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arXiv:1208.5193v1 [math.QA] 26 Aug 2012

Categorical aspects of compact quantum groups

Alexandru Chirvasitu∗

August 28, 2012

Abstract

We show that either of the two reasonable choices for the category of compact quantum

groups is nice enough to allow for a plethora of universal constructions, all obtained “by abstract

nonsense” via the adjoint functor theorem. This approach both recovers constructions which

have appeared in the literature, such as the quantum Bohr compactification of a locally compact

semigroup, and provides new ones, such as the coproduct of a family of compact quantum

groups, and the compact quantum group freely generated by a locally compact quantum space.

In addition, we characterize epimorphisms and monomorphisms in the category of compact

quantum groups.

Keywords: compact quantum group, CQG algebra, presentable category, SAFT category, adjoint

functor theorem

Contents

1 Preliminaries

1.1Compact quantum groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2CQG algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3 Locally compact and algebraic quantum spaces and semigroups . . . . . . . . . . . .

1.4 SAFT categories and the adjoint functor theorem . . . . . . . . . . . . . . . . . . . .

1.5 Finitely presentable categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

4

5

7

8

9

2 The category of CQG algebras is finitely presentable10

3

C*QG is SAFT 14

4 Applications

4.1 Limits in CQG and C*QG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2Quantum groups generated by quantum spaces . . . . . . . . . . . . . . . . . . . . .

4.3 Variations on the Bohr compactification theme . . . . . . . . . . . . . . . . . . . . .

4.4 Kac quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

16

20

21

23

5 Monomorphisms25

References26

∗UC Berkeley, chirvasitua@math.berkeley.edu

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Introduction

Compact quantum groups were introduced in essentially their present form in [Wor87] (albeit under

a different name), and the area has been expanding rapidly ever since. The subject can be viewed as

part of Connes’ general program [Con94] to make “classical” notions (spaces, topology, differential

geometry) non-commutative: One recasts compact groups as C∗-algebras via their algebras of

continuous functions, and then removes the commutativity assumption from the definition (see

§1.1 for details). Starting with the relatively simple resulting definition, all manner of compact-

group-related notions and constructions can then be generalized to the non-commutative setting:

Peter-Weyl theory ([Wor87, Wor98]), Tannaka-Krein duality and reconstruction ([Wor88, Wan97]),

Pontryagin duality ([PW90]), actions on operator algebras ([Boc95, Wan98, Wan99]) and other

structures, such as (classical or quantum) metric spaces ([Ban05, QS10]) or graphs ([Bic03]), and

so on. This list is not (and cannot be) exhaustive.

The goal of this paper is to analyze compact quantum groups from a category-theoretic per-

spective, with a view towards universal constructions.

Bits and pieces appear in the literature: In [Wan95], Wang constructs coproducts in the category

C*QG (see §1.1), opposite to that of compact quantum groups (it consists of C∗-algebras, which are

morally algebras of functions on the non-existent quantum groups). More generally, he constructs

other types of colimits (e.g. pushouts) of diagrams with one-to-one connecting morphisms. The

fact that this coproduct can be constructed simply at the level of C∗-algebras (forgetting about

comultiplications) parallels the fact that classically, the underlying space of a categorical product

?Gi of compact groups Gi is, as a set, just the ordinary Cartesian product. The category of

compact groups, however, also admits coproducts, and they are slightly more difficult to construct:

One endows the ordinary, discrete coproduct?Gi(i.e. coproduct in the category of discrete groups,

also known as the free product of the Gi) with the finest topology making the canonical inclusions

Gj →

group. It is natural, then, to ask whether or not coproducts of compact quantum groups exist,

or equivalently, whether the category C*QG opposite to that of compact quantum groups has

products. We will see in §4.1 that this is indeed the case, and moreover, the category is complete

(i.e. it has all small limits).

Another example of universal construction that fits well within the framework of this paper is

the notion of quantum Bohr compactification [So? l05]. One of the main results of that paper is,

essentially, that the forgetful functor from compact quantum groups to locally compact quantum

semigroups has a left adjoint; remembering that we are always passing from (semi)groups to algebras

of functions and hence reversing arrows, this amounts to the existence of a certain right adjoint

([So? l05, 3.1,3.2]). Section 4 recovers this as one among several right-adjoint-type constructions,

such as compact quantum groups “freely generated by a quantum space” (as opposed to quantum

semigroup; see §4.2).

Most compact-quantum-group-related universal constructions in the literature seem to be of a

“left adjoint flavor”: the already-mentioned colimits in C*QG, the quantum automorphism groups

of, say, [Wan98], which are basically initial objects in the category of C*QG objects endowed with a

coaction on a fixed C∗-algebra, etc. By contrast, apart from the Bohr compactification mentioned in

the previous paragraph, universal constructions of the right adjoint flavor (limits in C*QG, or right

adjoints to functors with domain C*QG) appear not to have received much attention. This is all the

more surprising for at least two reasons. First, they seem to be more likely to exist than the other

kind of universal construction; example: a (unital, say) C∗-algebra A endowed with a coassociative

map A → A⊗A into its minimal tensor square (this would be the object dual to a compact quantum

semigroup) always has a compact quantum group (meaning its dual object, as in Definition 1.1.1)

?Gi continuous, and then takes the Bohr compactification of the resulting topological

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mapping into it universally, but does not, in general, have a compact quantum group receiving

a universal arrow from it (Remark 4.3.3). Secondly, the representation-theoretic interpretation of

limits in C*QG is often simpler than that of colimits; see Proposition 4.1.3 (especially part (a))

and surrounding discussion.

The structure of the paper is as follows:

Section 1 recalls the machinery that will be used in the sequel and fixes notations and con-

ventions, introducing the two versions of the category opposite that of compact quantum groups:

CQG, consisting of so-called CQG algebras (these are like the algebra of representative functions

on a compact group; see Definition 1.2.2), and C*QG, whose objects are analogous to algebras of

continuous functions on compact groups (Definition 1.1.1).

In Section 2, Theorem 2.0.5 shows that the category CQG is finitely presentable (§1.5). This

technical property will later allow us to reduce the existence of right adjoints for functors defined

on CQG to checking that these functors are cocontinuous, i.e. preserve colimits. This is typically

an easy task, as the routine nature of most proofs in Section 4 shows.

Section 3 proves a property slightly weaker than finite presentability for the category C*QG

(Theorem 3.0.13). The nice features from the previous section are preserved however, and the

same types of results (existence of right adjoints to various functors defined on C*QG) follow.

In Section 4 we list some of the consequences of the previous two sections. These include the

automatic existence of limits in CQG and C*QG (§4.1), compact quantum groups freely generated

by quantum spaces and semigroups (§§ 4.2 and 4.3 respectively), and universal Kac type compact

quantum groups associated to any given compact quantum group (§4.4).

Finally, in Section 5 we characterize monomorphisms in the categories CQG and C*QG. It

turns out that in the former they have to be one-to-one, whereas in the latter being mono is

slightly weaker than injectivity (Proposition 5.0.3). The results are analogous to the fact ([Rei70,

Proposition 9]) that epimorphisms of compact groups are surjective.

Acknowledgements

This work is part of my PhD dissertation. I would like to thank my advisor Vera Serganova for all

the support, and Piotr So? ltan for helpful discussions on the contents of [So? l05].

1 Preliminaries

All algebraic entities in this paper (algebras, coalgebras, bialgebras, etc.)

algebra is, as usual, a complex algebra endowed with a conjugate linear, involutive, algebra anti-

automorphism ‘∗’. Unless we are dealing with non-unital C∗-algebras as in §1.3 below, in which case

the reader will be warned, algebras are assumed to be unital (and coalgebras are always counital).

Our main references for the necessary basics on coalgebra, bialgebra and Hopf algebra theory

are [Abe80, Mon93, Swe69]. The notation pertaining to coalgebras is standard: ∆ for antipodes

and ε for the counit, perhaps adorned with the name of the coalgebra if we want to be more precise

(example: ∆C, εC). The same applies to antipodes for Hopf algebras, which are usually denoted

by S. We use Sweedler notation both for comultiplication, as in ∆(c) = c1⊗ c2, and for comodule

structures: If ρ : M → M⊗C is a right C-comodule structure, it will be written as ρ(m) = m0⊗m1.

All comodules are right, and the category of right comodules over a coalgebra C is denoted by MC.

For any comodule V over any coalgebra H (the notation suggests that it will become a Hopf

algebra soon), there is a largest subcoalgebra H(V ) over which V is a comodule. If the comodule

structure map is ρ : V → V ⊗ H and (ei)i∈Iis a basis for V ,then H(V ) is simply the span of the

are complex.A ∗-

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elements uijdefined by

ρ(ej) =

?

i

ei⊗ uij.

We refer to uijas the coefficients of the basis (ei), to (uij) as the coefficient matrix of the basis,

and to H(V ) as the coefficient coalgebra of V . The coalgebra structure is particularly simple on

coefficients:

∆(uij) =uik⊗ ukj,

?

k

ε(uij) = δij.

Henceforth, the standing assumption whenever we mention coefficients and coefficient coalgebras

is that the comodule in question is finite-dimensional (that is, I is finite). When V is simple, the

coefficients uijwith respect to some basis are linearly independent, and the coefficient coalgebra is

a matrix coalgebra, in the sense that its dual is a matrix algebra.

1.0.1 Remark Note that maps V → V ⊗C are the same as elements of V ⊗V∗⊗C = End(V )⊗C.

If u = (uij) is the coefficient matrix of a basis ei, i = 1,n for V and End(V ) is identified with Mn

via the same basis ei, then the element of End(V ) ⊗ C∼= Mn(C) corresponding to the coaction

is exactly the coefficient matrix u. We will often blur the distinction between these two points of

view, and might refer to u itself as the comodule structure.

?

