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П•(g) П€(g в€’1 k) (...) .
(П• в€— П€)(k) =
gв€€О±в€’1 (О±(k))

But in order to integrate, we need measures on the О±-п¬Ѓbers. If {О»x }xв€€X is a family
of measures on the О±-п¬Ѓbers, then we deп¬Ѓne the convolution product of П• and П€
to be
П•(g) П€(g в€’1 k) dО»О±(k) .
(П• в€— П€)(k) =
gв€€О±в€’1 (О±(k))

Here, we assume that the family of measures {О»x } is continuous in x. To ensure
that this product is associative, we require left invariance of {О»x }, i.e. we require
that {О»x } be a Haar system (cf. Sections 13.4 and 11.4).
The vector space of bounded continuous functions П• on G for which the target
map О± restricted to support(П•) is a proper map, is closed under the convolution
product. Its completion under a suitable norm is called the groupoid C в€— -algebra
associated to the Haar system {О»x }. Since the multiplicative structure depends
14.3 Intrinsic Groupoid Algebras 99

on the choice of {О»x }, the groupoid algebra is sometimes denoted by AО» . We refer
to  for more details about the analytic aspects of this construction.
The groupoid algebra operates on functions on the base. Let П• be a function
on G, and u a function on X. Deп¬Ѓne

(Op П•)u(x) := П•(g) u(ОІ(g)) dО»x .
gв€€О±в€’1 (x)

Intuitively, if we think of the elements of G as вЂњarrowsвЂќ on the base space X, then
this integral tells us to look at all the arrows g going into a given point x в€€ X,
evaluate the function u at the tail of each of those arrows, then move back to x and
integrate over all arrows g with вЂњweightвЂќ given by П•.

Examples.

вЂў G = X Г— X вЂ“ The groupoid algebra is isomorphic to the вЂњmatrixвЂќ algebra
of functions on X Г— X (see Section 14.1). If X is п¬Ѓnite, it really is a matrix
algebra.

вЂў G = a group вЂ“ The groupoid algebra is isomorphic to a subalgebra of the
standard group algebra (see Chapter 11). A function on G acts on constant
functions via multiplication by its integral over G.

вЂў G = X, where (f в€— g)(x) = f (x)g(x) вЂ“ The groupoid algebra is the algebra of
functions on X (which operates on itself by pointwise multiplication).

в™¦

14.3 Intrinsic Groupoid Algebras

Suppose that G is a Lie groupoid over X. Denote the bundles over G of 1 -densities
2
1 1
along the О±- and ОІ-п¬Ѓbers by в„¦О± and в„¦ОІ , respectively. Letting
2 2

1 1
в„¦ = в„¦О± вЉ— в„¦ОІ ,
2 2

the intrinsic groupoid algebra of G is the completion of the space О“(в„¦) of
compactly supported sections of в„¦ under a suitable norm. The term вЂњintrinsicвЂќ
refers to the fact that it does not involve the arbitrary choice of a Haar system.
The multiplication on О“(в„¦) is deп¬Ѓned as follows.
Suppose that ОІ(g) = О±(h) = x в€€ X G(0) . There is a natural isomorphism

в„¦(g) вЉ— в„¦(h) в€’в†’ в„¦1 (g) вЉ— в„¦(gh)
О±

constructed using the identiп¬Ѓcations

1 вЉ— rhE
1 1 1 1
в„¦О± (g) вЉ— в„¦ОІ (g) в„¦О± (g) вЉ— в„¦ОІ (gh)
2 2 2 2

вЉ— 1E
1 1 1 1
g
в„¦О± (h) вЉ— в„¦ОІ (h) в„¦О± (gh) вЉ— в„¦ОІ (h)
2 2 2 2
100 14 GROUPOID ALGEBRAS

together with
lgв€’1
1 1
в„¦О± (g) E в„¦О± (x)
2 2

d
d
d
d
d
В‚
1 1
2 (T G/T G(0) ) = в„¦ 2 (N G(0) )
в„¦ x x x

В
В
В
В
В
1 1
h
в„¦ОІ (h) ' в„¦ОІ (x)
2 2

1 1
In general, there is no natural isomorphism between в„¦О± and в„¦ОІ at a given point
2 2

in G. However, on G(0) an isomorphism is provided by projection along the identity
1 1
section from X into G(0) : we can identify both в„¦О± and в„¦ОІ over x в€€ G(0) with the
2 2

1 (0)
to G(0) in G at x.
2 -densities on the normal space Nx G
We use these isomorphisms to determine the product of П•, П€ в€€ О“(в„¦). The
product section П•П€ в€€ О“(в„¦) is given at a point k в€€ G by the formula

П•(g) П€(g в€’1 k)
(П•П€)(k) =
{g|О±(g)=О±(k)}

where we regard П•(g)П€(g в€’1 k) as an element of в„¦1 (g) вЉ— в„¦(k), and we integrate the
О±
1-density factor over the О±-п¬Ѓber through k.

Exercise 48
Check that, if we instead use the maps
в„¦(g) вЉ— в„¦(h) в€’в†’ в„¦1 (h) вЉ— в„¦(gh) ,
ОІ

the resulting multiplicative structure on О“(в„¦) is the same.

Remark. The identiп¬Ѓcations above also provide a natural isomorphism
1 1
О±в€— (в„¦ 2 E) вЉ— ОІ в€— (в„¦ 2 E) ,
в„¦

в™¦
which we will use often.
Let E be the normal bundle of G(0) in G. The smooth groupoid algebra
О“(в„¦) acts on smooth sections of
1 1
в„¦ 2 E := | в€§top | 2 E в€— .
1
To see this left action, take П• в€€ О“(в„¦) and a section Оі of в„¦ 2 E. We can think of П•
at g в€€ G as a 1 -density on the normal space through x = О±(g), times a 1 -density
2 2
on the normal space through through y = ОІ(g):
1 1 1 1
в„¦О± (g) вЉ— в„¦ОІ (g) в„¦ 2 Ex вЉ— в„¦ 2 Ey
в„¦(g) = 2 2

E П•О± (x) вЉ— П•ОІ (y)
П•(g)
14.4 Groupoid Actions 101

Since П•ОІ (y)Оі(y) в€€ в„¦1 Ey в„¦1 (y) в„¦1 (g), we can consider П•(g)Оі(y) as an element
О± О±
1 1
1
of в„¦О± (g) вЉ— в„¦ x
2 E . The new section П• В· Оі of в„¦ 2 E is then given at a point x в€€ X by

(П• В· Оі)(x) = П•(g)Оі(ОІ(g))
gв€€О±в€’1 (x)
1
where we integrate the в„¦1 factor of П•(g)Оі(ОІ(g)) в€€ в„¦1 (g) вЉ— в„¦ 2 Ex over the О±-п¬Ѓber
О± О±
through x.

Exercise 49
Check that this is indeed a left action, i.e. П• В· (П€ В· Оі) = (П•П€) В· Оі, for any
1
П•, П€ в€€ О“(в„¦), and Оі в€€ О“(в„¦ 2 E).

