arXiv:1208.5193v1 [math.QA] 26 Aug 2012
Categorical aspects of compact quantum groups
August 28, 2012
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
Keywords: compact quantum group, CQG algebra, presentable category, SAFT category, adjoint
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 The category of CQG algebras is finitely presentable10
C*QG is SAFT 14
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
∗UC Berkeley, email@example.com
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
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
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.
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].
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 ∗-
elements uijdefined by
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
∆(uij) =uik⊗ ukj,
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
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
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
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=
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).
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.
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,
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?→?
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]).
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,
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
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
1.3 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
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.
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*
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*
The category A*
compatible with comultiplications as arrows.
0QS has algebraic quantum semigroups as objects and and Alg*
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.
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
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
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.
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-
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
structural morphisms into the colimit. We have to show that the canonical map
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
Alg*(A,Aj) → Alg*(A,B) (2)
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
B ⊗ B,Aj⊗ Aj
A ⊗ A
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
Alg*(A,Aj⊗ Aj) −→ Alg*(A, lim
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
have to show that the canonical map lim
(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
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
(Vi⊗ Vj)∼= lim
(Vi⊗ Vj)∼= lim
(Vi⊗ V )∼= V ⊗ V.(3)
The first one, on the other hand, is the usual Fubini-type separation of variables for colimits ([ML98,
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
(Vj⊗ Vj) = lim F|J−→ lim F = lim
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
Au(Qα) → A
i) of Vα.
ijdefines a CQG algebra morphisms Au(Qα) → A. Moreover, the resulting
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
ij) of A.
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
kl)ik,jlis a unitary
f : Y =
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
kl)ik,jlis the unitary coefficient matrix
kof the tensor product Hilbert space Vα⊗Vβ. But
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
ik,jldefined above lift uniquely to elements of C, and we slightly abusively denote these lifts by
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
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
ijthe same multipli-
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
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.
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
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
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
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
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?
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?
n ≥ 0 (tensor product of Aa-Aa-bimodules). Moreover, ι′identifies Bawith the summand Aa⊕M
Now consider the commutative diagram
Babreaks up as a direct sum of 2ncopies of M⊗nfor
where the right hand vertical map comes from the universality property of the pushout applied to
the two inclusions Ba→ Ba?
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
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.
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.
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-
AA ⊗ AAab⊗ Aab
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
A ⊗ AAab⊗ Aab
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
A ⊗ AAab⊗ Aab
B ⊗ B
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
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
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
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
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
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]).
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
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
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
provide precisely the same information, so we
, but this is it: The set of pairs of projections
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
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*
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
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:
By cofree-ness again, the loop ι ◦ π must be the identity; since π was a surjection, it must be an
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*
0QS sending a CQG algebra to its underlying algebraic quantum semi-
(b) forget : C*QG → C*
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
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
A ⊗ AB ⊗ B
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*
Setting H = cofree(A), the canonical map f : H → A factors as
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-
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
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.
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
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
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).
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
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.
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
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
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
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′.
[Abe80] Eiichi Abe. Hopf algebras, volume 74 of Cambridge Tracts in Mathematics. Cambridge
University Press, Cambridge, 1980. Translated from the Japanese by Hisae Kinoshita and
[AR94] Jiˇ r´ ı Ad´ amek and Jiˇ r´ ı Rosick´ y. Locally presentable and accessible categories, volume 189
of London Mathematical Society Lecture Note Series. Cambridge University Press, Cam-
[Ban99] Teodor Banica. Fusion rules for representations of compact quantum groups. Exposition.
Math., 17(4):313–337, 1999.
[Ban05] Teodor Banica. Quantum automorphism groups of small metric spaces. Pacific J. Math.,
[Bic03] Julien Bichon. Quantum automorphism groups of finite graphs. Proc. Amer. Math. Soc.,
131(3):665–673 (electronic), 2003.
[BKQ11] Erik B´ edos, S. Kaliszewski, and John Quigg. Reflective-coreflective equivalence. Theory
Appl. Categ., 25:No. 6, 142–179, 2011.
[BMT01] E. B´ edos, G. J. Murphy, and L. Tuset. Co-amenability of compact quantum groups. J.
Geom. Phys., 40(2):130–153, 2001.
[Boc95] Florin P. Boca.
Ast´ erisque, (232):93–109, 1995. Recent advances in operator algebras (Orl´ eans, 1992).
Ergodic actions of compact matrix pseudogroups on C∗-algebras.
[Chi11] A. Chirvasitu. Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras. ArXiv
e-prints, October 2011.
[Con94] Alain Connes. Noncommutative Geometry. Academic Press, San Diego, CA, 1994. available
for download at http://www.alainconnes.org/en/.
[DK94] Mathijs S. Dijkhuizen and Tom H. Koornwinder. CQG algebras: a direct algebraic ap-
proach to compact quantum groups. Lett. Math. Phys., 32(4):315–330, 1994.
[ES92] Michel Enock and Jean-Marie Schwartz. Kac algebras and duality of locally compact groups.
