Categorical Duality for Yetter–Drinfeld Algebras

Sergey Neshveyev^{∗} Makoto Yamashita^{†}

Received: August 5, 2014 Revised: September 9, 2014

Communicated by Joachim Cuntz

Abstract. We study tensor structures on (Rep G)-module categories defined by actions of a compact quantum group G on unital C^{∗}-algebras. We show that having a tensor product which defines the module structure is equivalent to enriching the action of G to the structure of a braided-commutative Yetter–Drinfeld algebra. This shows that the category of braided-commutative Yetter–Drinfeld G-C^{∗}-algebras is equivalent to the category of generating unitary tensor functors from Rep G into C^{∗}-tensor categories. To illustrate this equivalence, we discuss coideals of quotient type in C(G), Hopf–Galois extensions and noncommutative Poisson boundaries.

2010 Mathematics Subject Classification: Primary 20G42; Secondary 18D10, 46L53, 57T05. Keywords and Phrases: C^{∗}-tensor category, quantum group, Yetter– Drinfeld algebra, Poisson boundary.

Introduction

This paper is a contribution to the study of compact quantum group actions from the categorical point of view. The idea of this approach can be traced to works of Wassermann [Was88] and Landstad [Lan92] in the 1980s. From the modern point of view, they proved that there is a one-to-one correspondence between full multiplicity ergodic actions of a compact group G and unitary fiber functors Rep G → Hilb_{f}. The quantum analogue of this result in the purely algebraic setting was proved by Ulbrich [Ulb89] and Schauenburg [Sch96], and the corresponding result in the C^{∗}-algebraic setting was proved by Bichon, De Rijdt and Vaes [BDRV06]. Thus, for a compact quantum group G, there is a correspondence between unitary fiber functors on Rep G and full quantum multiplicity ergodic actions of G. It is natural to ask then what corresponds to unitary tensor functors from Rep G into arbitrary C^{∗}-tensor categories. We show that this is braided-commutative Yetter–Drinfeld algebras, which are algebras equipped with actions of G and Ĝ satisfying certain compatibility conditions. Therefore such algebras play the same role for general tensor functors as Hopf–Galois objects for fiber functors, at least in the C^{∗}-setting. This can also be interpreted as follows. As has recently been shown in [DCY13, Nes], actions of G can be described in terms of (Rep G)-module categories. Then our result says that such a module category structure is defined by a tensor functor if and only if we can also define an action of Ĝ to get a braidedcommutative Yetter–Drinfeld algebra. As an application, we strengthen a result of Tomatsu [Tom07] characterizing coideals of quotient type in C(G), and extend this characterization to Hopf– Galois objects. We also show that the correspondence between Yetter–Drinfeld algebras and tensor functors provides a rigorous link between Izumi’s theory of Poisson boundaries of discrete quantum groups [Izu02] and categorical Poisson boundaries we introduced in [NY14a].

∗The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement no. 307663.

†Supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92), and by JSPS KAKENHI Grant Number 25800058.

Documenta Mathematica 19 (2014) 1105–1139

Acknowledgement Part of this research was carried out while the authors were attending the workshop “Noncommutative Geometry” at Mathematisches Forschungsinstitut Oberwolfach in September 2013. We thank the organizers and the staff for their hospitality. M.Y. thanks P. Schauenburg and K. Shimizu for bringing his attention to [BN11] and [DMNO13].

1 Preliminaries

In this section we briefly summarize the theory of compact quantum groups and their actions on operator algebras in the C^{∗}-algebraic formulation, as well as discuss an algebraic approach to Yetter–Drinfeld C^{∗}-algebras. We mainly follow the conventions of [NT13]. When A and B are C^{∗}-algebras, A ⊗ B denotes their minimal tensor product. Unless said otherwise, we assume that C^{∗}-categories are closed under subobjects. On the other hand, for C^{∗}- tensor categories we do not assume that the unit object is simple. For objects U and V in a category C we denote by C(U, V) the set of morphisms U → V.

1.1 Compact quantum groups

A compact quantum group G is represented by a unital C^{∗}-algebra C(G) equipped with a unital ∗-homomorphism Δ: C(G) → C(G) ⊗ C(G) satisfying the coassociativity (Δ ⊗ ι)Δ = (ι ⊗ Δ)Δ and cancellation properties, meaning that (C(G) ⊗ 1)Δ(C(G)) and (1 ⊗ C(G))Δ(C(G)) are dense in C(G) ⊗ C(G). There is a unique state h satisfying (h ⊗ ι)Δ = h (and/or (ι ⊗ h)Δ = h) called the Haar state. If h is faithful, G is called a reduced quantum group, and we are mainly interested in such cases. A finite dimensional unitary representation of G is a unitary element U ∈ B(H_{U}) ⊗ C(G), where H_{U} is a finite dimensional Hilbert space, such that (ι ⊗ Δ)(U) = U_{12}U_{13}. The dense ∗-subalgebra of C(G) spanned by matrix coefficients of finite dimensional representations is denoted by ℂ[G]. The intertwiners between two representations U and V are the linear maps T from H_{U} to H_{V} satisfying V (T ⊗ 1) = (T ⊗ 1)U. The tensor product of two representations U and V is defined by U_{13}V_{23} and denoted by U ◯_{⊤} V. The category Rep G of finite dimensional unitary representations with intertwiners as morphisms and with tensor product ◯_{⊤} becomes a semisimple C^{∗}-tensor category. Using the monoidal structure on Rep G, for any W ∈ Rep G, we can define an endofunctor ι ⊗ W on Rep G which maps an object U to U ◯_{⊤} W and a morphism T to T ⊗ ι. A natural transformation between such functors ι ⊗ W and ι ⊗ V is given by a collection of morphisms η_{U}: U ◯_{⊤} W → U ◯_{⊤} V for U ∈ Rep G that are natural in U. Denote the Woronowicz character f_{1} ∈ U(G) = ℂ[G]^{∗} by ρ. The space U(G) has the structure of a ∗-algebra, defined by duality from the Hopf ∗-algebra (ℂ[G], Δ). Every finite dimensional unitary representation U of G defines a ∗- representation π_{U} of U(G) on H_{U} by π_{U}(ω) = (ι⊗ω)(U). We will often omit π_{U} in expressions. Using the element ρ the conjugate unitary representation to U is defined by

Ū = (j(ρ)^{1/2} ⊗ 1)(j ⊗ ι)(U^{∗})(j(ρ)^{−1/2} ⊗ 1) ∈ B(H̄_{U}) ⊗ ℂ[G],

where j denotes the canonical ∗-anti-isomorphism B(H_{U}) ≅ B(H̄_{U}) defined by j(T)ξ̄ = \overline{T^{∗}ξ}. We have morphisms R_{U}: 𝟙 → Ū ◯_{⊤} U and R̄_{U}: 𝟙 → U ◯_{⊤} Ū defined by

R_{U}(1) = ∑_{i} ξ̄_{i} ⊗ ρ^{−1/2}ξ_{i} and R̄_{U}(1) = ∑_{i} ρ^{1/2}ξ_{i} ⊗ ξ̄_{i},

where {ξ_{i}}_{i} is an orthonormal basis in H_{U}. They solve the conjugate equations for U and Ū, meaning that

(R_{U}^{∗} ⊗ ι)(ι ⊗ R̄_{U}) = ιŪ and (R̄_{U}^{∗} ⊗ ι)(ι ⊗ R_{U}) = ι_{U}.

Therefore Rep G is a rigid C^{∗}-tensor category. Woronowicz’s Tannaka–Krein duality theorem recovers the ∗-Hopf algebra ℂ[G] from the rigid semisimple C^{∗}-tensor category Rep G and the forgetful fiber functor U ↦ H_{U}.

1.2 G-algebras and (Rep G)-module categories

Given a compact quantum group G, a unital G-C^{∗}-algebra is a unital C^{∗}- algebra B equipped with a continuous left action α: B → C(G)⊗B of G. This means that α is an injective unital ∗-homomorphism such that (Δ ⊗ ι)α = (ι ⊗ α)α and such that the space (C(G) ⊗ 1)α(B) is dense in C(G) ⊗ B. The linear span of spectral subspaces,

B = {x ∈ B | α(x) ∈ ℂ[G] ⊗_{alg} B},

which is a dense ∗-subalgebra of B, is called the regular subalgebra of B, and the elements of B are called regular. More concretely, the algebra B is spanned by the elements of the form (h ⊗ ι)((x ⊗ 1)α(a)) for x ∈ ℂ[G] and a ∈ B. This algebra is of central importance for the categorical reconstruction of B. When D is a C^{∗}-category, the category End(D) of endofunctors of D, with bounded natural transformations as morphisms, forms a C^{∗}-tensor category. A C^{∗}-category D endowed with a unitary tensor functor from Rep G to the opposite of End(D) is called a right (Rep G)-module category. For U ∈ Rep G, we denote the induced functor on D by X ↦ X × U. An object X in a (Rep G)-module category D is said to be generating if any other object Y ∈ D is isomorphic to a subobject of X × U for some U ∈ Rep G. Let us summarize the categorical duality theory of continuous actions of reduced compact quantum groups on unital C^{∗}-algebras developed in [DCY13] and [Nes].

Theorem 1.1 ([DCY13, Theorem 6.4; Nes, Theorem 3.3]). Let G be a reduced compact quantum group. Then the following two categories are equivalent:

(i) The category of unital G-C^{∗}-algebras B with unital G-equivariant ∗-homo-morphisms as morphisms.