We could just as well deп¬Ѓne a right action by reversing the О± and ОІ roles, namely,

(Оі В· П•)(x) = П•(h)Оі(О±(h)) ,
hв€€ОІ в€’1 (x)
1 1
with П•(h)Оі(О±(h)) в€€ в„¦1 EО±(h) вЉ— в„¦ 2 Ex в„¦1 (h) вЉ— в„¦ 2 Ex .
ОІ

14.4 Groupoid Actions
Вµ
A groupoid G over X G(0) may act on sets M в†’ X that map to X. Let G в€— M
be the space
G в€— M := {(g, m) в€€ G Г— M | ОІ(g) = Вµ(m)} .
A (left) groupoid action of G on M is deп¬Ѓned to be a map G в€— M в†’ M , taking
the pair (g, m) to g В· m, with the properties:
1. Вµ(g В· m) = О±(g),
2. (gh) В· m = g В· (h В· m),
3. (ОµВµ(m)) В· m = m.
The map Вµ : M в†’ X is sometimes called the moment map, by analogy with
symplectic geometry.
Remark. The terms вЂњmoment mapвЂќ and вЂњmomentum mapвЂќ are usually used
interchangeably in the literature, with diп¬Ђerent authors preferring each of these
two translations of SouriauвЂ™s  French term, вЂњmomentвЂќ. By contrast, in these
notes, we have used the terms in diп¬Ђerent ways. Here, a вЂњmomentum mapвЂќ is a
Poisson map J : M в†’ gв€— to a Lie-Poisson manifold gв€— , generating a hamiltonian
action of an underlying Lie group G on M . On the other hand, a вЂњmoment mapвЂќ
is a map Вµ : M в†’ X to the base X of a groupoid G which is acting on M . в™¦

Example. A groupoid G over X acts on G by left multiplication with moment
в™¦
map О± and on X with moment map the identity.
Given additional structure on G or M , we can specify special types of actions.
For instance, groupoids act on vector bundles (rather than vector spaces). Suppose
that we have a groupoid G over X and a vector bundle V also over X,
G V
О± в†“в†“ ОІ в†“Вµ
X X
102 14 GROUPOID ALGEBRAS

A representation or linear action of G on V is a groupoid action of G on V
whose maps
gВ· : Вµв€’1 (ОІ(g)) в€’в†’ Вµв€’1 (О±(g))
are linear. For more on groupoid actions, see .

V
gВ· \$\$\$
\$\$
\$\$ Вµв€’1 (y)
W
Вµв€’1 (x)

ry
g \$\$\$
\$\$
r\$
W\$
x
X

We can think of a representation of a groupoid as a collection of representations
of the isotropy subgroups together with ways of identifying these representations
using diп¬Ђerent вЂњarrowsвЂќ in X.
Example. If X is a topological space, and G = О (X) is the fundamental
groupoid, then a representation of О (X) on a vector bundle V would be a п¬‚at
connection of V . By п¬‚at connection, we do not yet mean a diп¬Ђerential-geometric
notion, but rather a topological one, namely that parallel transport only depends
on the homotopy class of the base path.
To see the п¬‚at connection, recall that О (X) is the collection of homotopy classes
of paths in X. A representation of О (X) determines precisely how to parallel
translate along paths to deп¬Ѓne a connection.
With this п¬‚at connection, we can look at the isotropy subgroup of loops based
at a point. The fundamental group of X acts on each п¬Ѓber in the usual sense, and
we thus see that the representation of the fundamental groupoid on V includes the
action of the fundamental group on a п¬Ѓber of V .
For applications to the moduli spaces used in topological quantum п¬Ѓeld theory,
в™¦
see .
As with groups, the notion of groupoid representation can be formalized in
terms of the following deп¬Ѓnition. The general linear groupoid of a vector bundle
Вµ : V в†’ X is

GL(V ) = {(x, , y) | x, y в€€ X, : Вµв€’1 (y) в†’ Вµв€’1 (x) is a linear isomorphism} .

The isotropy subgroup over any point is the general linear group of the correspond-
ing п¬Ѓber of V . A representation of G in V is then a groupoid homomorphism
from G to GL(V ), covering the identity map on X.
14.5 Groupoid Algebra Actions 103

The general linear groupoid is a subset of a larger object

gl(V ) = {(x, , y) | x, y в€€ X, : Вµв€’1 (y) в†’ Вµв€’1 (x) is linear} ,

where is an arbitrary linear map between п¬Ѓbers. This is a generalization of the
Lie algebra gl(n; R) of the general linear group GL(n; R).

14.5 Groupoid Algebra Actions

Example. If G is a group, V is a vector space, and r : G в†’ End(V ) is a map,
then there is an induced map r : C (G) в†’ End(V ) deп¬Ѓned by the formula

П• в€’в†’ r(П•) := r(g) П•(g) dg .
G

If r is a representation, then r will be a homomorphism of algebras. Hence, group
в™¦
representations correspond to representations of the measure group algebra.
For a groupoid G, there is a similar correspondence. Given a representation
of a groupoid G on a vector bundle V and a Haar system {О»x } on G, there is an
action of the groupoid algebra AО» on sections of V deп¬Ѓned as follows. Let П• be any
continuous compactly supported function on G, and let u в€€ О“(V ). Deп¬Ѓne

(П• В· u)(x) = П•(g) g В· u(ОІ(g)) dО»x (g) .
gв€€О±в€’1 (x)

We can also describe the action of the intrinsic groupoid algebra.
Recall that, if E denotes the normal bundle to G(0) in G, then the intrinsic
groupoid algebra is (a suitable completion of) the set of sections of
1 1
в„¦ = О±в€— (в„¦ 2 E) вЉ— ОІ в€— (в„¦ 2 E) .

For a vector bundle V over G(0) , we deп¬Ѓne

End(V ) := О±в€— (V ) вЉ— ОІ в€— (V в€— ) ;

that is, End(V ) is the bundle over G whose п¬Ѓber over each point g в€€ G is
в€—
VО±(g) вЉ— VОІ(g) = Hom (VОІ(g) , VО±(g) ) .

Given a representation of G on V , the sections of

в„¦ вЉ— End(V )

act naturally on sections of V . We thus build a groupoid algebra with coeп¬ѓ-
cients in a vector bundle V ,

О“(в„¦ вЉ— End(V )) .

Remark. In Section 14.3, we found an action of the intrinsic groupoid algebra
1
on sections of в„¦ 2 E. However, this does not generally come from a representation
1
of G on в„¦ 2 E (see below).
104 14 GROUPOID ALGEBRAS

We would have liked that the groupoid algebra acted on 1 -densities on G(0) X
2
1
itself. However, in general, the algebra that acts on sections of в„¦ 2 T X is that of
sections of
1 1 1 1 1
в„¦ вЉ— End(в„¦ 2 T X) = О±в€— (в„¦ 2 E) вЉ— ОІ в€— (в„¦ 2 E) вЉ— О±в€— (в„¦ 2 T X) вЉ— ОІ в€— (в„¦в€’ 2 T X) .

In very special instances, there might be a natural trivialization of
1 1
О±в€— (в„¦ 2 T X) вЉ— ОІ в€— (в„¦в€’ 2 T X)

and we do obtain an action on 1 -densities on X. в™¦
2

Alternatively, the intrinsic groupoid algebra itself acts on sections of
1
V вЉ— в„¦2 E .

In order to obtain a representation of the groupoid algebra on sections of V , we
1
hence need a representation of G on V вЉ— в„¦в€’ 2 E.
Examples.

вЂў When G is a Lie group, then E = g is the Lie algebra, and there does exist a
natural adjoint action of G on g. This gives rise to a representation of G on
1 1 1 1
в„¦ 2 E = в„¦ 2 g (and also on в„¦в€’ 2 E = в„¦ 2 gв€— ).
вЂў At the other extreme, for the pair groupoid over a manifold X there is no
1
natural representation of G on в„¦ 2 E. The normal space E along the identity
section can be identiп¬Ѓed with T X, the tangent space to X.

