(• — ψ)(k) =

g∈±’1 (±(k))

But in order to integrate, we need measures on the ±-¬bers. If {»x }x∈X is a family

of measures on the ±-¬bers, then we de¬ne the convolution product of • and ψ

to be

•(g) ψ(g ’1 k) d»±(k) .

(• — ψ)(k) =

g∈±’1 (±(k))

Here, we assume that the family of measures {»x } is continuous in x. To ensure

that this product is associative, we require left invariance of {»x }, i.e. we require

that {»x } be a Haar system (cf. Sections 13.4 and 11.4).

The vector space of bounded continuous functions • on G for which the target

map ± restricted to support(•) is a proper map, is closed under the convolution

product. Its completion under a suitable norm is called the groupoid C — -algebra

associated to the Haar system {»x }. Since the multiplicative structure depends

14.3 Intrinsic Groupoid Algebras 99

on the choice of {»x }, the groupoid algebra is sometimes denoted by A» . We refer

to [143] for more details about the analytic aspects of this construction.

The groupoid algebra operates on functions on the base. Let • be a function

on G, and u a function on X. De¬ne

(Op •)u(x) := •(g) u(β(g)) d»x .

g∈±’1 (x)

Intuitively, if we think of the elements of G as “arrows” on the base space X, then

this integral tells us to look at all the arrows g going into a given point x ∈ 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 “weight” given by •.

Examples.

• G = X — X “ The groupoid algebra is isomorphic to the “matrix” algebra

of functions on X — X (see Section 14.1). If X is ¬nite, it really is a matrix

algebra.

• G = a group “ 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.

• G = X, where (f — g)(x) = f (x)g(x) “ The groupoid algebra is the algebra of

functions on X (which operates on itself by pointwise multiplication).

™¦

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 ±- and β-¬bers by „¦± and „¦β , respectively. Letting

2 2

1 1

„¦ = „¦± — „¦β ,

2 2

the intrinsic groupoid algebra of G is the completion of the space “(„¦) of

compactly supported sections of „¦ under a suitable norm. The term “intrinsic”

refers to the fact that it does not involve the arbitrary choice of a Haar system.

The multiplication on “(„¦) is de¬ned as follows.

Suppose that β(g) = ±(h) = x ∈ X G(0) . There is a natural isomorphism

„¦(g) — „¦(h) ’’ „¦1 (g) — „¦(gh)

±

constructed using the identi¬cations

1 — rhE

1 1 1 1

„¦± (g) — „¦β (g) „¦± (g) — „¦β (gh)

2 2 2 2

— 1E

1 1 1 1

g

„¦± (h) — „¦β (h) „¦± (gh) — „¦β (h)

2 2 2 2

100 14 GROUPOID ALGEBRAS

together with

lg’1

1 1

„¦± (g) E „¦± (x)

2 2

d

d

d

d

d

‚

1 1

2 (T G/T G(0) ) = „¦ 2 (N G(0) )

„¦ x x x

©

1 1

h

„¦β (h) ' „¦β (x)

2 2

1 1

In general, there is no natural isomorphism between „¦± and „¦β 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 „¦± and „¦β over x ∈ 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 •, ψ ∈ “(„¦). The

product section •ψ ∈ “(„¦) is given at a point k ∈ G by the formula

•(g) ψ(g ’1 k)

(•ψ)(k) =

{g|±(g)=±(k)}

where we regard •(g)ψ(g ’1 k) as an element of „¦1 (g) — „¦(k), and we integrate the

±

1-density factor over the ±-¬ber through k.

Exercise 48

Check that, if we instead use the maps

„¦(g) — „¦(h) ’’ „¦1 (h) — „¦(gh) ,

β

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

Remark. The identi¬cations above also provide a natural isomorphism

1 1

±— („¦ 2 E) — β — („¦ 2 E) ,

„¦

™¦

which we will use often.

Let E be the normal bundle of G(0) in G. The smooth groupoid algebra

“(„¦) acts on smooth sections of

1 1

„¦ 2 E := | §top | 2 E — .

1

To see this left action, take • ∈ “(„¦) and a section γ of „¦ 2 E. We can think of •

at g ∈ G as a 1 -density on the normal space through x = ±(g), times a 1 -density

2 2

on the normal space through through y = β(g):

1 1 1 1

„¦± (g) — „¦β (g) „¦ 2 Ex — „¦ 2 Ey

„¦(g) = 2 2

E •± (x) — •β (y)

•(g)

14.4 Groupoid Actions 101

Since •β (y)γ(y) ∈ „¦1 Ey „¦1 (y) „¦1 (g), we can consider •(g)γ(y) as an element

± ±

1 1

1

of „¦± (g) — „¦ x

2 E . The new section • · γ of „¦ 2 E is then given at a point x ∈ X by

(• · γ)(x) = •(g)γ(β(g))

g∈±’1 (x)

1

where we integrate the „¦1 factor of •(g)γ(β(g)) ∈ „¦1 (g) — „¦ 2 Ex over the ±-¬ber

± ±

through x.

