of imprimitivity » in representation theory [Zie96]. It is why the moment map μ
will be called the moment of imprimitivity. The diﬀeological framework gives it so
a full formal status.

Proof.

First of all let us check the variance of Taut by the action of C

∞

(X, R).

Let f be a smooth real function deﬁned on X, let Q

× P be a plot of X × Ω

(X).

We have ¯

∗

(Taut)(P

× Q)

= Taut( ¯

◦ (Q × P))

= (P(r) + df )(Q)

= P(r)(Q)

df (Q)

= Taut(Q

× P)

+ df (pr

◦ (Q × P))

= Taut(Q

× P)

+ pr

∗

(df )(Q

× P)

So, ¯

∗

(Taut) = Taut + pr

∗

(df ). Now let us check that this action is compatible

with the value relation. Let (x, α) and (x

, α

) be two elements of X

× Ω

(X) such

that value(α)(x) = value(α

)(x

). That is, x = x

and for every plot Q of X

This diﬀerential has nothing to do with the derivative of the Dirac distributions in the sense

of De Rham’s currents.

That is, f (x) = f (x

) for all smooth real function f if and only if x = x

11. EXAMPLES OF MOMENT MAPS IN DIFFEOLOGY

centered at x, α(Q)

= α

(Q)

. So, (α + df )(Q)

= (α

+ df )(Q)

and value(α +

df )(x) = value(α)(x) + value(df )(x), or (α + df )(x) = α(x) + df (x). Thus, the
action of C

∞

(X, R) projects on T

∗

X as the action ¯

f : (x, a)

→ a + df(x). Now,

since ¯

∗

(Taut) = Taut + pr

∗

(df ), clearly ¯

∗

(Liouv) = Liouv + π

∗

(df ). Or, in

another way, ¯

∗

(Liouv) = Liouv + dF(f ) with F

∈ C

∞

(X, R), C

∞

∗

X, R))

and F(f ) = π

∗

(f ) = f

◦ π.

Let us denote by R(x, a) the orbit map f

→ a + df(x). Let p be a path of T

∗

such that p(0) = (x

, a

) and p(1) = (x

, a

). We get

Ψ(p)

∗

(KdLiouv)

∗

(ˆ

∗

(Liouv)

− ˆ0

∗

(Liouv)

− dKLiouv)

(ˆ

◦ ˆp)

∗

(Liouv)

− (ˆ0 ◦ ˆp)

∗

(Liouv)

− d[(KLiouv) ◦ ˆp]

R(x

, a

)

∗

(Liouv)

− R(x

, a

)

∗

(Liouv)

− d[f → KLiouv(ˆp(f))].

Let us consider ﬁrst the term [f

→ KLiouv(ˆp(f))]. Let p(t) = (x

, a

), so ˆ

p(f ) =

→ (x

, a

+ df (x

))]. Thus,

KLiouv(ˆ

p(f )))

→ x

]

s=t

dt +

df [t

→ x

] dt

→ x

]

s=t

dt + f (x

)

− f(x

Thus,

d[f

→ KLiouv(ˆp(f))] = d[f →

→ x

]

s=t

dt + f (x

)

− f(x

)]

d[f

→ f(x

)

− f(x

)].

Let us compute now R(x, a)

∗

(Liouv), for any (x, a)

∈ T

∗

X. Let P : U

→ C

∞

(X, R)

be a plot. We have

R(x, a)

∗

(Liouv)(P)

Liouv(R(x, a)

◦ P)

Liouv(r

→ P(r)(x, a))

Liouv(r

→ a + d[P(r)](x))

(a + d[P(r)](x))(r

→ x)

because the 1-form a + d[P(r)](x) is evaluated on the constant plot r

→ x. And,

every form evaluated to a constant plot vanishes. So, we get ﬁnally

Ψ(p) = d[f

→ f(x

)]

− d[f → f(x

)].

Now, clearly Ψ() = 0 for every loop of T

∗

X, and the action of C

∞

(X, R) is

hamiltonian Γ =

{0}. So, ψ((x

, a

), (x

, a

)) = μ(x

, a

)

− μ(x

, a

), with the

moment map

μ : (x, a)

→ −d[f → f(x)] = −dδ

Let us check now the invariance of the moment map μ. Note that, for every h

∈

∞

(X, R), we have δ

◦ L(h) = [f → f(x) + h(x)]. So, for every h ∈ C

∞

(X, R) we

have ˆ

∗

(μ)(x, a) = ˆ

∗

(

−dδ

) =

−d(δ

◦ L(h)) = −d[f → f(x) + h(x)] = −d[f →

f (x)] =

−dδ

= μ(x, a). Hence, μ is invariant. The 2-points moment map ψ is

exact. Souriau’s class of the action of C

∞

(X, R) on T

∗

X vanishes.

PATRICK IGLESIAS-ZEMMOUR

11.2. On the intersection 2-form of a surface I. Let Σ be a closed surface

oriented by a 2-form Surf, chosen once and for all. Let us consider Ω

(Σ), the

inﬁnite dimensional vector space of 1-forms of Σ, equipped with the functional
diﬀeology. Let us consider the antisymmetric bilinear map deﬁned on Ω

(Σ) by

(α, β)

→

∧ β,

for all α, β in Ω

(Σ). Since the wedge-product α

∧ β is a 2-form of Σ, there exists a

real smooth function ϕ

∈ C

∞

(Σ, R) such that α

∧ β = ϕ × Surf. So, by deﬁnition,

∧ β =

× Surf.

1) To the above bilinear form is naturally associated a well deﬁned diﬀerential

2-form ω of Ω

(X). For every n-plot P : U

→ X, for all r ∈ U, δr and δ

r in R

ω(P)

(δr, δ

r) =

∂P(r)

∂r

(δr)

∧

∂P(r)

∂r

(δ

2) The 2-form ω is the diﬀerential of the 1-form λ deﬁned on Ω

(Σ) by,

λ(P)

(δr) = 1

P(r)

∧

∂P(r)

∂r

(δr)

and

ω = dλ.

3) Let us consider now the the additive group (C

∞

(Σ, R), +) of smooth real

functions of Σ. And, let us deﬁne the following action of C

∞

(Σ, R) on Ω

(Σ).

For all f

∈ C

∞

(Σ, R),

→ ¯

f = [α

→ α + df].

So, the additive group C

∞

(Σ, R) acts by automorphisms on the pair (Ω

(Σ), ω).

For all f in C

∞

(Σ, R),

∗

(ω) = ω.

Note that the kernel of the action f

→ ¯

f is the subgroup of constant maps. And,

the image of C

∞

(Σ, R) is just the group B

1
DR

(Σ) of exact 1-forms of Σ.

4) Let p

∈ Paths(Ω

(Σ)) be a path connecting α

to α

. The paths moment

map Ψ(p) is given by

Ψ(p) =

∗

(λ) + d

→ 1

× dα

−

∗

(λ) + d

→ 1

× dα

On this expression, we check immediately that the 2-points moment map is just
given by ψ(α

, α

) = Ψ(p), for any path p connecting α

to α

. Note that, since

(Σ) is contractible the holonomy of the action of C

∞

(Σ, R) vanishes, Γ =

{0},

the action of C

∞

(Σ, R) is hamiltonian.

5) The moment maps of this action of C

∞

(Σ, R) on Ω

(Σ) are, up to a constant,

equal to

μ : α

→ d

→

× dα

Moreover, the moment map μ is equivariant. That is, invariant, since the group
C

∞

(Σ, R) is abelian.

For all f

∈ C

∞

(Σ, R),

◦ ¯

f = μ.

So, the action of C

∞

(Σ, R) on Ω

(Σ) is exact and hamiltonian.

Note

— The moment map μ(α) is fully characterized by dα. This is why we

ﬁnd in the mathematical literature on the subject that, the moment map for this
action is the exterior derivative (or curvature, depending on the authors) α

→ dα.

11. EXAMPLES OF MOMENT MAPS IN DIFFEOLOGY

But, as we see again on this example, diﬀeology gives to this sketchy assertion a
precise meaning.

Let us remark also that, the moment map μ is linear, for all t, s reals and all α

and β in Ω

(Σ), μ(t α + s β) = t μ(α) + s μ(β). And, the kernel of μ is the subspace

of closed 1-forms,

ker(μ) = Z

1
DR

(Σ) =

∈ Ω

(Σ)

| dα = 0

If we consider the orbit of the zero form 0

∈ Ω

(Σ) by C

∞

(Σ, R), this is just

the subspace B

(Σ, R), which is included in ker(μ) = Z

1
DR

(Σ).