If in the above discussion H is a Hopf algebra, more can be said: The matrix S(uij)i,jis inverse

to (uij)i,j. Moreover, giving the dual V∗the usual right H-comodule structure

?f0,v?f1= ?f,v0?S(v1),v ∈ V, f ∈ V∗,

the coefficient matrix of the basis dual to (ei) is precisely (S(uji))i,j(note the flipped indices).

A word on tensor products: In this paper, the symbol ‘⊗’ means at least three things. When

appearing between purely algebraic objects, such as algebras or just vector spaces, it is the usual,

algebraic tensor product. Between C∗-algebras it always means the minimal, or injective tensor

product ([Tak02, IV.4]). Finally, on rare occasions, we use the so-called spatial tensor product

(referred to as W∗-tensor product in [Tak02, IV.5]) between von Neumann (or W∗) algebras. It

will always be made clear what the nature of the tensored objects is, so that no confusion is likely

to arise.

1.1Compact quantum groups

This is by now a very rich and well-referenced theory, so we will be very brief, and will refer the

reader to one of the many excellent sources (e.g. the papers and book cited below and the references

therein) for details on the topic.

No list of references would be complete without mentioning the seminal papers [Wor87, Wor88],

where Woronowicz laid the foundation of the subject, introducing the main characters under the

name “compact matrix pseudogroups”, while an exposition of the main features of the theory is

given by the same author in [Wor98]. Other good references are the survey paper [KT99], and

[KS97, 11.4].

As mentioned in the introduction, the main idea is that since one can study compact groups by

means of the algebras of continuous functions on them, which are commutative, unital C∗-algebras

with some additional structure, dropping the commutativity assumption but retaining the extra

structure should still lead to interesting objects, which are trying to be “continuous functions on

a quantum group”. The additional structure just alluded to is captured in the following definition

([KT99, 3.1.1]):

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1.1.1 Definition A compact quantum group is a pair (A,∆), where A is a unital C∗-algebra, and

∆ : A → A ⊗ A is a morphism of unital C∗-algebras satisfying the conditions

1. (Coassociativity) (∆ ⊗ id) ◦ ∆ = (id⊗∆) ◦ ∆;

2. (Antipode) The subspaces ∆(A)(1 ⊗ A) and ∆(A)(A ⊗ 1) are dense in A ⊗ A.

C*QG is the category whose objects are compact quantum groups, and whose morphisms f :

A → B are unital C∗-algebra maps preserving the comultiplication in the sense that (f ⊗f)◦∆A=

∆B◦ f.

?

The second condition needs some explanation. The space ∆(A)(1 ⊗ A) is defined as the linear

span of products of the form ∆(a)(1 ⊗ b) ∈ A ⊗ A, and similarly for ∆(A)(A ⊗ 1). The condition

is named ‘antipode’ because it has to do with the demand that A, regarded as a kind of bialgebra,

have something like an antipode. For comparison, consider the case of an ordinary, purely algebraic

bialgebra B. The condition that it have an antipode, i.e. that it be a Hopf algebra, is equivalent

to the requirement that the map

B ⊗ B → B ⊗ B,x ⊗ y ?→ x1⊗ x2y

be a bijection.

Since compact quantum groups as defined above are morally functions of algebras, representa-

tions of the group must be comodules of some sort over the corresponding algebras (endowed with

their comultiplication). Some care must be taken, as Definition 1.1.1 makes no mention of a counit,

and so the usual definition of comodule has to be modified slightly. The solution is (see [KT99,

discussion before Proposition 3.2.1]):

1.1.2 Definition Let (A,∆) be a compact quantum group. A finite-dimensional comodule over A

is a finite-dimensional complex vector space V together with a coassociative coaction ρ : V → V ⊗A

such that the corresponding element of End(V ) ⊗ A is invertible.

?

We will often drop the adjective ‘finite-dimensional’. Remark 1.0.1 applies, and we will often

refer to the coefficient comatrix of some basis as being the comodule structure. [KS97, 11.4.3,

Lemma 45] says that any comodule is unitarizable, in the sense that there is an inner product on

V making the coefficient matrix u ∈ Mn(A) of an orthonormal basis unitary (cf. Definition 1.2.1).

1.2CQG algebras

These are the algebraic counterparts of compact quantum groups. More precisely, if a compact

quantum group as in Definition 1.1.1 plays the role of the algebra of continuous functions on a

“quantum group”, then the associated CQG algebra wants to be the algebra of representative func-

tions of the quantum group, i.e. matrix coefficients of finite-dimensional unitary representations.

The main reference for this subsection is [KS97, 11.1-4].

Recall that a Hopf ∗-algebra H is a Hopf algebra with a ∗-structure making H into a ∗-algebra,

and such that the comultiplication and counit are morphisms of ∗-algebras. This is the kind of

structure that allows one to define what it means for a representations of a quantum group (i.e. a

comodule over the corresponding “function algebra”) to be unitary.

Let V be an n-dimensional comodule over a Hopf ∗-algebra H.

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1.2.1 Definition If ( | ) is an inner product on V , the pair (V,( | )) is said to be unitary provided

the coefficients uijof an orthonormal basis ei, i = 1,n form a unitary matrix in H.

A comodule V is said to be unitarizable if there exists an inner product making it unitary. This

is equivalent to saying that for any basis (ei), the coefficient matrix (uij)i,jcan be made unitary

by conjugating it with a scalar n × n matrix.

?

This is [KS97, Definition 5], and it is the correct compatibility condition for a comodule structure

and an inner product. See also [KS97, 11.1.5, Proposition 11] for alternative characterizations of

unitary comodules. We are now ready to recall the main definition of this subsection ([KS97, 11.3.1,

Definition 9]):

1.2.2 Definition A CQG algebra is a Hopf ∗-algebra which is the linear span of the coefficient

matrices of its unitarizable (or equivalently, unitary) finite-dimensional comodules.

The category having CQG algebras as objects and Hopf ∗-algebra morphisms as arrows will be

denoted by CQG.

?

1.2.3 Remark It is a simple but useful observation that a quotient Hopf ∗-algebra of a CQG

algebra is automatically CQG. Indeed, a morphism of Hopf ∗-algebras will turn a unitary coefficient

matrix into another such.

?

Let us recall that CQG algebras are automatically cosemisimple [KS97, 11.2], i.e. their cate-

gories of comodules are semisimple. Another way to say this is that a CQG algebra is the direct

sum of its matrix subcoalgebras.

The following example will play an important role in Section 2. It is a family of “universal” CQG

algebras, in a sense that will be made precise below (see [KS97, 11.3.1, Example 6], or [VDW96],

where these objects were introduced in their C∗-algebraic incarnation).

1.2.4 Example Let Q ∈ GLn(C) be a positive operator, and denote by Au(Q) the ∗-algebra freely

generated by elements uij, i,j = 1,n subject to the relations making both u = (uij)i,jand Q

unitary, where u = (u∗

ij)i,j. Strictly speaking, the main character here is the pair (Au(Q),u) rather

than just Au(Q): We always assume the uijare fixed as part of the structure, and refer to them as

the standard generators of Au(Q).

One way to state the universality property mentioned above is: For any CQG algebra A and

any unitary coefficient matrix v = (vij) satisfying S2(v) = QvQ−1, the map uij ?→ vij lifts to a

unique CQG algebra morphism Au(Q) → A.

Note that Au(Q) has a standard n-dimensional unitary comodule with orthonormal basis (ei)n

with the obvious structure ej?→?

1

2uQ−1

2

i=1,

iei⊗ uij.

?

There are various functors going back and forth between CQG and C*QG. First, since a CQG

algebra is generated by elements of unitary matrices, there is, for any element of the algebra, a

uniform bound on the norm that element can have when acting on any Hilbert space. It follows

that any CQG algebra A has an enveloping C∗-algebra A. The fact that the comultiplication

and counit of the CQG algebra lift to give A a compact quantum group structure follows from the

universality property of this envelope, as does the functoriality of this construction ([KS97, 11.3.3]).

This functor will be denoted by univ : CQG → C*QG.

On the other hand, for any compact quantum group B, the coefficients of all comodules

(Definition 1.1.2) span a sub-Hopf ∗-algebra of B in the obvious sense, and again, the construction

is easily seen to be functorial. We denote this functor by alg : C*QG → CQG. Moreover, alg(B)

is the only dense sub-Hopf ∗-algebra of B ([BMT01, A.1]).

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Functors constructed in some natural way and going in opposite directions are in the habit of

being adjoints, and this situation is no different: univ is the left adjoint. As it happens, alg almost

has right adjoint too. ‘Almost’ because only its restriction to the category of CQG algebras and

one-to-one morphisms has a right adjoint, red, associating to each CQG algebra A the so-called

reduced [BMT01, §2] compact quantum group having A as its dense sub-Hopf ∗-algebra. It is the

“smallest” such object, in the sense that any compact quantum group A admits a unique surjective

morphism A → red(alg(A)) which restricts to the identity on alg(A). A detailed discussion on

the interplay between the three functors mentioned in this paragraph can be found in [BKQ11,

6.2].

It is probably clear by now that (the opposites of) CQG and C*QG are “the two reasonable

choices for the category of compact quantum groups” of the abstract. Which one is most convenient

in any given case depends on which aspects of the theory one wishes to focus on. For representation-

theoretic purposes, CQG seems to be the correct choice, since the CQG algebra alg(B) discussed

above is tailor-made to capture all information about unitary B-comodules. On the other hand,

there are purely analytic concepts (coamenability [BMT01]) whose very definition requires the use

of C*QG.

In this paper, the CQG vs. C*QG distinction is a matter of technical necessity. For various

reasons having to do with the topological aspect of being a C∗rather than a ∗-algebra, CQG is

the easier category to work with, as should be apparent from the announced results once we review

the necessary category theory: The title of Section 2 (see §1.5) is stronger than that of Section 3

(§1.4).