(ii) The category of pairs (D, M), where D is a right (Rep G)-module C^{∗}- category and M is a generating object in D, with equivalence classes of unitary (Rep G)-module functors respecting the prescribed generating objects as morphisms.

We omit the precise definition of the equivalence relation on functors between pairs (D, M), since it will not be important to us, see [DCY13, Theorem 7.1] for details. Note also that, as follows from the proof, under the above correspondence the fixed point algebra B^{G} is isomorphic to End_{D}(M).

In the following subsections we overview the proof of the theorem.

1.3 From algebras to module categories

Given a G-C^{∗}-algebra (B, α), we consider the category D_{B} of G-equivariant finitely generated right Hilbert B-modules. In other words, objects of D_{B} are finitely generated right Hilbert B-modules X equipped with a linear map δ = δ_{X}: X → C(G) ⊗ X which satisfies the comultiplicativity property

(Δ ⊗ ι)δ = (ι ⊗ δ)δ,

such that (C(G) ⊗ 1)δ(X) is dense in C(G) ⊗ X, and such that δ is compatible with the Hilbert B-module structure in the sense that

δ(ξa) = δ(ξ)α(a),  ⟨δ(ξ), δ(ζ)⟩ = α(⟨ξ, ζ⟩),

for ξ, ζ ∈ X and a ∈ B. Here, C(G) ⊗ X is considered as a right Hilbert (C(G) ⊗ B)-module. For X ∈ D_{B} and U ∈ Rep G, we obtain a new object X × U in D_{B} given by the linear space H_{U} ⊗ X, which is a right Hilbert B-module such that

(ξ ⊗ x)a = ξ ⊗ xa and ⟨ξ ⊗ x, η ⊗ y⟩_{B} = (η, ξ)⟨x, y⟩_{B}

for ξ, η ∈ H_{U}, x, y ∈ X, a ∈ B, together with the compatible C(G)-coaction map

δ = δ_{H_{U} ⊗X}: H_{U} ⊗X → C(G)⊗H_{U} ⊗X, δ(ξ ⊗x) = U_{21}^{∗}(ξ ⊗δ_{X}(x))_{213}. (1.1)

This construction is natural both in X and U, and satisfies (X × U) × V ≅ X ×(U ◯_{⊤}V), with the obvious isomorphism mapping ζ ⊗ξ ⊗x ∈ H_{V} ⊗H_{U} ⊗X into ξ ⊗ ζ ⊗ x. We keep the notation H_{U} ⊗ X when we want to emphasize the realization of the object X × U as a Hilbert module. This way D_{B} becomes a right (Rep G)-module category. It is known that by the stabilization argument, any object in D_{B} is a direct summand of B ×U for some U ∈ Rep G. Thus we may, and often will, consider D_{B} as an idempotent completion of Rep G via the correspondence U ↦ B × U. To be precise, we start from a C^{∗}-category with the same objects as in Rep G, but with the new enlarged morphism sets

C_{B}(U, V) = Hom_{G,B}(H_{U} ⊗ B, H_{V} ⊗ B),

and form new objects from projections in the C^{∗}-algebras C_{B}(U, U), thus obtaining a C^{∗}-category C_{B}. Note that, more explicitly, the set C_{B}(U, V) consists of elements T ∈ B(H_{U}, H_{V}) ⊗ B such that

V_{12}^{∗}(ι ⊗ α)(T)U_{12} = T_{13}.

Note also that we automatically have C_{B}(U, V) ⊂ B(H_{U}, H_{V}) ⊗ B. For every W ∈ Rep G, the functor ι⊗W on Rep G extends to C_{B} in the obvious way: given a morphism T ∈ C_{B}(U, V) ⊂ B(H_{U}, H_{V}) ⊗ B the corresponding

morphism T ⊗ ι ∈ C_{B}(U ◯_{⊤} W, V ◯_{⊤} W) is T_{13} ∈ B(H_{U}, H_{V}) ⊗ B(H_{W}) ⊗ B. The

right (Rep G)-module C^{∗}-categories C_{B} and D_{B} are equivalent via the functor mapping U into B × U. Although the category C_{B} might appear somewhat ad hoc compared to D_{B}, it is more convenient for computations and some of the constructions become simpler for C_{B}. For example, suppose that f: B_{0} → B_{1} is a morphism of G-C^{∗}-algebras. Then, ι ⊗ f defines linear transformations f_{#,U,V} : C_{B_{0}}(U, V) → C_{B_{1}}(U, V), which together define a functor f_{#} : C_{B_{0}} → C_{B_{1}}. The pair (f_{#}, ι)_{U,V} gives a (Rep G)-module homomorphism in the sense of [DCY13, Definition 3.17]. Under the above equivalence D_{B} ≃ C_{B}, this obvious construction corresponds to the scalar extension functor D_{B_{0}} → D_{B_{1}}, mapping X into X ⊗_{B_{0}} B_{1}, discussed in [DCY13]. Note also that for the composition of Gequivariant maps we have the desired equality of functors f_{#}g_{#} = (fg)_{#} between the categories C_{B}, rather than a natural isomorphism of functors, which we would have for the categories D_{B}.

1.4 From module categories to algebras

We recall the construction of an action from a pair (D, M) following [Nes]. Without loss of generality we may assume that the (Rep G)-module category D is strict. Furthermore, in order to simplify the notation, by replacing D by an equivalent category we may assume that it is the idempotent completion of the category Rep G with larger morphism sets D(U, V) than in Rep G, such that M is the unit object 𝟙 in Rep G and the functor ι × U on D is an extension of the functor ι ⊗ U on Rep G. Namely, we simply define the new set of morphisms between U and V as D(M × U, M × V). Choose representatives U_{s} of isomorphism classes of irreducible representations of G, and assume that U_{e} = 𝟙 for some index e. We write H_{s} instead of H_{U_{s}}. Consider the linear space

B = ⨁_{s}(H̄_{s} ⊗ D(𝟙, U_{s})).  (1.2)

We may assume that Rep G is small and consider also the much larger linear space

B̃ = ⨁_{U}(H̄_{U} ⊗ D(𝟙, U)),  (1.3)

where the summation is over all objects in Rep G. Define a linear map π: B̃ → B as follows. Given a finite dimensional unitary representation U, choose isometries w_{i}: H_{s_{i}} → H_{U} defining a decomposition of U into irreducibles. Then, for ξ̄⊗ T ∈ H̄_{U} ⊗ D(𝟙, U), put

π(ξ̄⊗ T) = ∑_{i} \overline{w_{i}^{∗}ξ} ⊗ w_{i}^{∗}T.

This map is independent of any choices. The space B̃ is an associative algebra with product

(ξ̄⊗ T) · (ζ̄ ⊗ S) = \overline{(ξ ⊗ ζ)} ⊗ (T ⊗ ι)S.

This product defines a product on B such that π(x)π(y) = π(x · y) for all x, y ∈ B̃. In order to define the ∗-structure on B, first define an antilinear map • on B̃ by

(ξ̄⊗ T)^{•} = \overline{\overline{ρ^{−1/2}ξ}} ⊗ (T^{∗} ⊗ ι)R̄_{U} for ξ̄⊗ T ∈ H̄_{U} ⊗ D(𝟙, U).  (1.4)

This map does not define an involution on B̃, but on B we get an involution such that π(x)^{∗} = π(x^{•}) for all x ∈ B̃. The ∗-algebra B has a natural left ℂ[G]-comodule structure defined by the map α: B → ℂ[G] ⊗ B such that if U is a finite dimensional unitary representation of G, {ξ_{i}}_{i} is an orthonormal basis in H_{U} and u_{ij} are the matrix coefficients of U in this basis, then

α(π(ξ̄_{i} ⊗ T)) = ∑_{j} u_{ij} ⊗ π(ξ̄_{j} ⊗ T).  (1.5)

It is shown then that the action α is algebraic in the sense of [DCY13, Definition 4.2], meaning that the fixed point algebra A = B^{G} ≅ End_{D}(𝟙) is a unital C^{∗}-algebra and the conditional expectation (h ⊗ ι)α: B → A is positive and faithful. It follows that there is a unique completion of B to a C^{∗}-algebra B such that α extends to an action of the reduced form of G on B. This finishes the construction of an action from a module category. Example 1.2. Consider the action of G on itself by left translations, so we consider the coproduct map of C(G) as an action of G on C(G). It corresponds to the module category D = Hilb_{f} of finite dimensional Hilbert spaces, with the distinguished object ℂ, considered as a (Rep G)-module category using the forgetful tensor functor U ↦ H_{U}. Explicitly, identifying D(𝟙, H_{U}) with H_{U}, we get an isomorphism of the algebra B̃ constructed from the pair (Hilb_{f}, ℂ) onto { \atop ℂ[G]} = ⨁_{U} H̄_{U} ⊗ H_{U}, and then an isomorphism B ≅ ℂ[G] such that π: B̃ → B turns into the map π_{G}: { \atop ℂ[G]} → ℂ[G] that sends ξ̄⊗ ζ ∈ H̄_{U} ⊗ H_{U} into the matrix coefficient ((· ζ, ξ) ⊗ ι)(U) of U. ♢ Returning to the general case, consider the action α: B → C(G) ⊗ B defined by a pair (D, M) as described above. The equivalence between the (Rep G)- module categories D and D_{B} can be very concretely described as follows. First of all, as we have discussed, by replacing D by an equivalent category we may assume that it is the idempotent completion of Rep G with new morphisms sets. Similarly, instead of D_{B} we consider the category C_{B}. Then in order to define an equivalence we just have to describe the isomorphisms D(U, V) ≅ C_{B}(U, V). The equivalence between D and C_{B} constructed in the proof of [Nes, Theorem 2.3] (see also Section 3 there) has the property that a morphism T ∈ D(𝟙, V) is mapped into