The quotient

ker(μ)/C

∞

(Σ, R) is just Z

1
DR

(Σ)/B

1
DR

(Σ) = H

1
DR

(Σ), and the 2-form ω

ker(μ)

is just the pullback of the usual intersection form on H

1
DR

(Σ). I will discuss, in a

future work, the notion of « symplectic reduction » in diﬀeology.

Proof.

1) Let us check that ω deﬁnes a diﬀerential 1-form on Ω

(Σ). Note

that, for any r

∈ U = dom(P), P(r) is a section of the ordinary cotangent bundle

∗

Σ. That is, P(r) = [x

→ P(r)(x)] ∈ C

∞

(Σ, T

∗

Σ), where P(r)(x)

∈ T

∗

(Σ). So,

∂P(r)

∂r

(δr) = [x

→

∂P(r)(x)

∂r

(δr)]

and

∂P(r)(x)

∂r

(δr)

∈ T

∗

(Σ)

where ∂P(r)(x)/∂r denotes the tangent linear map D(r

→ P(r)(x)(r). And, the

formula giving ω is well deﬁned. Now, ω(P)

is clearly antisymmetric and depends

smoothly on r. So, ω(P) is a smooth 2-form of U. Let us check that P

→ ω(P)

deﬁnes a 2-form on Ω

(Σ). That is, satisﬁes the compatibility condition ω(P

◦ F) =

∗

(ω(P)), for all F

∈ C

∞

(V, U), where V is a numerical domain. Let s

∈ V, δs and

s two tangent vectors at s at V, let r = F(s):

ω(P

◦ F)

(δs, δ

∂P

◦ F(s)

∂s

(δs)

∧

∂P

◦ F(s)

∂s

(δ

∂P(r)

∂r

∂F(s)

∂s

(δs)

∧

∂P(r)

∂r

∂F(s)

∂s

(δ

ω(P)

F(s)

(DF

(δs), DF

(δ

s))

∗

(ω(P))

(δs, δ

Thus ω(P

◦ F) = F

∗

(ω(P)), and ω is a well deﬁned 2-form on Ω

(Σ).

2) First of all, the proof that the map P

→ λ(P) is a well deﬁned diﬀerential

1-form of Ω

(Σ) is analogous to the proof of the ﬁrst item. Now, let us remind that

ω = dλ is and only if d(λ(P)) = ω(P) for all plot P of Ω

(Σ). Let us apply the

usual formula of diﬀerentiation of 1-form on numerical domain,

(δr, δ

r) = δ(

(δ

r))

− δ

(

(δr))

where δ and δ

are to commuting variations. For the sake of simplicity let us denote

α = P(r),

δα =

∂P(r)

∂r

(δr),

α =

∂P(r)

∂r

(δ

r).

So,

d(λ(P))

(δr, δ

1
2

∧ δ

− δ

∧ δα

1
2

δα

∧ δ

α + α

∧ δδ

−

∧ δα + α ∧ δ

δα

PATRICK IGLESIAS-ZEMMOUR

but, δδ

α = δ

δα. So,

d(λ(P))

(δr, δ

1
2

δα

∧ δ

−

∧ δα

1
2

δα

∧ δ

α +

δα

∧ δ

δα

∧ δ

(δr, δ

r).

3) Let us compute the pullback of λ by the action of f

∈ C

∞

(Σ, R). Let

P : U

→ Ω

(Σ) be a n-plot, let r

∈ U and δr ∈ R

∗

(λ)(P)

(δr)

λ( ¯

◦ P)

(δr)

λ(r

→ P(r) + df)

(δr)

1
2

(P(r) + df )

∧

∂P(r)

∂r

(δr)

1
2

P(r)

∧

∂P(r)

∂r

(δr) + 1

∧

∂P(r)

∂r

(δr)

λ(P)

(δr) +

∂

∂r

1
2

∧ P(r)

(δr)

λ(P)

(δr)

−

∂

∂r

1
2

× d(P(r))

(δr)

So, for every f

∈ C

∞

(Σ, R), let us deﬁne the map ϕ(f ) : Ω

(Σ)

→ R by,

ϕ(f ) : α

→ 1

× dα.

So,

d(ϕ(f ))(P)

(δr) =

∂

∂r

1
2

× d(P(r))

(δr).

Thus,

∗

(λ)(P)

(δr) = λ(P)

(δr)

− (dϕ(f))(P)

(δr).

That is,

∗

(λ) = λ

− d(ϕ(f)).

Therefore, diﬀerential ω = dλ is invariant by the action of C

∞

(Σ, R).

4) Let p be a path of Ω

(Σ) connecting α

to α

. By deﬁnition Ψ(p) = ˆ

∗

(Kω).

Applying the property of the chain-homotopy operator d

◦ K + K ◦ d = ˆ1

∗

− ˆ0

∗

ω = dλ, we get

Ψ(p)

∗

(Kdλ)

∗

(ˆ

∗

(λ)

− ˆ0

∗

(λ)

− d(Kλ))

(ˆ

◦ ˆp)

∗

(λ)

− (ˆ0 ◦ ˆp)

∗

(λ)

− d[(Kλ) ◦ ˆp]

∗

(λ)

− ˆα

∗

(λ)

− d[f → Kλ(ˆp(f))]

11. EXAMPLES OF MOMENT MAPS IN DIFFEOLOGY

But, Kλ(ˆ

p(f )) = Kλ( ¯

◦ p) =

◦p

λ =

∗

(λ), and since ¯

∗

(λ) = λ

− d(ϕ(f)) we

have Kλ(ˆ

p(f )) =

−

d(ϕ(f )) =

− ϕ(f)(α

) + ϕ(f )(α

). Therefore,

Ψ(p)

∗

(λ)

− ˆα

∗

(λ)

− d[f → −ϕ(f)(α

) + ϕ(f )(α

)]

∗

(λ)

− ˆα

∗

(λ) + d

→ 1

× dα

− 1

× dα

And, ﬁnally we get the paths moment map Ψ given by

Ψ(p) =

∗

(λ) + d

→ 1

× dα

−

∗

(λ) + d

→ 1

× dα

For the the 2-points moment map ψ, we have clearly ψ(α

, α

) = Ψ(p) for any path

connecting α

to α

5) The 1-point moment maps are given by μ(α) = ψ(α

, α) for any origin α

Let us choose α

= 0. So,

μ(α) = ˆ

∗

(λ) + d

→ 1

× dα

− ˆ0

∗

(λ).

But ˆ

∗

(α) is not necessarily zero. Let us compute generally ˆ

∗

(λ). Let P : U

→

(Σ) be a n-plot. We have, ˆ

∗

(λ)(P) = λ( ˆ

◦ P) = λ(r → ˆα(P(r)) = λ(r →

α + d(P(r))). But,

λ(r

→ α + d(P(r))) = 1

(α + P(r))

∧

∂

∂r

(α + d(P(r)))

1
2

(α + P(r))

∧

∂d(P(r))

∂r

1
2

∧

∂d(P(r))

∂r

+ 1

P(r)

∧

∂d(P(r))

∂r

So,

( ˆ

∗

(λ)

− ˆ0

∗

(λ))(P) = 1

∧

∂d(P(r))

∂r

Therefore,

μ(α)(P)

( ˆ

∗

(λ)

− ˆ0

∗

(λ))(P)

+ d

→ 1

× dα

(P)

1
2

∧

∂d(P(r))

∂r

∂

∂r

1
2

P(r)

× dα

1
2

∂

∂r

∧ d(P(r)) + P(r) × dα

∂

∂r

P(r)

× dα

So, we get ﬁnally,

μ(α) = d

→

× dα

Now, let us express the variance of μ. Let f

∈ C

∞

(Σ, R), and let F(α) be the real

function F(α) : f

→

×dα, such that μ(α) = dF(α). We have, μ( ¯

f (α)) = μ(α +

df ) = dF(α + df ). But, for every h

∈ C

∞

(Σ, R), F(α + df )(h) =

× d(α + df) =

× dα = F(α)(h). So, for all f ∈ C

∞

(Σ, R), we have μ

◦ ˆ

f = μ. The moment

PATRICK IGLESIAS-ZEMMOUR

map μ is invariant by the group C

∞

(Σ, R). Souriau’s class vanishes. Thus, the

action of C

∞

(Σ, R) is exact and hamiltonian.

Let us compute ﬁnally the kernel of the moment map μ. We have: μ(α) = 0 if

and only if dF(α) = 0. But since C

∞

(Σ, R) is connected (actually contractible as

a diﬀeological vector space) dF(α) = 0 if and only if F(α) = const = F(α)(0) = 0.
But F(α) = 0 if and only if, for all f

∈ C

∞

(Σ, R),

× dα = 0. That is, if and

only if dα = 0.