G(0)
XT
В
В
В
В E
В
В
В
В
В  E
X

A representation of G on E consists of an identiп¬Ѓcation of Tx X with Ty X for
each (x, y) в€€ X Г— X. This amounts to a trivialization of the tangent bundle
to X вЂ“ that is, a global п¬‚at connection (with no holonomy). For an arbitrary
manifold X, such a thing will not exist; even if it exists, there is no natural
choice.
1
Similarly, to get a representation on в„¦ 2 E, we would need a global п¬Ѓeld of
1
2 -densities. This is equivalent to a global density on X, for which there is no
natural choice.

в™¦
15 Extended Groupoid Algebras
Extended groupoid algebras encompass bisections and sections of the normal bundle
to the identity section, just as distribution group algebras encompass Lie group
elements and Lie algebra elements.

15.1 Generalized Sections
Recall that for a Lie group G, the algebra C (G) of measures on the group sat
inside D (G), the distribution group algebra (see Section 11.5). Furthermore, we
saw that D (G) contained G itself as the set of evaluation maps, g as the dipoles at
the identity, and U(g) as the set of distributions supported at the identity element
of G (see Section 12.3).
More generally, we return to the case of a Lie groupoid G over X. The intrinsic
groupoid algebra is naturally identiп¬Ѓed (see Section 14.3) with the space of smooth
sections of
1 1
в„¦ =: О±в€— (в„¦ 2 E) вЉ— ОІ в€— (в„¦ 2 E) ,
where E is the normal bundle of X G(0) in G.
The extended (intrinsic) groupoid algebra, D (G), is the dual space of the
compactly supported smooth sections of
1 1
в„¦ =: О±в€— (в„¦ 2 E в€— ) вЉ— ОІ в€— (в„¦ 2 E в€— ) вЉ— в„¦1 T G .
The groupoid algebra is included in D (G), as we can pair в„¦ and в„¦ to get в„¦1 T G =
|в€§top | T в€— G, and then integrate a 1-density on G (that is, a section of |в€§top | T в€— G) to
obtain a number. Elements of D (G) are sometimes called generalized sections
of в„¦.
We may describe a typical section of в„¦ along the identity section X G(0) of
G. First, note that along X the bundle в„¦ reduces to
в„¦ |X = в„¦1 E в€— вЉ— в„¦1 T G|X .
Although the tangent space of G along X can be decomposed into the tangent space
of X and the normal space E, there is no natural choice of splitting. For densities,
however, we are able to make a natural construction. Using the exact sequence
0 в€’в†’ T X в€’в†’ T G|X в€’в†’ E в€’в†’ 0 ,
we see by Lemma 12.1 that
в„¦1 E в€— вЉ— в„¦1 T G|X
в„¦ |X =
в„¦1 E в€— вЉ— в„¦1 E вЉ— в„¦1 T X
в„¦1 T X .
Thus a section of в„¦ |X is just a 1-density on X. As a consequence, any measurable
function f : X в†’ R determines a generalized section, namely

П• в€€ О“c (в„¦ ) в€’в†’ f П•|X в€€ R .
X
The inclusion of measurable functions on X as generalized sections is in fact a
homomorphism.
We conclude that, in particular, all smooth functions on X belong to the ex-
tended intrinsic groupoid algebra:
C в€ћ (X) вЉ† D (G) .

105
106 15 EXTENDED GROUPOID ALGEBRAS

15.2 Bisections
The previous construction generalizes to other вЂњsectionsвЂќ besides the identity sec-
tion. A submanifold ОЈ of G such that the projections of ОЈ to X by О± and ОІ are
isomorphisms is called a bisection of G or an admissible section.

e e eВЎ eВЎ eВЎ eВЎ ВЎ ВЎ
e e ВЎe ВЎe ВЎe ВЎe ВЎ ВЎ
e eВЎeВЎeВЎeВЎeВЎ ВЎ
e eВЎ eВЎ eВЎ eВЎ eВЎ ВЎ
e ВЎe ВЎe ВЎe ВЎe ВЎe ВЎ
ОЈ
eВЎeВЎeВЎeВЎeВЎeВЎ
eВЎ eВЎ eВЎ eВЎ eВЎ eВЎ
ВЎe ВЎe ВЎe ВЎe ВЎe ВЎe
ВЎeВЎeВЎeВЎeВЎeВЎe
G(0)
X
ВЎ eВЎ eВЎ eВЎ eВЎ eВЎ e
ВЎ ВЎe ВЎe ВЎe ВЎe ВЎe e
ВЎ ВЎeВЎeВЎeВЎeВЎe e
ВЎ ВЎ eВЎ eВЎ eВЎ eВЎ e e
ВЎ ВЎ ВЎe ВЎe ВЎe ВЎe e e

Because we can identify the normal spaces of ОЈ with the tangent spaces of either
the О±- or the ОІ-п¬Ѓbers along ОЈ, we see that
в€’1 в€’1
в„¦О± 2 |ОЈ вЉ— в„¦ОІ 2 |ОЈ вЉ— в„¦1 T G|ОЈ
в„¦ |ОЈ =
в„¦в€’1 N ОЈ вЉ— в„¦1 T G|ОЈ
=
в„¦1 T ОЈ .
where N ОЈ is the normal bundle to ОЈ inside G. We can thus integrate sections of
в„¦ over ОЈ. Therefore, each bisection ОЈ determines an element of D (G). Let B(G)
denote the set of smooth bisections of G. We conclude that
B(G) вЉ† D (G) .

Remark. Before integrating we could have multiplied by any smooth function
on ОЈ (or X), thus obtaining other elements of D (G) (see the last exercise of this
в™¦
section).

Example. When G is a group, a bisection is a group element. The construction
above becomes evaluation at that element. The inclusion of bisections into D (G)
thus extends the identiп¬Ѓcation of elements of a group with elements of the distri-
bution group algebra, as evaluation maps. The objects generalizing the Lie algebra
в™¦
elements will be discussed in Sections 15.4 and 15.5.
The inclusion map from B(G) to D (G) is multiplicative if we deп¬Ѓne multipli-
cation of bisections as follows.
Given two subsets A and B of a groupoid G, we form their product by multi-
plying all possible pairs of elements in A Г— B,
AB = {xy в€€ G| (x, y) в€€ A Г— B в€© G(2) } .
This product deп¬Ѓnes a semigroup structure on the space 2G of subsets of G. There
are several interesting sub-semigroups of 2G :
15.3 Actions of Bisections on Groupoids 107

1. This multiplication deп¬Ѓnes a group structure on B(G). The identity element
of this group is just the identity section X G(0) .

Exercise 50
Show that:
(a) B(G) is closed under multiplication and that this multiplication satisп¬Ѓes
the group axioms.
(b) Multiplication of bisections in B(G) maps to convolution of distributions
in D (G) under the inclusion B(G) в†’ D (G).

2. There is a larger sub-semigroup Bloc (G) вЉ‡ B(G) of local bisections. A local
bisection is a subset of G for which the projection maps О±, ОІ are embeddings
onto open subsets. Bloc (G) is an example of an inverse semigroup (see [135,
143]).

For the pair groupoid over X, the group B(X Г— X) can be
Example.
identiп¬Ѓed with the group of diп¬Ђeomorphisms of X, since each bisection ОЈ is
the graph of a diп¬Ђeomorphism. Bloc (X Г— X) similarly corresponds to the
semigroup (sometimes called a pseudogroup) of local diп¬Ђeomorphisms of X.

Exercise 51
Show that the identiп¬Ѓcation B(X Г— X) в†’ Diп¬Ђ(X) is a group homomorphism
(or anti-homomorphism, depending on conventions).

в™¦

3. If we view G вЉ‚ 2G as the collection of one-element subsets, then G is not
closed under the multiplication above. But if we adjoin the empty set, then
{в€…} в€Є G вЉ† 2G is a sub-semigroup. This is the semigroup naturally associated
to a groupoid G, mentioned in Section 13.1.