∑_{j} ζ_{j} ⊗ π(ζ̄_{j} ⊗ T) ∈ C_{B}(𝟙, V) ⊂ B(ℂ, H_{V}) ⊗ B,

where {ζ_{j}}_{j} is an orthonormal basis in H_{V} and we identify B(ℂ, H_{V}) ⊗ B with H_{V} ⊗ B. Now assume that we have a morphism T ∈ D(U, V). We can write it as (ι ⊗ R_{U}^{∗})(S ⊗ ι), with S = (T ⊗ ι)R̄_{U} ∈ D(𝟙, V ⊗ Ū). Choose an orthonormal basis {ξ_{i}}_{i} in H_{U}. Then the morphism S ⊗ ι_{U} defines the element

∑_{i,j} ζ_{j}⊗ξ̄_{i}⊗1⊗π(\overline{(ζ_{j} ⊗ ξ̄_{i}})⊗S) ∈ C_{B}(U, V ⊗Ū⊗U) ⊂ B(H_{U}, H_{V} ⊗H̄_{U} ⊗H_{U})⊗B,

where we identify B(H_{U}, H_{V} ⊗ H̄_{U} ⊗ H_{U}) with H_{V} ⊗ H̄_{U} ⊗ B(H_{U}). It follows that T = (ι ⊗ R_{U}^{∗})(S ⊗ ι) is mapped into

∑_{ij} R_{U}^{∗}(ξ_{i} ⊗ ·)ζ_{j} ⊗ π(\overline{(ζ_{j} ⊗ ξ̄_{i})} ⊗ S) ∈ C_{B}(U, V) ⊂ B(H_{U}, H_{V}) ⊗ B.

Since R_{U}^{∗}(ξ_{i} ⊗ ξ) = (ρ^{−1/2}ξ, ξ_{i}), we conclude that the isomorphism D(U, V) ≅ C_{B}(U, V) is such that

D(U, V) → C_{B}(U, V), T ↦ ∑_{i,j} θ_{ζ_{j},ξ_{i}}π_{U}(ρ^{−1/2}) ⊗ π(\overline{(ζ_{j} ⊗ ξ̄_{i})} ⊗ (T ⊗ ι)R̄_{U}),

where θ_{ζ_{j},ξ_{i}} ∈ B(H_{U}, H_{V}) is the operator defined by θ_{ζ_{j},ξ_{i}}ξ = (ξ, ξ_{i})ζ_{j}. This can also be written as

T ↦ ∑_{i,j} θ_{ζ_{j},ξ_{i}} ⊗ π(\overline{(ζ_{j} ⊗ \overline{ρ^{−1/2}ξ_{i}})} ⊗ (T ⊗ ι)R̄_{U}).  (1.6)

1.5 Yetter–Drinfeld algebras

Assume we have a continuous left action of α: B → C(G) ⊗ B of a compact quantum group G on a unital C^{∗}-algebra B, as well as a continuous right action β: B → M(B ⊗ c_{0}(Ĝ)) of the dual discrete quantum group Ĝ. The action β defines a left ℂ[G]-module algebra structure ⊳: ℂ[G] ⊗ B → B on B by

x ⊳ a = (ι ⊗ x)β(x) for x ∈ ℂ[G] and a ∈ B.

Here we view c_{0}(Ĝ) as a subalgebra of U(G) = ℂ[G]^{∗}. This structure is compatible with involution, in the sense that

x ⊳ a^{∗} = (S(x)^{∗} ⊳ a)^{∗}.  (1.7)

We say that B is a Yetter–Drinfeld G-C^{∗}-algebra if the following identity holds for all x ∈ ℂ[G] and a ∈ B:

α(x ⊳ a) = x_{(1)}a_{(1)}S(x_{(3)}) ⊗ (x_{(2)} ⊳ a_{(2)}),  (1.8)

where we use Sweedler’s sumless notation, so we write Δ(x) = x_{(1)} ⊗ x_{(2)} and α(a) = a_{(1)} ⊗ a_{(2)}. Note that the above identity implies that B ⊂ B is a submodule over ℂ[G]. Yetter–Drinfeld G-C^{∗}-algebras can be regarded as D(G)-C^{∗}-algebras for the Drinfeld double D(G) of G, and they are studied in the more general setting of locally compact quantum groups by Nest and Voigt [NV10]. It is not difficult to see that our definition is equivalent to theirs,^{‡} but the case of compact quantum groups allows for the above familiar algebraic formulation, which is more convenient for our purposes. In the case of reduced compact quantum groups we can make it purely algebraic by getting rid of the right action β altogether.

‡It should also be taken into account that the definition of coproduct on C_{0}(Ĝ) used in the theory of locally compact quantum groups is opposite to the one usually used for compact quantum groups.

Documenta Mathematica 19 (2014) 1105–1139

Proposition 1.3. Assume that G is a reduced compact quantum group and α: B → C(G) ⊗ B is a continuous action of G on a unital C^{∗}-algebra B. Let B ⊂ B be the subalgebra of regular elements. Suppose that B is also a left ℂ[G]-module algebra such that conditions (1.7) and (1.8) are satisfied for all x ∈ ℂ[G] and a ∈ B. Then there exists a unique continuous right action β: B → M(B ⊗ c_{0}(Ĝ)) such that x ⊳ a = (ι ⊗ x)β(a) for all x ∈ ℂ[G] and a ∈ B.

Proof. Let us show first that for any finite dimensional unitary representation U = ∑_{i,j} m_{ij} ⊗ u_{ij} of G, where m_{ij} are matrix units in B(H_{U}), there exists a unital ∗-homomorphism β_{U}: B → B ⊗ B(H_{U}) such that

β_{U}(a) = ∑_{i,j}(u_{ij} ⊳ a) ⊗ m_{ij} for all a ∈ B.

From the assumption that B is a ℂ[G]-module algebra we immediately get that β_{U}: B → B ⊗ B(H_{U}) is a unital homomorphism. Condition (1.7) implies that this homomorphism is ∗-preserving. Thus, all we have to do is to show that β_{U} extends to a ∗-homomorphism B → B ⊗ B(H_{U}). For this observe that the Yetter–Drinfeld condition (1.8) implies that

(α ⊗ ι)β_{U}(a) = U_{31}(ι ⊗ β_{U})α(a)U_{31}^{∗}.

It follows that if we let B_{U} to be the norm closure of β_{U}(B) in B ⊗ B(H_{U}), then the restriction of the map

B ⊗ B(H_{U}) → C(G) ⊗ B ⊗ B(H_{U}), y ↦ U_{31}^{∗}(α ⊗ ι)(y)U_{31},

to B_{U} gives us a well-defined unital ∗-homomorphism γ: B_{U} → C(G) ⊗ B_{U}. Furthermore, since γ(β_{U}(a)) = (ι ⊗ β_{U})α(a) for a ∈ B, the map γ defines a continuous action of G on B_{U}. It follows that if we define a new C^{∗}-norm ∥ · ∥′ on B by

∥a∥′ = max{∥a∥, ∥β_{U}(a)∥},

then the action α of G on B extends to a continuous action on the completion of B in this norm. But according to [DCY13, Proposition 4.4] a C^{∗}-norm with such property is unique. Hence ∥a∥′ = ∥a∥ for all a ∈ B, and therefore the map β_{U} extends by continuity to B. Since c_{0}(Ĝ) ≅ c_{0}- ⨁_{s} B(H_{s}), the homomorphisms β_{U_{s}} define a unital ∗- homomorphism β: B → M(B ⊗ c_{0}(Ĝ)) = ℓ^{∞}- ⨁_{s}(B ⊗ B(H_{s})) such that (ι ⊗ x)β(a) = x ⊳ a for all x ∈ ℂ[G] and a ∈ B. It is then straightforward to check that β is a continuous action. The uniqueness is also clear.

2 Yetter–Drinfeld algebras and tensor functors

In this section we prove our main result, a categorical description of a class of Yetter–Drinfeld C^{∗}-algebras.

2.1 Two categories

A Yetter–Drinfeld G-C^{∗}-algebra B is said to be braided-commutative if for all a, b ∈ B we have

ab = b_{(2)}(S^{−1}(b_{(1)}) ⊳ a).  (2.1)

When b is in the fixed point algebra A = B^{G}, the right hand side reduces to ba, and we see that A is contained in the center of B. The following theorem is our principal result. A closely related result in the purely algebraic framework has been obtained by Bruguières and Natale [BN11].

Theorem 2.1. Let G be a reduced compact quantum group. Then the following two categories are equivalent:

(i) The category YD_{brc}(G) of unital braided-commutative Yetter–Drinfeld G-C^{∗}-algebras with unital G- and Ĝ-equivariant ∗-homomorphisms as morphisms.