11.3. On the intersection 2-form of a surface II. We continue with the

example of Subsection 11.2, using the same notations. Let us introduce the group
G of positive diﬀeomorphisms of (Σ, Surf). That is,

G =

∈ Diff(Σ)

∗

(Surf)

Surf

> 0

The group G acts by pushforward on Ω

(Σ). For all g

∈ G, for all α ∈ Ω

(Σ),

∗

(α)

∈ Ω

(Σ), and for all pair g, g

of elements of G, (g

◦ g

)

∗

= g

∗

◦ g

∗

. And, this

action is smooth. Now,

(1) The pushforward action of G on Ω

(Σ) preserves the 1-form λ, and thus

the 2-form ω. For all g

∈ G, (g

∗

)

∗

(λ) = λ, and (g

∗

)

∗

(ω) = ω. So, the

action of G is exact, σ = 0, and hamiltonian, Γ =

{0}.

(2) The moment maps are, up to a constant, equal to the moment μ, given

μ(α)(P)

(δr) = 1

∧ P(r)

∗

∂P(r)

∗

(α)

∂r

(δr)

for all α

∈ Ω

(Σ), for all n-plot P, where r

∈ dom(P) and δr ∈ R

. In

particular, applied to any 1-plot F centered at the identity 1

, that is

F(0) = 1

, we get the special expression

μ(α)(F)

(1) =

−1

∧ £

(α) =

−

(α)

× dα,

where £

(α) is the Lie derivative of α along F, and i

(α) the contraction

of α by F.

So, we ﬁnd again, through the diﬀeological formalism of the moment map, what is
asserted informally in the literature. The vague assertion « the moment map of the
group of diﬀeomorphism is the Lie derivative » makes here sense.

11. EXAMPLES OF MOMENT MAPS IN DIFFEOLOGY

Proof.

1) Let us compute the pullback of λ by the action of g

∈ G, that is

∗

)

∗

(λ). Let P : U

→ Ω

(Σ) be a n-plot, let r

∈ U, and δr ∈ R

. We have,

∗

)

∗

(λ)(P)

(δr)

λ(g

∗

◦ P)

(δr)

1
2

∗

(P(r))

∧

∂g

∗

(P(r))

∂r

(δr)

1
2

∗

(P(r))

∧ g

∗

∂P(r)

∂r

(δr)

1
2

∗

P(r)

∧

∂P(r)

∂r

(δr)

1
2

∗

(Σ)

P(r)

∧

∂P(r)

∂r

(δr)

1
2

P(r)

∧

∂P(r)

∂r

(δr)

λ(P)

(δr)

Thus, λ is invariant by G, and so do ω = dλ.

2) Since the 1-form λ is invariant by the action of G, we can use directly the

results of the exact case detailed in Subsection 6.1. Thus, the moment maps are,
up to a constant, equal to μ : α

→ ˆα

∗

(λ). So, let P : U

→ G be a n-plot, let r ∈ U

and δr

∈ R

. We have,

μ(α)(P)

(δr)

∗

(λ)(P)

(δr)

λ( ˆ

◦ P)

(δr)

λ(r

→ P(r)

∗

(α))

(δr)

1
2

P(r)

∗

(α)

∧

∂P(r)

∗

(α)

∂r

(δr)

1
2

∧ P(r)

∗

∂P(r)

∗

(α)

∂r

(δr)

Now, let P = F be a 1-plot centered at the identity, F(0) = 1

. Let us change

the variable r for the variable t. The previous expression, computed at t = 0 and
applied to the vector δt = 1 gives immediately

μ(α)(F)

(1)

1
2

∧

∂F(t)

∗

(α)

∂t

t=0

But, by deﬁnition of the Lie derivative, we have

∂F(t)

∗

(α)

∂t

t=0

∂(F(t)

−1

)

∗

(α)

∂t

t=0

−£

(α).

So, we get the ﬁrst expression of the moment map μ applied to F

μ(α)(F)

(1) =

−1

∧ £

(α).

Now, on a surface α

∧dα = 0, and i

(α

∧dα) = i

(α)

×dα−α∧i

(dα). So, i

(α)

dα = α

∧ i

(dα). Then, using the Cartan-Lie formula £

(α) = i

(dα) + d(i

(α)),

PATRICK IGLESIAS-ZEMMOUR

we get

∧ £

(α)

∧ [i

(dα) + d(i

(α))]

(α)dα +

∧ d(i

(α))

(α)dα +

(α)dα

−

d[α

∧ i

(α)]

(α)dα

And ﬁnally, we have the second expression for the moment map:

μ(α)(F)

(1) =

−

(α)

× dα,

for any 1-plot of the group of positive diﬀeomorphisms of the surface Σ, centered
at the identity.

11.4. On the intersection 2-form of a surface III. We continue again

with the example of Subsection 11.2, using the same notations. Let us consider the
space Ω

(Σ) as an additive group acting onto itself by translations. Let us denote

by t

the translation t

: α

→ α + β, where α and β belong to Ω

(Σ).

(1) The 2-form ω is invariant by translation. That is, t

∗

(ω) = ω for all

∈ Ω

(Σ). This action of Ω

(Σ) onto itself is hamiltonian but not exact.

(2) The moment maps of the additive action of Ω

(Σ) onto itself are equal,

up to a constant to

μ : α

→ d

→

∧ β

In other words, μ(α) = d[ω(α)], where ω is regarded as the smooth linear
function ω(α) : β

→ ω(α, β), deﬁned on Ω

(Σ). Moreover, the moment

map μ is linear and injective.

(3) The moment map μ is its own Souriau cocycle, θ = μ. The moment map

μ identiﬁes Ω

(Σ) with the θ-aﬃne coadjoint orbit of 0

∈ Ω

(Σ)

∗

. Be

aware that Ω

(Σ)

∗

denotes the space of invariant 1-forms of the abelian

group Ω

(Σ), and not its algebraic dual.

Note

— This situation is analogous to what happens for ﬁnite dimension

symplectic vector spaces. The 2-form ω can be regarded as a real 2-cocycle of the
additive group Ω

(Σ). This cocycle build up a central extension by R,

(α, t)

· (α

, t

) =

α + α

, t + t

∧ α

for all (α, t) and (α

, t

) in Ω

(Σ)

× R. This central extension acts on Ω

(Σ),

preserving ω. This action is hamiltonian, but now exact. The lack of equivariance,
characterized by Souriau’s class, has been absorbed in the extension. This group
could be named as the Heisenberg group of the oriented surface (Σ, Surf).

Note also that, according to Subsection 10.2, the space Ω

(Σ) equipped with the

2-form ω is an homogeneous symplectic space. Thus, we have a ﬁrst simple example
of inﬁnite dimensional symplectic diﬀeological space, avoiding any discussion on the
« kernel » of ω.

11. EXAMPLES OF MOMENT MAPS IN DIFFEOLOGY

Proof.

Let us compute the pullback of λ by a translation. Let P : U

→ X be

a n-plot, let r

∈ U, and δr ∈ R

. We have,

∗

(λ)(P)

(δr)

λ(t

◦ P)

(δr)

λ[r

→ P(r) + α]

(δr)

1
2

(P(r) + α)

∧

∂(P(r) + α)

∂r

(δr)

1
2

P(r)

∧

∂P(r)

∂r

(δr) + 1

∧

∂P(r)

∂r

(δr)

λ(P)

(δr) + d

→ 1

∧ β

(P)

(δr)

So, let us deﬁne, for all α

∈ Ω

(Σ), the smooth real function F(α) by

F(α) : β

→ 1

∧ β.

Such that

∗

(λ) = λ + d(F(α))

and

∗

(ω) = ω.

Then, Ω

(Σ), as an additive group, acts on itself by automorphisms. Let us compute

the moment maps. Let p be a path of Ω

(Σ), connecting α

to α

. We have

Ψ(p)

∗

(λ)

− ˆα

∗

(λ)

− d

→

d(F(β))

∗

(λ)

− ˆα

∗

(λ)

− d[β → F(β)(α

)

− F(β)(α

)]

{α

∗

(λ)

− d[β → F(β)(α

)]

} − {α

∗

(λ)

− d[β → F(β)(α

)]

}

{ˆα

∗

(λ) + d(F(α

))

} − {ˆα

∗

(λ) + d(F(α

))

So, the 2-points moment map Subsection 4.1 is given by ψ(α

, α

) = Ψ(p). Now,

the moment maps are, up to a constant equal to

μ(α) = ψ(0, α) = ˆ

∗

(λ) + d(F(α))

− ˆ0

∗

(λ).