Exercise 52
The subspaces B(G) and C в€ћ (X) of D (G) generate multiplicatively the larger
subspace of pairs (ОЈ, s) в€€ B(G) Г— C в€ћ (X). Here we identify functions on
a bisection ОЈ with functions on X via pull-back by О± (alternatively, ОІ). Let
ОЈ1 , ОЈ2 be bisections and si в€€ C в€ћ (ОЈi ). Find an explicit formula for the product
(ОЈ1 , s1 ) В· (ОЈ2 , s2 )
in D (G).

15.3 Actions of Bisections on Groupoids

The group of bisections B(G) acts on a groupoid G from the left (or from the
right). To see this left action, take elements g в€€ G and ОЈ в€€ B(G). Because ОЈ is
a bisection, there is a uniquely deп¬Ѓned element h в€€ ОЈ, such that ОІ(h) = О±(g). We
declare ОЈ В· g := hg в€€ G.
108 15 EXTENDED GROUPOID ALGEBRAS

rОЈ В· g = hg
ВЎe
ВЎe
ВЎ e
erg
ВЎ
ОЈ hr ВЎ
ВЎ
ВЎ ВЎ
e ВЎ
e ВЎ
eВЎ
G(0) er
ВЎ
ОІ(h) = О±(g)
Similarly, we can deп¬Ѓne a right action of B(G) on G by noting that there is
also a uniquely deп¬Ѓned element О±|в€’1 (ОІ(g)) в€€ ОЈ. These actions can be thought of
ОЈ
as вЂњslidingвЂќ by ОЈ. See .

rg В· ОЈ
ВЎe
ВЎe
ВЎ e
gr
ВЎ e
e e
e e
e e в€’1
erО±|ОЈ (ОІ(g)) ОЈ
e
e ВЎ
e ВЎ
eВЎ
G(0) r
eВЎ
ОІ(g)

Exercise 53
Check that this deп¬Ѓnes a group action and that the left and right actions
commute.

Remarks.

вЂў This construction generalizes the left (or right) regular representation of a
group on itself.

вЂў We can recover the bisection ОЈ from its left or right action on G since

ОЈ = ОЈ В· G(0) = G(0) В· ОЈ .

в™¦
The left action of B(G) preserves the ОІ-п¬Ѓbers of G, while the right action
preserves the О±-п¬Ѓbers. On the other hand, the left action of B(G) maps О±-п¬Ѓbers to
О±-п¬Ѓbers, while the right action of B(G) maps ОІ-п¬Ѓbers to ОІ-п¬Ѓbers.
The left (respectively, right) action respects the О±-п¬Ѓber (respectively, ОІ-п¬Ѓber)
structure even more, in the following sense. Note that B(G) acts on the base space
15.4 Sections of the Normal Bundle 109

X from the left (or from the right). For a bisection ОЈ в€€ B(G), the (left) action
on X is deп¬Ѓned by taking x в€€ X to О±(ОІ в€’1 (x)), where ОІ в€’1 (x) в€€ ОЈ is uniquely
determined.

в€’1
rОІ (x) ОЈ
ВЎeu
О±ВЎe
ВЎ e
X rВЎ er

ОЈВ·x x

It is easy to check that О±(ОЈ В· g) = ОЈ В· О±(g), and so О± is a left equivariant map
from G to X with respect to the B(G)-actions. Similarly, ОІ is a right equivariant
map.

15.4 Sections of the Normal Bundle

As we saw in Section 15.2, the concept of bisection of a Lie groupoid generalizes
the notion of Lie group element, both by its geometric deп¬Ѓnition, or when such an
element is regarded as an evaluation functional at that element. From this point
of view, we now explain how the objects corresponding to the Lie algebra elements
are the sections of the normal bundle E = T G|G(0) /T G(0) thought of as п¬Ѓrst order
perturbations of the submanifold G(0) .
By choosing a splitting of the tangent bundle over G(0) (for instance, with a
riemannian metric)
T G|G(0) T G(0) вЉ• E ,
we can identify the normal bundle E with a sub-bundle E вЉ† T G|G(0) . Under this
identiп¬Ѓcation, a section Пѓ в€€ О“(E) may be viewed as a vector п¬Ѓeld v : G(0) в†’
T G|G(0) . We can п¬Ѓnd, for suп¬ѓciently small Оµ, a path П€t : G(0) в†’ G deп¬Ѓned for
0 в‰¤ t < Оµ and such that

identity on G(0)
П€0 =

П€t в€’ П€0
dП€t
= lim =v .
dt t
tв†’0
t=0

At each time t, the image of П€t is a bisection ОЈt (restricted to the given compact
subset of G). In particular, ОЈ0 = G(0) is the identity section.
The one-parameter family of bisections {ОЈt } gives rise to an element, called Пѓ, of
the extended groupoid algebra D (G) by the following recipe. Let П• be a compactly
supported smooth section of в„¦ . Each individual bisection ОЈt в€€ D (G) = (О“c (в„¦ ))
pairs with П• to give a number ОЈt , П• as described in Section 15.2. We deп¬Ѓne the
new pairing by
ОЈt , П• в€’ ОЈ0 , П•
Пѓ, П• := lim .
t
tв†’0
110 15 EXTENDED GROUPOID ALGEBRAS

Exercise 54
Check that Пѓ, В· is a well-deп¬Ѓned linear functional on О“c (в„¦ ), independent of
the choice of E. (Hint: notice how vector п¬Ѓelds v в€€ О“(T G(0) ), i.e. tangent to
G(0) , yield a trivial pairing.)

We conclude that
О“(E) вЉ† D (G) .
Furthermore, these elements of the extended groupoid algebra have support in G(0) ,
that is, they vanish on test sections П• в€€ О“c (в„¦ ) with (support П•) в€© G(0) = в€….
If we think of Пѓ в€€ О“(E) as

ОЈt в€’ G(0)
Пѓ = lim ,
t
tв†’0

we can give an informal deп¬Ѓnition of a commutator bracket [В·, В·] on О“(E). Given
two sections of E
ОЈt в€’ G(0) О˜u в€’ G(0)
Пѓ = lim , Оё = lim ,
t u
tв†’0 uв†’0

we deп¬Ѓne
ОЈt в€’ G(0) О˜u в€’ G(0) О˜u в€’ G(0) ОЈt в€’ G(0)
В· в€’ В·
[Пѓ, Оё] = lim
t u u t
t,uв†’0

ОЈt О˜u в€’ О˜u ОЈt
= lim ,
tu
t,uв†’0

or, equivalently, the bracket evaluated on П• в€€ О“c (в„¦ ) is
ОЈt О˜u , П• в€’ О˜u ОЈt , П•
[Пѓ, Оё], П• = lim .
tu
t,uв†’0

Sections of E are in fact closed under the commutator bracket:

[О“(E), О“(E)] вЉ† О“(E) ,

as we will see in the next section where we deп¬Ѓne the bracket properly.
The distributions on G corresponding to sections of E are sometimes known as
dipole layers. (See the discussion of dipoles in Section 12.3.)

15.5 Left Invariant Vector Fields
Recall from Section 14.4 that there is a (left) action of the groupoid G on itself;
namely, each element g в€€ G acts on О±в€’1 (ОІ(g)) by left multiplication.
The ОІ-projection is invariant with respect to this action

ОІ(g В· h) = ОІ(gh) = ОІ(h) ,

while О±-п¬Ѓbers are mapped to О±-п¬Ѓbers
gВ·
О±в€’1 (ОІ(g)) в€’в†’ О±в€’1 (О±(g)) .