Exercise 49

Check that this is indeed a left action, i.e. • · (ψ · γ) = (•ψ) · γ, for any

1

•, ψ ∈ “(„¦), and γ ∈ “(„¦ 2 E).

We could just as well de¬ne a right action by reversing the ± and β roles, namely,

(γ · •)(x) = •(h)γ(±(h)) ,

h∈β ’1 (x)

1 1

with •(h)γ(±(h)) ∈ „¦1 E±(h) — „¦ 2 Ex „¦1 (h) — „¦ 2 Ex .

β

14.4 Groupoid Actions

µ

A groupoid G over X G(0) may act on sets M ’ X that map to X. Let G — M

be the space

G — M := {(g, m) ∈ G — M | β(g) = µ(m)} .

A (left) groupoid action of G on M is de¬ned to be a map G — M ’ M , taking

the pair (g, m) to g · m, with the properties:

1. µ(g · m) = ±(g),

2. (gh) · m = g · (h · m),

3. (µµ(m)) · m = m.

The map µ : M ’ X is sometimes called the moment map, by analogy with

symplectic geometry.

Remark. The terms “moment map” and “momentum map” are usually used

interchangeably in the literature, with di¬erent authors preferring each of these

two translations of Souriau™s [153] French term, “moment”. By contrast, in these

notes, we have used the terms in di¬erent ways. Here, a “momentum map” is a

Poisson map J : M ’ g— to a Lie-Poisson manifold g— , generating a hamiltonian

action of an underlying Lie group G on M . On the other hand, a “moment map”

is a map µ : M ’ X to the base X of a groupoid G which is acting on M . ™¦

Example. A groupoid G over X acts on G by left multiplication with moment

™¦

map ± 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

± ““ β “µ

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

g· : µ’1 (β(g)) ’’ µ’1 (±(g))

are linear. For more on groupoid actions, see [110].

V

g· $$$

$$

$$ µ’1 (y)

W

µ’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 di¬erent “arrows” in X.

Example. If X is a topological space, and G = Π(X) is the fundamental

groupoid, then a representation of Π(X) on a vector bundle V would be a ¬‚at

connection of V . By ¬‚at connection, we do not yet mean a di¬erential-geometric

notion, but rather a topological one, namely that parallel transport only depends

on the homotopy class of the base path.

To see the ¬‚at connection, recall that Π(X) is the collection of homotopy classes

of paths in X. A representation of Π(X) determines precisely how to parallel

translate along paths to de¬ne a connection.

With this ¬‚at connection, we can look at the isotropy subgroup of loops based

at a point. The fundamental group of X acts on each ¬ber 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 ¬ber of V .

For applications to the moduli spaces used in topological quantum ¬eld theory,

™¦

see [77].

As with groups, the notion of groupoid representation can be formalized in

terms of the following de¬nition. The general linear groupoid of a vector bundle

µ : V ’ X is

GL(V ) = {(x, , y) | x, y ∈ X, : µ’1 (y) ’ µ’1 (x) is a linear isomorphism} .

The isotropy subgroup over any point is the general linear group of the correspond-

ing ¬ber 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 ∈ X, : µ’1 (y) ’ µ’1 (x) is linear} ,

where is an arbitrary linear map between ¬bers. 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 ’ End(V ) is a map,

then there is an induced map r : C (G) ’ End(V ) de¬ned by the formula

• ’’ r(•) := r(g) •(g) dg .

G

If r is a representation, then r will be a homomorphism of algebras. Hence, group

™¦

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 {»x } on G, there is an

action of the groupoid algebra A» on sections of V de¬ned as follows. Let • be any

continuous compactly supported function on G, and let u ∈ “(V ). De¬ne

(• · u)(x) = •(g) g · u(β(g)) d»x (g) .

g∈±’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

„¦ = ±— („¦ 2 E) — β — („¦ 2 E) .

For a vector bundle V over G(0) , we de¬ne

End(V ) := ±— (V ) — β — (V — ) ;

that is, End(V ) is the bundle over G whose ¬ber over each point g ∈ G is

—

V±(g) — Vβ(g) = Hom (Vβ(g) , V±(g) ) .

Given a representation of G on V , the sections of

„¦ — End(V )

act naturally on sections of V . We thus build a groupoid algebra with coe¬-

cients in a vector bundle V ,

“(„¦ — End(V )) .

Remark. In Section 14.3, we found an action of the intrinsic groupoid algebra

1

on sections of „¦ 2 E. However, this does not generally come from a representation

1

of G on „¦ 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 „¦ 2 T X is that of

sections of

1 1 1 1 1

„¦ — End(„¦ 2 T X) = ±— („¦ 2 E) — β — („¦ 2 E) — ±— („¦ 2 T X) — β — („¦’ 2 T X) .

In very special instances, there might be a natural trivialization of

1 1

±— („¦ 2 T X) — β — („¦’ 2 T X)

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

2

Alternatively, the intrinsic groupoid algebra itself acts on sections of

1

V — „¦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 — „¦’ 2 E.