1.3 Locally compact and algebraic quantum spaces and semigroups

One of the themes that will be explored in Section 4 is, very roughly, the existence of “compact

quantum groups freely generated by quantum objects”. Here, ‘objects’ can be things like ‘semi-

groups’ or ‘spaces’. Keeping in mind that we are placing ourselves in the dual picture, where spaces

are explored through functions on them, we recall in this subsection how non-unital C∗-algebras or

plain ∗-algebras allow one to formalize such notions.

A good, brief account of more or less everything we need for the locally compact side of the

picture can be found in the ‘Notations and conventions’ section of [KV00] (assuming rudiments on

multiplier algebras of C∗-algebras [Tak02, III.6]).

Recall that for not-necessarily-unital C∗-algebras A and B, a morphism from A to B is by

definition a continuous ∗-algebra homomorphism f : A → M(B) into the multiplier algebra of B

which is non-degenerate, meaning that the space f(A)B is dense in B. It is then explained in

[KV00] how two such creatures can be composed, meaning that non-unital C∗-algebras together

with morphisms as defined above constitute a category denoted here by C*

as the category dual to that of locally compact quantum spaces. C*is the subcategory consisting

of unital C∗-algebras. Note that the non-degeneracy condition on morphisms automatically makes

a C*

For the definition of locally compact quantum semigroups we follow [So? l05], referring again

to the preliminary section of [KV00] for the missing details on compositions of morphisms in C*

(needed to make sense of the coassociativity condition below).

0. It is to be thought of

0arrow between objects of C*unital.

0

1.3.1 Definition A locally compact quantum semigroup is a pair (A,∆), where ∆ : A → A ⊗ A is

a morphism in C*

0, coassociative in the obvious sense.

The category C*

compatible with comultiplications as arrows.

0QS has locally compact quantum semigroups as objects, and C*

0-morphisms

?

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Turning now to the algebraic side, everything just said has a natural analogue. We again have

to deal with multiplier algebras, this time of not-necessarily-unital ∗-algebras; [VD94, Appendix]

provides sufficient background, and we will freely use the results and terminology therein. All

∗-algebras are assumed to be non-degenerate, in the sense that ab = 0, ∀b implies a = 0 (the

∗-structure makes this condition symmetric).

For ∗-algebras A and B, a morphism A → B is by definition a ∗-homomorphism f : A → M(B),

non-degenerate in the sense that f(A)B spans B. Composition goes through essentially as in the

C∗case, and we thus get a category Alg*

has only unital morphisms as arrows. The algebraic counterpart to Definition 1.3.1 is

0. As before, the full subcategory Alg*on unital ∗-algebras

1.3.2 Definition An algebraic quantum semigroup is a pair (A,∆), where ∆ : A → A ⊗ A is a

coassociative morphism in Alg*

0.

The category A*

compatible with comultiplications as arrows.

0QS has algebraic quantum semigroups as objects and and Alg*

0-morphisms

?

1.3.3 Remark Whatever results we prove below within the framework of Definition 1.3.2, close

analogues exist for plain complex algebras rather than ∗-algebras. I believe this one example is

sufficient to illustrate how the universal constructions of Section 4 go through in the algebraic, as

well as the C∗-algebraic setting.

?

1.4 SAFT categories and the adjoint functor theorem

As Section 4 below is all about showing that certain functors have adjoints, in this subsection and

the next we recall the categorical machinery involved in this. The main reference here is [ML98].

The set of morphisms x → y in a category C will be denoted by C(x,y). Recall that categories

with all (co)limits (always small in this paper) are said to be (co)complete, and functors preserving

those (co)limits are called (co)continuous (so ‘complete’ here means the same thing as Mac Lane’s

‘small-complete’ [ML98, V]).

A class S of objects in a category is said to be a generator (or a generating class) if any two

distinct parallel arrows f ?= g : y → z stay distinct upon composition with an arrow S ∋ x → y

([ML98, V.7]). We call category generated if there is a generating set (as opposed to a proper class).

An arrow f : x → y in a category C is an epimorphism if arrows out of y are uniquely determined

by their “restriction to x” via composition with f ([ML98, I.5]). The quotient objects of x are the

epimorphisms with source x, identified up to isomorphism in the comma category x ↓ C of arrows

with source x [ML98, II.6]. Finally, C is said to be co-wellpowered if for every object x, the class of

quotient objects of x is actually a set.

We explained above how the aim is to construct things like “the compact quantum group freely

generated by a quantum semigroup”. What this means, precisely, remembering that we are working

with algebra-of-functions-type objects, is that we want a right adjoint to, say, the inclusion functor

ι : CQG → A*

0QS (this is just one example; there is also a C∗version). Typically, when trying

to show that a functor ι is a left adjoint, one needs to check (1) that ι is cocontinuous (this is

certainly necessary, as left adjoints are always cocontinuous) and (2) that some kind of solution

set condition is satisfied [ML98, V.6.2]. For some categories, however, (2) is unnecessary: they are

such that any cocontinuous functor out of them is automatically a left adjoint. One sufficient set

of conditions that will ensure this is provided by the following result, due to Freyd and referred to

in the literature as the special adjoint functor theorem (dual to [ML98, V.8.2]):

1.4.1 Theorem Let C be a cocomplete, generated, and co-wellpowered category. Then, any cocon-

tinuous functor with domain C is a left adjoint.

?

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In view of this result, it is natural to isolate the hypotheses:

1.4.2 Definition A category is SAFT if it is cocomplete, generated, and co-wellpowered.

?

1.4.3 Remark Not-necessarily-cocomplete categories satisfying the adjoint functor theorem in the

sense that functors are only required to preserve those colimits which exist are called ‘compact’

in [Kel86]. One important property of compact (and hence SAFT) categories is that they are

automatically complete. This is, for example, the implication (ii) ⇒ (v) in [Kel86, Theorem 5.6].?

1.5Finitely presentable categories

One way to be SAFT is to be what in the literature is called ‘locally presentable’. We review the

main features of the theory here, and refer mainly to [AR94] for details. For brevity, in this paper

we drop the word ‘locally’.

A poset (J,≤) is said to be filtered if every finite subset is majorized by some element (this

is [AR94, 1.4], restricted to posets as opposed to arbitrary categories). In the sequel, a filtered

diagram in a category C is a functor a (J,≤) → C, and a filtered colimit is a colimit of such a

functor. Then, [AR94, 1.9] is (essentially, via [AR94, 1.5]):

1.5.1 Definition An object x ∈ C is finitely presentable if the functor C(x,−) preserves filtered

colimits.

The category C is finitely presentable if it is cocomplete, and there is a set S of finitely pre-

sentable objects such that every object in C is a filtered colimit of objects in S.

?

More rigorously, the last condition says that every object is the colimit of a functor F : J → C

taking values in S, with (J,≤) filtered.

1.5.2 Example All categories familiar from algebra, of the form ‘set with this or that kind of

structure’, such as groups, abelian groups, monoids, semigroups, algebras, ∗-algebras, modules over

a ring, etc. are finitely presentable. These are the so-called finitary varieties of algebras [AR94,

3.A], ‘finitary’ having to do with ‘finitely presentable’.

The terminology is also inspired by such examples. In the category of modules over a ring, say,

an object is finitely presentable in the above abstract sense if and only if it has a finite presentation

in the usual sense (in full generality, the result is [AR94, 3.11]).

?

What matters here is that as mentioned above, finitely presentable implies SAFT. Indeed,

cocompleteness is part of Definition 1.5.1, and the generating set required for SAFT-ness almost

is: S is easily seen to be a generator. Co-wellpowered-ness, on the other hand, is the difficult result

[AR94, 1.58].

The following proposition is the criterion of finite presentability we use in the proof of the main

result of Section 2. It is a consequence of [AR94, 1.11] (via 0.5, 0.6 of op. cit.), and in order to

state it, one more piece of terminology is needed.

1.5.3 Definition We will say that a generator S of a cocomplete category C is regular if every

object of C is the coequalizer of two parallel arrows f,g : y → z, where y and z are coproducts of

objects in S.

?

1.5.4 Proposition A cocomplete category with a regular generator consisting of finitely presentable

objects is finitely presentable.

?

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2 The category of CQG algebras is finitely presentable

This section is devoted to proving the result in the title:

2.0.5 Theorem The category CQG is finitely presentable.

The main tool in the proof is Proposition 1.5.4, according to which cocompleteness (by now

well known) is first on the agenda.

2.0.6 Proposition CQG is cocomplete.

Proof It is enough to show that the category has coproducts and coequalizers of parallel pairs of

arrows [ML98, V.2.1]. Both will be constructed as simply the colimits of the underlying diagrams

of ∗-algebras (i.e. coalgebra structures play no role in the construction of colimits).

Coproducts are essentially constructed in [Wan95, Theorem 1.1]. That result is concerned with

the C∗-algebraic version (constructing coproducts in the category C*QG), but as remarked by Wang

at the end of [Wan95, §1], the algebraic version holds as well. Given a set of CQG agebras, the

universal property of the coproduct of underlying algebras gives this coproduct the extra structure

that will make it into a CQG algebra; we leave the details to the reader.

To construct coequalizers, let f,g : A → B be morphisms of CQG algebras. The ideal I

generated by the elements f(a)−g(a), a ∈ A is in fact a coideal, as well as invariant under ∗. The

latter assertion is trivial, so let us focus on I being a coideal. Compatibiity of f and g with counits

says that f(a) − g(a) is annihilated by εB. On the other hand, the familiar computation

∆B(f(a) − g(a)) = (f ⊗ f)(∆A(a)) − (g ⊗ g)(∆A(a))

= f(a1) ⊗ f(a2) − g(a1) ⊗ g(a2)

= f(a1) ⊗ (f − g)(a2) + (f − g)(a1) ⊗ g(a2) ∈ B ⊗ I + I ⊗ B

shows that I plays well with the comultiplication. It follows that the coequalizer of f and g in Alg*

is a quotient Hopf ∗-algebra of B, and hence a CQG algebra by Remark 1.2.3.