Let
T О± G := ker T О± вЉ† T G
15.5 Left Invariant Vector Fields 111

be the distribution tangent to the О±-п¬Ѓbers. The action of g в€€ G induces a linear
map
T gВ·
T О± G|О±в€’1 (ОІ(g)) в€’в†’ T О± G|О±в€’1 (О±(g)) .

Exercise 55
The left action of the group of sections B(G) preserves the О±-п¬Ѓber structure
(see Section 15.3), and hence also induces an action on T О± G by diп¬Ђerentiation.
(a) Prove that a section of T О± G is G-left-invariant if and only if it is B(G)-
left-invariant.
(b) If a section of T G is B(G)-left-invariant, then do all of its values have
to lie in T О± G?

A left invariant section of T О± G is called a left invariant vector п¬Ѓeld on the
groupoid G. The set П‡L (G) of all left invariant vector п¬Ѓelds on G has the following
properties.

вЂў П‡L (G) is closed under the bracket operation

[П‡L (G), П‡L (G)] вЉ† П‡L (G) ,

and thus forms a Lie algebra.

вЂў An element of П‡L (G) is completely determined by its values along the identity
section G(0) . Equivalently, an element is determined by its values along any
other bisection.
О±
вЂў Every smooth section of TG(0) G := ker T О±|G(0) can be extended to an element
of П‡L (G).

Furthermore,
О±
TG(0) G E,

where E = TG(0) G/T G(0) is the normal bundle to G(0) in G.
Thus we have the identiп¬Ѓcations
О±
П‡L (G) О“(TG(0) G) О“(E) .

The bracket on П‡L (G) can therefore be considered as a bracket on О“(E); it agrees
with the one deп¬Ѓned informally in the previous section.
в€ћ
The left invariant vector п¬Ѓelds on G act by diп¬Ђerentiation on CL (G), the left
invariant functions on G. From the identiп¬Ѓcation
в€ћ
ОІ в€— C в€ћ (X) C в€ћ (X) ,
CL (G)

we get a map
О“(E) в€’в†’ П‡(X) := О“(T X) .
It is easy to see that this map is induced by the bundle map

ПЃ : E в€’в†’ T X
112 15 EXTENDED GROUPOID ALGEBRAS

given by composition of two natural maps:

E T О±(0) G
E G

d В
d В
d В
ПЃd В  TG(0) ОІ
d
В‚ В
TX
With this additional structure, E provides the typical example of a Lie algebroid.
We study these objects in the next chapter.
Example. When G is a Lie group (X is a point), both П‡L (G) g and E g are
the Lie algebra, and ПЃ : g в†’ {0} is the trivial map. в™¦
Part VII
Algebroids
16 Lie Algebroids
Lie algebroids are the inп¬Ѓnitesimal versions of Lie groupoids.

16.1 Deп¬Ѓnitions

A Lie algebroid over a manifold X is a (real) vector bundle E over X together
with a bundle map ПЃ : E в†’ T X and a (real) Lie algebra structure [В·, В·]E on О“(E)
such that:

1. The induced map О“(ПЃ) : О“(E) в†’ П‡(X) is a Lie algebra homomorphism.
2. For any f в€€ C в€ћ (X) and v, w в€€ О“(E), the following Leibniz identity holds

[v, f w]E = f [v, w]E + (ПЃ(v) В· f )w .

Remarks.

вЂў The map ПЃ is called the anchor of the Lie algebroid. By an abuse of notation,
the map О“(ПЃ) may be denoted simply by ПЃ and also called the anchor.
вЂў For each v в€€ О“(E), we deп¬Ѓne E-Lie derivative operations on both О“(E) and
C в€ћ (X) by
Lv w = [v, w]E ,
Lv f = ПЃ(v) В· f .
We can then view the Leibniz identity as a derivation rule

Lv (f w) = f (Lv w) + (Lv f ) w .

в™¦
When (E, ПЃ, [В·, В·]E ) is a Lie algebroid over X, the kernel of ПЃ is called the
isotropy. Each п¬Ѓber of ker ПЃ is a Lie algebra, analogous to the isotropy subgroups
of groupoids. To see this, let v and w в€€ О“(E) be such that ПЃ(v) and ПЃ(w) both
vanish at a given point x в€€ X. Then, for any function f в€€ C в€ћ (X),

[v, f w]E (x) = f (x)[v, w]E (x) .

So there is a well-deп¬Ѓned bracket operation on the vectors in any п¬Ѓber of ker ПЃ, and
ker ПЃ is a п¬Ѓeld of Lie algebras. These form a bundle when ПЃ has constant rank.
On the other hand, the image of ПЃ is an integrable distribution analogous
to the image of О  for Poisson manifolds. Therefore, X can be decomposed into
submanifolds, called orbits of the Lie algebroid, whose tangent spaces are the image
of ПЃ. There are are various proofs of this: one uses the corresponding (local) Lie
groupoid, another uses a kind of splitting theorem, and a third proof involves a more
general approach to integrating singular distributions. The articles of Dazord [37,
38] discuss this and related issues.

113
114 16 LIE ALGEBROIDS

16.2 First Examples of Lie Algebroids
1. A (п¬Ѓnite dimensional real) Lie algebra is a Lie algebroid over a one-point
space.
2. A bundle of Lie algebras over a manifold X (as in Section 16.3) is a Lie
algebroid over X, with ПЃ в‰Ў 0. Conversely, if E is any Lie algebroid with
ПЃ в‰Ў 0, the Leibniz identity says that the bracket in О“(E) is a bilinear map of
C в€ћ (X)-modules and not simply of R-modules, and hence that each п¬Ѓber is a
Lie algebra. (Such an E is all isotropy.)
3. We saw in Section 15.5 that the normal bundle E along the identity section of
a Lie groupoid G over X carries a bracket operation and anchor ПЃ : E в†’ T X
satisfying the Lie algebroid conditions. This is called the Lie algebroid of
the Lie groupoid G. The isotropy algebras of this Lie algebroid are the Lie
algebras of the isotropy groups of G. The orbits are the connected components
of the G-orbits.
As for the case of Lie groups and Lie algebras, it is natural to pose the
integrability problem (see also Sections 16.3 and 16.4):
вЂў When is a given Lie algebroid the Lie algebroid of a Lie groupoid?
вЂў If the Lie algebroid does come from a Lie groupoid, is the Lie groupoid
unique?
4. The tangent bundle T X of a manifold X, with ПЃ the identity map, is a Lie
algebroid over X. We can see it the Lie algebroid of the Lie groupoid X Г— X,
or of the fundamental groupoid О (X), or of yet other possibilities; near the
identity section, О (X) looks like X Г— X.
Generally, we can say that a Lie algebroid determines and is determined by
a neighborhood of the identity section in the groupoid, just as a Lie algebra
determines and is determined by a neighborhood of the identity element in
the corresponding Lie group.
5. Suppose that we have a right action of a Lie algebra g on X, that is, a Lie
Оі
algebra homomorphism g в†’ П‡(X). The associated transformation Lie
ПЃ
algebroid X Г— g has anchor X Г— g в†’ T X deп¬Ѓned by
ПЃ(x, v) = Оі(v)(x) .
Combining this with the natural projections X Г— g в†’ X and T X в†’ X, we
form the commutative diagram
ПЃE
X Г—g TX
В
В
В
В
В
X
A section v of X Г— g can be thought of as a map v : X в†’ g. We deп¬Ѓne the
bracket on sections of X Г— g by
[v, w](x) = [v(x), w(x)]g + (Оі(v(x)) В· w)(x) в€’ (Оі(w(x)) В· v)(x) .
16.2 First Examples of Lie Algebroids 115

When v, w are constant functions X в†’ g, we recover the Lie algebra bracket
of g.
It is easy to see in this example that the п¬Ѓbers of ker ПЃ are the usual isotropy
Lie algebras of the g-action. The orbits of the Lie algebroid are just the orbits
of the Lie algebra action.
If Оі comes from a О“-action on X, where О“ is a Lie group with Lie algebra g,
then X Г— g is the Lie algebroid of the corresponding transformation groupoid
GО“ .