But, for any plot P : U

→ Ω

(Σ), we have

∗

(λ)(P)

− ˆ0

∗

(λ)(P)

λ( ˆ

◦ P) − λ(ˆ0 ◦ P)

λ(r

→ P(r) + α) − λ(r → P(r))

→ 1

∧ β

(P)

d(F(α))(P).

Thus, ˆ

∗

(λ)(P)

− ˆ0

∗

(λ) = d(F(α)) and the moment map μ is ﬁnally given by

μ(α) = 2d(F(α)) = d

→

∧ β

The moment map μ is not equivariant, and Souriau’s cocycle θ is given by,

μ(t

∗

(β)) = μ(α + β) = μ(β) + θ(α)

with

θ(α) = μ(α).

So, the moment map μ is clearly smooth and linear. Let α

∈ ker(μ), μ(α) = 0 if

and only if d(F(α)) = 0, that is if and only if F(α) = const = F(α)(0) = 0. Thus,
F(α)(β) = 0 for any β

∈ Ω

(Σ), hence α = 0. Therefore, the moment map μ is

injective.

PATRICK IGLESIAS-ZEMMOUR

11.5. On symplectic irrational tori. Let us consider the numerical space

, for some integer n. For all u

∈ R

, let us denote by t

the translation by u.

That is, t

: x

→ x + u. Let ω be a 2-form of R

invariant by translations. That

is, for all u

∈ R

, t

∗

(ω) = ω. Thus, ω is a constant bilinear 2-form, necessarily

closed, dω = 0. Let us consider the moment maps associated to the translations
(R

, +). Since R

is simply connected, the holonomy vanishes, Γ =

{0}. Let p be

a path of R

connecting x = p(0) to y = p(1), the paths moment map Ψ(p), and

the 2-points moment map ψ(p) are given by

Ψ(p) = ψ(x, y) = ω(y

− x),

where ω(u) is regarded as the linear 1-form ω(u) : v

→ ω(u, v). So, the moment

maps are, up to constant, equal to the linear map

μ : x

→ ω(x).

And therefore, Souriau’s cocycle θ associated to μ is equal to μ. For all u

∈ R

θ(u) = μ(u) = ω(u).

Let us consider now a discrete diﬀeological subgroup K

⊂ R

. Let us denote by

Q the quotient Q = R

/K and by π : R

→ Q the canonical projection. Let us

continue to denote by t

the translation on Q, by u

∈ R

. That is t

(q) = π(x + u)

for any x such that q = π(x). Now, since ω is invariant by translations, ω is
invariant by K, and since K is discrete, ω projects on Q as a R

-invariant closed

2-form denoted by ω

. That is,

= π

∗

(ω)

ω = π

∗

(ω

Note that, the translation by any vector u of R

on Q is still an automorphism of

, that is t

∗

(ω

) = ω

(1) The holonomy Γ

of the action of (R

, +) on (Q, ω

) is the image of the

subgroup K by μ.

= μ(K),

⊂ R

∗

Thus, if ω

= 0 and if K is not reduce to {0}, then the action of (R

, +)

on (Q, ω

) is not hamiltonian and not exact.

(2) The moment map μ : R

→ R

∗

projects on a moment μ

such that the

following diagram commutes.

Q = R

∗

/μ(K)

∗

That is, for all q

∈ Q, μ

(q) = pr(ω(x)) for any x such that q = π(x).

Souriau’s cocycle θ

associated to μ

, for all u

∈ R

, is given by

(u) = μ

(π(u)).

So, if we consider the space Q as an additive group acting on itself by
translations, then the moment map μ

, of this action, coincide with its

Souriau cocycle θ

11. EXAMPLES OF MOMENT MAPS IN DIFFEOLOGY

(3) The map μ is a ﬁbration onto its image whose ﬁber is the kernel of μ. That

is val(μ)

/E, E = ker(μ). And, the map μ

is a ﬁbration onto its

image μ(R

)/μ(K) whose ﬁber is ker(μ

) = E/(K

∩ E). If ω : R

→ R

∗

is injective (which implies that n is even) then the moment map μ

is a

diﬀeomorphism which identiﬁes Q with its image R

∗

/μ(K).

Note 1

— Regarded as a group Q = R

/K acts onto itself by projection of

the translations of R

. Since the pullback by π : R

→ Q is an isomorphism from

∗

to R

∗

is the universal covering of Q), the moment maps computed above

give the moment maps associated to this action.

Note 2

— This construction applies to the torus T

= R

. The action

of (R

, +), is obviously not hamiltonian, but the moment map μ

is well deﬁned.

And, μ

identiﬁes T

with the quotient of R

∗

— the (Γ

, θ

)-coadjoint orbit

of the point 0 — by the holonomy Γ

= ω(Z

)

⊂ R

∗

. In the meaning we gave

above of the notion of coadjoint orbit, the torus T

, equipped with the standard

symplectic form ω, is a coadjoint orbit of R

, or even a coadjoint orbit of itself.

This is a special case of the the Subsection 9.3 discussion.

Note 3

— All this construction above can be also applied to situations which

are regarded as more singular that the simple quotient of R

by a lattice. It

can by applied as well to the product of any irrational tori. An (n-dimensional)
irrational torus T

is the quotient of R

by any generating discrete strict subgroup

K of R

. See for example [IL90] for an analysis of 1-dimensional irrational tori.

For example, we can consider the product of two 1-dimensional irrational torus
Q = T

× T

, quotient of R

= R

× R by the discrete subgroup α

)

× α

where α

: R

→ R and α

: R

→ R are two linear 1-forms. In this case, the

moment map μ

will also identify T

× T

with the quotient of R

∗

— (Γ

, θ

coadjoint orbit of 0 — by Γ

= ω(α

)

× α

)). This is the simplest example

of totally irrational symplectic space, and totally irrational coadjoint orbit. Note
that, these cases escape completely to the usual analysis, but also to the analysis
in terms of Sikorski’s or Fr¨

olicher’s spaces.

Proof.

First of all, the fact that there exists a closed 2-form ω

on R/K such

that π

∗

(ω

) = ω is an application of the criterion of pushing forward forms, in

the special case of a covering [Piz05]. Now, the computation of the moment map
of a linear antisymmetric form ω on R

is well know, and independently of the

method gives the same result μ(x) = ω(x). The additive constant is ﬁxed here by
the condition μ(0) = 0. But, the value of the paths moment map Ψ(p) can be found
as well by the method described above, applying the particular expression

Kω

(δp) =

p(t)

( ˙

p(t), δp(t))dt

with

p(t) =

dp(t)

of the chain-homotopy operator for manifold.

Where p is a path and δp is a

« variation » of p. So, since the result depends only on the ends of the path, let us
choose, for any points x and y in R

, the connecting path p : t

→ x + t(y − x). Let

us remind that Ψ(p) = ˆ

∗

(Kω). Let u and δu in R

. Note that ˆ

∗

) = t

◦ p =

PATRICK IGLESIAS-ZEMMOUR

→ p(t) + u]. So,

Ψ(p)

(δu)

∗

(Kω)

(δu)

(Kω)

◦p

(δ(t

◦ p)), with δp = 0

ω( ˙

p(t), δu) d t

ω(y

− x, δu)

So Ψ(p) = ψ(x, y) = ω(y

− x) = ω(y) − ω(x). And, μ : x → ω(x), for all x in R

Now, let us consider ω

. Since R

is the universal covering of Q, every loop

∈ Loops(Q, 0) can be lifted into a path p of R

starting at 0 and ending in K. In

other words,

Γ =

{Ψ() | ∈ Loops(Q)} = {Ψ(t → tk) | k ∈ K} = ω(K)

The other propositions are then a direct application of the functoriality of the
moment map described in Subsection 7.2, and standard analysis on quotients and
ﬁbrations.

11.6. The corner orbifold. Let us consider the quotient

Q of R

by the

action of the ﬁnite subgroup K

{±1}

, embedded in GL(2, R) by

K =

ε,ε

∈ {±1}

The space

Q = R

/K is an orbifold, according to [IKZ05]. It is diﬀeomorphic

to the quarter space [0,

∞[×[0, ∞[⊂ R

, equipped with the pushforward of the

standard diﬀeology of R

by the map π : R

→ [0, ∞[×[0, ∞[, deﬁned by,

π(x, y) = (x

, y

)

and

Q π

∗

So the letter

Q will denote indiﬀerently the quotient R

/K or the quarter space

Corner Orbifold

Plane

′

′′

′′′

Figure 1.