Examples.

• 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

„¦ 2 E = „¦ 2 g (and also on „¦’ 2 E = „¦ 2 g— ).

• At the other extreme, for the pair groupoid over a manifold X there is no

1

natural representation of G on „¦ 2 E. The normal space E along the identity

section can be identi¬ed with T X, the tangent space to X.

G(0)

XT

E

E

X

A representation of G on E consists of an identi¬cation of Tx X with Ty X for

each (x, y) ∈ X — X. This amounts to a trivialization of the tangent bundle

to X “ that is, a global ¬‚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 „¦ 2 E, we would need a global ¬eld of

1

2 -densities. This is equivalent to a global density on X, for which there is no

natural choice.

™¦

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 identi¬ed (see Section 14.3) with the space of smooth

sections of

1 1

„¦ =: ±— („¦ 2 E) — β — („¦ 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

„¦ =: ±— („¦ 2 E — ) — β — („¦ 2 E — ) — „¦1 T G .

The groupoid algebra is included in D (G), as we can pair „¦ and „¦ to get „¦1 T G =

|§top | T — G, and then integrate a 1-density on G (that is, a section of |§top | T — G) to

obtain a number. Elements of D (G) are sometimes called generalized sections

of „¦.

We may describe a typical section of „¦ along the identity section X G(0) of

G. First, note that along X the bundle „¦ reduces to

„¦ |X = „¦1 E — — „¦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 ’’ T X ’’ T G|X ’’ E ’’ 0 ,

we see by Lemma 12.1 that

„¦1 E — — „¦1 T G|X

„¦ |X =

„¦1 E — — „¦1 E — „¦1 T X

„¦1 T X .

Thus a section of „¦ |X is just a 1-density on X. As a consequence, any measurable

function f : X ’ R determines a generalized section, namely

• ∈ “c („¦ ) ’’ f •|X ∈ 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 ∞ (X) ⊆ D (G) .

105

106 15 EXTENDED GROUPOID ALGEBRAS

15.2 Bisections

The previous construction generalizes to other “sections” besides the identity sec-

tion. A submanifold Σ of G such that the projections of Σ to X by ± and β are

isomorphisms is called a bisection of G or an admissible section.

e e e¡ e¡ e¡ e¡ ¡ ¡

e e ¡e ¡e ¡e ¡e ¡ ¡

e e¡e¡e¡e¡e¡ ¡

e e¡ e¡ e¡ e¡ e¡ ¡

e ¡e ¡e ¡e ¡e ¡e ¡

Σ

e¡e¡e¡e¡e¡e¡

e¡ e¡ e¡ e¡ e¡ e¡

¡e ¡e ¡e ¡e ¡e ¡e

¡e¡e¡e¡e¡e¡e

G(0)

X

¡ e¡ e¡ e¡ e¡ e¡ e

¡ ¡e ¡e ¡e ¡e ¡e e

¡ ¡e¡e¡e¡e¡e e

¡ ¡ e¡ e¡ e¡ e¡ e e

¡ ¡ ¡e ¡e ¡e ¡e e e

Because we can identify the normal spaces of Σ with the tangent spaces of either

the ±- or the β-¬bers along Σ, we see that

’1 ’1

„¦± 2 |Σ — „¦β 2 |Σ — „¦1 T G|Σ

„¦ |Σ =

„¦’1 N Σ — „¦1 T G|Σ

=

„¦1 T Σ .

where N Σ is the normal bundle to Σ inside G. We can thus integrate sections of

„¦ over Σ. Therefore, each bisection Σ determines an element of D (G). Let B(G)

denote the set of smooth bisections of G. We conclude that

B(G) ⊆ D (G) .

Remark. Before integrating we could have multiplied by any smooth function

on Σ (or X), thus obtaining other elements of D (G) (see the last exercise of this

™¦

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 identi¬cation of elements of a group with elements of the distri-

bution group algebra, as evaluation maps. The objects generalizing the Lie algebra

™¦

elements will be discussed in Sections 15.4 and 15.5.

The inclusion map from B(G) to D (G) is multiplicative if we de¬ne 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 — B,

AB = {xy ∈ G| (x, y) ∈ A — B © G(2) } .

This product de¬nes 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 de¬nes 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 satis¬es

the group axioms.

(b) Multiplication of bisections in B(G) maps to convolution of distributions

in D (G) under the inclusion B(G) ’ D (G).

2. There is a larger sub-semigroup Bloc (G) ⊇ B(G) of local bisections. A local

bisection is a subset of G for which the projection maps ±, β 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 — X) can be

Example.

identi¬ed with the group of di¬eomorphisms of X, since each bisection Σ is

the graph of a di¬eomorphism. Bloc (X — X) similarly corresponds to the

semigroup (sometimes called a pseudogroup) of local di¬eomorphisms of X.

