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ON ENDOMORPHISM ALGEBRAS OF SEPARABLE MONOIDAL

FUNCTORS

BRIAN DAY AND CRAIG PASTRO

Abstract. _{We show that the (co)endomorphism algebra of a sufficiently separable}

“fibre” functor into Vectk, for k a field of characteristic 0, has the structure of what

we call a “unital” von Neumann core in Vectk. For Vectk, this particular notion of

algebra is weaker than that of a Hopf algebra, although the corresponding concept in

Set_{is again that of a group.}

1. Introduction

Let C _{= (}C_{,⊗, I, c) be a braided (or even symmetric) monoidal category. Recall that} an algebra in C _{is an object A ∈}C _{equipped with a multiplication µ: A⊗A → A and} a unit η : I → A satisfying µ3 = µ(1⊗µ) = µ(µ⊗1) : A⊗3 → A (associativity) and

µ(η⊗1) = 1 =µ(1⊗η) :A→A (unit conditions). Dually, a coalgebra inC _{is an object}

C ∈C _{equipped with a comultiplicationδ :C→C⊗C and a counitǫ:C →I satisfying}

δ3 = (1⊗δ)δ = (δ⊗1)δ :C →C⊗3 (coassociativity) and (ǫ⊗1)δ = 1 = (1⊗ǫ)δ:C →C

(counit conditions).

A very weak bialgebra inC _{is an object A∈}C _{with both the structure of an algebra} and a coalgebra in C _{related by the axiom}

δµ= (µ⊗µ)(1⊗c⊗1)(δ⊗δ) :A⊗A→A⊗A.

For example, when C ₌ Vect_{k, any k-bialgebra or weak k-bialgebra is a very weak}

bialgebra in this sense.

We note briefly that, if A is such a structure, but has no unit or counit, we simply call A a semibialgebra, or core for short. This minimal structure on A is then called a

von Neumann core in C _{if it also is equipped with an endomorphism S : A → A in} C satisfying the axiom

µ3(1⊗S⊗1)δ3 = 1 :A→A.

The first author gratefully acknowledges partial support of an Australian Research Council grant, while the second author is partially supported by GCOE, Kyoto University. Parts of this work were completed while the second author was a Ph.D. student at Macquarie University, Australia, and during that time was gratefully supported by an international Macquarie University Research Scholarship and a Scott Russell Johnson Memorial Scholarship.

The authors would like to thank Ross Street for several helpful comments. Received by the editors 2007-12-21 and, in revised form, 2009-04-17. Transmitted by Jean-Louis Loday. Published on 2009-04-20.

2000 Mathematics Subject Classification: 18D99, 16B50.

Key words and phrases: separable fibre functor, Tannaka reconstruction, bialgebra, von Neumann core.

c

A von Neumann regular semigroup is precisely a von Neumann core inSet_{, while the free}

k-vector space on it is a special type of von Neumann core in Vect_{k. However, within}

this article we shall always suppose that A has both a unit and a counit. For example,

when C ₌Vect_{k, a Hopf k-algebra or a weak Hopf k-algebra is a von Neumann core in}

this somewhat stronger sense.

Since groups A inSet _{are characterized by the (stronger) axiom}

1⊗η = (1⊗µ)(1⊗S⊗1)δ3 :A→A⊗A, (†)

a very weak bialgebra A satisfying (†), in the general C_{, will be called a unital von}

Neumann core inC_{. Such a unital von Neumann core A always has a left inverse to the} “fusion” operator [9]

(1⊗µ)(δ⊗1) : A⊗A→A⊗A,

namely

(1⊗µ)(1⊗S⊗1)(δ⊗1) :A⊗A →A⊗A.

Any Hopf algebra in C _{satisfies the stronger axiom (†), but a weak Hopf algebra does} not necessarily do so. In this article we are mainly interested in producing a unital von Neumann core, namely End∨

U, associated to a certain type of split monoidal functor

U into Vect_{k. It seems unlikely that all unital von Neumann cores in} Vect_{k may be}

reproduced as such.