The corner orbifold

∗

). And the meaning of the letter π follows. Now, let us remark that, the

decomposition of

Q in terms of point’s structure is given by,

Str(0, 0) =

{±1}

Str(x, 0) = Str(0, y) =

{±1} and Str(x, y) = {1},

where x and y are positive real numbers. So, since the structure of a point is
preserved by diﬀeomorphisms [IKZ05], there are at least three orbits of Diff(

Q),

11. EXAMPLES OF MOMENT MAPS IN DIFFEOLOGY

the point 0

= (0, 0), the regular stratum ˙

Q =]0, ∞[

and the union of the two

axes, ox and oy. So, in particular any diﬀeomorphism of

Q preserves the origin 0

Actually, these are exactly the orbits of Diff(

Q). Let us remark that, dim(Q) = 2

[Piz06-b]. So, every 2-form is closed. Now,

1) Every 2-form of

Q is proportional to the 2-form ω deﬁned on Q by

∗

(ω) :

x
y

→ 4xy × dx ∧ dy.

That is, for any other 2-form ω

there exists a smooth function φ

∈ C

∞

(

Q, R) such

that ω

= φ

× ω.

2) The space (

Q, ω) is hamiltonian Γ

{0}. And, the action of G

is exact,

that is σ

= 0. In particular, the universal moment map μ

deﬁned by μ

) = 0,

is equivariant.

3) The universal equivariant moment map μ

vanishes on the singular strata

{0}, ox and oy, and is injective on the regular stratum ˙Q. So, the image μ

(

Q) is

diﬀeomorphic to an open disc with a point attached on the boundary.

Proof.

1) Let ω

be a 2-form on

Q and let ˜ω

be its pullback by π, ˜

∗

(ω

). So, there exists a smooth real function F such that ˜

= F

× dx ∧ dy.

But, since π

◦ k = π, for all k ∈ K we get εε

F(εx, ε

y) = F(x, y), for all (x, y)

∈

and all ε, ε

{±1}. Thus, F(−x, y) = −F(x, y) and F(x, −y) = −F(x, y).

In particular, F(0, y) = 0 and F(x, 0) = 0. Therefore, since F is smooth, there
exists f

∈ C

∞

, R) such that F(x, y) = 4xyf (x, y), with f (εx, ε

y) = f (x, y).

Therefore, ˜

= f

× ˜ω, with ˜ω = 4xy × dx ∧ dy. Now ˜ω = d(x

)

∧ d(y

), but x

→ x

and y

→ y

are invariant by K so, they are the pullback by π of some smooth

real functions on

Q. Thus, d(x

) and d(y

) are the pullback of 1-forms on

Q, let

us say d(x

) = π

∗

(ds) and d(y

) = π

∗

(dt), so ˜

ω = π

∗

(ω), where ω = ds

∧ dt is a

well deﬁned 2-form on

Q. Now, since f(x,

y) = f (x, y) means just that f is the

pullback of a smooth real function φ on

Q, it follows that any 2-form ω

Q is

proportional to ω, that is ω

= φ

× ω, with φ ∈ C

∞

(

Q, R).

2) The orbifold is contractible. The deformation retraction (s, x, y)

→ (sx, sy)

of R

{(0, 0)} projects on a smooth deformation retraction of Q. So, there is no

holonomy, Γ =

{0}. Now, since the origin 0

is the only point with structure

{±1},

every diﬀeomorphism of

Q preserves the origin 0

. So, the 2-point moment map is

exact, see the note 2 of Subsection 5.2, Souriau’s cocycle vanishes, σ

= 0. Let q

be any point of

Q and let μ

(q) = ψ(0

, q). This is an equivariant moment map

and μ

) = ψ(0

, 0

) = 0.

3) Let q

∈ Q, thus μ

(q) = Ψ(p) for any path p connecting 0

to q. Now, let

q belongs to a semi-axis ox or oy, and let us choose p = t

→ λ(t)q, where λ is a

smashing function equal to 0 on ]

− ∞, 0] and equal to 1 on [1, +∞[. Thus for all

∈ R, p(t) belongs to the same semi-axis than q. Thanks to the expression ♥ of

Subsection 3.2, we have for any 1-plot φ of Diff(

Q, ω

), centered at the identity,

Ψ(p)(φ)

(1) =

s
r

→ φ(r)(λ(s + t)q)

(

0
0

)

1
0

0
1

dt,

But, now (s, r)

→ φ(r)(λ(s + t)q) is a plot of the semi-axis, and thanks to the

item 1, the form ω vanishes on the semi-axis.

So, the integrand vanishes and

PATRICK IGLESIAS-ZEMMOUR

Ψ(p)(φ)

(1) = 0. Now, since 1-forms are characterized by 1-plots and since mo-

menta are characterized by centered plots, μ

(q) = 0 for all q

∈ Q belonging to any

semi-axis.

On the other hand, let q and q

be two points of the regular stratum ˙

Q. Since

{(x, y) | x > 0 & y > 0} is a diﬀeomorphism, and since ˜ω {(x, y) | x > 0 & y >

} is symplectic there exists always a symplectomorphism φ with compact support

S ⊂ {(x, y) | x > 0 & y > 0} which exchange q and q

. So, the image of this

diﬀeomorphism on ˙

Q can be extended by the identity on the whole Q. Therefore,

the automorphisms of ω are transitive on the regular stratum.

11.7. The cone orbifold. Let

be the quotient of the smooth complex

plane C by the action of the cyclic subgroup

{ζ ∈ C | ζ

= 1

} with m > 1.

The space

is an orbifold, according to [IKZ05]. We identify

to the complex

plane C, equipped with the pushforward of the standard diﬀeology by the map
π

: z

→ z

. That is, a plot of

is any parametrization P of C which writes

locally P(r) = φ(r)

, where φ is a smooth parametrization of C. Let us remark

Plane

Cone Orbifold

′

′′

Figure 2.

The cone orbifold

ﬁrst that the decomposition of

, in terms of structure group, is given by

Str(0) = Z

and

Str(z) =

{1} if z = 0.

And secondly that there is two orbits of Diff(

), the point 0 and the regular

stratum ˙

= C

− {0}. In particular any diﬀeomorphism of Q

preserves the

origin 0. It is not diﬃcult to check that dim(

) = 2 [Piz06-b], so every 2-form

is closed. Now,

1) Every 2-form of

is proportional to the 2-form ω uniquely deﬁned by

∗

(ω) : z

→ dx ∧ dy with z = x + iy.

That is, for any other 2-form ω

there exists a smooth function f

∈ C

∞

(

, R)

such that ω

= f

× ω.

2) The space (

Q, ω) is hamiltonian Γ

{0}. And, the action of G

is exact,

that is σ

= 0. In particular, the universal moment map μ

deﬁned by μ

(0) = 0,

is equivariant.

11. EXAMPLES OF MOMENT MAPS IN DIFFEOLOGY

3) The universal moment map μ

is injective. Its image is the reunion of two

coadjoint orbits, the point 0

∈ G

∗

, value of the origin of

, and the image of the

regular stratum ˙

Proof.

Let us ﬁrst prove that the usual surface form Surf = dx

∧ dy is the

pullback of a 2-form ω deﬁned on

. We shall apply the standard criterion and

prove that for any two plots φ

and φ

of C such that π

◦ φ

= π

◦ φ

we have

Surf(φ

) = Surf(φ

). That is, φ

(r)

= φ

(r)

implies Surf(φ

) = Surf(φ

First of all let us recall that, since we are dealing with 2-forms, is is suﬃcient to
consider 2-plots. So, let the φ

be deﬁned on some numerical domain U

⊂ R

. Let

∈ U, we split the problem into 2 cases.

1) φ

)

= 0 — Thus φ

)

= 0, there exists a open disk B centered at r

on which the φ

do not vanishes. Thus, the map r

→ ζ(r) = φ

(r)/φ

(r) deﬁned

on B is smooth with values in Z

. But, since Z

is discrete there exists ζ

∈ Z

such that φ

(r) = ζ

× φ

(r) on B. Now, Surf is invariant by U(1)

⊃ Z

. Therefore

Surf(φ

) = Surf(φ

) on B.

2) φ

) = 0 — Thus, φ

) = 0. Now, we have Surf(φ

) = det(D(φ

))

×Surf,

where D(φ

) denotes the tangent map of φ

. We split this case into two sub-cases:

2.a) D(φ

)

is non-degenerate — Thus, thanks to the implicit function the-

orem, there exists a small open disk B around r

where φ

is a local diﬀeomor-

phisms onto its image. Since φ

(r)

= φ

(r)

, the common zero r

of both φ

and φ

is isolated. Thus, the map r

→ ζ(r) = φ

(r)/φ

(r) deﬁned on B

− {r

}

is smooth, and for the same reason than in the ﬁrst case, ζ is constant.