(ii) The category Tens(Rep G) of pairs (C, E), where C is a C^{∗}-tensor category and E: Rep G → C is a unitary tensor functor such that C is generated by the image of E. The set of morphisms (C, E) → (C′, E′) in this category is the set of equivalence classes of pairs (F, η), where F is a unitary tensor functor F: C → C′ and η is a natural unitary monoidal isomorphism η: FE → E′.

Moreover, given a morphism [(F, η)]: (C, E) → (C′, E′), the corresponding homomorphism of Yetter–Drinfeld C^{∗}-algebras is injective if and only if F is faithful, and it is surjective if and only if F is full.

The condition that C is generated by the image of E means that any object in C is isomorphic to a subobject of E(U) for some U ∈ Rep G. We remind the reader that we assume that C^{∗}-categories are closed under subobjects. We also stress that we do not assume that the unit in C is simple. In fact, as will be clear from the proof, the C^{∗}-algebra End_{C}(𝟙) is exactly the fixed point algebra B^{G} in the C^{∗}-algebra B corresponding to (C, E). We have to explain how we define the equivalence relation on pairs (F, η). Assume (F, η) is a pair consisting of a unitary tensor functor F: C → C′ and a natural unitary monoidal isomorphism η: FE → E′. Then, for all objects U and V in Rep G, we get linear maps

C(E(U), E(V)) → C′(E′(U), E′(V)), T ↦ η_{V} F(T)η_{U}^{−1}.

We say that two pairs (F, η) and (F̃, η̃) are equivalent, if the corresponding maps C(E(U), E(V)) → C′(E′(U), E′(V)) are equal for all U and V.

A somewhat more concrete way of thinking of the category Tens(Rep G) of pairs (C, E) is as follows. Assume (C, E) is such a pair. First of all observe that the functor E is automatically faithful by semisimplicity and existence of conjugates in Rep G. Then replacing the pair (C, E) by an isomorphic one, we may assume that C is a strict C^{∗}-tensor category containing Rep G and E is simply the embedding functor. Namely, similarly to our discussion in Section 1.4, define new sets of morphisms between objects U and V in Rep G as C(E(U), E(V)) and then complete the new category we thus obtain with respect to subobjects. Assume now that we have two strict C^{∗}-tensor categories C and C′ containing Rep G, and consider the embedding functors E: Rep G → C and E′: Rep G → C′. Assume [(F, η)]: (C, E) → (C′, E′) is a morphism. This means that the unitary isomorphisms η_{U}: F(U) → U in C′ are such that F(T) = η_{V}^{−1}T η_{U} for any morphism T: U → V in Rep G, and the morphisms

F_{2;U,V} : F(U) ⊗ F(V) → F(U ⊗ V)

defining the tensor structure of F are given by F_{2;U,V} = η_{U◯_{⊤}V}^{−1} (η_{U} ⊗ η_{V}). We can then define a new unitary tensor functor F̃ from the full subcategory of C formed by the objects in Rep G ⊂ C into C′ by letting F̃(U) = U, F̃(T) = η_{V} F(T)η_{U}^{−1} for T ∈ C(U, V), and F̃_{2;U,V} = ι. This functor can be extended to C, by sending any subobject X ⊂ U with corresponding projection p_{X} ∈ End_{C}(U) to an object corresponding to the projection F̃(p_{X}) ∈ End_{C′}(U). Such an extension is unique up to a natural unitary monoidal isomorphism. Then by definition [(F, η)] = [(F̃, ι)]. Therefore morphisms (C, E) → (C′, E′) are equivalence classes of unitary tensor functors F: C → C′ such that F is the identity functor on Rep G ⊂ C and F_{2;U,V} = ι for all objects U and V in Rep G. Two such functors F and G are equivalent, or in other words they define the same morphism, if F(T) = G(T) for all morphisms T ∈ C(U, V) and all objects U and V in Rep G.

The rest of this section is devoted to the proof of Theorem 2.1.

2.2 From Yetter–Drinfeld algebras to tensor categories

In this subsection the assumption that G is reduced will not be important. Assume that B is a braided-commutative Yetter–Drinfeld G-C^{∗}-algebra. Consider the category D_{B} of G-equivariant finitely generated right Hilbert Bmodules discussed in Section 1.3. Then D_{B} can be turned into a C^{∗}-tensor category. This construction is known for the D(G)-equivariant B-modules in the purely algebraic approach [CW94,CVOZ94], and our key observation is that the same formula works for the G-equivariant modules, see also [DMNO13, Section 3.7]. Let us say that, for X ∈ D_{B}, a vector ξ ∈ X is regular if δ_{X}(ξ) lies in the algebraic tensor product ℂ[G] ⊗_{alg} X.

Lemma 2.2. Assume that X is a G-equivariant finitely generated right Hilbert B-module, and X be its subspace of regular vectors. Then there exists a unique unital ∗-homomorphism π_{X}: B → End_{B}(X) such that π_{X}(a)ξ = ξ_{(2)}(S^{−1}(ξ_{(1)}) ⊳ a) for all a ∈ B and ξ ∈ X. Furthermore, we have δ_{X}(π_{X}(a)ξ) = (ι ⊗ π_{X})α(a)δ_{X}(ξ) for all a ∈ B and ξ ∈ X.

Proof. It suffices to consider the case X = H_{U} ⊗ B for an irreducible unitary representation U = U_{s} of G, since any other module embeds into a finite direct sum of such modules as a direct summand. Then, using the action β: B → M(B⊗c_{0}(Ĝ)) and the projection B⊗c_{0}(Ĝ) → B⊗B(H_{s}) ≅ B(H_{s})⊗B, we get a unital ∗-homomorphism π_{X}: B → End_{B}(H_{U} ⊗ B) = B(H_{U}) ⊗ B such that

π_{X}(a) = ∑_{i,j} m_{ij} ⊗ (u_{ij} ⊳ a),

where U = ∑_{i,j} m_{ij} ⊗ u_{ij} and m_{ij} are the matrix units in B(H_{U}) defined by an orthonormal basis {ξ_{i}}_{i} in H_{U}. In order to see that this gives the correct definition of π_{X}, take b ∈ B. Recalling definition (1.1) of δ_{H_{U} ⊗B}, we get

(ξ_{i} ⊗ b)_{(1)} ⊗ (ξ_{i} ⊗ b)_{(2)} = ∑_{j} u_{ij}^{∗}b_{(1)} ⊗ (ξ_{j} ⊗ b_{(2)}).

Hence

(ξ_{i} ⊗ b)_{(2)}(S^{−1}((ξ_{i} ⊗ b)_{(1)}) ⊳ a) = ∑_{j} ξ_{j} ⊗ b_{(2)}(S^{−1}(u_{ij}^{∗}b_{(1)}) ⊳ a)

= ∑_{j} ξ_{j} ⊗ (S^{−1}(u_{ij}^{∗}) ⊳ a)b,

where the last equality follows by braided commutativity. Since S^{−1}(u_{ij}^{∗}) = u_{ji}, we see that π_{X}(a) acts as stated in the formulation of the lemma. In order to show that δ_{X}(π_{X}(a)ξ) = (ι⊗π_{X})α(a)δ_{X}(ξ) we take an arbitrary X. It suffices to consider a ∈ B. Then for ξ ∈ X we have

δ_{X}(π_{X}(a)ξ) = δ_{X}(ξ_{(2)}(S^{−1}(ξ_{(1)}) ⊳ a))
= ξ_{(2)}(S^{−1}(ξ_{(1)}) ⊳ a)_{(1)} ⊗ ξ_{(3)}(S^{−1}(ξ_{(1)}) ⊳ a)_{(2)}.

Applying the Yetter–Drinfeld condition (1.8) we see that the last expression equals

ξ_{(4)}S^{−1}(ξ_{(3)})a_{(1)}ξ_{(1)} ⊗ ξ_{(5)}(S^{−1}(ξ_{(2)}) ⊳ a_{(2)})

= a_{(1)}ξ_{(1)} ⊗ ξ_{(3)}(S^{−1}(ξ_{(2)}) ⊳ a_{(2)}),

and this is exactly (ι ⊗ π_{X})α(a)δ_{X}(ξ).

If X is as in the lemma, we conclude that X has the structure of an Gequivariant B-B-correspondence. If f is a G-equivariant endomorphism of the right Hilbert B-module X, it is automatically a B-bimodule map because of the way the left action of B is defined. Therefore the category D_{B} can be considered as a full subcategory of the C^{∗}-category of G-equivariant B-B-correspondences. The latter category has a natural C^{∗}-tensor structure. In order to show that D_{B} forms a C^{∗}-tensor subcategory it suffices to show that, given objects X and Y in D_{B}, we have:

(i) X ⊗_{B} Y is a finitely generated right B-module;

(ii) the left B-module structure on X ⊗_{B} Y induced by that on X coincides with the left B-module structure given by Lemma 2.2 using the action of G and the right B-module structure on X ⊗_{B} Y.

The second property is a routine computation similar to the one in the proof of the second part of Lemma 2.2, so we omit it. In order to check (i) it suffices to consider modules of the form H_{U} ⊗ B. For such modules we have the following more precise result.

