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.
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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