We will tacitly assume throughout the article that the ground category [8] is Vect₌

Vect_{k, for k a field of characteristic 0, so that the categories and functors considered}

here are allk-linear (although any reasonable category [D_,Vect_{] of parameterized vector}

spaces would suffice). We denote by Vect_{f the full subcategory of}Vect_{consisting of the}

finite dimensional vector spaces, and we further suppose thatC _{= (}C_{,⊗, I, c) is a braided} monoidal category with a “fibre” functor

U :C _→Vect_,

with both a monoidal structure (U, r, r0) and a comonoidal structure (U, i, i0), which need

not be inverse to one another. We callU separable1 if ri = 1 and i0r0 = dim(U I)·1; i.e.,

for all A, B ∈C_{, the diagrams}

U(A⊗B) U A⊗U B

U(A⊗B) i

/

/

r

1

'

'

O O O O O O O O O O O

O k U I

k

r0

/

/

i0

dimU I·1

#

#

G G G G G G G G G G G G

1_{Strictly, we should also require the conditions (cf. [1, 10])}

commute.

First we produce an algebra structure (µ, η) on

End∨

U =

Z C

U(C)∗ ⊗U C

using the monoidal and comonoidal structures on U. Secondly, we suppose that C _has a suitable small generating set A _{of objects, and produce a coalgebra structure (δ, ǫ) on} End∨

U when each value U A, A∈A_{, is finite dimensional. Finally, we assume that each}

A∈A _{has a⊗-dualA}∗

which also lies inA_{, and thatU is equipped with an isomorphism}

U(A∗

)∼=U(A)∗

for all A ∈ A_{. This isomorphism should be suitably related to the evaluation and} coevaluation maps ofC _andVect_{f which then allows us to define a natural non-degenerate}

form

U(A∗

)⊗U A→k.

This last assumption is sufficient to provide End∨

U with an automorphism S so that it becomes a unital von Neumann core in the above sense whenever (U, r, r0) is a braided

monoidal functor.

By way of examples, we note that many separable monoidal functors are constructable from separable monoidal categories, i.e., from monoidal categoriesC _{for which the tensor} product map

⊗:C_{(A, B)⊗}C_{(C, D)→}C_{(A⊗C, B⊗D)}

is a naturally split epimorphism (as is the case for some finite cartesian products such as

Vectn_{f). A closely related source of examples is the notion of a weak dimension functor}

onC _{(cf. [6]); this is a comonoidal functor}

(d, i, i0) :C →Setf

for which the comonoidal transformation components

i=iC,D :d(C⊗D)→dC×dD

are injective functions, while the unique map i0 : dI →1 is surjective. Various examples

are described at the conclusion of the paper.

We suppose the reader is familiar to some extent with the standard Tannaka recon-struction problem when restricted to the case ofU strong monoidal (see [7] for example).

2. The very weak bialgebra End

∨

U

If C _{is a (k-linear) monoidal category and}

has a monoidal structure (U, r, r0) and a comonoidal structure (U, i, i0), then End∨U,

when it exists, has an associative and unital k-algebra structure whose multiplication µ

is the composite map

Z C

while the unit η is given by

k

The associativity and unit axioms for (End∨

U, µ, η) now follow directly from the corre-sponding associativity and unit axioms for (U, r, r0) and (U, i, i0). An augmentation ǫ is

inVect_{, where e denotes evaluation in}Vect_.

We also observe that the coend

End∨

U =

Z C

U(C)∗ ⊗U C

actually exists in Vect _if C _{contains a small full subcategory}A _{with the property that} the family

{U f :U A→U C |f ∈C_{(A, C), A∈}A_}

is epimorphic in Vect_{for each objectC ∈}C_{. In fact, we shall use the stronger condition} that the maps

αC : Z A∈A

should be isomorphisms, not just epimorphisms. This stronger condition implies that we can effectively replace RC∈C

byRA∈A

since by the Yoneda lemma

Z C

If we furthermore ask that each value U A be finite dimensional for A in A_{, then}

End∨

U ∼=

Z A∈A

U(A)∗ ⊗U A

is canonically ak-coalgebra with counit the augmentation ǫ, and comultiplication δgiven by

where n denotes the coevaluation morphism in Vect_f.

