∆

c i—i c

C—C E A—A

commutes, where the left arrow is c ’ c — 1,

7. the co-unit is a homomorphism of algebras, i.e.

µ

A

C'

T T

m

µ—µ

C—C ' A—A

commutes, where the left arrow is multiplication of complex numbers, and

8. the antipode is an anti-homomorphism of algebras, i.e.

± —E

±

A—A A—A

m m

±

c c

A EA

commutes, where m(•, ψ) = m(ψ, •).

9. the antipode is an anti-homomorphism of coalgebras, i.e.

±—±

A—A ' A—A

T T

∆ ∆

±

A' A

commutes, where ∆ is ∆ composed with the map • — ψ ’ ψ — •.

10. the following diagram involving the antipode commutes2

µE i

A EA

C

T

∆ m

± — id

c

A—A E A—A

2 It is not generally true that the square of the antipode equals the identity map.

72 11 GROUP ALGEBRAS

and the similar diagram with the bottom arrow being id — ± also commutes.

This is sometimes regarded as the de¬ning axiom for the antipode.

Hopf came across the structure just described while studying the cohomology

rings of topological groups.

11.2 Commutative and Noncommutative Hopf Algebras

When A = C(G) is the algebra of continuous real functions on a locally compact

topological group G, the (pointwise) multiplication of functions extends to a prod-

uct on a topological completion “ — ” of the standard algebraic tensor product for

which C(G — G) C(G)“ — ”C(G) (see [159]). The algebra C(G) is a commuta-

tive Hopf algebra (commutativity here refers to the ¬rst multiplication) where

1. the multiplication is pointwise multiplication of functions,

2. the comultiplication is the pull-back m— of the multiplication on G,

3. the identity is the function identically equal to 1, or, equivalently, the homo-

morphism C ’ C(G), c ’ • ≡ c,

4. the coidentity is given by evaluation at the identity of G, and

5. the antipode is the pull-back by the inversion map on G.

Exercise 33

Show that the associativity of group multiplication on G translates to coasso-

ciativity on C(G).

Commutative Hopf algebras are closely related to groups: if A = C(X) is the

set of continuous functions on a locally compact Hausdor¬ space X, then a Hopf

algebra structure on A (with C(X — X) playing the role of A — A) de¬nes a (not

necessarily commutative) multiplication on X which can be shown to satisfy the

group axioms.

A noncommutative Hopf algebra is thus to be thought of as “the algebra

of functions on a quantum group”. There is no universally accepted de¬nition

of a quantum group. Many people restrict the name to the objects obtained by

deforming a Hopf algebra of functions on a Lie group.

Between commutative and noncommutative Hopf algebras lies the category of

Poisson Hopf algebras. A Poisson Hopf algebra A is a commutative Hopf algebra

equipped with a bracket operation making A into a Poisson algebra. We then

require the comultiplication and co-unit to be Poisson algebra homomorphisms,

while the antipode is an anti-homomorphism. When A = C ∞ (P ) for some Poisson

manifold P , the comultiplication gives P the structure of a Poisson Lie group;

i.e. the multiplication map P — P ’ P is a Poisson map. Poisson Lie groups can

be regarded as the transitional objects between groups and quantum groups, or as

classical limits of quantum groups. A comprehensive reference on quantum groups

and Poisson Lie groups is [25].

11.3 Algebras of Measures on Groups 73

11.3 Algebras of Measures on Groups

Let G be a locally compact topological group G, and let C(G) be its algebra of

continuous real functions. The dual space C (G) consists of compactly supported

measures on G. (The Lie group version of this construction will be presented in

Section 11.5).

Denoting by m the multiplication map on G, we described in Section 11.2 a

coproduct

∆ = m—

m— (•)(g, h) = •(gh) .

C(G)“ — ”C(G) ,

C(G) E

On C (G) we obtain a map

∆—

' = m— C (G)“ — ”C (G)

C (G)

de¬ned by

∆— (µ)(S) = µ(m’1 S) ,

where µ ∈ C (G)“ — ”C (G) C (G—G) is a measure on G—G, S is any measurable

subset of G, and m’1 S = {(g, h) ∈ G — G | gh ∈ S}. The map ∆— is just the

push-forward of measures by the multiplication map.

Composing ∆— with the natural bilinear map (µ, ν) ’ µ—ν from C (G)—C (G)

to the tensor product, we obtain a multiplication of measures on G. By a simple

diagram chase through the axioms, we can check that ∆— is associative. This

multiplication is called convolution, and we will denote ∆— (µ — ν) by µ — ν. The

following (abusive) notation is commonly used

f (x) d(µ — ν)(x) = for f ∈ C(G) .

f (yz) dµ(y) dν(z)

The space C (G) (or a suitable completion, such as the integrable signed mea-

sures) with the convolution operation is known as the measure group algebra of

G.

The diagonal map in the group

G—G

D: G E

’’

g (g, g)

induces by push-forward a coproduct on measures

D—

E C (G)“ — ”C (G)

C (G)

de¬ned by

D— (µ)(S) = µ(D’1 S) ,

where D’1 S = {g ∈ G | (g, g) ∈ S}.

The space C (G) becomes a Hopf algebra for the convolution product ∆— and

this coproduct D— ; the unit is the delta measure at the identity e of G (or rather,

it is the map C c ’ cδe ), the co-unit is evaluation of measures on the total set

G, and the antipode of a measure is its pushforward by the group inversion map.

In summary, we see that the group structure on G gives rise to:

• a (pull-back of group multiplication) coproduct on C(G), and its dual

74 11 GROUP ALGEBRAS

• a (convolution) multiplication on C (G).

Independent of the group structure we have:

• a (pointwise) multiplication on C(G), and its dual

• a (push-forward by the diagonal map) coproduct on C (G).

Remark. Each element g ∈ G de¬nes an evaluation functional δg on G by

δg (f ) := f (g) .

This identi¬cation allows us to think of G as sitting inside C (G). Note that δgh =

δg — δh . Moreover, the push-forward of the diagonal map behaves nicely on G ⊆

C (G):

D— (δg ) = δg — δg .

An element of C (G) is called group(-element)-like if it satis¬es the property

™¦

above.