Exercise 51

Show that the identi¬cation B(X — X) ’ Di¬(X) is a group homomorphism

(or anti-homomorphism, depending on conventions).

™¦

3. If we view G ‚ 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

{…} ∪ G ⊆ 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 ∞ (X) of D (G) generate multiplicatively the larger

subspace of pairs (Σ, s) ∈ B(G) — C ∞ (X). Here we identify functions on

a bisection Σ with functions on X via pull-back by ± (alternatively, β). Let

Σ1 , Σ2 be bisections and si ∈ C ∞ (Σi ). Find an explicit formula for the product

(Σ1 , s1 ) · (Σ2 , 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 ∈ G and Σ ∈ B(G). Because Σ is

a bisection, there is a uniquely de¬ned element h ∈ Σ, such that β(h) = ±(g). We

declare Σ · g := hg ∈ G.

108 15 EXTENDED GROUPOID ALGEBRAS

rΣ · g = hg

¡e

¡e

¡ e

erg

¡

Σ hr ¡

¡

¡ ¡

e ¡

e ¡

e¡

G(0) er

¡

β(h) = ±(g)

Similarly, we can de¬ne a right action of B(G) on G by noting that there is

also a uniquely de¬ned element ±|’1 (β(g)) ∈ Σ. These actions can be thought of

Σ

as “sliding” by Σ. See [2].

rg · Σ

¡e

¡e

¡ e

gr

¡ e

e e

e e

e e ’1

er±|Σ (β(g)) Σ

e

e ¡

e ¡

e¡

G(0) r

e¡

β(g)

Exercise 53

Check that this de¬nes a group action and that the left and right actions

commute.

Remarks.

• This construction generalizes the left (or right) regular representation of a

group on itself.

• We can recover the bisection Σ from its left or right action on G since

Σ = Σ · G(0) = G(0) · Σ .

™¦

The left action of B(G) preserves the β-¬bers of G, while the right action

preserves the ±-¬bers. On the other hand, the left action of B(G) maps ±-¬bers to

±-¬bers, while the right action of B(G) maps β-¬bers to β-¬bers.

The left (respectively, right) action respects the ±-¬ber (respectively, β-¬ber)

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 Σ ∈ B(G), the (left) action

on X is de¬ned by taking x ∈ X to ±(β ’1 (x)), where β ’1 (x) ∈ Σ is uniquely

determined.

’1

rβ (x) Σ

¡eu

±¡e

¡ e

X r¡ er

Σ·x x

It is easy to check that ±(Σ · g) = Σ · ±(g), and so ± is a left equivariant map

from G to X with respect to the B(G)-actions. Similarly, β 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 de¬nition, 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 ¬rst 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) • E ,

we can identify the normal bundle E with a sub-bundle E ⊆ T G|G(0) . Under this

identi¬cation, a section σ ∈ “(E) may be viewed as a vector ¬eld v : G(0) ’

T G|G(0) . We can ¬nd, for su¬ciently small µ, a path ψt : G(0) ’ G de¬ned for

0 ¤ t < µ and such that

identity on G(0)

ψ0 =

ψt ’ ψ0

dψt

= lim =v .

dt t

t’0

t=0

At each time t, the image of ψt is a bisection Σt (restricted to the given compact

subset of G). In particular, Σ0 = G(0) is the identity section.

The one-parameter family of bisections {Σt } gives rise to an element, called σ, of

the extended groupoid algebra D (G) by the following recipe. Let • be a compactly

supported smooth section of „¦ . Each individual bisection Σt ∈ D (G) = (“c („¦ ))

pairs with • to give a number Σt , • as described in Section 15.2. We de¬ne the

new pairing by

Σt , • ’ Σ0 , •

σ, • := lim .

t

t’0

110 15 EXTENDED GROUPOID ALGEBRAS

Exercise 54

Check that σ, · is a well-de¬ned linear functional on “c („¦ ), independent of

the choice of E. (Hint: notice how vector ¬elds v ∈ “(T G(0) ), i.e. tangent to

G(0) , yield a trivial pairing.)

We conclude that

“(E) ⊆ D (G) .

Furthermore, these elements of the extended groupoid algebra have support in G(0) ,

that is, they vanish on test sections • ∈ “c („¦ ) with (support •) © G(0) = ….

If we think of σ ∈ “(E) as

Σt ’ G(0)

σ = lim ,

t

t’0

we can give an informal de¬nition of a commutator bracket [·, ·] on “(E). Given

two sections of E

Σt ’ G(0) ˜u ’ G(0)

σ = lim , θ = lim ,

t u

t’0 u’0

we de¬ne

Σt ’ G(0) ˜u ’ G(0) ˜u ’ G(0) Σt ’ G(0)

· ’ ·

[σ, θ] = lim

t u u t

t,u’0

Σt ˜u ’ ˜u Σt

= lim ,

tu

t,u’0

or, equivalently, the bracket evaluated on • ∈ “c („¦ ) is

Σt ˜u , • ’ ˜u Σt , •

[σ, θ], • = lim .

tu

t,u’0

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

[“(E), “(E)] ⊆ “(E) ,

as we will see in the next section where we de¬ne 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 ∈ G acts on ±’1 (β(g)) by left multiplication.