?

The plan now is to show that the set S consisting of the CQG algebras Au(Q) of Example 1.2.4

(for all possible positive operators Q, of all possible sizes) satisfies the hypotheses of Proposition 1.5.4:

Every Au(Q) is finitely presentable in CQG in the sense of Definition 1.5.1, and S is a regular gen-

erator. We start with the former.

2.0.7 Proposition For any positive Q ∈ GLn(C), the object A = Au(Q) ∈ CQG is finitely pre-

sentable.

Proof Let (J,≤) be a filtered poset, and let Aj, j ∈ J implement a functor J → CQG by means

of CQG algebra morphisms ιj′j: Aj→ Aj′ for j ≤ j′. Denote also by ιi: Ai→ B = lim

j

Aj the

structural morphisms into the colimit. We have to show that the canonical map

lim

j

CQG(A,Aj) → CQG(A,B) (1)

is a bijection.

It is clear from the description in Example 1.2.4 that A is finitely presented as a ∗-algebra,

and is hence a finitely presentable object in Alg*([AR94, 3.11] with λ = ℵ0). In conclusion, the

canonical map

lim

j

Alg*(A,Aj) → Alg*(A,B) (2)

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is bijective, and the injectivity of (1) follows from the commutative square

lim CQG(A,Aj) CQG(A,B)

lim Alg*(A,Aj) Alg*(A,B),

where the vertical arrows are the obvious inclusions.

To prove that (1) is surjective, fix a CQG algebra morphism f : A → B = lim Aj. It is, in

particular, a morphism in Alg*, so by the surjectivity of (2), it factors through a unital ∗-algebra

morphism fi: A → Aifor some i ∈ J, and hence through fj= ιji◦fi: A → Ajfor j ≥ i. For such

j, consider the diagram

AAj

B

B ⊗ B,Aj⊗ Aj

A ⊗ A

fj

ιj

fj⊗fj

ιj⊗ιj

where the vertical maps are comultiplications. The commutativity of the outer rectangle is nothing

but the preservation of coproducts by f = ιj◦ fj, while the right hand square commutes because

ιj : Aj → B is by definition a colimit in CQG. It follows that the two J-indexed systems of

morphisms ∆Aj◦ fjand (fj⊗ fj) ◦ ∆Abecome equal upon composing further with ιj⊗ ιj. The

fact that they must then be equal for sufficiently large j follows from the next lemma, which says

essentially that ιj⊗ ιj make B ⊗ B the colimit of the diagram consisting of the maps ιj′j⊗ ιj′j,

together with the injectivity of

lim

j

Alg*(A,Aj⊗ Aj) −→ Alg*(A, lim

j

Aj⊗ Aj)

resulting from the finite presentability of A in Alg*.

?

2.0.8 Lemma The tensor square endofunctor A ?→ A ⊗ A on Alg*preserves filtered colimits.

Proof Working with ∗-algebras is of little importance here: The forgetful functor from any finitary

variety of algebras to the category Set of sets creates filtered colimits. We refer again to [AR94,

3.A] for background on varieties of algebras. The claim just made can be proven either by realizing

the variety as the Eilenberg-Moore category of a finitary monad (i.e. one which preserves filtered

colimits) on Set [AR94, 3.18], or directly. It follows that it is enough to prove the analogous

statement in the category Vec of complex vector spaces.

Let (J,≤) be a filtered poset, ιj′j: Vj→ Vj′ a functor from it to Vec, and V = lim

j

Vj. We

have to show that the canonical map lim

j

(Vj⊗ Vj) → V ⊗ V is an isomorphism.

Let (J ×J,≤) be the cartesian square of the category (J,≤); it is simply the poset structure on

the set J×J defined by (i,j) ≤ (i′,j′) iff i ≤ i′and j ≤ j′. Consider the functor F : (J×J,≤) → Vec

given by (i,j) ?→ Vi⊗ Vj (with the obvious action on morphisms). For any vector space W, the

11

Page 12

endofunctor W ⊗ • : Vec → Vec is left adjoint to Hom(W,•), and hence cocontinuous. Applying

this observation first to Viand then to V , we get the last two isomorphisms in the chain

lim

i,j

(Vi⊗ Vj)∼= lim

i

lim

j

(Vi⊗ Vj)∼= lim

i

(Vi⊗ V )∼= V ⊗ V.(3)

The first one, on the other hand, is the usual Fubini-type separation of variables for colimits ([ML98,

IX.8]).

The original poset J sits diagonally inside J × J as the set of pairs (j,j). Moreover, the fact

that J is filtered translates to J being cofinal in J × J in the sense that everyone in the latter is

majorized by someone in the former. But it then follows [ML98, IX.3.1] that the canonical map

lim

j

(Vj⊗ Vj) = lim F|J−→ lim F = lim

i,j

(Vi⊗ Vj)

is an isomorphism. Composing it with (3) finishes the proof.

?

2.0.9 Remark We mentioned briefly at the end of §1.2 that the category C*QG is not as friendly

as CQG. Lemma 2.0.8, for example, is somewhat problematic. The problem with the above proof

is that it hinges on functors of the form A ⊗ • : Alg*→ Alg*preserving filtered colimits; I do not

know whether the analogous result holds for the category C*of unital C∗-algebras with the minimal

tensor product (which is what would be needed to make the proof work verbatim for C*QG).

Some partial results (which we do not prove here) are that (a) for a C∗-algebra A, the functor

A⊗• does preserve filtered colimits of injections, (b) the same functor preserves all filtered colimits

provided A is an exact C∗-algebra in the sense of [Was94] (this simply means that minimal tensoring

with A preserves short exactness of sequences in C*

does preserve all filtered colimits. This suggests that trying to adapt Lemma 2.0.8 to C*adds a

layer of difficulty, in that one has to deal with issues like nuclearity and exactness.

0), and (c) the maximal tensor product with A

?

The last piece of the puzzle is

2.0.10 Proposition The set S of all Au(Q) is a regular generator in CQG.

Proof According to Definition 1.5.3, we have to show that an arbitrary CQG algebra A is the

coequalizer of two arrows f,g : Y → Z between coproducts of Au(Q)’s.

Let Vαbe representatives for the set? A of unitary simple comodules of A. Then, A is the

direct sum of the matrix coalgebras Cαspanned by the unitary coefficient matrices vα= (vα

respect to orthonormal bases (eα

The squared antipode S2conjugates every matrix vαby some positive operator Qα[KS97,

11.2.3, Lemma 30], and hence, by the universality property of the Au(Q)’s as cited in Example 1.2.4,

the assignment uij?→ vα

map

π :

Au(Qα) → A

ij) with

i) of Vα.

ijdefines a CQG algebra morphisms Au(Qα) → A. Moreover, the resulting

?

? A

(4)

is surjective. The left hand side of this expression will be our Z. For α ∈? A, we denote by uα

the standard coefficient matrix in Au(Qα) (earlier in this paragraph, where we reasoned one α at

a time, it was denoted simply by u).

The two morphisms f,g : Y → Z that we are looking for (and whose coequalizer (4) should be)

ought to somehow recover the relations of A, i.e. the multiplication table with respect to the basis

(vα

ij) of A.

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Page 13

To define f, fix α,β ∈? A. A simple calculation shows that uαuβ= (uα

coefficient matrix of Z =?Au(Qα), which the squared antipode of this CQG algebra conjugates

by Qαβ= Qα⊗Qβ(this is just notation). It follows that there is a unique morphism Au(Qαβ) → Z

defined by uik,jl?→ uα

kl. Putting all of these together for all pairs of comodules, we get

?

? A×? A

ijuβ

kl)ik,jlis a unitary

ijuβ

f : Y =

Au(Qαβ) →

?

Au(Qα) = Z.

As before, since we need to distinguish between the various coefficient matrices in Y , we denote

them by uαβin the obvious way.

We now start on our way towards constructing g : Y → Z. The same game as in the previous

paragraph can be played in A: For any α,β ∈? A, vαvβ= (vα

with respect to the tensor product basis eα

now, since we are in A, the elements of this matrix can be expressed as linear combinations of vγ

In order to avoid cumbersome indices on the coefficients of such linear combinations, we simply

write

vα

ik,jl∈

ijvβ

kl)ik,jlis the unitary coefficient matrix

kof the tensor product Hilbert space Vα⊗Vβ. But

i⊗eβ

ij’s.

ijvβ

kl= ℓαβ

?

γ

Cγ,

where Cγis the matrix coalgebra corresponding to γ ∈? A, and γ ranges over the simple comodules

appearing in the decomposition of Vα⊗ Vβ.

Now, because the restriction of π to the direct sum C ≤ Z of matrix coalgebras spanned by

uα⊂ Z is one-to-one (in fact, this restriction is by definition an isomorphism onto A), the elements

ℓαβ

ik,jldefined above lift uniquely to elements of C, and we slightly abusively denote these lifts by

π−1(ℓαβ

conjugates by Qα⊗Qβ. Indeed, all of these properties can be stated inside C (without appealing to

multiplication), using only the antipode and the ∗-structure (being unitary, for example, amounts

to the antipode turning π−1(ℓαβ

jl,ik)∗); since π preserves both the antipode and the ∗

structure and its restriction to C is a coalgebra isomorphism, the properties all lift from the ℓ’s to

the π−1(ℓ)’s.

Finally, the claim just proven allows us to construct g : Y → Z by sending uαβ

The coequalizer of f and g is the quotient of Z by the relations imposing on uα

cation table as that of the vα

ij’s, so it is now clear that this coequalizer is precisely π : Z → A.

ik,jl). I claim that for fixed α and β, these elements form a unitary coefficient matrix which S2

ik,jl) into π−1(ℓαβ

ik,jlto π−1(ℓαβ

ijthe same multipli-

ik,jl).