2.1. Proposition_{. If U is separable then End}∨

U satisfies the k-bialgebra axiom ex-pressed by the commutativity of

End∨

Proof_{. Let} B _{denote the monoidal full subcategory of} C _{generated by} A _{(we will} essentially replace C _{by this small category} B_{). Then, for all C, D in} B_{, we have, by} induction on the tensor lengths of C and D, that U(C⊗D) is finite dimensional since it is a retract of U C⊗U D. Moreover, we have

by the Yoneda lemma, since the natural transformation

α=αB :

Z A∈A

is an isomorphism for all B ∈B_{. Since ri= 1, the triangle}

commutes in Vect_{f, where n denotes the coevaluation maps. The asserted bialgebra}

axiom then holds on End∨

U since it reduces to the following diagram on filling in the definitions of µand δ (where, for the moment, we have dropped the symbol “⊗”):

U C U(C)∗

Notably the bialgebra axiom expressed by the commutativity of

End∨

does not hold in general, while the form of the axiom expressed by

k

which is the endomorphism k-algebra of the functor

U(−)∗

:Cop _→Vect_.

If obA _{is finite, so that}

Z A

U(A)∗ ⊗U A

is finite dimensional, then

Z

C

[U(C)∗

, U(C)∗

]∼= Z

A

[U(A)∗

, U(A)∗

]

is also a k-coalgebra.

3. The unital von Neumann core End

∨

U

We now take C _{= (}C_{,⊗, I, c) to be a braided monoidal category and} A _⊂ C _{to be a} small full subcategory ofC _{for which the monoidal and comonoidal functorU :}C _→Vect

induces

U :A _→Vect_f

on restriction to A_{. We suppose that} A _{is such that}

• the identity I of ⊗ lies in A_{, and each object of A ∈} A _{has a ⊗-dual A}∗

lying in A_.

With respect to U, we supposeA _{has the properties}

• “U-irreducibility”: A_{(A, B)= 0 implies dim6 U A= dimU B for all A, B ∈}A_,

• “U-density”: the canonical map

αC : Z A∈A

C_{(A, C)⊗U A→U C}

is an isomorphism for all C ∈C_,

• “U-trace”: each object of A _{has a U-trace in} C_{(I, I), where by U-trace of A ∈}A we mean an isomorphism d(A) in C_{(I, I) such that the following two diagrams} commute.

I

A⊗A∗

n

A∗ ⊗A

c

/

/

I

e

O

O

d(A)

/

/ k

U I

r0

U I

dimU I·U(d(A))

/

/

k

dimU A

/

/

r0

We require also a natural isomorphism

commutes. This means that U “preserves duals” when restricted to A_. An endomorphism

σ : End∨

U →End∨

U

may be defined by components

U(A)∗

each σA being given by commutativity of

U(A)∗

3.1. Theorem_{. Let} C_, A_{, and U be as above, and suppose that U is braided and}

separable as a monoidal functor. Then there is an automorphism S on End∨

U such that

(End∨

Then, by the U-irreducibility assumption on the category A_{, this family induces an} automorphism S on the coend

End∨ to be the prospective core endomorphism on End∨

U and check that

1⊗η = (1⊗µ)(1⊗S⊗1)δ3.

From the definition of µ and δ, we require commutativity of the exterior of the following diagram (where, again, we have dropped the symbol “⊗”):

The region labelled by (1) commutes on composition with 1⊗n⊗1 since

commutes, so that the exterior of

Finally, the region labelled by (3) commutes on examination of the following diagram

k∗

order for

4. The fusion operator

The unital von Neumann axiom on End∨

U implies that the fusion operator

f = (1⊗µ)(δ⊗1) : End∨

5. Examples of separable monoidal functors in the present context

Unless otherwise indicated, categories, functors, and natural transformations shall be

k-linear, for k a field of characteristic 0.

For these examples we recall that a (small) k-linear promonoidal category (A_{, p, j)} (previously called “premonoidal” in [2]) consists of ak-linear category A _{and twok-linear} functors

p:Aop_⊗Aop_⊗A _→Vect

equipped with associativity and unit constraints satisfying axioms (as described in [2]) analogous to those used to define a monoidal structure onA_{. The notion of a symmetric} promonoidal category (also introduced in [2]) was extended in [4] to that of a braided promonoidal category.