ПЃ(E) вЉ† T X being
6. Suppose that ПЃ is injective. This is equivalent to E
an integrable distribution, as the bracket on E is completely determined by
that on T X. A universal choice of a Lie groupoid with this Lie algebroid
is the holonomy groupoid of the corresponding foliation. (It might not be
Hausdorп¬Ђ.)
The case when ПЃ is surjective will be discussed in Section 17.1.

Exercise 56
Let (v1 , . . . , vn ) be a basis of sections for a Lie algebroid E such that [vi , vj ] =
k cijk vk where the cijk вЂ™s are constants. Show that E is isomorphic to a
transformation Lie algebroid.

Historical Remark. Already in 1963, Rinehart  noted that, if a Lie algebra
О“ over a п¬Ѓeld k is a module over a commutative k-algebra C, and if there is a
homomorphism ПЃ from О“ into the derivations of C, then there is a semidirect product
Lie bracket on the sum О“ вЉ• C deп¬Ѓned by the formula

[(v, g), (w, h)] = ([v, w], ПЃ(v) В· h в€’ ПЃ(w) В· g) .

Furthermore, this bracket satisп¬Ѓes the Leibniz identity:

[(v, g), f (w, h)] = f [(v, g), (w, h)] + (ПЃ(v) В· f )(w, h) for f в€€ C .

In the special case where C = C в€ћ (X), the C в€ћ (X)-module О“, if projective, is
the space of sections of some vector bundle E over X. The homomorphism ПЃ and
the Leibniz identity imply that ПЃ is induced by a bundle map ПЃ : E в†’ T X. The
Leibniz identity for О“(E) вЉ• C в€ћ (X) also encodes the Leibniz identity for the bracket
on О“(E) alone.
In 1967, Pradines  coined the term вЂњLie algebroidвЂќ and proved that every
Lie algebroid comes from a (local) Lie groupoid. He asserted that the local condition
was not needed, but this was later shown by Almeida and Molino  to be false.
(See Section 16.4.)
Rinehart  proved (in a more algebraic setting) an analogue of the PoincarВґ-e
Birkhoп¬Ђ-Witt theorem for Lie algebroids. He showed that there is a linear iso-
morphism between the graded version of a universal object for the actions of
О“(E) вЉ• C в€ћ (X) on vector bundles V over X, and the polynomials on the dual
of the Lie algebroid E. As a result, the dual bundle of a Lie algebroid carries a
Poisson structure. This Poisson structure is described abstractly in  as the base
of the cotangent groupoid T в€— G of a Lie groupoid G; it is described more explicitly
in . (See Section 16.5.)
116 16 LIE ALGEBROIDS

The most basic instance of this phenomenon is when E = T X. The dual to
the Lie algebroid is T в€— X with its standard (symplectic) Poisson structure (see
Section 6.5). The universal object is the algebra of diп¬Ђerential operators on X, and
в™¦
the Rinehart isomorphism is a вЂњsymbol mapвЂќ.

16.3 Bundles of Lie Algebras

For a п¬Ѓrst look at the integrability problem, we examine Lie algebroids for which
the anchor map is zero.
A bundle of Lie groups is a bundle of groups (see Section 13.4) for which each
п¬Ѓber is a Lie group. Bundles of Lie algebras are vector bundles for which each п¬Ѓber
has a Lie algebra structure which varies continuously (or smoothly). Every bundle
of Lie groups deп¬Ѓnes a bundle of Lie algebras: the Lie algebras of the individual
п¬Ѓbers. More problematic is the question of whether we can integrate a bundle of
Lie algebras to get a bundle of Lie groups.

Theorem 16.1 (Douady-Lazard ) Every bundle of Lie algebras can be inte-
grated to a (not necessarily Hausdorп¬Ђ ) bundle of Lie groups. (Fibers and base are
Hausdorп¬Ђ, but the bundle itself might not be.)

Example. Given a Lie algebra g with bracket [vi , vj ]= cijk vk , we deп¬Ѓned in
Section 1.2 a family of Lie algebras gОµ = (g, [В·, В·]Оµ ), Оµ в€€ R, by the structure equations
[vi , vj ]Оµ = Оµ cijk vk . This can be thought of as a bundle of Lie algebras over R.
There is a bundle of Lie groups corresponding to this bundle: the п¬Ѓber over 0 в€€ R
is an abelian Lie group (either euclidean space, a cylinder or a torus), while the
п¬Ѓber over any other point Оµ в€€ R can be chosen to be a п¬Ѓxed manifold. The п¬Ѓber
dimensions cannot jump, but the topology may vary drastically. In the particular
case of g = su(2), the bundle of groups corresponding to the deformation gОµ has
п¬Ѓber SU(2) S 3 for Оµ = 0, and п¬Ѓber R3 at Оµ = 0. Here the total space is Hausdorп¬Ђ,
since it is homeomorphic to R Г— S 3 with a point removed from {0} Г— S 3 . в™¦

Example. [48, p.148] Consider now the bundle of Lie algebras over R with п¬Ѓbers
gОµ = (R3 , [В·, В·]Оµ ), Оµ в€€ R, where the brackets are deп¬Ѓned by

[xОµ , zОµ ]Оµ = в€’yОµ ,
[xОµ , yОµ ]Оµ = zОµ , [yОµ , zОµ ]Оµ = ОµxОµ .

Here xОµ , yОµ , zОµ denote the values at Оµ of a given basis of sections x, y, z for the bundle
g := R Г— R3 .
The corresponding simply connected Lie groups GОµ are as follows for Оµ в‰Ґ 0.
G1 is the group of unit quaternions, if we identify the basis x1 , y1 , z1 of g1 with
11 1
2 i, 2 j, 2 k, respectively. Consequently, exp(4ПЂx1 ) = e1 is the identity element of
G1 . For Оµ > 0, gОµ g1 under the isomorphism
в€љ в€љ
xОµ в†’ x1 , yОµ в†’ Оµ y1 , zОµ в†’ Оµ z1 .

Taking GОµ G1 , we still have that exp(4ПЂxОµ ) = eОµ is the identity of GОµ , Оµ > 0. At
Оµ = 0, G0 is the semidirect product R Г— R2 , where the п¬Ѓrst factor R acts on R2 by
rotations. Here exp(tx0 ) = (t, 0), thus, in particular, exp(4ПЂx0 ) = e0 .
Therefore, the set of points Оµ в€€ R where the two continuous sections exp(4ПЂВ·)
and e coincide is not closed, hence G is not Hausdorп¬Ђ.
16.4 Integrability and Non-Integrability 117

Following this example, Douady and Lazard show that, if we replace the group
G0 by the semidirect product S 1 Г—R2 (the group of euclidean motions of the plane),
the resulting bundle of groups is Hausdorп¬Ђ. They then go on to show that a certain
C в€ћ bundle of semidirect product Lie algebras admits no Hausdorп¬Ђ bundle of Lie
groups. They conclude by asking whether there is an analytic example. The next
в™¦
example answers this question.