11.4 Convolution of Functions

If we choose a reference Borel measure » on G, we can identify locally integrable

functions • on G with measures by • ’ •». The map from compactly supported

locally integrable functions to C (G) is neither surjective (its image is the set of

compactly supported measures which are absolutely continuous with respect to

» [146]), nor injective (if two functions di¬er only on a set of »-measure 0, then

they will map to the same measure). In any case, we can use this rough identi¬cation

together with convolution of measures to describe a new product on functions on

G.

Before we can do this, we need to make a digression through measures on groups.

We de¬ne a measure » to be quasi-invariant if, for each g ∈ G, the measure ( g )— »

induced by left translation is absolutely continuous with respect to »; in other words,

there is a locally integrable function • such that ( g )— » = •». We de¬ne » to be

left-invariant if ( g )— » = » for all g ∈ G.

Theorem 11.1 If G is locally compact, then there exists a left-invariant measure

which is unique up to multiplication by positive scalars.

Such a measure is called a Haar measure.

Remarks.

• For Lie groups, this theorem can be proven easily using a left-invariant volume

form, which can be identi¬ed with a non-zero element of the highest dimen-

sional exterior power §top g— of g— : use left translation to propagate such an

element to the entire group.

• For general locally compact groups, this theorem is not trivial [149].

11.4 Convolution of Functions 75

• For some quantum groups, an analogous result holds; the study of Haar mea-

sures on quantum groups is still in progress (see [25], Section 13.3B).

™¦

Observe that if » is a left-invariant measure, then (rg )— » is again left-invariant

for any g ∈ G. Thus by Theorem 11.1, there is a function δ : G ’ R+ such that

(rg )— » = δ(g)» .

It is easy to check the following lemma.

Lemma 11.2 δ(gh) = δ(g)δ(h).

δ is known as the modular function or the modular character of G. Due

to the local compactness of G, we also know that δ is continuous. If G is compact,

then we see that δ ≡ 1. Any group with δ ≡ 1 is called unimodular. Notice that

δ is independent of the choice for ». Also, when G is a Lie group, we can compute

δ(g)» = ( g’1 )— (rg )— »

(Ad g ’1 )— »

=

|(det ad g)|» .

=

Thus we interpret the modular function of a Lie group as (the absolute value of)

the determinant of the adjoint representation on the Lie algebra.

Exercise 34

1. Compute the modular function for the group of a¬ne transformations,

x ’ ax + b, of the real line.

2. Prove that GL(n) is unimodular.

3. To check the formula above for δ(g) on a Lie group, see whether δ(g) is

greater or smaller than 1 when Ad g ’1 is expanding. Is det ad g greater

or smaller than 1?

Let » be a Haar measure and δ : G ’ R+ the modular function. Given functions

•, ψ ∈ C(G), their convolution • —» ψ with respect to » is

(• —» ψ)» := •» — ψ» ,

or, equivalently,

•(gh’1 ) ψ(h) δ(h) d»(h) .

(• —» ψ)(g) =

h∈G

When G is unimodular, the δ factor drops out, and we recover the familiar formula

for convolution of functions. We can rewrite this formula in terms of a kernel for

the convolution (cf. Sections 14.1 and 14.2):

(• —» ψ)(g) = •(k) ψ( ) K(g, k, ) d»(k) d»( ) .

We think of K as a generalized function, and we interpret the expression K(g, k, )d»(k)d»( )

as a measure on G — G — G supported on {(g, k, ) | g = k }, which is the graph of

multiplication.

76 11 GROUP ALGEBRAS

Remark. There is a *-operation on complex-valued measures which is the push-

forward by the inversion map on the group, composed with complex conjugation. To

transfer this operation to functions • ∈ C(G), we need to incorporate the modular

function:

•— (x) = •(x’1 ) δ(x’1 ) .

The map • ’ •— is an anti-isomorphism of C(G). ™¦

11.5 Distribution Group Algebras

In Sections 11.2 and 11.3, we realized both C(G) and its dual C (G) as Hopf

algebras, with products and coproducts naturally induced from the group structure

of G.

There is a smooth version of this construction. If G is a Lie group, then D(G) =

∞

C (G) is a Hopf algebra. The product is pointwise multiplication of functions,

while the coproduct is again the pull-back of group multiplication. The “tensor”

here needs to be a smooth kind of completion, so that C ∞ (G)“ — ”C ∞ (G) becomes

C ∞ (G — G).

The dual space D (G) of compactly supported distributions [148] is called the

distribution group algebra of G. The space D (G) is larger than the measure

group algebra: an example of a distribution that is not a measure is evaluation

of a second derivative at a given point. As in the case of C (G), we can de¬ne a

product (convolution) and coproduct (push-forward of the diagonal map) on D (G)

to provide a Hopf algebra structure.

Remark. At the end of Section 11.3, we noted how the group G was contained

in C (G) as evaluation functionals:

g ∈ G ’’ δg ∈ C (G) .

Inside D (G), the evaluation functionals can be used to de¬ne left and right trans-

lation maps:

δg — · : D (G) ’’ D (G)

(δg — •)(x) = •(g ’1 x)

’’

•(x)

and

· — δg : D (G) ’’ D (G)

(• — δg )(x) = •(xg ’1 )

’’

•(x)

™¦

Exercise 35

Show that the algebra of di¬erential forms on a Lie group forms a Hopf algebra.

What is its dual?

12 Densities

As we have seen, group algebras, measure group algebras and distribution group

algebras encode much, if not all, of the structure of the underlying group. There are

counterparts of these algebras for the case of manifolds, as algebras of “generalized

functions”.

12.1 Densities

To construct spaces of distributions which behave as generalized functions, rather

than measures, we need the notion of density on a manifold.

Let V be a ¬nite dimensional vector space (over R), and let B(V ) be the set of

bases of V . An ±-density on V is a function σ : B(V ) ’ C such that, for every

A ∈ GL(V ) and β ∈ B(V ), we have the relation

σ(β · A) = |det A|± σ(β) ,

where

(β · A)i = βj Aji

j

when A is written as A = (Aij ) and β is the basis β = (β1 , . . . , βn ).

Remarks.

• When ± = 1, σ is equal up to signs to the function on bases given by an

element of §top V — . We often denote the space of ±-densities on V by

±

V— .

§top

In fact, if θ is an element of §top V — (§top V )— , there is an ±-density |θ|±

de¬ned by

±

|θ|± (β1 , . . . , βn ) = |θ(β1 § · · · § βn )| .