The β-projection is invariant with respect to this action

β(g · h) = β(gh) = β(h) ,

while ±-¬bers are mapped to ±-¬bers

g·

±’1 (β(g)) ’’ ±’1 (±(g)) .

Let

T ± G := ker T ± ⊆ T G

15.5 Left Invariant Vector Fields 111

be the distribution tangent to the ±-¬bers. The action of g ∈ G induces a linear

map

T g·

T ± G|±’1 (β(g)) ’’ T ± G|±’1 (±(g)) .

Exercise 55

The left action of the group of sections B(G) preserves the ±-¬ber structure

(see Section 15.3), and hence also induces an action on T ± G by di¬erentiation.

(a) Prove that a section of T ± 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 ± G?

A left invariant section of T ± G is called a left invariant vector ¬eld on the

groupoid G. The set χL (G) of all left invariant vector ¬elds on G has the following

properties.

• χL (G) is closed under the bracket operation

[χL (G), χL (G)] ⊆ χL (G) ,

and thus forms a Lie algebra.

• An element of χL (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.

±

• Every smooth section of TG(0) G := ker T ±|G(0) can be extended to an element

of χL (G).

Furthermore,

±

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 identi¬cations

±

χL (G) “(TG(0) G) “(E) .

The bracket on χL (G) can therefore be considered as a bracket on “(E); it agrees

with the one de¬ned informally in the previous section.

∞

The left invariant vector ¬elds on G act by di¬erentiation on CL (G), the left

invariant functions on G. From the identi¬cation

∞

β — C ∞ (X) C ∞ (X) ,

CL (G)

we get a map

“(E) ’’ χ(X) := “(T X) .

It is easy to see that this map is induced by the bundle map

ρ : E ’’ T X

112 15 EXTENDED GROUPOID ALGEBRAS

given by composition of two natural maps:

E T ±(0) G

E G

d

d

d

ρd TG(0) β

d

‚

©

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 χL (G) g and E g are

the Lie algebra, and ρ : g ’ {0} is the trivial map. ™¦

Part VII

Algebroids

16 Lie Algebroids

Lie algebroids are the in¬nitesimal versions of Lie groupoids.

16.1 De¬nitions

A Lie algebroid over a manifold X is a (real) vector bundle E over X together

with a bundle map ρ : E ’ T X and a (real) Lie algebra structure [·, ·]E on “(E)

such that:

1. The induced map “(ρ) : “(E) ’ χ(X) is a Lie algebra homomorphism.

2. For any f ∈ C ∞ (X) and v, w ∈ “(E), the following Leibniz identity holds

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

Remarks.

• The map ρ is called the anchor of the Lie algebroid. By an abuse of notation,

the map “(ρ) may be denoted simply by ρ and also called the anchor.

• For each v ∈ “(E), we de¬ne E-Lie derivative operations on both “(E) and

C ∞ (X) by

Lv w = [v, w]E ,

Lv f = ρ(v) · f .

We can then view the Leibniz identity as a derivation rule

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

™¦

When (E, ρ, [·, ·]E ) is a Lie algebroid over X, the kernel of ρ is called the

isotropy. Each ¬ber of ker ρ is a Lie algebra, analogous to the isotropy subgroups

of groupoids. To see this, let v and w ∈ “(E) be such that ρ(v) and ρ(w) both

vanish at a given point x ∈ X. Then, for any function f ∈ C ∞ (X),

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

So there is a well-de¬ned bracket operation on the vectors in any ¬ber of ker ρ, and

ker ρ is a ¬eld of Lie algebras. These form a bundle when ρ has constant rank.

On the other hand, the image of ρ is an integrable distribution analogous

to the image of Π for Poisson manifolds. Therefore, X can be decomposed into

submanifolds, called orbits of the Lie algebroid, whose tangent spaces are the image

of ρ. 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 (¬nite 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 ρ ≡ 0. Conversely, if E is any Lie algebroid with

ρ ≡ 0, the Leibniz identity says that the bracket in “(E) is a bilinear map of

C ∞ (X)-modules and not simply of R-modules, and hence that each ¬ber 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 ρ : E ’ 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):

• When is a given Lie algebroid the Lie algebroid of a Lie groupoid?

• 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 ρ the identity map, is a Lie

algebroid over X. We can see it the Lie algebroid of the Lie groupoid X — X,

or of the fundamental groupoid Π(X), or of yet other possibilities; near the

identity section, Π(X) looks like X — 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

γ

algebra homomorphism g ’ χ(X). The associated transformation Lie

ρ

algebroid X — g has anchor X — g ’ T X de¬ned by

ρ(x, v) = γ(v)(x) .