Springer-Verlag, Berlin, 1992. With a preface by Alain Connes, With a postface by Adrian
[GLR85] P. Ghez, R. Lima, and J. E. Roberts. W∗-categories. Pacific J. Math., 120(1):79–109,
[Kel86] G. M. Kelly. A survey of totality for enriched and ordinary categories. Cahiers Topologie
G´ eom. Diff´ erentielle Cat´ eg., 27(2):109–132, 1986.
[Kor12] A. Kornell. Quantum Collections. ArXiv e-prints, February 2012.
[KS97] Anatoli Klimyk and Konrad Schm¨ udgen. Quantum groups and their representations. Texts
and Monographs in Physics. Springer-Verlag, Berlin, 1997.
[KT99] Johan Kustermans and Lars Tuset. A survey of C∗-algebraic quantum groups. I. Irish
Math. Soc. Bull., (43):8–63, 1999.
[KV00] Johan Kustermans and Stefaan Vaes. Locally compact quantum groups. Ann. Sci.´Ecole
Norm. Sup. (4), 33(6):837–934, 2000.
[Lac10] Stephen Lack. A 2-categories companion. In Towards higher categories, volume 152 of
IMA Vol. Math. Appl., pages 105–191. Springer, New York, 2010.
[ML98] Saunders Mac Lane. Categories for the working mathematician, volume 5 of Graduate
Texts in Mathematics. Springer-Verlag, New York, second edition, 1998.
[Mon93] Susan Montgomery. Hopf algebras and their actions on rings, volume 82 of CBMS Re-
gional Conference Series in Mathematics.
Mathematical Sciences, Washington, DC, 1993.
Published for the Conference Board of the
[Por11] Hans-E. Porst. Limits and colimits of Hopf algebras. J. Algebra, 328:254–267, 2011.
[PW90] P. Podle´ s and S. L. Woronowicz. Quantum deformation of Lorentz group. Comm. Math.
Phys., 130(2):381–431, 1990.
[QS10] J. Quaegebeur and M. Sabbe. Isometric coactions of compact quantum groups on compact
quantum metric spaces. ArXiv e-prints, July 2010.
[Rei70] G. A. Reid. Epimorphisms and surjectivity. Invent. Math., 9:295–307, 1969/1970.
[Sch92] Peter Schauenburg. Tannaka duality for arbitrary Hopf algebras. 66:ii+57, 1992.
[Sch11] D. Sch¨ appi. The formal theory of Tannaka duality. ArXiv e-prints, December 2011.
[So? l05] Piotr M. So? ltan. Quantum Bohr compactification.
Illinois J. Math., 49(4):1245–1270
[So? l09] Piotr M. So? ltan. Quantum families of maps and quantum semigroups on finite quantum
spaces. J. Geom. Phys., 59(3):354–368, 2009.
[Sol10] P. M. Soltan. On quantum maps into quantum semigroups. ArXiv e-prints, October 2010.
[Swe69] Moss E. Sweedler. Hopf algebras. Mathematics Lecture Note Series. W. A. Benjamin, Inc.,
New York, 1969.
[Tak02] M. Takesaki. Theory of operator algebras. I, volume 124 of Encyclopaedia of Mathematical
Sciences. Springer-Verlag, Berlin, 2002.
Algebras and Non-commutative Geometry, 5.
Reprint of the first (1979) edition, Operator
[Tam90] D. Tambara. The coendomorphism bialgebra of an algebra. J. Fac. Sci. Univ. Tokyo Sect.
IA Math., 37(2):425–456, 1990.
[Tom07] Reiji Tomatsu. A characterization of right coideals of quotient type and its application to
classification of Poisson boundaries. Comm. Math. Phys., 275(1):271–296, 2007.
[VD94] A. Van Daele. Multiplier Hopf algebras. Trans. Amer. Math. Soc., 342(2):917–932, 1994.
[VDW96] Alfons Van Daele and Shuzhou Wang. Universal quantum groups. Internat. J. Math.,
[Ver12] J. Vercruysse. Hopf algebras—Variant notions and reconstruction theorems. ArXiv e-
prints, February 2012.
[Wan95] Shuzhou Wang.
Free products of compact quantum groups.Comm. Math. Phys.,
[Wan97] Shuzhou Wang. Krein duality for compact quantum groups. J. Math. Phys., 38(1):524–
[Wan98] Shuzhou Wang.
Quantum symmetry groups of finite spaces.Comm. Math. Phys.,
[Wan99] Shuzhou Wang. Ergodic actions of universal quantum groups on operator algebras. Comm.
Math. Phys., 203(2):481–498, 1999.
[Was94] Simon Wassermann. Exact C∗-algebras and related topics, volume 19 of Lecture Notes
Series. Seoul National University Research Institute of Mathematics Global Analysis Re-
search Center, Seoul, 1994.
[Wor87] S. L. Woronowicz. Compact matrix pseudogroups. Comm. Math. Phys., 111(4):613–665,
[Wor88] S. L. Woronowicz.
SU(N) groups. Invent. Math., 93(1):35–76, 1988.
Tannaka-Kre˘ ın duality for compact matrix pseudogroups. Twisted
[Wor98] S. L. Woronowicz. Compact quantum groups. In Sym´ etries quantiques (Les Houches,
1995), pages 845–884. North-Holland, Amsterdam, 1998.