±

• A density σ is completely determined by its value on one basis, so |§top | V —

is a one-dimensional vector space.

™¦

Lemma 12.1

1. For any vector space A and any ±, β ∈ R, there is a natural isomorphism

± β ±+β

§top A — §top §top

A A.

2. For any vector space A and any ± ∈ R, there are natural isomorphisms

’± ± ±

A)— A— .

§top ( §top §top

A

3. Given an exact sequence

0 ’’ A ’’ B ’’ C ’’ 0

of vector spaces, there is a natural isomorphism

± ± ±

§top A — §top §top

C B.

77

78 12 DENSITIES

Exercise 36

Prove the lemma above.

Now suppose that E is a vector bundle over a smooth manifold M with ¬ber

V . Letting B(E) be the bundle of bases of E, a C k ±-density on E is a C k map

σ : B(E) ’ C which satis¬es

σ(β · A) = |det A|± σ(β) .

In other words, σ must be GL(V )-equivariant with respect to the natural GL(V )-

action on the ¬bers of B(E), and the action of GL(V ) on C where A ∈ GL(V )

acts by multiplication by |det A|± . Hence, we can think of an ±-density on a vector

bundle E as a section of an associated line bundle B(E) — C/ ∼, where (β · A, z) ∼

(β, |det A|± z), for A ∈ GL(V ). Equivalently, an ±-density is a section of the

top ± — ±—

bundle |§ | E , whose ¬ber at a point p is |§top | Ep . Therefore, a density on

±

E is a family of densities on the ¬bers. When E = T M , we write |§top | M :=

± 1

|§top | T — M and |§top | M := |§top | T — M .

±

All the bundles |§top | E — are trivializable. However, they have no

Remark.

™¦

natural trivialization.

Example. A riemannian manifold carries for each ± a natural ±-density which

assigns the value 1 to every orthonormal basis. The orientation of the basis is not

™¦

relevant to the density.

Intrinsic Lp Spaces

12.2

Suppose that σ is a compactly supported C 0 1-density on a manifold M . In [42],

de Rham referred to such objects as odd di¬erential forms. The integral

σ

M

can be given a precise meaning (whether or not M is orientable!). To do so, use a

partition of unity to express σ as a sum of densities supported in local coordinate

systems. Thus we can restrict to the case

σ = f (x1 , . . . , xn ) |dx1 § · · · § dxn | .

Expressed in this way, the density can be integrated as

f (x1 , . . . , xn ) dx1 . . . dxn .

This integral is well-de¬ned because the jacobian of a coordinate change is the

absolute value of the determinant of the transformation.

1

If » is a compactly supported p -density on M , then

1/p

p

|»|

12.3 Generalized Sections 79

is well-de¬ned. Thus there is an intrinsic Lp norm on the space of compactly

1

supported p -densities on a manifold M . Also, if »1 , »2 are two compactly supported

1

2 -densities, then we can de¬ne a hermitian inner product

»1 »2 .

M

Completion with respect to the norm given by the inner product produces an in-

trinsic Hilbert space L2 (M ). The group of di¬eomorphisms of M acts on L2 (M )

by unitary transformations.

Trivializing | §top | M amounts to choosing a positive (smooth) density σ0 , or

equivalently, to choosing a nowhere vanishing (smooth) measure on M . Given such

a trivialization, which also trivializes |§top |± M for each ±, we can identify functions

with densities and hence obtain Lp spaces of functions.

Exercise 37

Show that the Lp spaces obtained in this way are the usual Lp spaces of

functions with respect to the given measure.

12.3 Generalized Sections

Let E be a vector bundle over M . De¬ne E to be

E := E — — | §top | M .

There is a natural pairing σ, „ between compactly supported smooth sections

of E, σ ∈ “c (E), and smooth sections of E , „ ∈ “(E ), given by the pairing

between E and E — and by integration of the remaining density.

Sections σ ∈ “c (E) de¬ne by σ, · continuous linear functionals with respect to

the C ∞ -topology on “(E ). (Recall that a sequence converges in the C ∞ -topology

if and only if it converges uniformly with all its derivatives on compact subsets of

domains of coordinate charts and bundle trivializations.)

Denoting the space dual to “(E ) by D (M, E), we conclude that there is a

natural embedding

“c (E) ⊆ D (M, E) .

For this reason, arbitrary elements of D (M, E) are called compactly supported

generalized sections of E. Occasionally, they are called compactly supported dis-

tributional sections or (less accurately) compactly supported distribution-valued

sections. Similarly, generalized sections of E which are not necessarily compactly

supported are de¬ned as the dual space to compactly supported smooth sections of

E , “c (E ). In this case, we have

“(E) ⊆ “c (E ) .

If E = | §top | M , then E is the trivial line bundle over M , and we recover the

usual compactly supported distributions on M :

D (M ) := D (M, | §top | M ) ⊇ “c (| §top | M ) .

Similarly, if E is the trivial line bundle µ, then E = | §top | M , and so

∞

D (M, µ) ⊇ Cc (M ) .

80 12 DENSITIES

Any di¬erential operator D on “c (E) has a unique formal adjoint D— , that

is, a di¬erential operator on “(E ) such that

Dσ, „ = σ, D— „

for all σ ∈ “c (E) and „ ∈ “(E ). These di¬erential operators are continuous with

respect to the C ∞ -topology, and we can thus extend them to operators on D (M, E)

by the same formula

Dσ, „ = σ, D— „ ,

where σ now lies in D (M, E).

‚

Example. It is easy to check that on Rn , the operator ‚xi has formal adjoint

‚δ

‚

’ ‚xi . To see how this extends to generalized sections, note that, for instance, ‚x0i

is de¬ned by

‚δ0 ‚f ‚f

, f = δ0 , ’ =’ (0) .

‚xi ‚xi ‚xi

™¦

We have shown that we can regard any Lie group G as sitting inside D (G) (see

Section 11.5). Similarly, on any manifold M , we can view a tangent vector as a

generalized density, i.e. a generalized section of | §top | M . Let X ∈ Tm M be

any tangent vector. Then, for • ∈ C ∞ (M ), the map

X·

C ∞ (M ) R

E

’’ X ·•

•

is continuous with respect to the C ∞ -topology, and thus X de¬nes an element of

D (M ). In particular, for M = G, we see that both G and g sit in D (G).