The main point is that (braided) promonoidal structures on A _{correspond to} cocon-tinuous (braided) monoidal structures on the functor category [A_,Vect_{]. This latter}

monoidal structure is often called the convolution product of A _and Vect _{and is given}

explicity by the coend formula

(f∗g)(c) = Z a,b

p(a, b, c)⊗f a⊗gb

inVect_{. The unit of this convolution product is given by j.}

5.1. Example_{. Let (}A_{, p, j) be a small braided promonoidal category with}

A_{(I, I)}∼_{=k where j =}A_(I,−),

and suppose that each hom-space A_{(a, b) is finite dimensional. Let f :}A _→Vect_{f be a}

very weak bialgebra in the convolution [A_,Vect_{] so that we have maps}

µ:f∗f →f and η:j →f

and

δ:f →f ∗f and ǫ:f →j,

satisfying associativity and unital axioms, plus the very weak bialgebra axiom. Suppose also that A _⊂C _whereC _{is a separable braided monoidal category, with}

p(a, b, c)∼=C_{(a⊗b, c) and j(a)}∼₌C_{(I, a)}

naturally, and suppose the induced maps

Z c∈A

p(a, b, c)⊗C_{(c, C)→}C_{(a⊗b, C)}

are isomorphisms (e.g., A _{monoidal). We also suppose that each a ∈} A _{has a dual}

a∗ ∈A_.

Define a functor U :C _→Vect_by

U C = Z a∈A

f a⊗C_{(a, C);}

then, by the Yoneda lemma, U(a∗

)∼=U(a)∗

if f(a∗

)∼=f(a)∗

for a∈A_{. Furthermore, by} the Yoneda lemma,

U I = Z a∈A

so that, by our assumption A_{(I, I)}∼_{=k the maps η and ǫ induce respectively maps}

r0 :k →U I and i0 :U I →k.

Mapsr and i are described in the following diagram.

U C ⊗U D

Z a,b

f a⊗f b⊗C_{(a, C)⊗}C_{(b, D)}

∼ = _/

/

Z a,b

f a⊗f b⊗C_{(a⊗b, C ⊗D)}

Z a,b

f a⊗f b⊗

Z c

p(a, b, c)⊗C_{(c, C⊗D)}

Z c

f c⊗C_{(c, C⊗D)}

U(C⊗D)oo 1

r

i

O

O

C separable

O

O

∼ =

µ

δ

O

O

These then produce a braided monoidal and comonoidal structure on U. Moreover, we have i0r0 = dimU I·1 if and only if ǫIηI = dimf I ·1, and if f is a separable very weak bialgebra, then U is separable sinceri = 1 ifµδ = 1.

Therefore, Theorem 3.1 may be applied when A _{and U satisfy the “U-irreducibility”} and “U-trace” criteria.

5.2. Example_{. Suppose that (}Aop_{, p, j) is a small braided promonoidal category with}

I ∈ A _{such that j} ∼₌ A_{(−, I) and with each x ∈} A _{an “atom” in} C _{(i.e., an object}

x∈C _{for which}C_{(x,−) preserves all colimits) where}C _{is a cocomplete and cocontinuous} braided monoidal category containing A _{and each x ∈} A _{has a dual x}∗

∈ A_{. Suppose} that the inclusion A _⊂C _{is dense over}Vect_{(that is, the canonical evaluation morphism}

Z a

C_{(a, C)·a→C}

is an isomorphism for all C ∈C_{), and}

x⊗y∼= Z z

p(x, y, z)·z (naturally in x, y ∈A₎

so that

C_{(a, x⊗y) =}C_(a, Z z

p(x, y, z)·z)

∼₌Z z_{p(x, y, z)⊗}C_{(a, z) sincea∈}A _{is an atom in} C_,

Let W :A _→Vect_{be a strong braided promonoidal functor on} A_{. This means that} we have structure isomorphisms

W x⊗W y ∼= Z z

C_{(z, x⊗y)⊗W z and}

k ∼=W I

satisfying suitable associativity and unital coherence axioms. Define a functor U : C _→

Vect_by

U C = Z a

C_{(a, C)⊗W a.}

Then, if we suppose that W(x∗

)∼=W(x)∗

for all x∈A_{, we have}

U(x∗

) = Z a

C_{(a, x}∗

)⊗W a

∼

=W(x∗

)