Lemma 2.3. For any finite dimensional unitary representations U and V of G, the map

T_{U,V} : (H_{V} ⊗ B) ⊗_{B} (H_{U} ⊗ B) → H_{U◯_{⊤}V} ⊗ B,

(ζ ⊗ b) ⊗ (ξ ⊗ a) ↦ ξ_{(2)} ⊗ ζ ⊗ (S^{−1}(ξ_{(1)}) ⊳ b)a,

is a G-equivariant unitary isomorphism of right Hilbert B-modules. Furthermore, the isomorphisms T_{U,V} have the property T_{U◯_{⊤}V,W} (ι ⊗ T_{U,V}) =

T_{U,V ◯_{⊤}W} (T_{V,W} ⊗ ι).

Recall that the ℂ[G]-comodule structure on H_{U} is given by ξ ↦ ξ_{(1)} ⊗ ξ_{(2)} = U_{21}^{∗}(1 ⊗ ξ).

Proof of Lemma 2.3. Since ξ_{(2)} ⊗ (S^{−1}(ξ_{(1)}) ⊳ b) = b(ξ ⊗ 1), it is clear that the map T_{U,V} defines a right B-module isomorphism

(H_{V} ⊗ B) ⊗_{B} (H_{U} ⊗ B) ≅ H_{U◯_{⊤}V} ⊗ B.

It is also obvious that T_{U,V} is isometric on the subspace spanned by vectors of the form (ζ ⊗ 1) ⊗ (ξ ⊗ 1). Since such vectors generate (H_{V} ⊗ B) ⊗_{B} (H_{U} ⊗ B) as a right B-module, and this module is dense in (H_{V} ⊗ B) ⊗_{B} (H_{U} ⊗ B), it follows that T_{U,V} extends by continuity to a unitary isomorphism of right Hilbert B-modules. Next let us check the G-equivariance. The ℂ[G]-comodule structure on

(H_{V} ⊗ B) ⊗_{B} (H_{U} ⊗ B)

is given by

δ((ζ ⊗ b) ⊗ (ξ ⊗ a)) = (ζ ⊗ b)_{(1)}(ξ ⊗ a)_{(1)} ⊗ (ζ ⊗ b)_{(2)} ⊗ (ξ ⊗ a)_{(2)}
= ζ_{(1)}b_{(1)}ξ_{(1)}a_{(1)} ⊗ (ζ_{(2)} ⊗ b_{(2)}) ⊗ (ξ_{(2)} ⊗ a_{(2)}).

Applying ι ⊗ T_{U,V} we get

ζ_{(1)}b_{(1)}ξ_{(1)}a_{(1)} ⊗ ξ_{(3)} ⊗ ζ_{(2)} ⊗ (S^{−1}(ξ_{(2)}) ⊳ b_{(2)})a_{(2)}.

On the other hand, using the same symbol δ for the comodule structure on

H_{U◯_{⊤}V} ⊗ B, since (U ◯_{⊤} V)^{∗} = V_{23}^{∗}U_{13}^{∗} we get

(ι ⊗ δ)T_{U,V} ((ζ ⊗ b) ⊗ (ξ ⊗ a))

= δ(ξ_{(2)} ⊗ ζ ⊗ (S^{−1}(ξ_{(1)}) ⊳ b)a)
= ζ_{(1)}ξ_{(2)}(S^{−1}(ξ_{(1)}) ⊳ b)_{(1)}a_{(1)} ⊗ ξ_{(3)} ⊗ ζ_{(2)} ⊗ (S^{−1}(ξ_{(1)}) ⊳ b)_{(2)}a_{(2)}.

Applying (1.8) we see that the last expression equals ζ_{(1)}ξ_{(2)}S^{−1}(ξ_{(1)})_{(1)}b_{(1)}S(S^{−1}(ξ_{(1)})_{(3)})a_{(1)} ⊗ ξ_{(3)} ⊗ ζ_{(2)} ⊗ (S^{−1}(ξ_{(2)}) ⊳ b_{(2)})a_{(2)}

= ζ_{(1)}ξ_{(4)}S^{−1}(ξ_{(3)})b_{(1)}ξ_{(1)}a_{(1)} ⊗ ξ_{(5)} ⊗ ζ_{(2)} ⊗ (S^{−1}(ξ_{(2)}) ⊳ b_{(2)})a_{(2)}
= ζ_{(1)}b_{(1)}ξ_{(1)}a_{(1)} ⊗ ξ_{(3)} ⊗ ζ_{(2)} ⊗ (S^{−1}(ξ_{(2)}) ⊳ b_{(2)})a_{(2)}.

Therefore the map T_{U,V} is indeed G-equivariant. Finally, in order to prove the equality T_{U◯_{⊤}V,W} (ι ⊗ T_{U,V}) = T_{U,V ◯_{⊤}W} (T_{V,W} ⊗ ι) it suffices to check it on tensor products of vectors of the form ξ ⊗ 1, since such tensor products generate a dense subspace of triple tensor products as right B-modules. But for such vectors the statement is obvious.

Therefore the category D_{B} can be considered as a full C^{∗}-tensor subcategory of the category of G-equivariant B-B-correspondences. In view of the previous lemma, it is convenient to replace the tensor product by the opposite one, so we put X × Y = Y ⊗_{B} X. Furthermore, the functor E_{B}: Rep G → C_{B} mapping U into the module H_{U} ⊗ B, together with the unitary isomorphisms T_{U,V} : E_{B}(U) ⊗ E_{B}(V) → E_{B}(U ◯_{⊤} V) from Lemma 2.3, is a unitary tensor functor. We have thus proved the following result.

Theorem 2.4. Let G be a compact quantum group and B be a unital braidedcommutative Yetter–Drinfeld G-C^{∗}-algebra. Then the G-equivariant finitely generated right Hilbert B-modules form a C^{∗}-tensor category D_{B} with tensor product X × Y = Y ⊗_{B} X. Furthermore, there is a unitary tensor functor E_{B}: Rep G → D_{B} mapping U to the module H_{U} ⊗ B.

Up to an isomorphism, the pair (D_{B}, E_{B}) can be more concretely described as follows. As we discussed in Section 1.3, the category D_{B} is equivalent to the category C_{B}, which is the idempotent completion of the category with the same objects as in Rep G, but with the new morphism sets

C_{B}(U, V) ⊂ B(H_{U}, H_{V}) ⊗ B

consisting of elements T such that V_{12}^{∗}(ι⊗α)(T)U_{12} = T_{13}. We define the tensor product of objects in C_{B} as in Rep G, and in order to completely describe the tensor structure it remains to write down a formula for the linear maps

C_{B}(U, V) ⊗ C_{B}(W, Z) → C_{B}(U ◯_{⊤} W, V ◯_{⊤} Z).

This can be done using Lemma 2.3. First, note that by the proof of Lemma 2.2, if U = ∑_{i,j} m_{ij} ⊗ u_{ij} then for any ξ ∈ H_{U} and b ∈ B we have

ξ_{(2)} ⊗ (S^{−1}(ξ_{(1)}) ⊳ b) = ∑_{ij}(m_{ij} ⊗ (u_{ij} ⊳ b))(ξ ⊗ 1).

Therefore the map T_{U,V} from Lemma 2.2 can be written as

(ζ ⊗ b) ⊗ (ξ ⊗ a) ↦ ∑_{i,j} m_{ij}ξ ⊗ ζ ⊗ (u_{ij} ⊳ b)a.

It follows that given T = ∑_{l} T_{l} ⊗ b_{l} ∈ C_{B}(W, Z), the morphism ι ⊗ T ∈

C_{B}(U ◯_{⊤} W, U ◯_{⊤} Z) considered as a map H_{U◯_{⊤}W} ⊗ B → H_{U◯_{⊤}Z} ⊗ B acts by

ξ ⊗ ζ ⊗ 1 ↦ T_{U,Z} (∑_{l}(T_{l}ζ ⊗ b_{l}) ⊗ (ξ ⊗ 1)) = ∑_{i,j,l} m_{ij}ξ ⊗ T_{l}ζ ⊗ (u_{ij} ⊳ b_{l}).

On the other hand, if S = ∑_{k} S_{k} ⊗ a_{k} ∈ C_{B}(U, V), then the morphism S ⊗ ι ∈

C_{B}(U ◯_{⊤} Z, V ◯_{⊤} Z) considered as a map H_{U◯_{⊤}W} ⊗ B → H_{U◯_{⊤}Z} ⊗ B acts by

ξ ⊗ ζ ⊗ 1 ↦ T_{V,Z} (∑_{k}(ζ ⊗ 1) ⊗ (S_{k}ξ ⊗ a_{k})) = ∑_{k} S_{k}ξ ⊗ ζ ⊗ a_{k}.

To summarize, the tensor structure on C_{B} is described by the following rules:

ι_{U} ⊗ T = ∑_{i,j,l} m_{ij} ⊗ T_{l} ⊗ (u_{ij} ⊳ b_{l}), if T = ∑_{l} T_{l} ⊗ b_{l} ∈ C_{B}(W, Z); (2.2)
S ⊗ ι_{Z} = S_{13}, if S ∈ C_{B}(U, V).  (2.3)

In this picture the functor E_{B}: Rep G → D_{B} becomes the strict tensor functor F_{B}: Rep G → C_{B} which is the identity map on objects, while on morphisms it is T ↦ T ⊗ 1.

2.3 From tensor categories to Yetter–Drinfeld algebras

Let us turn to the construction of a Yetter–Drinfeld algebra from a pair (C, E) ∈ Tens(Rep(G)). The category C can be considered as a right (Rep G)-module category with the distinguished object 𝟙. Therefore by Theorem 1.1 we can construct a C^{∗}-algebra B = B_{C} together with a left continuous action α: B → C(G) ⊗ B. Our goal is to prove the following.