Alternatively, let X ∈ Tm M and let X ∈ “(T M ) be a vector ¬eld on M whose

value at m is X. Then X acts on densities by the Lie derivative. Its formal adjoint

can be shown to be the negative of the usual action of X on functions, in the

following manner.

Exercise 38

For a density ±, use Stokes™ theorem to verify

(LX ±) —¦ • LX (±•) ’ ±(X —¦ •)

=

’ ±(X —¦ •) .

=

Let δm be the functional of evaluation at m. Then

’LX δm , • = δm , X•

= X(•)(m)

= X• ,

and thus we again see X as a generalized density. It is known as a dipole, since

•(mµ ) ’ •(m)

X• = lim

µ

µ’0

1 1

δmµ , • ’

= lim δm , • ,

µ’0 µ µ

12.4 Poincar´-Birkho¬-Witt Revisited

e 81

where µ ’ mµ is a path through m with tangent vector X at µ = 0.

If we apply di¬erential operators to δm , then the additional distributions ob-

tained are all supported at m; that is, the action of each of the distributions on a

test function • depends on • only in a neighborhood of m and thus can be obtained

by a ¬nite initial segment of the Taylor series of • at m.

Example. For the case of a Lie group G, inside the distribution group algebra,

D (G), we have all of the following spaces:

C (G) “ the measure group algebra

as the set of measures,

C(G) “ the group algebra

as the set of continuous functions,

G “ the group itself

as evaluation functionals,

“ the Lie algebra

g

as vector ¬elds applied to δe , and

U(g) “ the universal enveloping algebra

as arbitrary di¬erential operators applied to δe .

We will next see how U(g) sits in D (G); notice already that g is not closed under

the convolution multiplication in D (G). ™¦

12.4 Poincar´-Birkho¬-Witt Revisited

e

If two distributions on a Lie group G are supported at the identity e, so is their

convolution, and so the distributions supported at e (all derivatives of δe ) form a

subalgebra of D (G). For each such distribution σ, the convolution operation σ — ·

is a di¬erential operator on C ∞ (G). These operators commute with all translation

operators ·—δg , hence the distributions supported at the identity realize the universal

enveloping algebra U(g) as a subalgebra of D (G).

Remark. There is a general theorem that any distribution supported at a

point comes from applying a di¬erential operator to the evaluation function at that

™¦

point [148, p.100].

The following remarks are due to Berezin and can be found in [13]. Consider the

exponential map exp : g ’ G on a Lie group G. In general, distributions cannot be

pulled back by this map, since it can have singularities. If we are only interested in

distributions supported at e, though, then we can use the fact that the exponential

map is a di¬eomorphism near e to pull back such distributions.

—

'exp

generalized densities on g generalized densities on G

U(g) =

supported at 0 supported at e

The Fourier transform F maps ±-densities on a vector space g to (1’±)-densities

on its dual g— . The Fourier transform of a generalized 1-density supported at 0 ∈ g

82 12 DENSITIES

will be a polynomial on g— :

FE

generalized densities on g

Pol(g— ) S(g)

supported at 0

’’

δ0 1

‚δ0

’’ vi

‚xi

where (v1 , . . . , vn ) is a basis of g, and xi is the coordinate function on g correspond-

ing to vi .

Theorem 12.2 (Berezin [13]) The composite map

F —¦ exp—

U(g) S(g)

E

is the symmetrization map (see Section 1.3).

Exercise 39

Prove the theorem. To do so, ¬rst prove the theorem for powers of elements of

g and then extend to all of U(g) by “polarization”. See [13] and Chapter 2.

To review our construction, if G is a Lie group, then its di¬erential structure

provides an algebra C ∞ (G) with pointwise multiplication. On the other hand,

diagonal insertion gives rise to a coproduct on the measure group algebra

∆ : D (G) ’’ D (G)“ — ”D (G) D (G — G) .

On U(g) ⊆ D (G), this restricts to a map where the “tensor” is the usual algebraic

tensor product

∆ : U(g) ’’ U(g) — U(g) .

For X ∈ g ⊆ U(g), the map ∆ is de¬ned by

∆(X) = X — 1 + 1 — X ,

and this condition uniquely determines the algebra homomorphism ∆. This co-

product is co-commutative, which means that P —¦ ∆ = ∆, where

P : U(g) — U(g) ’’ U(g) — U(g)

is the permutation linear map de¬ned on elementary tensors by P (u — v) = v — u.

Using our isomorphisms of vector spaces S(g) U (gµ ) (Section 2.1), we obtain

deformed coproducts

∆µ : S(g) ’’ S(g) — S(g)

satisfying, for X ∈ g ‚ S(g),

∆µ (X) = X — 1 + 1 — X .

In general, the map ∆µ will be an algebra homomorphism with respect to the

algebra structure of U(gµ ), but not with respect to the algebra structure of S(g).

Whenever g is not abelian, these two algebra structures are di¬erent.

12.4 Poincar´-Birkho¬-Witt Revisited

e 83

Letting µ approach 0, we ask what ∆0 should be. It turns out that if we identify

S(g) with Pol(g— ), then ∆0 is the coproduct coming from the addition operation

on g— : ∆0 (Σ monomials) = Σ∆0 (monomials). For instance,

∆0 (µ4 µ2 + µ3 ) (∆0 (µ1 ))4 (∆0 (µ2 )) + ∆0 (µ3 )

=

1

(µ1 — 1 + 1 — µ1 )4 (µ2 — 1 + 1 — µ2 )

=

+µ3 — 1 + 1 — µ3

So the product and coproduct of U(g) are deformations of structures on g— ; thus

U(g) can be interpreted as the algebra of “functions on” a quantization of g— .

In summary, U(g) is a non-commutative, co-commutative Hopf algebra, while

S(g) is a Hopf algebra which is both commutative and co-commutative. Deforma-

tions Uq (sl(2)) of the Hopf algebra U(sl(2)) were among the earliest known (algebras

of “functions on”) quantum groups (see [25, 88]).

Part VI

Groupoids

13 Groupoids

A groupoid can be thought of as a generalized group in which only certain multi-

plications are possible.

13.1 De¬nitions and Notation

A groupoid over a set X is a set G together with the following structure maps:

1. A pair of maps

G

± ““ β

X

The map ± is called the target while β is called the source. 3 An element

g ∈ G is thought of as an arrow from x = β(g) to y = ±(g) in X:

g

r© r

y = ±(g) x = β(g)

2. A product m : G(2) ’ G, de¬ned on the set of composable pairs:

G(2) := {(g, h) ∈ G — G | β(g) = ±(h)} .