Combining this with the natural projections X — g ’ X and T X ’ X, we

form the commutative diagram

ρE

X —g TX

c©

X

A section v of X — g can be thought of as a map v : X ’ g. We de¬ne the

bracket on sections of X — g by

[v, w](x) = [v(x), w(x)]g + (γ(v(x)) · w)(x) ’ (γ(w(x)) · v)(x) .

16.2 First Examples of Lie Algebroids 115

When v, w are constant functions X ’ g, we recover the Lie algebra bracket

of g.

It is easy to see in this example that the ¬bers of ker ρ 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 γ comes from a “-action on X, where “ is a Lie group with Lie algebra g,

then X — g is the Lie algebroid of the corresponding transformation groupoid

G“ .

ρ(E) ⊆ T X being

6. Suppose that ρ 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

Hausdor¬.)

The case when ρ 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 ™s are constants. Show that E is isomorphic to a

transformation Lie algebroid.

Historical Remark. Already in 1963, Rinehart [145] noted that, if a Lie algebra

“ over a ¬eld k is a module over a commutative k-algebra C, and if there is a

homomorphism ρ from “ into the derivations of C, then there is a semidirect product

Lie bracket on the sum “ • C de¬ned by the formula

[(v, g), (w, h)] = ([v, w], ρ(v) · h ’ ρ(w) · g) .

Furthermore, this bracket satis¬es the Leibniz identity:

[(v, g), f (w, h)] = f [(v, g), (w, h)] + (ρ(v) · f )(w, h) for f ∈ C .

In the special case where C = C ∞ (X), the C ∞ (X)-module “, if projective, is

the space of sections of some vector bundle E over X. The homomorphism ρ and

the Leibniz identity imply that ρ is induced by a bundle map ρ : E ’ T X. The

Leibniz identity for “(E) • C ∞ (X) also encodes the Leibniz identity for the bracket

on “(E) alone.

In 1967, Pradines [139] coined the term “Lie algebroid” 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 [4] to be false.

(See Section 16.4.)

Rinehart [145] proved (in a more algebraic setting) an analogue of the Poincar´-e

Birkho¬-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

“(E) • C ∞ (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 [34] as the base

of the cotangent groupoid T — G of a Lie groupoid G; it is described more explicitly

in [35]. (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 — X with its standard (symplectic) Poisson structure (see

Section 6.5). The universal object is the algebra of di¬erential operators on X, and

™¦

the Rinehart isomorphism is a “symbol map”.

16.3 Bundles of Lie Algebras

For a ¬rst 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

¬ber is a Lie group. Bundles of Lie algebras are vector bundles for which each ¬ber

has a Lie algebra structure which varies continuously (or smoothly). Every bundle

of Lie groups de¬nes a bundle of Lie algebras: the Lie algebras of the individual

¬bers. 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 [48]) Every bundle of Lie algebras can be inte-

grated to a (not necessarily Hausdor¬ ) bundle of Lie groups. (Fibers and base are

Hausdor¬, but the bundle itself might not be.)

Example. Given a Lie algebra g with bracket [vi , vj ]= cijk vk , we de¬ned in

Section 1.2 a family of Lie algebras gµ = (g, [·, ·]µ ), µ ∈ R, by the structure equations

[vi , vj ]µ = µ 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 ¬ber over 0 ∈ R

is an abelian Lie group (either euclidean space, a cylinder or a torus), while the

¬ber over any other point µ ∈ R can be chosen to be a ¬xed manifold. The ¬ber

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 gµ has

¬ber SU(2) S 3 for µ = 0, and ¬ber R3 at µ = 0. Here the total space is Hausdor¬,

since it is homeomorphic to R — S 3 with a point removed from {0} — S 3 . ™¦

Example. [48, p.148] Consider now the bundle of Lie algebras over R with ¬bers

gµ = (R3 , [·, ·]µ ), µ ∈ R, where the brackets are de¬ned by

[xµ , zµ ]µ = ’yµ ,

[xµ , yµ ]µ = zµ , [yµ , zµ ]µ = µxµ .

Here xµ , yµ , zµ denote the values at µ of a given basis of sections x, y, z for the bundle

g := R — R3 .

The corresponding simply connected Lie groups Gµ are as follows for µ ≥ 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(4πx1 ) = e1 is the identity element of

G1 . For µ > 0, gµ g1 under the isomorphism

√ √

xµ ’ x1 , yµ ’ µ y1 , zµ ’ µ z1 .

Taking Gµ G1 , we still have that exp(4πxµ ) = eµ is the identity of Gµ , µ > 0. At

µ = 0, G0 is the semidirect product R — R2 , where the ¬rst factor R acts on R2 by

rotations. Here exp(tx0 ) = (t, 0), thus, in particular, exp(4πx0 ) = e0 .

Therefore, the set of points µ ∈ R where the two continuous sections exp(4π·)

and e coincide is not closed, hence G is not Hausdor¬.

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 —R2 (the group of euclidean motions of the plane),

the resulting bundle of groups is Hausdor¬. They then go on to show that a certain

C ∞ bundle of semidirect product Lie algebras admits no Hausdor¬ bundle of Lie

groups. They conclude by asking whether there is an analytic example. The next

™¦

example answers this question.