?

We can now put the last few results together:

Proof of Theorem 2.0.5 We know from Proposition 2.0.6 that CQG is cocomplete, and from

Propositions 2.0.7 and 2.0.10 that a set of finitely presentable objects forms a regular generator.

The conclusion follows from Proposition 1.5.4.

?

2.0.11 Remark In essentially the same way, we can show that the category CQGabof commutative

CQG algebras is finitely presentable. In this case, all distinctions between the algebraic and the

C∗-algbraic vanish: The restriction of univ to CQGabis an equivalence onto the full subcategory

C*QGabof C*QG consisting of commutative algebras. Moreover, CQGab(or C*QGab) is nothing

but the opposite of the category of compact groups, with a compact group G corresponding to the

CQG algebra of representative functions on it.

The only changes we need to make to the proofs in order to adapt the presentability result

to CQGabare (a) substitute tensor products (of perhaps infinite families) for coproducts, and (b)

use the set of CQG algebras associated to all unitary groups Unfor a generator, instead of the

Au(Q)’s.

?

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Page 14

2.0.12 Remark Although, strictly speaking, SAFT-ness would have sufficed for the purposes

of Section 4, the finite presentability of CQG is interesting in its own right, as it is somewhat

surprising: Given the close relationship between CQG and C*QG, discussed a little in §1.2 above,

one might think that the former category should look more or less like “unital C∗-algebras with a

lot of extra structure”, and hence should be at least as reluctant to be finitely presentable as the

category C*of unital C∗-algebras. However, this is not the case.

There is a more general notion of presentability for categories (local presentability in the lit-

erature, e.g. [AR94]) parametrized by a regular cardinal number, so that the technical term for

‘finitely presentable’ is ‘ℵ0-presentable’; the larger the cardinal, the weaker the notion. Now, it can

be shown that C*is ℵ1-presentable but not finitely presentable. Worse still, the same is true in the

commutative setting: Although the previous remark notes that CQGabis finitely presentable, the

category of commutative unital C∗-algebras is ℵ1, but not finitely presentable.

?

3

C*QG is SAFT

The main result of the section is the one just stated:

3.0.13 Theorem The cateory C*QG is SAFT.

We prove the three properties required for SAFT-ness (Definition 1.4.2) separately.

3.0.14 Proposition C*QG is cocomplete.

Proof This parallels the proof of Proposition 2.0.6 by constructing coequalizers and coproducts,

so we will be brief.

As noted in the proof just mentioned, coproducts are constructed in [Wan95, Theorem 1.1].

As for coequalizers, they are constructed as before. The coequalizer of two morphisms f,g :

A → B in C*QG is the quotient of B by the closed ideal I generated by f(a) − g(a), and the

argument from the proof of Proposition 2.0.6 can be paraphrased to show this.

Although Sweedler notation is not available anymore (because we are working with C∗tensor

products rather than algebraic ones), the computation carried out there can be written down in a

Sweedler-notation-free manner as saying that ∆B◦(f −g) equals (f ⊗(f −g)+(f −g) ⊗g)◦∆A.

It follows that B/I inherits a coassociative comultiplication and B → B/I respects it, while the

(Antipode) condition of Definition 1.1.1 follows immediately from that of B. In conclusion, the

quotient B → B/I is naturally a map in C*QG.

?

3.0.15 Remark Filtered colimits and pushouts in C*QG are constructed in [Wan95, 3.1,3.4]

in the case when the morphisms in the diagram are one-to-one.

Proposition 3.0.14, injectivity is not necessary in order to conclude that the colimit of a diagram

in C*QG in the category of unital C∗-algebras is automatically endowed with a compact quantum

group structure.

According to (the proof of)

?

Next in line is the generation condition of Definition 1.4.2. It turns out that Au(Q) will once

more come in handy. We need them as objects of C*QG, so recall the enveloping C∗-algebra functor

univ : CQG → C*QG.

3.0.16 Proposition The set univ(Au(Q)) for Q ranging over all positive matrices generates

C*QG.

14

Page 15

Proof That the Au(Q) form a generator in CQG is part of the statement of Theorem 2.0.5. It is

a simple exercise that left adjoints, such as univ, turn generators into generators provided their

right adjoints are faithful. In our case, the faithfulness of the right adjoint alg to univ follows

from the density of the inclusion alg(A) ⊂ A: Any arrow f : A → B in C*QG is the extension by

continuity of alg(f) : alg(A) → alg(B), and hence alg(f) = alg(g) implies f = g.

?

The only ingredient of Definition 1.4.2 still to be addressed is co-wellpoweredness. Recall (§1.4)

that this meant that every object has only a set of quotient objects. It will help, then, to know

exactly which morphisms in C*QG are epimorphisms; this is what the following result does.

3.0.17 Proposition A morphism f : A → B in C*QG is an epimorphism if and only if it is

surjective.

Proof As usual in categories where objects are sets with some kind of structure and morphisms

are maps preserving that structure, the implication surjective ⇒ epimorphism is immediate.

To prove the other implication, we will show that if f is an epimorphism, then alg(f) is sur-

jective (the conclusion follows from the denseness of alg(B) ⊂ B). Since we can always substitute

the image of f for A, we can (and will) assume that f is injective.

First, recall the construction CQG ∋ X ?→ red(X) ∈ C*QG mentioned in §1.2. It is functorial

when restricted to the category CQGinjof CQG algebras and injective morphisms, (this is the

essence of [BKQ11, 6.2.4]). In order to keep the notation manageable, indicate the functors alg

and red by superscripts, as in Xafor alg(X), Xrfor red(X), Xarfor red(alg(X)), etc.; the

same conventions are in place for morphisms.

Let ι : B → BB be the left hand canonical inclusion into the pushout of f along itself

?

A

in the category C*QG, or equivalently (by the proof of Proposition 3.0.14), in the category C*of

unital C∗-algebras. One condition equivalent to f being epimorphic is that ι be an isomorphism.

Similarly, we denote by ι′the left hand inclusion Ba→ Ba?

Aa

Bainto the pushout in Alg*. Note

that ι and ι′are both injective, as they have left inverses by the universality property of pushouts.

Assume now that fais not surjective. Then, I claim that (a) ι′cannot be surjective (equivalently,

an isomorphism), and hence (b) neither can (ι′)r. That (a) does indeed imply (b) follows from the

fact [BKQ11, 6.2.12] that the functor red : CQGinjreflects isomorphisms.

To prove (a), recall that an inclusion K ⊆ H of cosemisimple Hopf algebras always splits

as a K-K-bimodule map (e.g. as argued in the proof of [Chi11, 2.0.4]). Applied to the inclusion

fa: Aa→ Ba, this observation yields a direct sum decomposition Ba= Aa⊕M as Aa-Aa-bimodules

for some non-zero M, and the pushout Ba?

Aa

n ≥ 0 (tensor product of Aa-Aa-bimodules). Moreover, ι′identifies Bawith the summand Aa⊕M

therein.

Now consider the commutative diagram

Babreaks up as a direct sum of 2ncopies of M⊗nfor

BB?

ABBar?

Aar Bar

Bar

(Ba?

AaBa)r

ι

(ι′)r

15

Page 16

where the right hand vertical map comes from the universality property of the pushout applied to

the two inclusions Ba→ Ba?

Aa

B → Bar. We have just argued that if fais not surjective, then the lower left corner path is

not surjective. But then the upper right corner path isn’t either. However, the right hand upper

horizontal arrow is surjective, as is the right hand vertical arrow. In conclusion, the only morphism

in this path which can fail to be surjective (under the assumption that fais not surjective) is ι.?

Ba, and the other two unnamed maps are induced by the surjection

3.0.18 Remark The proof makes it clear that the analogous result is true for CQG, i.e. epimor-

phisms are surjective.

?

Since for any compact quantum group A there is only a set of quotients of alg(A) and hence only

a set of compact quantum groups having such quotients as dense CQG subalgebras, the previous

result implies:

3.0.19 Proposition C*QG is co-wellpowered.

?

Proof of Theorem 3.0.13 Propositions 3.0.14, 3.0.16 and 3.0.19 are precisely what is required

by Definition 1.4.2.

?

4 Applications

The goal of this section is to apply Theorems 2.0.5 and 3.0.13, together with the adjoint functor

theorem and abstract properties of presentable or SAFT categories, to the construction of compact

quantum groups with various universal properties. These constructions fall roughly into two cate-

gories: right adjoints to functors defined on CQG or C*QG, as direct applications of Theorem 1.4.1,

and left adjoints arising in a slightly more roundabout way in §4.4.

Note that limits in the categories CQG and C*QG, discussed in the next subsection, fit in

this framework as right adjoints: Given a small category J and a category C, the limits of functors

J → C, if they exist, can be obtained as images of a right adjoint to the diagonal functor ∆ : C → CJ

(the latter is notation for the category of all functors J → C) sending an object c ∈ C to the functor

∆(c) : J → C constant at c.

4.1Limits in CQG and C*QG

We now know from Theorems 2.0.5 and 3.0.13 that both CQG and C*QG are SAFT. Remembering

that SAFT-ness implies completeness (Remark 1.4.3), we get:

4.1.1 Theorem The categories CQG and C*QG are complete.

?

Limits in these categories are quantum analogues of colimits of compact groups. It is natural to

ask whether functors J → CQG or C*QG taking commutative values have commutative limits, or

in other words, whether Theorem 4.1.1 recovers ordinary colimits of compact groups. Since CQGab

is complete (by Remark 2.0.11 or simply constructing coequalizers and coproducts in the category

of compact groups), the next result confirms this:

4.1.2 Proposition The inclusions CQGab→ CQG and C*QGab→ C*QG are right adjoints.

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Page 17

Proof In both cases the left adjoint is abelianization, associating to an object A ∈ CQG (or C*QG)

its largest commutative quotient ∗-algebra (resp. C∗-algebra) Aab. This is all rather routine, so we

omit most of the details.