We will usually write gh for m(g, h). If h is an arrow from x = β(h) to

y = ±(h) = β(g) and g is an arrow from y to z = ±(g), then gh is the

composite arrow from x to z.

gh

©

r% r% r

g h

±(g) = ±(gh) β(g) = ±(h) β(h) = β(gh)

The multiplication m must have the properties4

• ±(gh) = ±(g), β(gh) = β(h), and

• associativity: (gh)k = g(hk).

3. An embedding µ : X ’ G, called the identity section, such that µ(±(g))g =

g = gµ(β(g)). (In particular, ± —¦ µ = β —¦ µ is the identity map on X.)

3 Some authors prefer the opposite convention for ± and β.

4 Whenever we write a product, we are assuming that it is de¬ned.

85

86 13 GROUPOIDS

4. An inversion map ι : G ’ G, also denoted by ι(g) = g ’1 , such that for all

g ∈ G,

ι(g)g = µ(β(g))

gι(g) = µ(±(g)) .

g

r© r

β(g ’1 ) = ±(g) β(g) = ±(g ’1 )

g ’1

By an abuse of notation, we shall simply write G to denote the groupoid above.

A groupoid G gives rise to a hierarchy of sets:

G(0) := µ(X) X

G(1) := G

G(2) {(g, h) ∈ G — G | β(g) = ±(h)}

:=

G(3) {(g, h, k) ∈ G — G — G | β(g) = ±(h), β(h) = ±(k)}

:=

.

.

.

The following picture can be useful in visualizing groupoids.

esgh e¡

e e e e¡ e¡ ¡ e¡ ¡ ¡ ¡

e e e ¡e ¡e ¡e ¡e ¡e ¡ ¡ ¡

β-¬berse ±-¬bers ¡

e e¡e¡e¡e¡e¡e¡ ¡

s

ge¡

e e e¡ e¡ e¡ e¡ e¡ ¡ ¡

e e ¡e ¡e ¡e ¡e ¡e ¡e ¡ ¡

e e¡e¡e¡e¡e¡e¡e¡ ¡

es h 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¡ s

e¡ e¡ e¡ e¡ e¡ e¡ e¡

¡eβ(g) ¡e ±(h) 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

¡ ¡e¡e¡e¡e¡e¡e¡e e

e¡ g ’1 s

¡ ¡ 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 e

¡ ¡ ¡ ¡e ¡e ¡e ¡e ¡e e e e

There are various equivalent de¬nitions for groupoids and various ways of think-

ing of them. For instance, a groupoid G can be viewed as a special category whose

objects are the elements of the base set X and whose morphisms are all invertible,

or as a generalized equivalence relation in which elements of X can be “equivalent

13.1 De¬nitions and Notation 87

in several ways” (see Section 13.2). We refer to Brown [19, 20], as well as [171], for

extensive general discussion of groupoids.

Examples.

1. A group is a groupoid over a set X with only one element.

2. The trivial groupoid over the set X is de¬ned by G = X, and ± = β =

identity.

3. Let G = X — X, with the groupoid structure de¬ned by

X —X

π1 ““ π2

±(x, y) := π1 (x, y) = x , β(x, y) := π2 (x, y) = y ,

(x, y)(y, z) = (x, z) ,

µ(x) = (x, x) ,

(x, y)’1 = (y, x) .

This is often called the pair groupoid, or the coarse groupoid, or the

Brandt groupoid after work of Brandt [17], who is generally credited with

introducing the groupoid concept.

' π2

X

µ(X)

π1

c

X

™¦

Remarks.

1. Given a groupoid G, choose some φ ∈ G. The groupoid multiplication on G

extends to a multiplication on the set G ∪ {φ} by

gφ = φg = φ

if (g, h) ∈ (G — G) \ G(2) .

gh = φ ,

The new element φ acts as a “receptacle” for any previously unde¬ned prod-

uct. This endows G ∪ {φ} with a semigroup structure. A groupoid thus

becomes a special kind of semigroup as well.

2. There is a natural way to form the product of groupoids:

88 13 GROUPOIDS

Exercise 40

If Gi is a groupoid over Xi for i = 1, 2, show that there is a naturally de¬ned

direct product groupoid G1 — G2 over X1 — X2 .

3. A disjoint union of groupoids is a groupoid.

™¦

13.2 Subgroupoids and Orbits

A subset H of a groupoid G over X is called a subgroupoid if it is closed under

multiplication (when de¬ned) and inversion. Note that

h ∈ H ’ h’1 ∈ H ’ both µ(±(h)) ∈ H and µ(β(h)) ∈ H .

Therefore, the subgroupoid H is a groupoid over ±(H) = β(H), which may or may

not be all of X. When ±(H) = β(H) = X, H is called a wide subgroupoid.

Examples.

1. If G = X is the trivial groupoid, then any subset of G is a subgroupoid, and

the only wide subgroupoid is G itself.

2. If X is a one point set, so that G is a group, then the nonempty subgroupoids

are the subgroups of G, but the empty set is also a subgroupoid of G.

3. If G = X — X is the pair groupoid, then a subgroupoid H is a relation on X

which is symmetric and transitive. A wide subgroupoid H is an equivalence

relation. In general, H is an equivalence relation on the set ±(H) = β(H) ⊆

X.

™¦

Given two groupoids G1 and G2 over sets X1 and X2 respectively, a morphism

of groupoids is a pair of maps G1 ’ G2 and X1 ’ X2 which commute with all

the structural functions of G1 and G2 . We depict a morphism by the following

diagram.

G1 E G2

±1 β1 ±2 β2

cc cc

X1 E X2

If we consider a groupoid as a special type of category, then a morphism between

groupoids is simply a covariant functor between the categories.

For any groupoid G over a set X, there is a morphism

(±, β)

X —X

G E

± β π1 π2

cc cc

X = X

13.3 Examples of Groupoids 89

from G to the pair groupoid over X. Its image is a wide subgroupoid of X — X,

and hence de¬nes an equivalence relation on X. The equivalence classes are called

the orbits of G in X. In category language, the orbits are the isomorphism classes

of the objects of the category. We can also think of a groupoid as an equivalence

relation where two elements might be equivalent in di¬erent ways, parametrized by

the kernel of (±, β). The groupoid further indicates the structure of the set of all

ways in which two elements are equivalent.