∼

=W(x)∗

∼

= Z a

C_{(a, x)⊗W a}

∗

=U(x)∗

,

so that U(x∗

)∼=U(x)∗

, and

i0 :U I =

Z a

C_{(a, I)⊗W a}

∼

=W I

∼

=k,

so that i0r0 = 1 and r0i0 = 1. Also there are mutually inverse composite maps r and i

given by:

r :U C ⊗U D ∼= Z x,y

C_{(x, C)⊗}C_{(y, D)⊗U x⊗U y}

∼

= Z x,y

C_{(x, C)⊗}C_{(y, D)⊗W x⊗W y}

∼

= Z x,y

C_{(x, C)⊗}C_{(y, D)⊗} Z z

C_{(z, x⊗y)⊗W z}

∼

= Z z

C_{(z, C⊗D)⊗W z}

∼

=U(C⊗D),

which uses the assumptions thatC _{is cocontinuous monoidal and}A _⊂C _{is dense. Thus,}

5.3. Example_{. (See [6] Proposition 3.) Let}C _{be a braided compact monoidal category} and let A _⊂ C _{be a full finite discrete Cauchy generator of} C _{which contains I and is} closed under dualization inC_{. As in the H¨aring-Oldenburg case [6], we suppose that each} hom-spaceC_{(C, D) is finite dimensional with a chosen natural isomorphism}C_(C∗

, D∗

)∼= C_{(C, D)}∗

.

Then we have a separable monoidal functor

U C = M

a,b∈A

C_{(a, C⊗b),}

whose structure maps are given by the composites

U C⊗U D ∼= M a,b,c,d

C_{(c, C ⊗b)⊗}C_{(a, D⊗d)}

c=d

/

/

adjoint

o

o

M

a,b,c

C_{(c, C⊗b)⊗}C_{(a, D⊗c)}

∼

= M

a,b

C_{(a, D⊗(C⊗b))}

∼

= M

a,b

C_{(a,(D⊗C)⊗b)}

∼

= M

a,b

C_{(a,(C⊗D)⊗b)}

= U(C⊗D),

and r0 :k →U I the diagonal, with i0 its adjoint. Moreover

U(C∗

) = M a,b

C_{(a, C}∗ ⊗b)

∼

=M

a,b C_(a∗

, C∗ ⊗b∗

)

∼

=M

a,b

C_{(a, C⊗b)}∗

∼

=U C∗

for all C ∈C_.

5.4. Example_{. Let (}A_{, p, j) be a finite braided promonoidal category over} Set_{f with}

I ∈A _{such that j} ∼₌A_{(I,−) and with a braided promonoidal functor}

d:Aop _→Set_f

for which each structure map

u: Z z

is an injection, and u0 :dI →1 is a surjection. Then we have corresponding maps

Z z

k[p(x, y, z)]⊗k[dz] oooo// // k[dx]⊗k[dy]

and

k[dI] oo oo//// k[1],

where k[s] denotes the free k-vector space on the (finite) set s, in Vect_{f. Define the}

functor U :C _→Vect_{f by}

U f = Z x

f x⊗k[dx]

forf ∈C _{= [k∗}A_,Vect_{f], with the convolution braided monoidal closed structure, where}

k∗A is the freek-linear category on A so that

r:U f ⊗U g = Z x

f x⊗k[dx]

⊗

Z y

gx⊗k[dy]

∼

= Z x,y

f x⊗gy⊗(k[dx]⊗k[dy])

⇄

Z x,y

f x⊗gy⊗

Z z

k[p(x, y, z)]⊗k[dz]

∼

=

Z zZ x,y

f x⊗gy⊗k[p(x, y, z)]

⊗k[dz]

= Z z

(f ⊗g)(z)⊗k[dz]

= Z z

U(f ⊗g)

and

i0 :U I =

Z x

k[A_{(I, x)]⊗k[dx]}

∼

=k[dI]

⇄k[1]∼=k.

Hencei0r0 = dimU I·1 =|dI| ·1. Thus,U becomes a braided separable monoidal functor.

5.5. Example_{. Let}A _{be a finite (discrete) set and give the cartesian product} A _×A the Set_{f-promonoidal structure corresponding to bimodule composition (i.e., to matrix}

multiplication). If

is a braided promonoidal functor, then its associated structure maps

X

z,z′

p((x, x′

),(y, y′

),(z, z′

))×d(z, z′

) = X

z,z′

A_{(z, x)×}A_(x′

, y)×A_(y′

, z′

)×d(z, z′

)

∼

=A_(x′

, y)×d(x, y′

)

→d(x, x′

)×d(y, y′

),

and

X

z,z′

j(z, z′

)×d(z, z′

) = X

z,z′

A_{(z, z}′

)×d(z, z′

)

∼

= X

z

d(z, z)

→ 1,

are determined by components

d(x, y′

)֌d(x, y)×d(y, y′

)

d(z, z)։1

which give A _{the structure of a discrete cocategory over} Set_f.