The universality property of the canonical quotient map π : A → Aabensures that the compo-

sition

∆

AA ⊗ AAab⊗ Aab

π⊗π

factors through a map ∆ab : Aab → Aab⊗ Aab.

uniqueness of the factorization of π⊗3◦ (∆ ⊗ id) ◦ ∆ : A → A⊗3

get a commutative square

The coassociativity of ∆ab follows from the

abthrough π. By construction, we

AAab

A ⊗ AAab⊗ Aab

π

π⊗π

∆

∆ab

The rest of the structure and properties (e.g. counit εab : Aab → C and counitality of ∆abin

the algebraic case) follow similarly, as do the functoriality and the desired universality property of

A ?→ Aab. To see the latter, for example, let f : A → B be a morphism in CQG or C*QG with B

commutative. Then, f factors uniquely as fab◦ π for an algebra map fab: Aab→ B. The outer

rectangle of

AAab

B

A ⊗ AAab⊗ Aab

B ⊗ B

π

fab

π⊗π

fab⊗fab

is commutative because f = fab◦π is compatible with comultiplications, and we have just observed

that the left hand square commutes. It follows that the precomposition of the right hand square

with π commutes also, and since π is onto, the right hand square must be commutative. We

again skip the entirely similar arguments for compatibility of fabwith counits and antipodes in the

algebraic case.

?

Outside of the general categorical framework provided by Theorems 2.0.5 and 3.0.13, one can

also arrive at limits in the categories CQG and C*QG by means of the Tannakian formalism

introduced in [Wor88] (CQG is better suited for this, so we focus on it). There, Woronowicz

associates a compact quantum group (or rather a compact quantum group of the form univ(A)

for A ∈ CQG, so effectively, he recovers a CQG algebra) from a so-called concrete monoidal W∗-

category with complex conjugation. These are basically just rigid, monoidal W∗-categories endowed

with a faithful, monoidal ∗-functor [GLR85] to the category Hilb of finite-dimensional Hilbert

spaces.

This is a version of Tannaka duality for Hopf algebras (e.g. as in [Sch92, Ver12] and the many

references therein): The CQG algebra constructed in [Wor88, 1.3] given a concrete monoidal W∗-

category C is what in [Sch92] would be called the coendomorphism Hopf algebra of the functor

C → Hilb that is implicitly part of Woronowicz’s definition.

Now, if one starts with the category of unitary comodules of a CQG algebra A and performs

the above construction, the resulting CQG algebra is again A. In other words, unitary comodules

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know all there is to know about a CQG algebra (hence the name ‘Tannaka reconstruction’). It

follows from this that the construction of new CQG algebras out of old (such as, say, the limit of

some functor F : J → CQG out of the values of F) has a chance of being carried out categorically:

Identify the category of unitary comodules, and you know the CQG algebra.

To get some insight into what limits in CQG look like, we do what the previous paragraph

suggests, for products (but also state the result for pullbacks, as it will be useful in Section 5):

Given a family Ai, i ∈ I of objects in CQG, what does the category of finite-dimensional, unitary

comodules of the product A =

Aiin CQG look like in terms of the categories of comodules of

?

i

the individual Ai? The answer turns out to be quite simple; arguably simpler, in fact, than the

description of the category of unitary comodules for the (more familiar, in the literature) coproduct

?

Aifrom [Wan95]. All comodules below are understood to be finite-dimensional and unitary.

Putting an A-comodule structure on an n-dimensional Hilbert space V is the same as giving

a CQG algebra morphism f : Au(Q) → A for some positive Q ∈ GLn(C): In one direction,

the comodule structure induces a morphism by sending the standard generators uij ∈ Au(Q)

(for an appropriate Q) to the coefficients vij with respect to an orthonormal basis of V ; in the

opposite direction, make A coact on the standard Au(Q)-comodule by “scalar corestriction” via

the coalgebra morphism f. In turn, by the defining property of the categorical product, a morphism

f : Au(Q) → A means a family of morphisms fi: Au(Q) → Ai, i ∈ I. Going through this comodule

structure - morphism correspondence in reverse for each i, the data consisting of the fi’s is equivalent

to putting an Ai-comodule structure on the canonical n-dimensional comodule of Au(Q) for every

i. A moment’s thought will show how do modify this argument to take care pullbacks, and all in

all, we have the following result:

4.1.3 Proposition Let Ai∈ CQG, i ∈ I be a set of objects, and A =?Aithe product in CQG.

Then, the category of A-comodules has as objects finite-dimensional Hilbert spaces admitting an

Ai-comodule structure for each Ai, and as morphisms linear maps respecting all of these structures.

Let f : B → C and f′: B′→ C be morphisms in CQG, and A = B×CB′the pullback in CQG.

The category of A-comodules has

(a) as objects, triples (V,V′,ϕ) where V and V′are B and B′-comodules respectively, and ϕ :

V → V′is a unitary map identifying V and V′as C-comodules;

(b) as morphisms from (V,V′,ϕ) to (W,W′,ψ), pairs (ξ,ξ′), where ξ : V → W and ξ′: V′→ W′

are morphisms in MBand MB′respectively, making the diagram

VV′

WW′

ϕ

ψ

ξ

ξ′

commutative.

?

4.1.4 Remark This statement describes the sought-after categories of unitary comodules very

explicitly. There is a more abstract, but also more elegant way to phrase all of this. We need some

basic 2-categorical notions to say it all (as in [Lac10]).

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Rigid, monoidal W∗-categories C endowed with monoidal ∗-functors C → Hilb form a bicategory

in a natural way, while CQG can be regarded as a bicategory with only identity 2-cells. Then,

sending a CQG algebra to its category of finite-dimensional, unitary comodules (together with

its forgetful functor to Hilb) is a pseudofunctor from the latter to the former. The essence of

Proposition 4.1.3 is that this pseudofunctor preserves limits. This is a familiar theme in Tannaka

duality: Woronowicz’s construction of a CQG algebra out of a functor C → Hilb is in fact nothing

but the left adjoint of the pseudofunctor mentioned above. This sort of situation is treated in

[Sch11], with a biadjunction analogous to the one just discussed appearing in Theorem 3.1.1.

?

4.1.5 Remark The references to [Wor88] in the above discussion are somewhat of a paraphrase,

as Woronowicz works with what are called compact matrix quantum groups (or CMQG algebras

on the algebraic side [DK94, 2.5]). These are basically compact quantum groups finitely generated

as C∗-algebras, and are analogous to compact Lie groups (the latter being precisely those compact

groups which embed in some unitary group). He also distinguishes a comodule whose coefficients

generate the algebra, and so works with pairs (A,u), where u ∈ Mn(A) is a unitary coefficient

matrix. Adapting the results of that paper to the general setting is straightforward.

A CMQG algebra always has a countable set of (isomorphism classes of) simple comodules. As

we will see in the next example, abandoning this restriction is absolutely necessary if we are going

to discuss limits in CQG, since products, for example, are very unlikely to satisfy this property. ?

4.1.6 Example One does not even have to go “quantum” to give an example of a very large (that

is, non-CMQG) product in CQG. Indeed, the smallest possible example will do: a coproduct of

two copies of Z/2 in the category of compact groups.

Denote this coproduct by G. According to the first part of Proposition 4.1.3, a unitary repre-

sentation of G consists of a finite-dimensional Hilbert space V and two involutive unitary operators

x and y on V . The projections p =1+x

2

work with them instead. V is irreducible preciely when p and q have no common proper, non-zero

invariant subspace. By the discussion carried out prior to the statement of [Tak02, Theorem 1.41],

this is equivalent to the four pairwise infima p ∧ q, p ∧ (1 − q) etc. all being zero (where ‘∧’ means

orthogonal projection on the intersection of the ranges of the two projections).

If dimV ≥ 2, the vanishing of all wedges p ∧ q, etc. makes it necessary that V be even-

dimensional and that p and q both have rankdimV

2

satisfying the requirements is open dense in the set of all pairs of projections of rankdimV

Hence, if dimV = 2n for some n ≥ 1, the set of pairs (p,q) that will make V into an irreducible

unitary G-representation is a manifold of dimension 4n2(twice the dimension of the Grassmannian

variety of n-dimensional subspaces of V as a real manifold).

the equivalence relation (p,q) ∼ (upu∗,uqu∗) for unitary u, which accounts for isomorphic G-

representations induced by different pairs of projections. Since the unitary group of V has dimension

n(2n − 1), this still leaves continuum many isomorphism classes of simples.

and q =1+y

2

provide precisely the same information, so we

, but this is it: The set of pairs of projections

2

on V .

We now have to quotienting by

?

We end this subsection with a note on terminology. As observed in [Sol10, §3], the name ‘free

product of compact quantum groups’ from [Wan95] is somewhat inconsistent with the prevailing

point of view that compact quantum groups form a category opposite to C*QG. The problem is

that in universal algebra, ‘free product’ is often synonymous to ‘coproduct’. Even though Wang’s

construction is a coproduct in C*QG rather than its opposite, and hence ‘product of compact

quantum groups’ would perhaps be a better fit, ‘free product’ seems to have been established

through use (besides, ‘product’ would clash with the interpretation of A ⊗ A as the Cartesian

square of a compact quantum group, implicit in Definition 1.1.1). On the other hand, products

19

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in C*QG (whose existence Theorem 4.1.1 proves) are probably best referred to as ‘coproducts of

compact quantum groups’.

4.2Quantum groups generated by quantum spaces

The idea here is that functors of the form “forget the comultiplication”, regarded as quantum

analogues of forgetting the multiplication on a group, have right adjoints. As most of the previous

discussion, all of this works both algebraically and C∗-algebraically:

4.2.1 Theorem The functors

(a) forget : CQG → Alg*

0sending a CQG algebra to its underlying ∗-algebra and

(b) forget : C*QG → C*

0sending a compact quantum group to its underlying C∗-algebra

are left adjoints.