Inside the groupoid X — X there is a diagonal subgroupoid ∆ = {(x, x) | x ∈

X}. We call (±, β)’1 (∆) the isotropy subgroupoid of G.

(±, β)’1 (∆) = {g ∈ G | ±(g) = β(g)} = Gx ,

x∈X

where Gx := {g | ±(g) = β(g) = x} is the isotropy subgroup of x.

If x, y ∈ X are in the same orbit, then any element g of

Gx,y := (±, β)’1 (x, y) = {g ∈ G | ±(g) = x and β(g) = y}

induces an isomorphism h ’ g ’1 hg from Gx to Gy . On the other hand, the groups

Gx and Gy have natural commuting, free transitive actions on Gx,y , by left and

right multiplication, respectively. Consequently, Gx,y is isomorphic (as a set) to Gx

(and to Gy ), but not in a natural way.

A groupoid is called transitive if it has just one orbit. The transitive groupoids

are the building blocks of groupoids, in the following sense. There is a natural

decomposition of the base space of a general groupoid into orbits. Over each orbit

there is a transitive groupoid, and the disjoint union of these transitive groupoids

is the original groupoid.

Historical Remark. Brandt [17] discovered groupoids while studying quadratic

forms over the integers. Groupoids also appeared in Galois theory in the description

of relations between sub¬elds of a ¬eld K via morphisms of K [108]. The isotropy

groups of the constructed groupoid turn out to be the Galois groups. Groupoids

occur also as generalizations of equivalence relations in the work of Grothendieck

on moduli spaces [75] and in the work of Mackey on ergodic theory [113]. For recent

™¦

applications in these two areas, see Keel and Mori [94] and Connes [32].

13.3 Examples of Groupoids

1. Let X be a topological space and let G = Π(X) be the collection of homotopy

classes of paths in X with all possible ¬xed endpoints. Speci¬cally, if γ :

[0, 1] ’ X is a path from x = γ(0) to y = γ(1), let [γ] denote the homotopy

class of γ relative to the points x, y. We can de¬ne a groupoid

Π(X) = {(x, [γ], y) | x, y ∈ X, γ is a path from x to y} ,

where multiplication is concatenation of paths. (According to our convention,

if γ is a path from x to y, the target is ±(x, [γ], y) = x and the source is

β(x, [γ], y) = y.) The groupoid Π(X) is called the fundamental groupoid

of X. The orbits of Π(X) are just the path components of X. See Brown™s

text on algebraic topology [20] for more on fundamental groupoids.

90 13 GROUPOIDS

There are several advantages of the fundamental groupoid over the fundamen-

tal group. First notice that the fundamental group sits within the fundamen-

tal groupoid as the isotropy subgroup over a single point. The fundamental

groupoid does not require a choice of base point and is better suited to study

spaces that are not path connected. Additionally, many of the algebraic prop-

erties of the fundamental group generalize to the fundamental groupoid, as

illustrated in the following exercise.

Exercise 41

Show that the Seifert-Van Kampen theorem on the fundamental group of a

union U ∪ V can be generalized to groupoids [20], and that the connectedness

condition on U © V is then no longer necessary.

2. Let “ be a group acting on a space X. In the product groupoid “—(X —X)

X — “ — X over {point} — X X, the wide subgroupoid

G“ = {(x, γ, y) | x = γ · y}

is called the transformation groupoid or action groupoid of the “-action.

The orbits and isotropy subgroups of the transformation groupoid are pre-

cisely those of the “-action.

(±,β)

A groupoid G over X is called principal if the morphism G ’’ X — X

is injective. In this case, G is isomorphic to the image (±, β)(G), which is

an equivalence relation on X. The term “principal” comes from the analogy

with bundles over topological spaces.

If “ acts freely on X, then the transformation groupoid G“ is principal, and

(±, β)(G“ ) is the orbit equivalence relation on X. In passing to the transfor-

mation groupoid, we have lost information on the group structure of “, as we

no longer see how “ acts on the orbits: di¬erent free group actions could have

the same orbits.

3. Let “ be a group. There is an interesting ternary operation

t

(x, y, z) ’’ xy ’1 z .

It is invariant under left and right translations (check this as an exercise), and

it de¬nes 4-tuples (x, y, z, xy ’1 z) in “ which play the role of parallelograms.

The operation t encodes the a¬ne structure of the group in the sense that, if

we know the identity element e, we recover the group operations by setting

x = z = e to get the inversion and then z = e to get the multiplication.

However, the identity element of “ cannot be recovered from t.

Denote

= set of subgroups of “

S(“)

= set of subsets of “ closed under t .

B(“)

Proposition 13.1 B(“) is the set of cosets of elements of S(“).

The sets of right and of left cosets of subgroups of “ coincide because gH =

(gHg ’1 )g, for any g ∈ G and any subgroup H ¤ G.

13.3 Examples of Groupoids 91

Exercise 42

Prove the proposition above.

We call B(“) the Baer groupoid of “, since much of its structure was

formulated by Baer [10]. We will next see that the Baer groupoid is a groupoid

over S(“).

For D ∈ B(“), let ±(D) = g ’1 D and β(D) = Dg ’1 for some g ∈ D. From

basic group theory, we know that ± and β are maps into S(“) and are inde-

pendent of the choice of g. Furthermore, we see that β(D) = g±(D)g ’1 is

conjugate to ±(D).

B(“)

± ““ β

S(“)

Exercise 43

’1 ’1

Show that if β(D1 ) = ±(D2 ), i.e. D1 g1 = g2 D2 for any g1 ∈ D1 , g2 ∈ D2 ,

then the product in this groupoid can be de¬ned by

D1 D2 := g2 D1 = g1 D2 = {gh | g ∈ D1 h ∈ D2 } .

Observe that the orbits of B(“) are the conjugacy classes of subgroups of

“. In particular, over a single conjugacy class of subgroups is a transitive

groupoid, and thus we see that the Baer groupoid is a re¬nement of the

conjugacy relation on subgroups.