Define the functor U :C _{= [k∗(}A _×A_),Vect_f]→Vect_{f by}

U f =M

x,y

(f(x, y)⊗k[d(x, y)]).

Then we obtain monoidal and comonoidal structure maps

U(f ⊗g)

i //

r

o

o _{U f ⊗U g}

U I

i0 //

r0

o

o _k∼_=k[1]

from the canonical maps

M

x,y,z

f(x, z)⊗g(z, y)⊗k[d(x, y)]

z=u=v //

adjoint

o

o M

x,u

f(x, u)⊗k[d(x, u)]

⊗M

v,y

g(v, y)⊗k[d(v, y)]

and

M

z

k[d(z, z)]⇄k ∼=k[1].

6. Concluding remarks

If the original “fibre” functor U is faithful and exact then the Tannaka equivalence (du-ality)

Lex(Cop_,Vect_)≃Comod_(End∨

U)

is available, where Lex(Cop_,Vect_{) is the category ofk-linear left exact functors from}Cop toVect_{. (See [3] for example.) Thus, since}C _{is braided monoidal, so is}Comod_(End∨

U) with the tensor product and unit induced by the convolution product on Lex(Cop_,Vect);

for convenience we recall [3] that, forC _{compact, this convolution product is given by the} restriction to Lex(Cop_,Vect_{) of the coend}

F ∗G= Z C,D

F C ⊗GD⊗C_{(−, C ⊗D)}

∼

= Z C

F C⊗G(C∗ ⊗ −)

computed in the whole functor category [Cop_,Vect_{]. Moreover, when U is separable}

monoidal, the category Co_(End∨

U) of cofree coactions of End∨

U (as constructed in [7] for example) also has a monoidal structure (Co_(End∨_{U),⊗, k), this time obtained from}

the algebra structure of End∨

U. The forgetful inclusion

Comod_(End∨

U)⊂Co_(End∨

U)

preserves colimits while Comod_(End∨_{U) has a small generator, namely {U C | C ∈} C_}, and thus, from the special adjoint functor theorem, this inclusion has a right adjoint. The value of the adjunction’s counit at the functor F ⊗G inCo_(End∨

U) is then a split monomorphism and, in particular, the monoidal forgetful functor

Comod_(End∨

U)→Vect_,

which is the composite Comod_(End∨_{U) ⊂}Co_(End∨_{U) →}Vect_{, is a separable braided}

monoidal functor extension of the given functor U :C _→Vect_.

References

[1] Aurelio Carboni. Matrices, relations, and group representations, J. Algebra 136 no. 2 (1991): 497–529.

[2] Brian Day. On closed categories of functors, in Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics 137 (1970): 1–38.

[4] B. Day, E. Panchadcharam, and R. Street. Lax braidings and the lax centre, inHopf Algebras and Generalizations, Contemporary Mathematics 441 (2007): 1–17.

[5] Brian Day and Ross Street. Quantum categories, star autonomy, and quantum groupoids, in Galois Theory, Hopf Algebras, and Semiabelian Categories, Fields In-stitute Communications 43 (2004): 187–226.

[6] Reinhard H¨aring-Oldenburg. Reconstruction of weak quasi-Hopf algebras, J. Algebra 194 no. 1 (1997): 14–35.

[7] Andr´e Joyal and Ross Street. An Introduction to Tannaka Duality and Quantum Groups, Lecture Notes in Mathematics 1488 (Springer-Verlag, Berlin, 1991): 411– 492.

[8] G. M. Kelly. Basic concepts of enriched category theory, London Mathematical So-ciety Lecture Note Series 64. Cambridge University Press, Cambridge, 1982. Also at

http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html

[9] Ross Street. Fusion operators and cocycloids in monoidal categories, Appl. Categ. Struct. 6 (1998): 177–191.

[10] Korn´el Szlach´anyi. Finite quantum groupoids and inclusions of finite type, Fields Inst. Comm. 30 (2001): 393–407.

Department of Mathematics Macquarie University

New South Wales 2109 Australia

Research Institute for Mathematical Sciences Kyoto University, Kyoto 606-8502 Japan

Email: craig@kurims.kyoto-u.ac.jp

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