Theorem 2.5. The G-C^{∗}-algebra B corresponding to the (Rep G)-module category C with the distinguished object 𝟙 has a natural structure of a braidedcommutative Yetter–Drinfeld C^{∗}-algebra.

The construction of the Yetter–Drinfeld structure can be described for any pair (C, E), but in order to simplify the notation we assume that C is strict, Rep G is a C^{∗}-tensor subcategory of C and E is simply the embedding functor. This is enough by the discussion following the formulation of Theorem 2.1. Recall from Section 1 that the subalgebra B ⊂ B of regular elements is given by (1.2). By Proposition 1.3, to prove the theorem we have to define a ℂ[G]- module algebra structure on B satisfying properties (1.7), (1.8), and (2.1). In Section 1 we also defined a ‘universal’ algebra B̃ = ⨁_{U}(H̄_{U} ⊗ C(𝟙, U)), together with a homomorphism π: B̃ → B. Recall from Example 1.2 that we denote by { \atop ℂ[G]} = ⨁_{U}(H̄_{U} ⊗ H_{U}) the algebra B̃ corresponding to the forgetful fiber functor Rep G → Hilb_{f}, and then the corresponding homomorphism

π_{G}: { \atop ℂ[G]} → ℂ[G] maps ξ̄⊗ ζ ∈ H̄_{U} ⊗ H_{U} into ((· ζ, ξ) ⊗ ι)(U).

Define a linear map

˜⊳: { \atop ℂ[G]} ⊗ B̃ → B̃

by letting, for ξ̄⊗ ζ ∈ H̄_{U} ⊗ H_{U} and η̄ ⊗ T ∈ H̄_{V} ⊗ C(𝟙, V),

(ξ̄⊗ ζ) ˜⊳ (η̄ ⊗ T) = \overline{(ξ ⊗ η ⊗ \overline{ρ^{−1/2}ζ})} ⊗ (ι ⊗ T ⊗ ι)R̄_{U}

∈ H̄_{U◯_{⊤}V ◯_{⊤}Ū} ⊗ C(𝟙, U ◯_{⊤} V ◯_{⊤} Ū). (2.4)

We remind that R̄_{U}: 𝟙 → U ◯_{⊤} Ū is given by R̄_{U}(1) = ∑_{i} ρ^{1/2}ξ_{i} ⊗ ξ̄_{i} for an orthonormal basis {ξ_{i}} in H_{U}. Identifying ℂ[G] with the subspace

⨁_{s}(H̄_{s} ⊗ H_{s}) ⊂ { \atop ℂ[G]},

we define a linear map

⊳: ℂ[G] ⊗ B → B by letting x ⊳ a = π(x ˜⊳ a) for x ∈ ℂ[G] and a ∈ B.

Lemma 2.6. The map ⊳ defines a left ℂ[G]-module algebra structure on B, and we have

π_{G}(x) ⊳ π(a) = π(x ˜⊳ a) for all x ∈ { \atop ℂ[G]} and a ∈ B̃.

Proof. We start with the second statement. We have to show that

π_{G}(x) ⊳ π(a) = π(x ˜⊳ a)

for x ∈ { \atop ℂ[G]} and a ∈ B̃. Take x = ξ̄ ⊗ ζ ∈ H̄_{U} ⊗ H_{U} and a = η̄ ⊗ T ∈

H̄_{V} ⊗ C(𝟙, V). Choose isometries u_{i}: H_{s_{i}} → H_{U} and v_{j}: H_{s_{j}} → H_{V} defining decompositions of U and V into irreducibles. Then

π_{G}(x) ⊳ π(a) = π (∑_{i,j}(\overline{u_{i}^{∗}ξ} ⊗ u_{i}^{∗}ξ) ˜⊳ (\overline{v_{j}^{∗}η} ⊗ v_{j}^{∗}T))

= π (∑_{i,j} \overline{(u_{i}^{∗}ξ ⊗ v_{j}^{∗}η ⊗ \overline{ρ^{−1/2}u_{i}^{∗}ζ})} ⊗ (ι ⊗ v_{j}^{∗}T ⊗ ι)R̄_{s_{i}} ) , (2.5)

Documenta Mathematica 19 (2014) 1105–1139

where R̄_{s_{i}} = R̄_{U_{s_{i}}}. On the other hand,

π(x ˜⊳ a) = π(\overline{(ξ ⊗ η ⊗ \overline{ρ^{−1/2}ζ})} ⊗ (ι ⊗ T ⊗ ι)R̄_{U})

= π (∑_{i,j,k} \overline{(u_{i}^{∗}ξ ⊗ v_{j}^{∗}η ⊗ ū_{k}^{∗}\overline{ρ^{−1/2}ζ})} ⊗ (u_{i}^{∗} ⊗ v_{j}^{∗}T ⊗ ū_{k}^{∗})R̄_{U} ) , (2.6)

where the morphism ū_{k}: HŪ_{s_{k}} = H̄_{s_{k}} → HŪ = H̄_{U} is defined by ū_{k}ξ̄ = \overline{u_{k}ξ}. Since u_{k}^{∗}π_{U}(ρ) = π_{U_{s_{k}}} (ρ)u_{k}^{∗}, R̄_{U} = ∑_{i}(u_{i} ⊗ū_{i})R̄_{s_{i}} and the partial isometries u_{i} have mutually orthogonal images, we see that expressions (2.5) and (2.6) are equal. In order to show that ⊳ defines a left ℂ[G]-module structure, take x = ξ̄⊗ ζ ∈ H̄_{U} ⊗ H_{U}, y = μ̄ ⊗ ν ∈ H̄_{W} ⊗ H_{W} and a = η̄ ⊗ T ∈ H̄_{V} ⊗ C(𝟙, V). Then

x ˜⊳ (y ˜⊳ a) = \overline{(ξ ⊗ μ ⊗ η ⊗ \overline{ρ^{−1/2}ν} ⊗ \overline{ρ^{−1/2}ζ})} ⊗ (ι ⊗ ι ⊗ T ⊗ ι ⊗ ι)(ι ⊗ R̄_{W} ⊗ ι)R̄_{U}

is an element in H̄_{U◯_{⊤}W ◯_{⊤}V ◯_{⊤}W̄ ◯_{⊤}Ū} ⊗ C(𝟙, U ◯_{⊤} W ◯_{⊤} V ◯_{⊤} W̄ ◯_{⊤} Ū), and

(x · y) ˜⊳ a = \overline{(ξ ⊗ μ ⊗ η ⊗ \overline{(ρ^{−1/2}ζ ⊗ ρ^{−1/2}ν)})} ⊗ (ι ⊗ T ⊗ ι)R̄_{U◯_{⊤}W}

is an element in H̄_{(U◯_{⊤}W)◯_{⊤}V ◯_{⊤}\overline{(U◯_{⊤}W)}} ⊗ C(𝟙, (U ◯_{⊤} W) ◯_{⊤} V ◯_{⊤} \overline{(U ◯_{⊤} W)}). The only

reason why these two elements are different is that the representations W̄ ◯_{⊤} Ū and \overline{U ◯_{⊤} W} are equivalent, but not equal. The map σ: H̄_{W} ⊗H̄_{U} → \overline{H_{U} ⊗ H_{W}}, σ(μ̄ ⊗ ξ̄) = \overline{ξ ⊗ μ} defines such an equivalence, and we have

R̄_{U◯_{⊤}W} = (ι ⊗ ι ⊗ σ)(ι ⊗ R̄_{W} ⊗ ι)R̄_{U}.

It follows that upon projecting to B we get an honest equality

π(x ˜⊳ (y ˜⊳ a)) = π((x · y) ˜⊳ a),

that is, π_{G}(x) ⊳ (π_{G}(y) ⊳ π(a)) = (π_{G}(x)π_{G}(y)) ⊳ π(a). It remains to show that ⊳ respects the algebra structure on B, that is, x⊳(ab) = (x_{(1)} ⊳ a)(x_{(2)} ⊳ b) for x ∈ ℂ[G] and a, b ∈ B. Take elements a = η̄ ⊗ T ∈ H̄_{V} ⊗ C(𝟙, V) and b = ζ̄ ⊗ S ∈ H̄_{W} ⊗ C(𝟙, W) in B̃. Let U be a finite dimensional unitary representation of G. Choose an orthonormal bases {ξ_{i}}_{i} in H_{U} and denote by u_{ij} the corresponding matrix coefficients of U. Since u_{ij} = π_{G}(ξ̄_{i} ⊗ ξ_{j}) and Δ(u_{ij}) = ∑_{k} u_{ik} ⊗ u_{kj}, we then have to show that

π((ξ̄_{i} ⊗ ξ_{j}) ˜⊳(a · b)) = ∑_{k} π(((ξ̄_{i} ⊗ ξ_{k}) ˜⊳a) · ((ξ̄_{k} ⊗ ξ_{j}) ˜⊳b)).