The isotropy subgroup of a subgroup H of “ consists of all left cosets of H

which are also right cosets of H. Any left coset gH is a right coset (gHg ’1 )g

of gHg ’1 . Thus gH is also a right coset of H exactly when gHg ’1 = H, or,

equivalently, when β(gH) = ±(gH). Thus the isotropy subgroup of H can be

identi¬ed with N (H)/H, where N (H) is the normalizer of H.

4. Let “ be a compact connected semisimple Lie group. An interesting conjugacy

class of subgroups of “ is

T = {maximal tori of “} ,

where a maximal torus of “ is a subgroup

Tk (S 1 )k = S 1 • · · · • S 1

of “ which is maximal in the sense that there does not exist an ≥ k such

that Tk < T ¤ “ (here, S 1 R/Z is the circle group). A theorem from

Lie group theory (see, for instance, [18]) states that any two maximal tori of

a connected Lie group are conjugate, so T is an orbit of B(“). We call the

transitive subgroupoid B(“)|T = W(“) the Weyl groupoid of “.

Remarks.

• For any maximal torus T ∈ T , the quotient N (T)/T is the classical Weyl

group. The relation between the Weyl groupoid and the Weyl group is

analogous to the relation between the fundamental groupoid and the

fundamental group.

92 13 GROUPOIDS

• There should be relevant applications of Weyl groupoids in the repre-

sentation theory of a group “ which is acted on by a second group, or in

studying the representations of groups that are not connected.

™¦

13.4 Groupoids with Structure

Ehresmann [53] was the ¬rst to endow groupoids with additional structure, as he

applied groupoids to his study of foliations. Rather than attempting to describe

a general theory of “structured groupoids,” we will simply mention some useful

special cases.

1. Topological groupoids: For a topological groupoid, G and X are required

to be topological spaces and all the structure maps must be continuous.

Examples.

• In the case of a group, this is the same as the concept of topological

group.

• The pair groupoid of a topological space has a natural topological struc-

ture derived from the product topology on X — X.

™¦

For analyzing topological groupoids, it is useful to impose certain further

axioms on G and X. For a more complete discussion, see [143]. Here is a

sampling of commonly used axioms:

(a) G(0) X is locally compact and Hausdor¬.

(b) The ±- and β-¬bers are locally compact and Hausdor¬.

(c) There is a countable family of compact Hausdor¬ subsets of G whose

interiors form a basis for the topology.

(d) G admits a Haar system, that is, admits a family of measures on the

±-¬bers which is invariant under left translations. For any g ∈ G, left

translation by g is a map between ± ¬bers

±’1 (β(g)) ±’1 (±(g))

’’

g

’’

h gh .

±’1 (±(g)) ±’1 (β(g))

s g (h)

gs

sh

s

β(g)

13.5 The Holonomy Groupoid of a Foliation 93

Example. For the pair groupoid, each ¬ber can be identi¬ed with the

base space X. A family of measures is invariant under translation if and

only if the measure is the same on each ¬ber. Hence, a Haar system on

™¦

a pair groupoid corresponds to a measure on X.

2. Measurable groupoids: These groupoids, also called Borel groupoids,

come equipped with a σ-algebra of sets and a distinguished subalgebra (called

the null sets); see [113, 120]. On each ±-¬ber, there is a measure class, which

is simply a measure de¬ned up to multiplication by an invertible measurable

function.

3. Lie groupoids or di¬erentiable groupoids: The groupoid G and the

base space X are manifolds and all the structure maps are smooth. It is not

assumed that G is Hausdor¬, but only that G(0) X is a Hausdor¬ manifold

and closed in G.5 Thus we can require that the identity section be smooth.

Recall that multiplication is de¬ned as a map on G(2) ⊆ G. To require that

multiplication be smooth, ¬rst G(2) needs to be a smooth manifold. It is

convenient to make the stronger assumption that the map ± (or β) be a

submersion.

Exercise 44

Show that the following conditions are equivalent:

(a) ± is a submersion,

(b) β is a submersion,

(c) the map (±, β) to the pair groupoid is transverse to the diagonal.

4. Bundles of groups: A groupoid for which ± = β is called a bundle of

groups. This is not necessarily a trivial bundle, or even a locally trivial bundle

in the topological case, as the ¬bers need not be isomorphic as groups or as

topological spaces. The orbits are the individual points of the base space, and

the isotropy subgroupoids are the ¬ber groups of the bundle.

13.5 The Holonomy Groupoid of a Foliation

Let X be a (Hausdor¬) manifold. Let F ⊆ T X be an integrable subbundle, and

F the corresponding foliation (F is the decomposition of X into maximal integral

manifolds called leaves). The notion of holonomy can be described as follows.

An F -path is a path in X whose tangent vectors lie within F . Suppose that

γ : [0, 1] ’ O is an F -path along a leaf O. Let Nγ(0) and Nγ(1) be cross-sections

for the spaces of leaves near γ(0) and γ(1), respectively, i.e. they are two small

transversal manifolds to the foliation at the end points of γ. There is an F -path

near γ from each point near γ(0) in Nγ(0) to a uniquely determined point in Nγ(1) .

This de¬nes a local di¬eomorphism between the two leaf spaces. The holonomy

of γ is de¬ned to be the germ, or direct limit, of such di¬eomorphisms, between the

local leaf spaces Nγ(0) and Nγ(1) .

The notion of holonomy allows us to de¬ne an equivalence relation on the set

of F -paths from x to y in X. Let [γ]H denote the equivalence class of γ under the

relation that two paths are equivalent if they have the same holonomy.

5 Throughout these notes, a manifold is assumed to be Hausdor¬, unless it is a groupoid.

94 13 GROUPOIDS

The holonomy groupoid [32], also called the graph of the foliation [175], is

H(F) = {(x, [γ]H , y) | x, y ∈ X, γ is an F -path from x to y} .

Given a foliation F, there are two other related groupoids obtained by changing

the equivalence relation on paths:

1. The F-pair groupoid “ This groupoid is the equivalence relation for which

the equivalence classes are the leaves of F, i.e. we consider any two F -paths

between x, y ∈ O to be equivalent.

2. The F-fundamental groupoid “ For this groupoid, two F -paths between

x, y are equivalent if and only if they are F -homotopic, that is, homotopic

within the set of all F -paths. Let [γ]F denote the equivalence class of γ under

F -homotopy. The set of this groupoid is

Π(F) = {(x, [γ]F , y) | x, y ∈ X, γ is an F -path from x to y} .