Example. Coppersmith [33] constructed an analytic family of 4-dimensional Lie

algebras parametrized by R2 which cannot be integrated to a Hausdor¬ family of

Lie groups. The ¬ber gµ over µ = (µ1 , µ2 ) ∈ R2 has basis xµ , yµ , zµ , wµ and bracket

given by

[wµ , xµ ]µ = [wµ , yµ ]µ = [wµ , zµ ]µ = 0 ,

[xµ , zµ ]µ = ’yµ , [yµ , zµ ]µ = xµ , [xµ , yµ ]µ = µ1 zµ + µ2 wµ .

™¦

16.4 Integrability and Non-Integrability

To ¬nd Lie algebroids which are not integrable even by non-Hausdor¬ groupoids,

we must look beyond bundles of Lie algebras.

Example. As a ¬rst attempt, take the transformation Lie algebroid X — 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 “, then the “-action

on X de¬nes a transformation groupoid G“ with Lie algebroid X — g. Now we can

make the g-action non-integrable by restricting to an open set U ⊆ X not invariant

under “. The Lie algebroid X — g restricts to a Lie algebroid U — 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 =

±’1 (U ) © β ’1 (U ) is a subgroupoid of G with base space U .

We conclude that this restriction H of the transformation groupoid X — “ has Lie

algebroid U — g. ™¦

Example. We could instead look for incomplete vector ¬elds that cannot even be

completed by inserting into a bigger manifold. One example begins with X = R2

‚

and U = R2 \ {0}, where the vector ¬eld ‚x is incomplete. If R2 is identi¬ed with

C, then the subspace U has a double cover de¬ned by the map z ’ z 2 . If we

‚

pull ‚x back to the double cover, there is no way to smoothly “¬ll in the hole”

to a complete vector ¬eld. Could this then give an example of a non-integrable

Lie algebroid? Unfortunately, this type of construction is also doomed to fail, if

™¦

non-Hausdor¬ groupoids are allowed.

Theorem 16.2 (Dazord [40]) 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 [134] in 1957. In that manuscript, he proved results close to Dazord™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 [48]. Mackenzie [110] had already developed an obstruction

™¦

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 ’’ L ’’ T X — R ’’ T X ’’ 0 ,

E

where L is the trivial real line bundle over X. We de¬ne a bracket on sections of

E, “(E) = χ(X) — C ∞ (X), by

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

where „¦ is a given 2-form on X. The bracket [·, ·]E,„¦ satis¬es the Jacobi identity

if and only if „¦ is closed.

Each integral 2-cycle γ ∈ H2 (X; Z) gives rise to a period

„¦∈Λ.

γ

If the set of periods of „¦ is not cyclic in R, and if X is simply connected, then

one can show that E does not come from a groupoid [3].6 In this way we obtain a

™¦

non-integrable Lie algebroid.

Remark. There is still a sort of Lie groupoid corresponding to this Lie algebroid.

As a bundle over X — X, it has structure group R/Λ, where Λ is generated by

two numbers which are linearly independent over Q. There are no nonconstant

di¬erentiable functions on R/Λ, but there is a notion of smooth curves, if one uses

Souriau™s notion of di¬eological space [154]. In general, a map M ’ R/Λ is said to

be smooth if it (locally) lifts to a smooth map M ’ R.

R

c

M E R/Λ

Examples of Lie algebroids which are “even more” non-integrable can also be con-

™¦

structed [39].

6 For instance, on X = S 2 — S 2 with projections

S2 — S2

π1 ““ π2

S2

— —

de¬ne the 2-form „¦ = c1 π1 ω + c2 π2 ω, where ω is the standard volume on S 2 and c1 , c2 are

rationally independent constants, Then the periods of „¦ 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 , · · · , xn be local coordinates on a manifold X, and let e1 , · · · , er be a local

basis of sections of a Lie algebroid (E, ρ, [·, ·]E ) over it. With respect to these

coordinates and basis, the Lie bracket and anchor map are described by structure

functions cijk , bij ∈ C ∞ (X) as

[ei , ej ]E = cijk ek

k

‚

ρ(ei ) = bij .

‚xj

j

Exercise 59

The Leibniz identity and Jacobi identity translate into di¬erential equations

for the cijk and bij . Write out these di¬erential equations.

Let x1 , · · · , xn , ξ1 , · · · , ξn be the associated coordinates on the dual bundle E — ,

where ξ1 , · · · , ξn are the linear functions on E — de¬ned by evaluation at e1 , · · · , er .

We de¬ne a bracket {·, ·}E on C ∞ (E — ) by setting

{xi , xj }E =0

{ξi , ξj }E = cijk ξk

k

{ξi , xj }E ’bij

=

Proposition 16.3 The bracket {·, ·}E de¬nes a Poisson structure on E — .

Exercise 60

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

for E.