We have

(ξ̄_{i} ⊗ ξ_{j}) ˜⊳(a · b) = \overline{(ξ_{i} ⊗ η ⊗ ζ ⊗ \overline{ρ^{−1/2}ξ_{j}})} ⊗ (ι ⊗ T ⊗ S ⊗ ι)R̄_{U}.  (2.7)

Documenta Mathematica 19 (2014) 1105–1139

On the other hand,

∑_{k}((ξ̄_{i} ⊗ ξ_{k}) ˜⊳a) · ((ξ̄_{k} ⊗ ξ_{j}) ˜⊳b)

= ∑_{k} (\overline{(ξ_{i} ⊗ η ⊗ \overline{ρ^{−1/2}ξ_{k}})} ⊗ (ι ⊗ T ⊗ ι)R̄_{U}))

· (\overline{(ξ_{k} ⊗ ζ ⊗ \overline{ρ^{−1/2}ξ_{j}})} ⊗ (ι ⊗ S ⊗ ι)R̄_{U})

= ∑_{k} \overline{(ξ_{i} ⊗ η ⊗ \overline{ρ^{−1/2}ξ_{k}} ⊗ ξ_{k} ⊗ ζ ⊗ \overline{ρ^{−1/2}ξ_{j}})}

⊗ (ι ⊗ T ⊗ ι ⊗ ι ⊗ S ⊗ ι)(R̄_{U} ⊗ R̄_{U}).

Since ∑_{k} \overline{ρ^{−1/2}ξ_{k}} ⊗ ξ_{k} = R_{U}(1), and R_{U} is, up to a scalar factor, an isomeric embedding of 𝟙 into Ū ◯_{⊤} U, by applying π to the above expression we get

π(\overline{(ξ_{i} ⊗ η ⊗ ζ ⊗ \overline{ρ^{−1/2}ξ_{j}})} ⊗ (ι ⊗ T ⊗ R_{U}^{∗} ⊗ S ⊗ ι)(R̄_{U} ⊗ R̄_{U})).

Since (R_{U}^{∗} ⊗ι)(ι⊗R̄_{U}) = ι, this is exactly the expression we obtain by applying π to (2.7).

We next check compatibility (1.7) of ⊳ with the ∗-structure.

Lemma 2.7. For all x ∈ ℂ[G] and a ∈ B we have x ⊳ a^{∗} = (S(x)^{∗} ⊳ a)^{∗}.

Proof. Recall that the involution on B arises from the map • on B̃ defined by (1.4), so for a = η̄ ⊗ T ∈ H̄_{V} ⊗ C(𝟙, V) we have

a^{•} = \overline{\overline{ρ^{−1/2}η}} ⊗ (T^{∗} ⊗ ι)R̄_{V} ∈ H̄V̄ ⊗ C(𝟙, V̄).

Let us also define an antilinear map † on { \atop ℂ[G]} by letting, for x = ξ̄ ⊗ ζ ∈ H̄_{U} ⊗ H_{U},

x^{†} = ζ̄ ⊗ ξ.

We then have π_{G}(x^{†}) = S(π_{G}(x))^{∗}. Indeed, using that (ι ⊗ S)(U) = U^{∗}, we compute:

S(π_{G}(x))^{∗} = S(((· ζ, ξ) ⊗ ι)(U))^{∗} = ((· ζ, ξ) ⊗ ι)(U^{∗}))^{∗}

= ((· ξ, ζ) ⊗ ι)(U) = π_{G}(x^{†}).

Turning now to the proof of the lemma, we have to show that

π(x ˜⊳ a^{•}) = π((x^{†} ˜⊳ a)^{•}).

We compute:

x ˜⊳ a^{•} = (ξ̄⊗ ζ) ˜⊳ (\overline{\overline{ρ^{−1/2}η}} ⊗ (T^{∗} ⊗ ι)R̄_{V} )
= \overline{(ξ ⊗ \overline{ρ^{−1/2}η} ⊗ \overline{ρ^{−1/2}ζ})} ⊗ (ι ⊗ T^{∗} ⊗ ι ⊗ ι)(ι ⊗ R̄_{V} ⊗ ι)R̄_{U} (2.8)

Documenta Mathematica 19 (2014) 1105–1139

and

(x^{†} ˜⊳ a)^{•} = ((ζ̄ ⊗ ξ) ˜⊳ a)^{•} = (\overline{(ζ ⊗ η ⊗ \overline{ρ^{−1/2}ξ})} ⊗ (ι ⊗ T ⊗ ι)R̄_{U})^{•}

= \overline{\overline{(ρ^{−1/2}ζ ⊗ ρ^{−1/2}η ⊗ ξ̄)}} ⊗ (((ι ⊗ T ⊗ ι)R̄_{U})^{∗} ⊗ ι\overline{U◯_{⊤}V ◯_{⊤}Ū})R̄_{U◯_{⊤}V ◯_{⊤}Ū}, (2.9)

where we used that πŪ(ρ) = j(π_{U}(ρ))^{−1}, that is, ρξ̄ = \overline{ρ^{−1}ξ}. Similarly to the proof of the previous lemma, the main reason why expressions (2.8) and (2.9) are not equal is that the representations U ◯_{⊤} V̄ ◯_{⊤} Ū and \overline{U ◯_{⊤} V ◯_{⊤} Ū} are equivalent, but not equal. The map σ(ξ ⊗ η̄ ⊗ ζ̄) = \overline{ζ ⊗ η ⊗ ξ̄} defines such an equivalence, and then

R̄_{U◯_{⊤}V ◯_{⊤}Ū} = (ι ⊗ ι ⊗ ι ⊗ σ)(ι ⊗ ι ⊗ R_{U} ⊗ ι ⊗ ι)(ι ⊗ R̄_{V} ⊗ ι)R̄_{U}.

Since (R̄_{U}^{∗} ⊗ ι)(ι ⊗ R_{U}) = ι, we get

(((ι ⊗ T ⊗ ι)R̄_{U})^{∗} ⊗ ι)R̄_{U◯_{⊤}V ◯_{⊤}Ū} = σ(ι ⊗ T^{∗} ⊗ ι ⊗ ι)(ι ⊗ R̄_{V} ⊗ ι)R̄_{U}.

From this we see that upon applying π expressions (2.8) and (2.9) indeed become equal.

Our next goal is to check the Yetter–Drinfeld condition (1.8).

Lemma 2.8. For all x ∈ ℂ[G] and a ∈ B we have

α(x ⊳ a) = x_{(1)}a_{(1)}S(x_{(3)}) ⊗ (x_{(2)} ⊳ a_{(2)}).

Proof. Let U and V be finite dimensional unitary representations of G. Choose orthonormal bases {ξ_{i}}_{i} in H_{U} and {η_{k}}_{k} in H_{V}, and let u_{ij} and v_{kl} be the matrix coefficients of U and V, respectively. In order to simplify the computations assume that the vectors ξ_{i} are eigenvectors of ρ, so ρξ_{i} = ρ_{i}ξ_{i} for some positive number ρ_{i}. Then the matrix coefficients of Ū in the basis {ξ̄_{i}}_{i} are given by

ū_{ij} = ρ_{i}^{1/2}ρ_{j}^{−1/2}u_{ij}^{∗} = ρ_{i}^{1/2}ρ_{j}^{−1/2}S(u_{ji}).  (2.10)

Consider elements x = u_{i_{0}j_{0}} ∈ ℂ[G] and a = π(η_{k_{0}} ⊗ T) ∈ B for some T ∈ C(𝟙, V). Recalling definition (1.5) of the action α, we have

α(a) = ∑_{k} v_{k_{0}k} ⊗ π(η̄_{k} ⊗ T).

It follows that

x_{(1)}a_{(1)}S(x_{(3)}) ⊗ (x_{(2)} ⊳ a_{(2)}) = ∑_{i,j,k} u_{i_{0}i}v_{k_{0}k}S(u_{jj_{0}}) ⊗ (u_{ij} ⊳ π(η̄_{k} ⊗ T))

= ∑_{i,j,k} ρ_{j}^{−1/2}u_{i_{0}i}v_{k_{0}k}S(u_{jj_{0}}) ⊗ π(\overline{(ξ_{i} ⊗ η_{k} ⊗ ξ̄_{j})} ⊗ (ι ⊗ T ⊗ ι)R̄_{U}). (2.11)

Documenta Mathematica 19 (2014) 1105–1139

On the other hand,

α(x ⊳ a) = ρ_{j_{0}}^{−1/2}α(π(\overline{(ξ_{i_{0}} ⊗ η_{k_{0}} ⊗ ξ̄_{j_{0}})} ⊗ (ι ⊗ T ⊗ ι)R̄_{U}))

= ρ_{j_{0}}^{−1/2} ∑_{i,j,k} u_{i_{0}i}v_{k_{0}k}ū_{j_{0}j} ⊗ π(\overline{(ξ_{i} ⊗ η_{k} ⊗ ξ̄_{j})} ⊗ (ι ⊗ T ⊗ ι)R̄_{U}). (2.12)

Since ρ_{j}^{−1/2}S(u_{jj_{0}}) = ρ_{j_{0}}^{−1/2}ū_{j_{0}j}, we see that expressions (2.11) and (2.12) are equal.

It remains to check the braided commutativity condition (2.1).

Lemma 2.9. For all and a, b ∈ B we have ab = b_{(2)}(S^{−1}(b_{(1)}) ⊳ a).

Proof. Let U, V, {ξ_{i}}_{i}, u_{ij}, ū_{ij} be as in the proof of the previous lemma. Note that by swapping the roles of U and Ū in (2.10) we get

S^{−1}(u_{ij}) = ρ_{i}^{−1/2}ρ_{j}^{1/2}ū_{ji}.