So,

(r) = ζ

× φ

(r) on B

− {r

}. But, since φ

) = 0, this equality extends on

B. Therefore Surf(φ

) = Surf(φ

) on B.

2.b) D(φ

)

is degenerate — Let u be in the kernel of D(φ

)

. We have φ

su)

= φ

+ su)

for enough small real s. Then, diﬀerentiating this equality

m times with respect to s, for s = 0 we get 0 = D(φ

)

(u)

= D(φ

)

(u)

Therefore, D(φ

)

is also degenerate at r

and thus 0 = Surf(φ

)

= Surf(φ

)

So, we have proved that for any r

∈ U, Surf(φ

)

= Surf(φ

)

. Therefore, there

exists a 2-form ω on

such that π

∗

(ω) = Surf, and this form ω is completely

deﬁned by its pullback. Now, since the pullback by π

of any other 2-form ω

is proportional to Surf, the form ω

is proportional to ω.

Now, for the same reasons than in Subsection 11.6 the universal holonomy

and Souriau’s class σ

vanish, and the universal moment map μ

deﬁned by

(0) = 0

∗

is equivariant. Moreover, the regular stratum ˙

Q is just a symplectic

manifold for the restriction of ω. Any symplectomorphism with compact support
which doesn’t contain 0 can be extended to an automorphism of (

Q, ω). Thus, since

the compactly supported symplectomorphisms of a connected symplectic manifold
are transitive, the regular stratum ˙

Q is an orbit of Diff(Q, ω). Therefore, the

moment map μ

maps

Q onto two orbits, {0

∗

} and μ

( ˙

Q).

11.8. The inﬁnite projective space. This example of the symplectic struc-

ture of the inﬁnite projective space is extracted from [Piz06-a], everything not
proved here can be found there. Let

H be the Hilbert space of the square summa-

ble complex series.

H =

Z = (Z

)

∞

i=1

· Z

∞

PATRICK IGLESIAS-ZEMMOUR

Where the dot denotes the hermitian product. The space

H is equipped with

the ﬁne structure of complex diﬀeological vector space. That is, its diﬀeology is
generated by the linear injections from C

H, or if we prefer, let P : U → H be

a plot, then for every r

∈ U, there exists an integer n, an open superset V ⊂ U of

, a ﬁnite family

F = {(λ

, Z

)

}

∈A

, where the Z

∈ H, and the λ

∈ C

∞

(V, C

)

such that P

V : r →

∈A

(r)

× Z

. Such a family

{(λ

, Z

)

}

∈A

is called a

local family of P at the point r

. We deﬁned the symbol dZ which associates to

every local family

F = {(λ

, Z

)

}

∈A

deﬁned on the domain V, the complex valued

1-form of V

dZ(

F) : r →

∈A

dλ

(r)Z

For every λ

= x

+ iy

, where x

and y

are real smooth parametrizations, dλ

+ idy

. Now, there exists on

H a 1-form α deﬁned by

α = 1

· dZ − dZ · Z].

1) As an additive group (

H, +) acts on itself, preserving dα. Let Z ∈ H and let

be the translation by Z, then t

∗

(dα) = dα. This action is hamiltonian but not

exact. Let μ be the moment map of the translations (

H, +), deﬁned by μ(0

) = 0.

μ(Z) = 2d[w(Z)]

with

w(ζ) : Z

→ 1

[ζ

· Z − Z · ζ] ∈ C

∞

(

H, R).

The moment map μ is injective and (

H, dα) is an homogeneous symplectic space.

2) Let U(

H) be the group of unitary transformations of H, equipped with the

functional diﬀeology. The group U(

H) acts on H preserving α. The action of U(H)

on (

H, dα) is exact and hamiltonian. Let P : U → U(H) be a n-plot. The value of

the moment map μ of the action of U(

H) on (H, dα), evaluated on P is given by

μ(Z)(P)

(δr) = 1

P(r)(Z)

∂P(r)(Z)

∂r

(δr)

−

∂P(r)(Z)

∂r

(δr)

· P(r)(Z)

where, r

∈ U , δr ∈ R

and:

P(r)(Z) =

loc

∈A

(r)Z

then

∂P(r)(Z)

∂r

(δr) =

loc

∈A

∂λ

(r)

∂r

(δr)Z

3) The unit sphere

S ⊂ H is an homogeneous space of U(H). The ﬁbers of the

equivariant moment map μ of the action of U(

H) on (S, dα S) are the ﬁbers of

the inﬁnite Hopf ﬁbration π :

S → P = S/S

, where S

∈ C acts multiplicatively

S. There exists a symplectic form ω on P, such that π

∗

(ω) = dα

S. The

equivariant moment map of the induced action of U(

H) on P is injective. So, the

inﬁnite projective space

P, equipped with the Fubini-Study form, is an homogeneous

symplectic space and can be regarded as a coadjoint orbit of U(

H).

Proof.

Many of what is asserted here has been proved in [Piz06-a]. So, we

shall just check what is not in this paper.

1) Since

H is contractible, there is no holonomy. Now, let ζ ∈ H and t

the translation t

(Z) = Z + ζ. A direct computation shows that, t

∗

(α) = α +

d[w(ζ)]. Thus, dα is invariant by translation t

∗

(dα) = dα. Now, let p be any path

connecting 0

to Z, we have μ(Z) = Ψ(p) = ˆ

∗

K(dα) = ˆ

∗

(α)

− ˆ0

∗

(α)

− d[Kα ◦ ˆp].

But, on one hand we have ˆ

Z = t

, thus ˆ

∗

(α)

− ˆ0

∗

(α) = t

∗

(α)

− 1

∗

(α) = α +

d[w(Z)]

− α = d[w(Z)]. And, on the other hand we have, ˆp(ζ) = t

◦ p, and

11. EXAMPLES OF MOMENT MAPS IN DIFFEOLOGY

thus Kα

◦ ˆp =

◦p

α =

∗

(α) =

α +

d[w(ζ)] =

α + w(ζ)(Z), since

w(ζ)(0

) = 0. So, μ(Z) = d[w(Z)]

− d[ζ → w(ζ)(Z)]. But, w(ζ)(Z) = −w(Z)(ζ)

so μ(Z) = d[w(Z)]

− d[ζ → −w(Z)(ζ)] = 2d[w(Z)]. Now, let Z be in the kernel of

μ, so w(Z) = const = w(0

) = 0. But w(Z)(Z

) = 0 for all Z

∈ H if and only if

Z = 0

, we have just to decompose Z into real and imaginary parts and use the

fact that the hermitian norm on

H is not degenerated. Therefore, μ is injective.

2) Since the 1-form α is invariant by U(

H), this statement is a direct application

of Subsection 6.1.

11.9. The Virasoro coadjoint orbits. Let Imm(S

, R

) be the space of all

the immersions of the circle S

= R/2πZ into R

, equipped with the functional

diﬀeology. For every n-plot P : U

→ Imm(S

, R

) let us deﬁned the 1-form α(P)

on U by

α(P)

(δr) =

2π

P(r)

(t)

P(r)

(t)

∂P(r)

(t)

∂r

(δr)

dt.

for every r

∈ U and δr ∈ R

. Where the prime denotes the derivative with respect

to the parameter t, and the bracket

· | · denotes the ordinary scalar product of

the vector space R

1. As deﬁned above, α is a 1-form of Imm(S

, R

Let us consider now the group Diff

) of positive diﬀeomorphisms of the circle,

and its action on Imm(S

, R

) by re-parametrization. For every ϕ

∈ Diff

for every x

∈ Imm(S

, R

), let us denote by ¯

ϕ(x) the pushforward of x by ϕ,

ϕ(x) = ϕ

∗

(x) = x

◦ ϕ

−1

And, let F : Diff

)

→ C

∞

(Imm(S

, R

), R) be the map deﬁned, for all ϕ

∈

Diff

), by

F(ϕ) : x

→

2π

log

(t)

d log(ϕ

(t))

2. The map F is smooth and for every ϕ

∈ Diff(S

∗

(α) = α

− d[F(ϕ)].

So, the 2-form ω = dα, deﬁned on Imm(S

, R

), is closed and invariant by

the action of Diff(S

). Moreover, the action of Diff(S

) is hamiltonian.

3. Let x

: class(t)

→ (cos(t), sin(t)) be the standard immersion from S

R/2πZ to R

. The moment maps for ω, of Diff

) on the connected

component of x

∈ Imm(S

, R

), are translated by a constant from

μ(x)(r

→ ϕ)

(δr) =

2π

(u)

−

log

(u)

δu du.