Proof We already know from the proofs of Propositions 2.0.6 and 3.0.14 that the forgetful functors

from CQG and C*QG to unital ∗-algebras and C∗-algebras respectively are cocontinuous. I claim

that so are the inclusions Alg*→ Alg*

cocontinuous; that they are left adjoints then follows from the SAFT-ness of CQG and C*QG

(Theorems 2.0.5 and 3.0.13) and the adjoint functor theorem.

Finally, to prove the claim made above that the two inclusions Alg*→ Alg*

are cocontinuous, note that they are in fact left adjoints: In both cases, the right adjoint is the

multiplier algebra construction A ?→ M(A).

0and C*→ C*

0. Assuming this for now, (a) and (b) are

0and C*→ C*

0

?

4.2.2 Remark Von Neumann algebras would be an alternative way to formalize the idea of “quan-

tum space”. This is the point of view espoused in [Kor12], where the category W*of unital von

Neumann algebras and unital normal homomorphisms is opposite to the category of so-called

quantum collections. The idea here is that a von Neumann algebra is a quantum analogue of a set,

ordinary sets X corresponding to ℓ∞(X).

Adopting this perspective, the enveloping W∗-algebra functor env : C*→ W*is a kind of

forgetful functor, disregarding the topological side of a compact quantum space and remembering

only the underlying quantum collection; similarly, the forgetful functor W*→ C*(which is right

adjoint to env) is a kind of quantum Stone-Cech compactification.

Composing (b) of Theorem 4.2.1 further with env is again a left adjoint (the composition of

two left adjoints), and its right adjoint could be thought of as the functor associating to every

quantum ollection the compact quantum group freely generated by it.

?

We refer to the image of a ∗ or C∗-algebra A through the respective right adjoint as the cofree

CQG algebra or C*QG object on A (‘co’ because it universally maps into A, as opposed to being

mapped into). The notation cofree stands for either of the two functors, and we rely on context

to distinguish between the possibilities.

It is to be expected in such cofree-Hopf-algebra-on-an-algebra type constructions that com-

mutativity will be preserved (as explained, for instance, in the introduction of [Por11]). In other

words, one would like the right adjoint from part (b), say, when applied to the algebra of functions

vanishing at infinity on a locally compact space X, to yield precisely the compact group freely gen-

erated by X: Construct the abstract group G freely generated by X, endow it with the strongest

topology making the canonical map X → G continuous, and take the Bohr compactification of the

resulting topological group. That this is indeed the case is essentially the content of the following

proposition:

20

Page 21

4.2.3 Proposition If A is a commutative ∗ or C∗-algebra, then cofree(A) is commutative.

Proof To fix ideas, we prove the algebraic statement regarding ∗-algebras, and leave the simple

modifications that will adapt the proof to the other cases to the reader.

Let H = cofree(A), and recall the CQG algebra structure on Habfrom Proposition 4.1.2. The

universality property of the abelianization factorizes the left hand diagonal arrow in the following

diagram through the right hand one, while the cofree-ness gives the other commutative triangle,

passing through ι:

π

HHab

A

ι

By cofree-ness again, the loop ι ◦ π must be the identity; since π was a surjection, it must be an

isomorphism.

?

4.3 Variations on the Bohr compactification theme

A right adjoint to the inclusion functor C*QG → C*

construction is referred to as the quantum Bohr compactification. For any A ∈ C*

an object H ∈ C*QG mapping universally into A so as to preserve the comultiplication; remem-

bering the arrow reversal inherent to passing from spaces to functions on them, this should indeed

be thought of as a compact quantum group into which the locally compact quantum semigroup

maps universally. Moreover, when A is commutative and hence the algebra of functions vanishing

at infinity on a locally compact semigroup X, the construction returns precisely the algebra of

functions on the ordinary Bohr compactification of X ([So? l05, 4.1]).

We recover these results and their algebraic counterparts below (Theorem 4.3.1 and Proposition 4.3.4),

as applications of the categorical machinery already in place.

0QS is constructed directly in [So? l05], and this

0QS, it provides

4.3.1 Theorem The inclusion functors

(a) forget : CQG → A*

group and

0QS sending a CQG algebra to its underlying algebraic quantum semi-

(b) forget : C*QG → C*

tum group

0QS sending a compact quantum group to its underlying compact quan-

are left adjoints.

Proof As before, Theorems 2.0.5 and 3.0.13 and the adjoint functor theorem ensure that we only

need to prove the two functors cocontinuous.

coequalizers of pairs and coproducts. The four arguments (coequalizers and coproducts, (a) and

(b)) follow essentially the same path, so let us focus on coproducts for part (a).

Let I be a set, and Ai, i ∈ I objects in CQG. Let also fi: Ai→ B be morphisms in A*

(strictly speaking, they are morphisms forget(Ai) → B, but since forget really is just an

inclusion, we omit it in the rest of the proof). Since forgetting further to Alg*

comultiplications) is, according to part (a) of Theorem 4.2.1, cocontinuous, the fiaggregate into

Equivalently, this means showing they preserve

0QS

0(i.e. disregarding

21

Page 22

a unique ∗-algebra morphism f : A =?Ai→ B. We are done if we can show that f preserves

comultiplications. To see this, consider the diagram

Ai

AB

Ai⊗ Ai

A ⊗ AB ⊗ B

ιi

f

ιi⊗ιi

f⊗f

where the vertical arrows are comultiplications, and ιi : Ai → A are the structure maps of the

coproduct. The commutativity of the outer rectangle is the preservation of comultiplications by

the fi= f ◦ιi, while the left hand square commutes because A was defined as the coproduct of the

Aiin CQG. It follows that precomposing the two possible ways to get from A to B ⊗ B with ιi

yields the same morphism Ai→ B ⊗ B; then, by the universality in Alg*

right hand square must also be commutative.

0of the coproduct A, the

?

4.3.2 Remark Part (b) of the proposition can be tweaked slightly in the spirit of Remark 4.2.2.

The enveloping von Neumann algebra functor env : C*

called env) from C*

of von neumann algebras M endowed with a coassociative morphism ∆ : M → M ⊗ M in W*

(remember that the tensor product of von Neumann algebras here is the spatial one) and with

W*maps which preserve these comultiplications as morphisms. It is to be thought of as a kind of

forgetful functor, ignoring the topology of a locally compact quantum semigroup and remembering

only the underlying quantum collection, together with the “multiplication”.

It can be shown further that env◦forget : C*QG → W*QS is cocontinuous, and hence a left

adjoint. In other words, every W∗quantum semigroup has a quantum Bohr compactification.

0→ W*can be lifted to a functor (again

0QS to the category W*QS of von Neumann quantum semigroups, consisting

?

4.3.3 Remark In a C∗-algebraic variant of the Tambara construction [Tam90], (a particular case

of) [So? l09, Theorem 3.3] constructs, for every finite-dimensional C∗-algebra A, an object B of C*QS

(the full subcategory of C*

0QS consisting of unital algebras) coacting universally on A. In other

words, there is a coassociative C∗-algebra map A → A ⊗ B making B an initial object in the

category of objects of C*QS endowed with such maps.

If B were to map universally into an object B′∈ C*QG, the latter would be the quantum

automorphism group of A in the sense of [Wan98]. However, we know from [Wan98, Theorem 6.1

(1)] that finite-dimensional C∗-algebras do not have compact quantum automorphism groups, in

general. In conclusion, although C*QG → C*QS is a left adjoint (by a variant of Theorem 4.3.1),

it is not a right adjoint.

?

As in §4.2, we denote the right adjoints of Theorem 4.3.1 by cofree. Once more, it turns out

that they preserve commutativity.

4.3.4 Proposition If the object A of A*

0QS or C*

0QS is commutative, so is cofree(A).

Proof We focus on the A*

0QS case.

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Setting H = cofree(A), the canonical map f : H → A factors as

HHab

A

π

f

fab

for some morphism fabin Alg*

Proposition 4.1.2 shows that fabis actually a morphism in A*

proof of Proposition 4.2.3 to conclude that π is in fact an isomorphism, and hence H is commuta-

tive.

0. Essentially the same argument as the one used in the proof of

0QS. Finally, we can now repeat the

?

4.3.5 Remark Proposition 4.3.4 goes through in the setting of Remark 4.3.2: If M is a commuta-

tive von Neumann algebra, then it can be shown in much the same way as above that the quantum

Bohr compactification is commutative.

?

4.4 Kac quotients

Recall that a compact quantum group A ∈ C*QG is said to be of Kac type if the antipode on alg(A)

lifts to a continuous map A → A. Equivalently, the antipode of alg(A) is involutive (S2= id),

or commutes with the ∗ operation. This definition extends in the obvious way to CQG algebras;

in that case, ‘of Kac type’ or simply ‘Kac’ will be synonymous to ‘having involutive antipode’.

Alternate terms are ‘Kac algebra’ (under which these objects and their relatives were introduced;

e.g. [ES92] and the references therein) or sometimes ‘Woronowicz-Kac algebra’ (as in [Ban99]). For

brevity, we will sometimes simply use ‘Kac’ as an adjective. A k subscript on either CQG of C*QG

indicates the full subcategory on objects of Kac type.

In [So? l05, Appendix], So? ltan constructs what in that paper is called the canonical Kac quotient

of an object A ∈ C*QG (notion originally due to of Stefaan Vaes). It is obtained by quotienting

out all elements of A killed by some trace (meaning, as usual, that the trace sends x∗x to zero).

This, however, seems to be a bit of a misnomer: While it is shown in [So? l05, A.1] that the result is

indeed a compact quantum group of Kac type, it is not clear that a Kac compact quantum group

is its own canonical Kac quotient1! A more appropriate term might be, perhaps, canonical tracial

quotient: One quotients out as much as one needs to in order to ensure that the result has enough

traces.