Example. Coppersmith  constructed an analytic family of 4-dimensional Lie
algebras parametrized by R2 which cannot be integrated to a Hausdorп¬Ђ family of
Lie groups. The п¬Ѓber gОµ over Оµ = (Оµ1 , Оµ2 ) в€€ R2 has basis xОµ , yОµ , zОµ , wОµ and bracket
given by
[wОµ , xОµ ]Оµ = [wОµ , yОµ ]Оµ = [wОµ , zОµ ]Оµ = 0 ,
[xОµ , zОµ ]Оµ = в€’yОµ , [yОµ , zОµ ]Оµ = xОµ , [xОµ , yОµ ]Оµ = Оµ1 zОµ + Оµ2 wОµ .
в™¦

16.4 Integrability and Non-Integrability

To п¬Ѓnd Lie algebroids which are not integrable even by non-Hausdorп¬Ђ groupoids,
we must look beyond bundles of Lie algebras.
Example. As a п¬Ѓrst attempt, take the transformation Lie algebroid X Г— g for
an action of the Lie algebra g on X (see Example 5 of Section 16.2). If the action
of the Lie algebra can be integrated to an action of the group О“, then the О“-action
on X deп¬Ѓnes a transformation groupoid GО“ with Lie algebroid X Г— g. Now we can
make the g-action non-integrable by restricting to an open set U вЉ† X not invariant
under О“. The Lie algebroid X Г— g restricts to a Lie algebroid U Г— g. We might
hope that the corresponding groupoid does not restrict. However, one property of
groupoids is that they can always be restricted to open subsets of the base space.

Exercise 57
Let G be any groupoid over X and U an open subset of X. Then H =
О±в€’1 (U ) в€© ОІ в€’1 (U ) is a subgroupoid of G with base space U .

We conclude that this restriction H of the transformation groupoid X Г— О“ has Lie
algebroid U Г— g. в™¦

Example. We could instead look for incomplete vector п¬Ѓelds that cannot even be
completed by inserting into a bigger manifold. One example begins with X = R2
в€‚
and U = R2 \ {0}, where the vector п¬Ѓeld в€‚x is incomplete. If R2 is identiп¬Ѓed with
C, then the subspace U has a double cover deп¬Ѓned by the map z в†’ z 2 . If we
в€‚
pull в€‚x back to the double cover, there is no way to smoothly вЂњп¬Ѓll in the holeвЂќ
to a complete vector п¬Ѓeld. Could this then give an example of a non-integrable
Lie algebroid? Unfortunately, this type of construction is also doomed to fail, if
в™¦
non-Hausdorп¬Ђ groupoids are allowed.

Theorem 16.2 (Dazord ) Every transformation Lie algebroid is integrable.

Exercise 58
Find a groupoid which integrates the Lie algebroid in the previous example.
118 16 LIE ALGEBROIDS

Historical Remark. Important work on integrability of Lie algebra actions was
done by Palais  in 1957. In that manuscript, he proved results close to DazordвЂ™s
theorem, but without the language of groupoids.
The following example of a non-integrable Lie algebroid is due to Almeida and
Molino [3, 4]. It is modeled on an example of a non-integrable Banach Lie algebra
due to Douady-Lazard . Mackenzie  had already developed an obstruction
в™¦
theory to integrating Lie algebroids, but never wrote a non-zero example.

Example. We will construct a Lie algebroid E which has the following form as
a bundle
0 в€’в†’ L в€’в†’ T X Г— R в€’в†’ T X в€’в†’ 0 ,
E

where L is the trivial real line bundle over X. We deп¬Ѓne a bracket on sections of
E, О“(E) = П‡(X) Г— C в€ћ (X), by

[(v, f ), (w, g)]E,в„¦ = ([v, w]T X , v В· g в€’ w В· f + в„¦(v, w)) ,

where в„¦ is a given 2-form on X. The bracket [В·, В·]E,в„¦ satisп¬Ѓes the Jacobi identity
if and only if в„¦ is closed.
Each integral 2-cycle Оі в€€ H2 (X; Z) gives rise to a period

в„¦в€€О›.
Оі

If the set of periods of в„¦ is not cyclic in R, and if X is simply connected, then
one can show that E does not come from a groupoid .6 In this way we obtain a
в™¦
non-integrable Lie algebroid.

Remark. There is still a sort of Lie groupoid corresponding to this Lie algebroid.
As a bundle over X Г— X, it has structure group R/О›, where О› is generated by
two numbers which are linearly independent over Q. There are no nonconstant
diп¬Ђerentiable functions on R/О›, but there is a notion of smooth curves, if one uses
SouriauвЂ™s notion of diп¬Ђeological space . In general, a map M в†’ R/О› is said to
be smooth if it (locally) lifts to a smooth map M в†’ R.
R

В
В
В
В
В  c
M E R/О›

Examples of Lie algebroids which are вЂњeven moreвЂќ non-integrable can also be con-
в™¦
structed .

6 For instance, on X = S 2 Г— S 2 with projections
S2 Г— S2
ПЂ1 в†“в†“ ПЂ2
S2
в€— в€—
deп¬Ѓne the 2-form в„¦ = c1 ПЂ1 П‰ + c2 ПЂ2 П‰, where П‰ is the standard volume on S 2 and c1 , c2 are
rationally independent constants, Then the periods of в„¦ do not lie in a cyclic subgroup of R.
16.5 The Dual of a Lie Algebroid 119

16.5 The Dual of a Lie Algebroid

Let x1 , В· В· В· , xn be local coordinates on a manifold X, and let e1 , В· В· В· , er be a local
basis of sections of a Lie algebroid (E, ПЃ, [В·, В·]E ) over it. With respect to these
coordinates and basis, the Lie bracket and anchor map are described by structure
functions cijk , bij в€€ C в€ћ (X) as

[ei , ej ]E = cijk ek
k
в€‚
ПЃ(ei ) = bij .
в€‚xj
j

Exercise 59
The Leibniz identity and Jacobi identity translate into diп¬Ђerential equations
for the cijk and bij . Write out these diп¬Ђerential equations.

Let x1 , В· В· В· , xn , Оѕ1 , В· В· В· , Оѕn be the associated coordinates on the dual bundle E в€— ,
where Оѕ1 , В· В· В· , Оѕn are the linear functions on E в€— deп¬Ѓned by evaluation at e1 , В· В· В· , er .
We deп¬Ѓne a bracket {В·, В·}E on C в€ћ (E в€— ) by setting

{xi , xj }E =0
{Оѕi , Оѕj }E = cijk Оѕk
k
{Оѕi , xj }E в€’bij
=

Proposition 16.3 The bracket {В·, В·}E deп¬Ѓnes a Poisson structure on E в€— .

Exercise 60
Show that the Jacobi identity for {В·, В·}E follows from the Lie algebroid axioms
for E.

Remark. Although the Poisson bracket {В·, В·}E is deп¬Ѓned in terms of coordinates
and a basis, it is independent of these choices. Hence, the passage between the Lie
algebroid structure on E and the Poisson structure on E в€— is intrinsic. в™¦

Examples.

1. When X is a point, and E = g is a Lie algebra, then the Poisson bracket on
E в€— = gв€— regarded as the dual of a Lie algebroid, coincides with the Lie-Poisson
bracket deп¬Ѓned in Section 3.1.
In general, the Poisson bracket {В·, В·}E on the dual of a Lie algebroid is some-
times also called a Lie-Poisson bracket.
в€‚
2. When E = T X, we can choose ei = в€‚xi to give the standard basis of vector
п¬Ѓelds induced by the choice of coordinates on X, so that cijk = 0 and bij = Оґij .
The Poisson structure on the dual bundle E в€— = T в€— X as a dual of a Lie
algebroid is the one induced by the canonical symplectic structure dxi в€§dОѕi
because
{xi , Оѕj }E = в€’{Оѕj , xi }E = Оґij .
120 16 LIE ALGEBROIDS

3. When E = T в€— X is the Lie algebroid of a Poisson manifold X (see Sec-
tion 17.3), we obtain on the tangent bundle E в€— = T X the tangent Poisson
structure; see .