Where r

→ ϕ is any plot of Diff

) deﬁned on some n-domain U, r is

a point of U, δr

∈ R

, u = ϕ

−1

(t), and δu = D(r

→ u)(r)(δr).

4. With the same conventions as in item 3,Souriau’s cocycles of the Diff

)

action on Imm(S

, R

) are cohomologous to θ deﬁned by,

θ(g)(r

→ ϕ)

(δr) =

2π

3γ

(u)

− 2γ

(u)γ

(u)

δu du,

where g

∈ Diff

) and γ = g

−1

. We recognize the integrand of the

right hand side as the so-called Schwartzian derivative of γ.

PATRICK IGLESIAS-ZEMMOUR

5. Let β be the function for all g and h in Diff

) by

β(g, h) =

2π

log(g

◦ h)

(t) d log h

(t).

So, for all g and h in Diff

) we have

F(g

◦ g

) = F(g)

◦ ¯g

+ F(g

)

− β(g, g

This function β is known as Bott’s cocycle [Bot78]. The central exten-
sion of Diff

) by β is the so-called Virasoro group. Its action on

Imm(S

, R

), through Diff

), is still hamiltonian, but now exact.

This is a well known construction which will be not more developed here.

This example which has been built on purpose [Igl95], gathers the main ingredients
found in the literature on the construction of Virasoro’s group. I regard this example
as a nice illustration of the whole theory.

Proof.

The proof is actually a long and tedious series of computations. To

make it as clear as possible, we shall split the computations in a few steps.

The 1-form α — We prove ﬁrst that α is a well deﬁned 1-form on Imm(S

, R

Let F : U

→ U be a smooth m-parametrization. We have, for all s ∈ V and all

δs

∈ R

α(P

◦ F)(s)(δs) =

2π

(P ◦ F)(s)

(t)

◦ F)(s)

(t)

∂(P

◦ F)(s)

(t)

∂s

(δs)

That is,

α(P

◦ F)(s)(δs) =

2π

P(F(s))

(t)

P(F(s))

(t)

∂P(F(s))

(t)

∂s

(δs)

dt.

Let us denote by r the point F(s). We get,

α(P

◦ F)(s)(δs) =

2π

P(r)

(t)

P(r)

(t)

∂P(r)

(t)

∂r

∂F(s)

∂s

(δs)

α(P)

r=F(s)

∂F(s)

∂s

(δs)

∗

(α(P))

(δs).

So, α(P

◦ F) = F

∗

(α(P)), and α satisﬁes the diﬀerential form axiom.

Let us consider now the action of Diff

) on Imm(S

, R

). This action is

obviously smooth from the very deﬁnition of the functional diﬀeology of Diff

Let us denote ϕ

−1

by φ such that

∗

(α)(P) = α( ¯

◦ P) = α[r → P(r) ◦ ϕ

−1

] = α[r

→ P(r) ◦ φ].

Note that Diff

) acts on speed and acceleration of any immersion x, by

(

♥)

◦ φ)

(t)

(φ(t))

· φ

(t)

◦ φ)

(t)

(φ(t))

· φ

(t)

+ x

(φ(t))

· φ

(t).

Let us denote by Q the plot ¯

◦ P, that is Q = [r → P(r) ◦ φ]. Such that,

α( ¯

◦ P)

(δr) =

2π

Q(r)

(t)

Q(r)

(t)

∂Q(r)

(t)

∂r

(δr)

11. EXAMPLES OF MOMENT MAPS IN DIFFEOLOGY

for all r

∈ U and all δr ∈ R

. But, from

♥,

Q(r)

(t)

(P(r)

◦ φ)

(t) = P(r)

(φ(t))

· φ

(t)

Q(r)

(t)

(P(r)

◦ φ)

(t) = P(r)

(φ(t))

· φ

(t)

+ P(r)

(φ(t))

· φ

(t)

So, α( ¯

◦ P)

(δr) is equal to the sum A + B of the two following integrals, related

to the decomposition of Q(r)

(t),

A =

2π

P(r)

(φ(t))

· φ

(t)

P(r)

(φ(t))

· φ

(t)

∂P(r)

(φ(t))

· φ

(t)

∂r

(δr)

dt,

B =

2π

P(r)

(φ(t))

· φ

(t)

P(r)

(φ(t))

· φ

(t)

∂P(r)

(φ(t))

· φ

(t)

∂r

(δr)

dt.

The ﬁrst integral is equal to

A =

2π

P(r)

(φ(t))

P(r)

(φ(t))

∂P(r)

(φ(t))

∂r

(δr)

(t)dt.

And, since ϕ, and thus φ, is a positive diﬀeomorphism, after the change of variable
t

→ φ(t), we get

A = α(P)

(δr).

The second integral is given by

B =

2π

P(r)

(φ(t))

P(r)

(φ(t))

∂P(r)

(φ(t))

∂r

(δr)

(t)

Let us denote for short,

x = P(r),

= P(r)

and

δx

→

∂P

(r)(t)

∂r

(δr)

such that the last integral writes

B =

2π

(φ(t))

| δx

(φ(t))

(t)

dt.

Let us remind that, for any variation δ

v =

v | δv ⇒ δ log v =

v =

v | δv.

So, the integrand in the last expression of B writes,

(φ(t))

| δx

(φ(t))

= δ log x

(φ(t))

Thus, the term B becomes

2π

δ log

(φ(t))

d log(φ

(t))

2π

log

(φ(t))

d log(φ

(t))

2π

log

(ϕ

−1

(t))

d log((ϕ

−1

)

(t))

PATRICK IGLESIAS-ZEMMOUR

Let us make the change of variable s = ϕ

−1

(t), we get,

+ δ

2π

log

(s)

d log[(ϕ

−1

)

(ϕ(s))]

− δ

2π

log

(s)

d log(ϕ

(s))

−

∂

∂r

2π

log

P(r)

(s)

d log(ϕ

(s))

(δr)

−

∂

∂r

F(ϕ)(P(r))

(δr)

− d[F(ϕ)](P)

(δr).

Coming back to α( ¯

◦ P)

(δr) we get ﬁnally,

α( ¯

◦ P)

(δr) = α(P)

(δr)

− d[F(ϕ)](P)

(δr)

that is

∗

(α) = α

− d[F(ϕ)].

Thus, the exterior diﬀerential ω = dα is invariant by the action of Diff

). And

since the diﬀerence ¯

∗

(α)

− α is exact, this action is hamiltonian.

The 2-point moment map — Now, let us compute the 2-points moment maps

ψ of the action of Diff

) on (Imm(S

, R

), ω). Let p be a path connecting two

immersions x

and x

. We have Ψ(p) = ˆ

∗

(Kω) = ˆ

∗

(Kdα) = ˆ

∗

(ˆ

∗

(α)

− ˆ0

∗

(α)

−

d(Kα)) = ˆ

∗

(α)

− ˆx

∗

(α)

− d(Kα ◦ ˆp). But, for all ϕ ∈ Diff

Kα

◦ ˆp(ϕ) =

ϕ(p)

α =

∗

(α) =

−

dF(ϕ) =

− F(ϕ)(x

) + F(ϕ)(x

So, we get ﬁnally

Ψ(p) = ψ(x

, x

) =

{ˆx

∗

(α) + d[ϕ

→ F(ϕ)(x

)]

} − {ˆx

∗

(α) + d[ϕ

→ F(ϕ)(x

)]

But notice that, ˆ

∗

(α) + d[ϕ

→ F(ϕ)(x) is not a momentum of Diff

The 1-point moment maps — Let us compute the moment map ψ(x

, x). Let

m =

{ˆx

∗

(α) + d[ϕ

→ F(ϕ)(x)]}(r → ϕ)

(δr).

And, let us denote for short

∗

(α)(r

→ ϕ)

(δr)

d[ϕ

→ F(ϕ)(x)](r → ϕ)

(δr) =

∂F(ϕ)(x)

∂r

δr.

We shall use the notation m

, A

and B

for the immersion x

. Thus,

ψ(x

, x)(r

→ ϕ)

(δr) = m

− m

= A + B

− A

− B

We have, ˆ

∗

(α)(r

→ ϕ) = α(ˆx ◦ [r → ϕ]) = α(r → x ◦ ϕ

−1

). Let φ = ϕ

−1

, so

A =

2π

(x ◦ φ)

(t)

◦ φ)

(t)

∂(x

◦ φ)

(t)

∂r

(δr)

Let us introduce now,

u = φ(t),

= φ(t)

and

= φ

(t).