If two paths γ1 , γ2 are F -homotopic with ¬xed endpoints, then they give the

same holonomy, so the holonomy groupoid is intermediate between the F-pair

groupoid and the F-fundamental groupoid:

[γ1 ]F = [γ2 ]F =’ [γ1 ]H = [γ2 ]H .

The pair groupoid may not be a manifold. With suitably de¬ned di¬erentiable

structures, though, we have:

Theorem 13.2 H(F) and Π(F) are (not necessarily Hausdor¬ ) Lie groupoids.

For a nice proof of this theorem, and a comparison of the two groupoids,

see [137]. Further information can be found in [102].

Exercise 45

Compare the F -pair groupoid, the holonomy groupoid of F , and the F -

fundamental groupoid for the M¨bius band and the Reeb foliation, as described

o

below.

1. The M¨bius band. Take the quotient of the unit square [0, 1] — [0, 1] by

o

the relation (1, x) ∼ (0, 1 ’ x). De¬ne the leaves of F to be images of the

horizontal strips {(x, y) | y = constant}.

2. The Reeb foliation [142]. Consider the family of curves x = c + sec y on the

strip ’π/2 < y < π/2 in the xy-plane. If we revolve about the axis y = 0,

then this de¬nes a foliation of the solid cylinder by planes. Noting that the

foliation is invariant under translation, we see that this de¬nes a foliation

of the open solid torus D2 — S 1 by planes. The foliation is smooth because

its restriction to the xy-plane is de¬ned by the 1-form cos2 y dx + sin y dy,

which is smooth even when y = ± π . We close the solid torus by adding one

2

exceptional leaf “ the T2 boundary.

Let ± be a vanishing cycle on T2 , that is, [±] ∈ π1 (T2 ) generates the kernel of

the natural map π1 (T2 ) ’ π1 (D2 — S 1 ). Although ± is not null-homotopic on

the exceptional leaf, any perturbation of ± to a nearby leaf results in a curve

13.5 The Holonomy Groupoid of a Foliation 95

that is F -homotopically trivial. On the other hand, the transverse curve (the

cycle given by (c, y) ∈ D2 — S 1 for some ¬xed c ∈ ‚D2 ) cannot be pushed

onto any of the nearby leaves.

A basic exercise in topology shows us that we can glue two solid tori together

so that the resulting manifold is the 3-sphere S 3 . For this gluing, the trans-

verse cycle of one torus is the vanishing cycle of the other. (If we instead

glued the two vanishing cycles and the two transverse cycles together, we

would obtain S 2 — S 1 .)

It is interesting to compute the holonomy on each side of the gluing T2 . Each

of the two basic cycles in T2 has trivial holonomy on one of its sides (holonomy

given by the germ of the identity di¬eomorphism), and non-trivial holonomy

on the other side (given by the germ of an expanding di¬eomorphism).

Nγ(0) Nγ(1)

γ inside T2

This provides an example of one-sided holonomy, a phenomenon that cannot

happen for real analytic maps. The leaf space of this foliation is not Hausdor¬;

in fact, any function constant on the leaves must be constant on all of S 3 ,

since all leaves come arbitrarily close to the exceptional leaf T2 . This foliation

and its holonomy provided the inspiration for the following theorems.

Theorem 13.3 (Hae¬‚iger [79]) S 3 has no real analytic foliation of codimension-

1.

Theorem 13.4 (Novikov [132]) Every codimension-1 foliation of S 3 has a

compact leaf that is a torus.

14 Groupoid Algebras

Groupoid algebras include matrix algebras, algebras of functions, and group alge-

bras. We refer the reader to [101, 135, 143] for extensive discussion of groupoid

algebras as sources of noncommutative algebras in physics and mathematics.

14.1 First Examples

Let X be a locally compact space with a Borel measure µ. Let Cc (X — X) be the

space of compactly supported continuous functions on X — X. We de¬ne multi-

plication of two functions f, g ∈ Cc (X — X) by the following integral, representing

“continuous matrix multiplication”

(f — g)(x, y) = f (x, z) g(z, y) dµ(z) .

X

Exercise 46

Check that this multiplication is associative and that the *-operation

f (x, y) ’’ f — (x, y) := f (y, x)

is compatible with multiplication:

f — — g — = (g — f )— .

To de¬ne our multiplication without the choice of a measure on X, we replace

Cc (X — X) by the space whose elements are objects of the form f (x, y)dy. Such an

object assigns to each point of X a measure on X.

T

y

f (x, y)dy

E

x

These objects have a “matrix” multiplication as written above. Furthermore,

they operate on functions on X by

u(·) ’’ f (·, y) u(y) dy .

X

However, the *-operation can no longer be described in this language.

When X is a manifold, there is a related algebra on which the *-operation can

be de¬ned intrinsically. Let A be the space of compactly supported 1 -densities on

2

X — X. A typical element of A is of the form

f (x, y) |dx| |dy| .

We multiply two elements

|dx| |dz| , |dz| |dy| ,

f (x, z) g(z, y)

97

98 14 GROUPOID ALGEBRAS

by integrating over z:

|dx| |dy| .

f (x, z)g(z, y)|dz|

z∈X

1

This algebra no longer acts on functions, but rather on 2 -densities on X. The

*-operation is now de¬ned by

f — (x, y) |dx| |dy| = f (y, x) |dy| |dx| .

Exercise 47

1

Give a precise de¬nition of a generalized -density which serves as an identity

2

element for this algebra.

Implicit in these formulations is the multiplication law for the pair groupoid

(x, z)(z, y) = (x, y) .

From this point of view, our multiplication operation becomes convolution in the

groupoid algebra, as we shall see in the next section.

14.2 Groupoid Algebras via Haar Systems

Let G be a locally compact groupoid over X, and let • and ψ be compactly sup-

ported continuous functions on G. A product function • — ψ might be obtained in

the following way: for its value at k ∈ G, we evaluate • and ψ on all possible pairs

(g, h) ∈ G — G satisfying gh = k, and then integrate the products of the values.

That is, we write the integral

(• — ψ)(k) = •(g) ψ(h) (...) ,

{(g,h)|gh=k}

where we need a measure (...) on the set {(g, h) ∈ G — G | gh = k}. If we rewrite

gh = k as h = g ’1 k, we see that the domain of integration is all g ∈ G such that

β(g ’1 ) = ±(g) = ±(k). In other words, the product above equals