Exercise 61

(a) Let E1 and E2 be Lie algebroids over X. Show that a bundle map
П• : E1 в†’ E2 is a Lie algebroid morphism (i.e. compatible with
brackets and anchors) if and only if П•в€— : E2 в†’ E1 is a Poisson map.
в€— в€—

(b) Show that the dual of the anchor map of a Lie algebroid is a Poisson
map from T в€— X to E в€— .
(c) Use the result of part (a) to suggest a deп¬Ѓnition of morphism between
Lie algebroids over diп¬Ђerent base manifolds. See Proposition 6.1 in .

Exercise 62
в€‚ в€‚
Let x1 , В· В· В· , xn be coordinates on X, в€‚x , В· В· В· , в€‚x and dx1 , В· В· В· , dxn the in-
n
1
duced local bases of T X and T в€— X, x1 , В· В· В· , xn , v1 , В· В· В· , vn associated coordinates
on T X, and x1 , В· В· В· , xn , Оѕ1 , В· В· В· , Оѕn associated coordinates on T в€— X.
Express the Poisson bracket for the tangent Poisson structure on T X in terms
of the Poisson bracket on X given by ПЂij (x) = {xi , xj }.
Лњ
Check that, although T X в†’ X is not a Poisson map, the map О  : T в€— X в†’ T X
is a Poisson map.

в™¦

16.6 Complex Lie Algebroids

It can be interesting to work over C even if X is a real manifold. To deп¬Ѓne a
complex Lie algebroid geometrically, take a complex vector bundle E over X
and a complex bundle map ПЃ : E в†’ TC X to the complexiп¬Ѓed tangent bundle.7 The
immediate generalization of our deп¬Ѓnition in Section 16.1 amounts to imposing that
the space of sections of E be a complex Lie algebra satisfying the (complex versions
of the) two axioms.
Example. Let X be a (real) manifold with an almost complex structure J :
T X в†’ T X, i.e. J is a bundle map such that J 2 = в€’id. The graph of в€’iJ in
TC X = T X вЉ• iT X is the sub-bundle

E = {v в€’ iJ(v)| v в€€ T X} вЉ† TC X .

The bracket operation on T X extends by linearity to a bracket on TC X. To endow
E with a Lie algebroid structure, we need the sections of E to be closed under that
bracket:
[О“(E), О“(E)] вЉ† О“(E) .
This holds if and only if J is an integrable structure. That is, by the Newlander-
Nirenberg theorem , we have a complex Lie algebroid structure on E if and
only if J comes from a complex structure on X. A complex structure on a
we have changed our вЂњground ringвЂќ from C в€ћ (X) to C в€ћ (X; C).
7 Algebraically,
16.6 Complex Lie Algebroids 121

manifold is, in this way, a typical example of a complex Lie algebroid. The natural
questions arise: When does such a Lie algebroid come from a complex Lie groupoid?
в™¦
What is a complex Lie groupoid?

Example. Let X be a manifold of dimension 2n в€’ 1. Suppose that F вЉ† T X is a
codimension-1 sub-bundle with an almost complex structure J : F в†’ F (J is linear
and J 2 = в€’id). As before, deп¬Ѓne a sub-bundle E of the complexiп¬Ѓed F to be the
graph of в€’iJ
E = {v в€’ iJ(v)| v в€€ F } вЉ‚ FC = F вЉ• iF .
If О“(E) is closed under the bracket operation, i.e. if E is a Lie algebroid over X,
then (F, J) is called a Cauchy-Riemann structure or CR-structure on X.
To explain the motivation behind this construction, we consider the special
situation when X 2nв€’1 вЉ‚ Y 2n is a real submanifold of a complex n-manifold Y .
At a point x в€€ X вЉ‚ Y , the tangent space Tx Y is a vector space over C. We
denote by JY the complex multiplication by i in this space. Because X has odd
real dimension, Tx X cannot be equal to JY (Tx X) as subspaces of Tx Y , and thus
the intersection Fx := Tx X в€© JY (Tx X) must have codimension 1 in Tx X. Then F
is the maximal complex sub-bundle of T X.
Functions on a CR-manifold annihilated by the sections of E are called CR-
functions. In the case where X is a hypersurface in a complex manifold Y , they
include (and sometimes coincide with) the restrictions to X of holomorphic func-
tions on one side of X in Y .
This construction opens several questions, including:

вЂў What is the Lie groupoid in this case? However, at this point it is not clear
what it means to integrate a complex Lie algebroid.

вЂў What does the analytic theory of complex Lie algebroids look like? It seems
to be at least as complicated as that of CR-structures, which is already very
delicate .

вЂў The cohomology theory of Lie algebroids can be applied to complex Lie al-
gebroids. In the examples above, we recover the usual в€‚ cohomology and
boundary в€‚ cohomology on complex and CR-manifolds, respectively. What
does complex Lie algebroid cohomology look like in more general cases?

в™¦

Remark. When X is a complex manifold, it is tempting to impose the Lie alge-
broid axioms on the space of holomorphic sections of a holomorphic vector bundle
E. This idea fails in general, for the following reason. In the real case, sections of
E always exist. On the other hand, the only holomorphic functions on a compact
complex manifold are the constant functions. Similarly, it is possible that there
are no non-zero holomorphic sections for a complex vector bundle. It is therefore
more appropriate to look instead at the sheaf of local sections. AtiyahвЂ™s (see  and
Section 17.1) study of the obstructions to the existence of holomorphic connections
on principal GL(n; C)-bundles over complex manifolds used this approach to the
в™¦
вЂњAtiyah algebroidвЂќ.
17 Examples of Lie Algebroids
Lie algebroids with surjective anchor map are called transitive Lie algebroids,
Atiyah algebras, or Atiyah sequences because of AtiyahвЂ™s work mentioned
below. When a corresponding groupoid exists, it will be (locally) transitive, in the
sense that its orbits are open.

17.1 Atiyah Algebras
In 1957, Atiyah  constructed in the setting of vector bundles the Lie algebroid of
the following key example of a locally transitive groupoid. Suppose that we have a
principal bundle P over a manifold X
в†ђ
P H
ПЂв†“
X

with structure group H acting on the right. The quotient G = (P Г— P )/H of the
product groupoid by the diagonal action of H is a groupoid over X. An element
g = [p, q] of this groupoid is an equivalence class of pairs of points p в€€ ПЂ в€’1 (x), q в€€
ПЂ в€’1 (y) in P ; it is the graph of an equivariant map from the п¬Ѓber ПЂ в€’1 (y) to ПЂ в€’1 (x).
A bisection of this groupoid corresponds to a gauge transformation, that is, an
automorphism (i.e. an H-equivariant diп¬Ђeomorphism) of the principal bundle. For
this reason, we call G the gauge groupoid of P . The group of bisections B(G) and
the gauge group G are thus isomorphic. The inп¬Ѓnitesimal generators of G are the
H-invariant vector п¬Ѓelds. Since H acts on the п¬Ѓbers of ПЂ freely and transitively,
H-invariant vector п¬Ѓelds are determined by their values on one point of each п¬Ѓber,
so they can be identiп¬Ѓed with sections of

E = T P/H

considered as a bundle over X. The bracket on E is that induced from П‡(P ); this
is well-deп¬Ѓned because the bracket of two H-invariant vector п¬Ѓelds is H-invariant.

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