So, the decomposition given by

♥, writes

◦ φ)

(t) = x

(u)

· u

and

◦ φ)

(t) = x

(u)

· u

+ x

(u)

· u

11. EXAMPLES OF MOMENT MAPS IN DIFFEOLOGY

Then, we shall use the preﬁx δ for every variation associated to δr, that is δ =
D(r

→ )(r)(δr). So,

∂(x

◦ φ)

(t)

∂r

(δr) = δ[x

(u)

· u

] = x

(u)

· δu · u

+ x

(u)

· δu

Thus,

2π

(u)

(u)u

+ x

(u)u

| x

(u)u

δu + x

(u)δu

2π

(u)

δu u

dt +

2π

(u), x

(u)

δu

dt +

2π

δu

Now,

B =

∂F(ϕ)(x)

∂r

δr =

−

∂ ¯

F(φ)(x)

∂r

δr =

−δ[¯F(φ)(x)],

with

F(φ)(x) =

2π

log

(φ(t))

d log φ

(t) =

2π

log

(u)

d log(u

So, after the variation with respect to δr and an integration by part, we get

−

2π

(u), x

(u)

δu

−

2π

log

(u)

δd log(u

)

−

2π

(u), x

(u)

δu

dt +

2π

(u), x

(u)

δ log(u

) dt

−

2π

(u), x

(u)

δu

dt +

2π

(u), x

(u)

δu

Therefore, grouping the terms and integrating again by part, we get

A + B

2π

(u)

δu du + 2

2π

(u), x

(u)

δu

dt +

2π

δu

2π

(u)

δu du

− 2

2π

log

(u)

δu du +

2π

δu

2π

(u)

−

log

(u)

δu du +

2π

δu

Now, since

(t)

= 1 we get the value of the 2-point moment map,

ψ(x

, x)(r

→ ϕ)

(δr) =

2π

(u)

−

log

(u)

δu du

−

2π

δu du.

The second term of the right hand side of the equality is a constant momentum
of Diff

), so it can be avoided. And, every moment map is, up to a constant,

equal to the moment μ announced.

Souriau’s cocycles — Souriau’s cocycle associated to immersion x

is deﬁned

by θ(g) = ψ(x

, ¯

g(x

)), see Subsection 5.2. So, we have to replace, in the expression

of ψ above, x by ¯

g(x

) = x

◦ g

−1

, that is x = x

◦ γ. So, θ(g)(r → ϕ)

(δr) =

ψ(x

, x

◦ γ). So, note ﬁrst that

◦ γ)

(u) = x

(γ(u))γ

(u)

and

◦ γ)

(u) = x

(γ(u))γ

(u)

+ x

(u)γ

(u).

PATRICK IGLESIAS-ZEMMOUR

And, let us remind that

= x

= 1 and x

| x

= 0. We get,

(u)

= γ

(u)

and

(u)

= γ

(u)

+ γ

(u)

This gives

(u)

= γ

(u)

and

log

(u)

= 2

(u)γ

(u)

− γ

(u)

Thus,

θ(g)(r

→ ϕ)

(δr)

2π

3γ

(u)

− 2γ

(u)γ

(u)

δu du

2π

(u)

δu du

−

2π

δu du.

But, after a change of variable u

→ v = γ(u), we get

2π

(u)

δu du =

2π

(δuγ

(u)) γ

(u)du =

2π

δv dv.

So the two last terms cancel each other, and we get the value announced for
Souriau’s cocycle θ.

Bott’s cocycle — The real function F(g

◦h)−F(g)◦ ¯h −F(h) is constant since X

is connected, and its diﬀerential is equal to (¯

◦ ¯h)

∗

(α)

− ¯h

∗

(¯

∗

(α)), that is 0. Now,

to explicit β(g, g

) = F(g)

◦ ¯g

+ F(g

)

−β(g, g

)

−F(g ◦g

), it is suﬃcient to compute

the right hand member on the standard immersion x

, for which the speed norm is

equal to 1, and thus log

(t)

= 0 for all real t. So we get,

β(g, h)

F(g)(x

◦ h

−1

)

− F(h)(x

)

− F(g ◦ h)(x

)

2π

log

◦ h

−1

)

(t)

d log g

(t)

2π

log(h

−1

)

(t) d log g

(t)

−

2π

log h

−1

(t)) d log g

(t)

−

2π

log h

(s) d log g

(h(s))

2π

log(g

◦ h)

(t) d log h

(t)

And this is the standard expression of Bott’s cocycle.

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nonorientable surfaces, 2009

947 W. Turner, Rock blocks, 2009

946 Jay Jorgenson and Serge Lang, Heat Eisenstein series on SL

(C), 2009

945 Tobias H. J¨

ager, The creation of strange non-chaotic attractors in non-smooth

saddle-node bifurcations, 2009

TITLES IN THIS SERIES

944 Yuri Kifer, Large deviations and adiabatic transitions for dynamical systems and Markov

processes in fully coupled averaging, 2009

943 Istv´

an Berkes and Michel Weber, On the convergence of

f (n

x), 2009

942 Dirk Kussin, Noncommutative curves of genus zero: Related to ﬁnite dimensional

algebras, 2009

941 Gelu Popescu, Unitary invariants in multivariable operator theory, 2009

940 G´

erard Iooss and Pavel I. Plotnikov, Small divisor problem in the theory of

three-dimensional water gravity waves, 2009

939 I. D. Suprunenko, The minimal polynomials of unipotent elements in irreducible

representations of the classical groups in odd characteristic, 2009

938 Antonino Morassi and Edi Rosset, Uniqueness and stability in determining a rigid

inclusion in an elastic body, 2009

937 Skip Garibaldi, Cohomological invariants: Exceptional groups and spin groups, 2009

936 Andr´

e Martinez and Vania Sordoni, Twisted pseudodiﬀerential calculus and

application to the quantum evolution of molecules, 2009

935 Mihai Ciucu, The scaling limit of the correlation of holes on the triangular lattice with

periodic boundary conditions, 2009

934 Arjen Doelman, Bj¨

orn Sandstede, Arnd Scheel, and Guido Schneider, The

dynamics of modulated wave trains, 2009

933 Luchezar Stoyanov, Scattering resonances for several small convex bodies and the

Lax-Phillips conjuecture, 2009

932 Jun Kigami, Volume doubling measures and heat kernel estimates of self-similar sets,

2009

931 Robert C. Dalang and Marta Sanz-Sol´

e, H¨

older-Sobolv regularity of the solution to

the stochastic wave equation in dimension three, 2009

930 Volkmar Liebscher, Random sets and invariants for (type II) continuous tensor product

systems of Hilbert spaces, 2009

929 Richard F. Bass, Xia Chen, and Jay Rosen, Moderate deviations for the range of

planar random walks, 2009

928 Ulrich Bunke, Index theory, eta forms, and Deligne cohomology, 2009

927 N. Chernov and D. Dolgopyat, Brownian Brownian motion-I, 2009

926 Riccardo Benedetti and Francesco Bonsante, Canonical wick rotations in

3-dimensional gravity, 2009

925 Sergey Zelik and Alexander Mielke, Multi-pulse evolution and space-time chaos in

dissipative systems, 2009

924 Pierre-Emmanuel Caprace, “Abstract” homomorphisms of split Kac-Moody groups,

2009

923 Michael J¨

ollenbeck and Volkmar Welker, Minimal resolutions via algebraic discrete

Morse theory, 2009

922 Ph. Barbe and W. P. McCormick, Asymptotic expansions for inﬁnite weighted

convolutions of heavy tail distributions and applications, 2009

921 Thomas Lehmkuhl, Compactiﬁcation of the Drinfeld modular surfaces, 2009

920 Georgia Benkart, Thomas Gregory, and Alexander Premet, The recognition

theorem for graded Lie algebras in prime characteristic, 2009

919 Roelof W. Bruggeman and Roberto J. Miatello, Sum formula for SL

over a totally

real number ﬁeld, 2009

918 Jonathan Brundan and Alexander Kleshchev, Representations of shifted Yangians

and ﬁnite W -algebras, 2008

For a complete list of titles in this series, visit the

AMS Bookstore at www.ams.org/bookstore/.

ISBN 978-0-8218-4709-1

9 780821 847091

MEMO/207/972

Document Outline

Introduction
Chapter 1. Few words about diffeology
Chapter 2. Diffeological groups and momenta
Chapter 3. The paths moment map
Chapter 4. The 2-points moment map
Chapter 5. The moment maps
Chapter 6. The moment maps for exact 2-forms
Chapter 7. Functoriality of the moment maps
Chapter 8. The universal moment maps
Chapter 9. About symplectic manifolds
Chapter 10. The homogeneous case
Chapter 11. Examples of moment maps in diffeology
Bibliography

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