Remark. Although the Poisson bracket {·, ·}E is de¬ned 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 — is intrinsic. ™¦

Examples.

1. When X is a point, and E = g is a Lie algebra, then the Poisson bracket on

E — = g— regarded as the dual of a Lie algebroid, coincides with the Lie-Poisson

bracket de¬ned in Section 3.1.

In general, the Poisson bracket {·, ·}E on the dual of a Lie algebroid is some-

times also called a Lie-Poisson bracket.

‚

2. When E = T X, we can choose ei = ‚xi to give the standard basis of vector

¬elds induced by the choice of coordinates on X, so that cijk = 0 and bij = δij .

The Poisson structure on the dual bundle E — = T — X as a dual of a Lie

algebroid is the one induced by the canonical symplectic structure dxi §dξi

because

{xi , ξj }E = ’{ξj , xi }E = δij .

120 16 LIE ALGEBROIDS

3. When E = T — X is the Lie algebroid of a Poisson manifold X (see Sec-

tion 17.3), we obtain on the tangent bundle E — = T X the tangent Poisson

structure; see [5].

Exercise 61

(a) Let E1 and E2 be Lie algebroids over X. Show that a bundle map

• : E1 ’ E2 is a Lie algebroid morphism (i.e. compatible with

brackets and anchors) if and only if •— : E2 ’ E1 is a Poisson map.

— —

(b) Show that the dual of the anchor map of a Lie algebroid is a Poisson

map from T — X to E — .

(c) Use the result of part (a) to suggest a de¬nition of morphism between

Lie algebroids over di¬erent base manifolds. See Proposition 6.1 in [111].

Exercise 62

‚ ‚

Let x1 , · · · , xn be coordinates on X, ‚x , · · · , ‚x and dx1 , · · · , dxn the in-

n

1

duced local bases of T X and T — X, x1 , · · · , xn , v1 , · · · , vn associated coordinates

on T X, and x1 , · · · , xn , ξ1 , · · · , ξn associated coordinates on T — X.

Express the Poisson bracket for the tangent Poisson structure on T X in terms

of the Poisson bracket on X given by πij (x) = {xi , xj }.

˜

Check that, although T X ’ X is not a Poisson map, the map Π : T — X ’ T X

is a Poisson map.

™¦

16.6 Complex Lie Algebroids

It can be interesting to work over C even if X is a real manifold. To de¬ne a

complex Lie algebroid geometrically, take a complex vector bundle E over X

and a complex bundle map ρ : E ’ TC X to the complexi¬ed tangent bundle.7 The

immediate generalization of our de¬nition 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 ’ T X, i.e. J is a bundle map such that J 2 = ’id. The graph of ’iJ in

TC X = T X • iT X is the sub-bundle

E = {v ’ iJ(v)| v ∈ T X} ⊆ 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:

[“(E), “(E)] ⊆ “(E) .

This holds if and only if J is an integrable structure. That is, by the Newlander-

Nirenberg theorem [131], 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 “ground ring” from C ∞ (X) to C ∞ (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?

™¦

What is a complex Lie groupoid?

Example. Let X be a manifold of dimension 2n ’ 1. Suppose that F ⊆ T X is a

codimension-1 sub-bundle with an almost complex structure J : F ’ F (J is linear

and J 2 = ’id). As before, de¬ne a sub-bundle E of the complexi¬ed F to be the

graph of ’iJ

E = {v ’ iJ(v)| v ∈ F } ‚ FC = F • iF .

If “(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 2n’1 ‚ Y 2n is a real submanifold of a complex n-manifold Y .

At a point x ∈ X ‚ 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 © 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:

• 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.

• 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 [87].

• The cohomology theory of Lie algebroids can be applied to complex Lie al-

gebroids. In the examples above, we recover the usual ‚ cohomology and

boundary ‚ cohomology on complex and CR-manifolds, respectively. What

does complex Lie algebroid cohomology look like in more general cases?

™¦

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. Atiyah™s (see [9] 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

™¦

“Atiyah algebroid”.

17 Examples of Lie Algebroids

Lie algebroids with surjective anchor map are called transitive Lie algebroids,

Atiyah algebras, or Atiyah sequences because of Atiyah™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 [9] 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

←

P H

π“

X

with structure group H acting on the right. The quotient G = (P — 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 ∈ π ’1 (x), q ∈

π ’1 (y) in P ; it is the graph of an equivariant map from the ¬ber π ’1 (y) to π ’1 (x).

A bisection of this groupoid corresponds to a gauge transformation, that is, an

automorphism (i.e. an H-equivariant di¬eomorphism) 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 in¬nitesimal generators of G are the

H-invariant vector ¬elds. Since H acts on the ¬bers of π freely and transitively,

H-invariant vector ¬elds are determined by their values on one point of each ¬ber,

so they can be identi¬ed with sections of

E = T P/H

considered as a bundle over X. The bracket on E is that induced from χ(P ); this

is well-de¬ned because the bracket of two H-invariant vector ¬elds is H-invariant.