The notion has also received attention in [Tom07], where Tomatsu shows in Theorem 4.8 that

a compact quantum group A has a largest quotient of Kac type (he uses dual phrasing, thinking

of the latter as a largest compact quantum subgroup of Kac type) provided A is coamenable and

the Grothendieck ring of its category of comodules (its so-called fusion algebra) is commutative.

Regarding the terminology problem from the previous paragraph, note that for the reason pointed

out there, it is not clear, a priori, that Tomatsu’s quotient is the same as So? ltan’s. The two do

coincide, however, if the Haar measure of the compact quantum group is faithful (which is the

standing assumption of [Tom07]), hence [Tom07, Remark 4.9].

In conclusion, the question of whether or not every A ∈ CQG or C*QG has a largest Kac

quotient seems to be an interesting one. One of the aims of this subsection is to show that this is

indeed the case: The desired quotient map is precisely the reflection of A in the subcategory CQGk

1I am grateful to Piotr So? ltan for pointing this out.

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or C*QGk(i.e. the image of A through the left adjoint to the inclusion of the subcategory into

CQG or C*QG, respectively).

4.4.1 Theorem The inclusions CQGk→ CQG and C*QGk→ C*QG each have a left, as well as

a right adjoint.

Proof That the inclusions are both left adjoints is shown in much the same way in which we

have proven all results asserting the existence of various right adjoints so far. The arguments of

Sections 2 and 3 can be repeated to show that CQGkis finitely presentable and C*QGkis SAFT.

The only difference that is even remotely significant is the fact that now the regular generator to

go into an analogue of Proposition 2.0.10 consists of Au(In), n ≥ 1 rather than all Au(Q). Colimits

are again constructed simply at the level of ∗ or C∗-algebras, making it clear that the inclusions

are cocontinuous and hence left adjoints.

The interesting problem, then, is the one discussed before the statement of the theorem: con-

structing left adjoints to the two inclusions. For each A ∈ CQG or C*QG we want an arrow

κ : A → Akinto a Kac type object, universal in the sense that any morphism from A into a Kac

object factors uniquely through κ. In other words, we have to show that the comma category

A ↓ CQG (resp. A ↓ C*QG) consisting of arrows from A into Kac objects has an initial object. To

do this, we apply Freyd’s initial object theorem [ML98, V.6.1]. It says that a complete category

has an initial object as soon as it has a weakly initial set of objects; this simply means a set S of

objects such that any object y admits at least one arrow S ∋ x → y (not necessarily unique).

In our case, a weakly initial set is easy to come by: All surjections A → B for Kac type B

will do, since any map of A into a Kac type object will certainly factor through the image of that

map. On the other hand completeness follows quickly if we show that CQGkand C*QGkare closed

under limits in CQG and C*QG respectively: Limits would then be created by the forgetful functor

A ↓ CQGk→ CQGk→ CQG (and similarly for C*QG).

In conclusion, it is enough to show that products of Kac objects in CQG (or C*QG) are again

Kac, and similarly, equalizers of parallel pairs of arrows between Kac objects are Kac. Since (a)

alg : C*QG → CQG is a right adjoint and hence preserves limits, and (b) by definition, an object

B ∈ C*QG is Kac if and only if alg(B) is, it is enough to restrict ourselves to CQG.

Equalizers are easy: If f,g : B → C are arrows between Kac CQG algebras, the equalizer

injects into B, so it is again Kac. To prove the statement about products, let Bi, i ∈ I be a set

of Kac CQG algebras, and denote the structure maps of their product in CQG by πi: B → Bi.

Throughout the rest of this proof, for a CQG algebra C, we denote by C′the CQG algebra with the

same underlying set, but reversed multiplication and comultiplication. Note that the product of the

objects B′

from the functoriality of products. By this same functoriality, S is involutive (strictly speaking,

this means S′◦ S = id, where S′: B′→ B is S as a map, but we have switched the domain and

codomain). If we show that S is the antipode SBof B, we are done (Kac means involutive). To

prove this, note that the two squares in the diagram

iis precisely B′. Let Sibe the antipodes of Bi, and S =?Si: B → B′the map obtained

BB′

Bi

B′

i

S

SB

Si

πi

πi

are both commutative (the S-square by the definition of S as the product of the Si, and the

SB-square because πiare Hopf algebra morphisms and hence preserve antipodes).

?

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Page 25

Denote by A ?→ Akthe left adjoint to either of the inclusions CQGk→ CQG or C*QGk→

C*QG. It is a simple observation now that the canonical map A → Akis a surjection, and hence

warrants the name ‘Kac quotient’:

4.4.2 Proposition Let A ∈ CQG or C*QG and κ : A → Akthe universal arrow resulting from

the unit of the adjunction that the Kac quotient functor is part of. Then, κ is onto.

Proof To keep things streamlined, let us focus on CQG. We have seen this sort of argument

before, in a dual form, in Proposition 4.2.3. Let ι : A′

denote the corestriction by κ′: A → A′

k. We have the diagram

k→ Akbe the inclusion of A′

k= Im(κ), and

A′

k

Ak

A

ι

f

κ′

κ

where f is the unique arrow κ → κ′in A ↓ CQGk. Both triangles are commutative, and by the

universality of κ, the loop ι ◦ f must be the identity. Since ι was by definition one-to-one, it must

be an isomorphism.

?

5 Monomorphisms

One subject is conspicuously absent from Proposition 3.0.17: what about monomorphisms? The

main result of this section is just such a characterization:

5.0.3 Proposition A morphism in CQG is a monomorphism if and only if it is one-to-one. Sim-

ilarly, a morphism f in C*QG is mono if and only if alg(f) is one-to-one.

5.0.4 Remark Morphisms in C*QG that are injective at the algebraic level play an important role

in [BKQ11, 6.2]. The proposition gives a nice interpretation: They are precisely the monomorphisms

of C*QG.

?

5.0.5 Remark In the commutative setting, where all distinctions between the algebraic and the

C∗-algebraic sides of the picture disappear (Remark 2.0.11), the analogous result would be that

epimorphisms of compact groups are surjective. This is [Rei70, Proposition 9], and in fact, the

proof below is a paraphrase of Reid’s.

?

Proof First, let’s reduce the second part of the statement to the first. Recall that right adjoints

(such as alg) send monomorphisms to monomorphisms, so a morphism f in C*QG can only be

mono if alg(f) is. On the other hand, the faithfulness of alg (a consequence of the density of

alg(A) ⊂ A for A ∈ C*QG) implies the converse. Indeed, if alg(f) is mono and f ◦g = f ◦h, the

series

alg(f) ◦ alg(g) = alg(f) ◦ alg(h)⇒

alg(g) = alg(h)⇒g = h

of equalities does the trick (the first implication says that alg(f) is a monomorphism, while the

second one is faithfulness).

We are now left with the first statement, on CQG algebras. Just as in the case of epimorphisms

treated in Proposition 3.0.17, the implication injective ⇒ mono is the easy part. Focusing on the

25

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converse, let f : A → B be a monomorphism in CQG. Let also π : P = A×BA → A be one of the

defining projections of the pullback (in the category CQG). The functor CQG(•,P) represents the

functor sending C ∈ CQG to the set of pairs of morphisms g,h : C → A satisfying the condition

fg = fh. Since the latter implies g = h by f being a monomorphism, this functor is isomorphic

to CQG(•,A), and it follows that the natural transformation CQG(•,P) → CQG(•,A) induced by

π (and hence π itself) is an isomorphism; we have made use of this sort of result, in its dual form

having to do with epimorphisms, in Proposition 3.0.17. It will actually be more convenient to say

it like this: The diagonal map d : A → A ×BA = P is an isomorphism; indeed, its very definition

implies that πd = id.

Now transport all of the above to the level of (finite-dimensional, unitary) comodules. Recalling

what the category of P-comodules looks like from part (b) of Proposition 4.1.3, the fact d is an

isomorphism says that V ?→ (V,V,id) is an equivalence from A-comodules to P-comodules. In

particular, the essential surjectivity of this functor implies that for any P-comodule (V,V′,ϕ), the

a priori B-comodule isomorphism ϕ : V → V′is actually an A-comodule map.

According to the previous paragraph, it is enough, assuming that f is not one-to-one, to find

finite-dimensional, unitary A-comodules V and V′together with a unitary isomorphism ϕ : V → V′

as B-comodules which does not preserve the A-comodule structures. Let us simplify the situation

further. Suppose V is a non-trivial, simple, unitary A-comodule (non-trivial meaning not isomor-

phic to the monoidal unit of the cateory of comodules) which has trivial components when regarded

as a B-comodule via scalar corestriction through f : A → B. This means that there is some non-

zero vector v ∈ V fixed by B in the sense that v1⊗f(v2) = v ⊗1, but not fixed by A. The unitary

reflection across the orthogonal complement of v would then be a morphism in MBbut not in MA,

and we would be done.

In conclusion, it suffices to find V as above. Since the Peter-Weyl theorem for CQG algebras

expresses each as a direct sum of W⊕dimWfor W ranging over its set of unitary simple comodules,

the only ways in which f can be non-injective are if (a) some simple A-comodule becomes non-

simple as a B-comodule, or (b) there are two distinct simples over A which become isomorphic over

B (we will see soon that in fact (a) always happens).

In case (a), choose some such simple W ∈? A. Then, the trivial comodule has multiplicity one in

W∗⊗W as an A-comodule, but strictly larger than one over B. Hence, there is a simple V ≤ W∗⊗W

that will satisfy the sought-for conditions. In case (b), let W and W′be non-isomorphic, unitary

simple A-comodules which bcome isomorphic over B. Then, W∗⊗ W′does not contain the trivial

comodule over A, but it does over B. So as before, we can find our desired V among the simple

summands of W∗⊗ W′.

?

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