(Recall again that ρξ̄_{i} = \overline{ρ^{−1}ξ_{i}}.) Using this, take P ∈ C(𝟙, U), η̄ ⊗ T ∈ H̄_{V} ⊗ C(𝟙, V) and for a = π(η̄ ⊗ T) and b = π(ξ̄_{i} ⊗ P) compute:

b_{(2)}(S^{−1}(b_{(1)}) ⊳ a) = ∑_{j} π(ξ̄_{j} ⊗ P)(S^{−1}(u_{ij}) ⊳ π(η̄ ⊗ T))

= ∑_{j} ρ_{i}^{−1/2}ρ_{j}^{1/2}π(ξ̄_{j} ⊗ P)π((¯ξ̄_{j} ⊗ ξ̄_{i}) ˜⊳ (η̄ ⊗ T))

= ∑_{j} ρ_{i}^{−1/2}ρ_{j}^{1/2}π(ξ̄_{j} ⊗ P)

π(\overline{(ξ̄_{j} ⊗ η ⊗ \overline{ρ^{−1/2}ξ̄_{i}})} ⊗ (ι ⊗ T ⊗ ι)R̄Ū)

= ∑_{j} ρ_{j}^{1/2}π(\overline{(ξ_{j} ⊗ ξ̄_{j} ⊗ η ⊗ ¯ ξ̄_{i})} ⊗ (P ⊗ (ι ⊗ T ⊗ ι)R̄Ū)).

Denote by w the map ξ ↦ ¯ ξ̄ defining an equivalence between U and ¯ Ū. Then R̄Ū = (ι ⊗ w)R_{U}. Hence the above expression equals

∑_{j} ρ_{j}^{1/2}π(\overline{(ξ_{j} ⊗ ξ̄_{j} ⊗ η ⊗ ξ_{i})} ⊗ (P ⊗ (ι ⊗ T ⊗ ι)R_{U}))

= π(\overline{(R̄_{U}(1) ⊗ η ⊗ ξ_{i})} ⊗ (ι ⊗ ι ⊗ T ⊗ ι)(ι ⊗ R_{U})P ).

Since R̄_{U} is, up to a scalar factor, an isometric embedding of 𝟙 into U ◯_{⊤} Ū, the last expression equals

π(\overline{(η ⊗ ξ_{i})} ⊗ (R̄_{U}^{∗} ⊗ T ⊗ ι)(ι ⊗ R_{U})P ) = π(\overline{(η ⊗ ξ_{i})} ⊗ (T ⊗ P)).

But this is exactly ab.

This finishes the proof of Theorem 2.5.

2.4 Functoriality

Consider the category YD_{brc}(G) of unital braided-commutative Yetter–Drinfeld G-C^{∗}-algebras. For every object B we have constructed isomorphic pairs (D_{B}, E_{B}) and (C_{B}, F_{B}). Using the extension of scalars functor discussed at the end of Section 1.3, either of this constructions extends to a functor, giving us two naturally isomorphic functors T : YD_{brc}(G) → Tens(Rep G) and T̃: YD_{brc}(G) → Tens(Rep G). Namely, giving a morphism f: B_{0} → B_{1} we have a functor f_{#} : D_{B_{0}} → D_{B_{1}} which maps a G-equivariant finitely generated right Hilbert B_{0}-module X into X ⊗_{B_{0}} B_{1}. We define a tensor structure on this functor by using the isomorphisms

(X ⊗_{B_{0}} B_{1}) ⊗_{B_{1}} (Y ⊗_{B_{0}} B_{1}) ≅ (X ⊗_{B_{0}} Y) ⊗_{B_{0}} B_{1}

such that (x ⊗ a) ⊗ (y ⊗ b) ↦ x ⊗ y_{(2)} ⊗ (S^{−1}(y_{(1)}) ⊳ a)b. That these maps are indeed well-defined and that they give us a tensor structure on f_{#}, is not difficult to check using arguments similar to those in the proof of Lemma 2.3. The tensor functor f_{#} together with the obvious isomorphisms

η_{U}: (H_{U} ⊗ B_{0}) ⊗_{B_{0}} B_{1} → H_{U} ⊗ B_{1}

define a morphism (D_{B_{0}}, E_{B_{0}}) → (D_{B_{1}}, E_{B_{1}}). If we consider the map B ↦ (C_{B}, F_{B}) instead of B ↦ (D_{B}, E_{B}), then the situation is even better: in this case the functor f_{#} : C_{B_{0}} → C_{B_{1}} defined by a morphism f: B_{0} → B_{1} is a strict tensor functor, meaning that f_{#}(T ⊗ S) = f_{#}(T) ⊗ f_{#}(S) on morphisms. This follows immediately from equations (2.2) and (2.3) describing the tensor structure on the categories C_{B}. Let us now construct a functor S in the opposite direction. It is possible to define this functor on the whole category Tens(Rep G), but in order to simplify notation we will construct it only on the full subcategory Tens_{si}(Rep G) consisting of pairs (C, E) such that C is a strict C^{∗}-tensor category containing Rep G, C is generated by Rep G, and that E is the embedding functor. Since the embedding functor Tens_{si}(Rep G) → Tens(Rep G) is an equivalence of categories, any functor Tens_{si}(Rep G) → YD_{brc}(G) extends to Tens(Rep G), and this extension is unique up to a natural isomorphism. Given two objects (C_{0}, E_{0}) and (C_{1}, E_{1}) in Tens_{si}(Rep G), consider the corresponding Yetter–Drinfeld C^{∗}-algebras B_{0} and B_{1}, and take a morphism [(F, η)]: (C_{0}, E_{0}) → (C_{1}, E_{1}). As we discussed after the formulation of Theorem 2.1, we may assume that the restriction of F to Rep G ⊂ C_{0} is the identity tensor functor and η_{U} = ι. In this case it is obvious from the construction of the algebras B_{i} that the maps H̄_{U} ⊗ C_{0}(𝟙, U) → H̄_{U} ⊗ C_{1}(𝟙, U), ξ̄⊗ T ↦ ξ̄⊗ F(T), define a unital ∗-homomorphism B_{0} → B_{1} that respects the ℂ[G]-comodule and ℂ[G]-module structures. It extends to a homomorphism f of C^{∗}-algebras by [DCY13, Proposition 4.5]. It is also clear by our definition of morphisms in the category of pairs (C, E) that f depends only on the equivalence class of (F, ι). We thus get a functor S: Tens_{si}(Rep G) → YD_{brc}(G).

Furthermore, it is clear from the construction that the morphism f: B_{0} → B_{1} defined by a morphism [(F, ι)]: (C_{0}, E_{0}) → (C_{1}, E_{1}) is injective if and only if the maps C_{0}(𝟙, U_{s}) → C_{1}(𝟙, U_{s}), T ↦ F(T), are injective for all s, and f is surjective if and only if these maps are surjective. Using Frobenius reciprocity it is easy to see that the maps C_{0}(𝟙, U_{s}) → C_{1}(𝟙, U_{s}) are injective, resp. surjective, for all s if and only if the maps C_{0}(U, U_{V}) → C_{1}(U, V) are injective, resp. surjective, for all objects U and V in Rep G ⊂ C_{0}, C_{1}. Since the categories C_{i} are generated by Rep G, it follows that f is injective if and only if F is faithful, and f is surjective if and only if F is full. It is also worth noting that since a morphism T ∈ C(𝟙, U) is zero if and only if T^{∗}T = 0 in End_{C}(𝟙), we have, given a morphism [(F, η)]: (C_{0}, E_{0}) → (C_{1}, E_{1}), that F is faithful if and only if the homomorphism End_{C_{0}}(𝟙) → End_{C_{1}}(𝟙) is injective. On the C^{∗}-algebra level this corresponds to the simple property that a morphism B_{0} → B_{1} of G-C^{∗}-algebras for a reduced compact quantum group G is injective if and only if its restriction to the fixed point algebra B_{0}^{G} is injective.

2.5 Equivalence of categories

To finish the proof of Theorem 2.1 it remains to show that the functors T : YD_{brc}(G) → Tens(Rep G) or T̃: YD_{brc}(G) → Tens(Rep G), and S: Tens(Rep G) → YD_{brc}(G) are inverse to each other up to an isomorphism.

Let us start with a strict C^{∗}-tensor category C containing Rep G and construct a braided-commutative Yetter–Drinfeld C^{∗}-algebra B as described in Section 2.3. By Theorem 1.1 the (Rep G)-module C^{∗}-categories C and C_{B} are equivalent. We will use the concrete form of this equivalence explained in Section 1.4. Recall that C_{B} is the idempotent completion of the category Rep G with morphisms C_{B}(U, V) ⊂ B(H_{U}, H_{V}) ⊗ B, and we have a unitary equivalence F: C → C_{B} such that F(U) = U for U ∈ Rep G, while the action of F on morphisms is given by (1.6), so

C(U, V) ∋ T ↦ ∑_{i,j} θ_{ζ_{j},ξ_{i}} ⊗ π(\overline{(ζ_{j} ⊗ \overline{ρ^{−1/2}ξ_{i}})} ⊗ (T ⊗ ι)R̄_{U}),

where {ξ_{i}}_{i} and {ζ_{j}}_{j} are orthonormal bases in H_{U} and H_{V}, respectively. We claim that F is a strict tensor functor on the full subcategory of C consisting of objects U ∈ Rep G. This tensor functor extends then to a unitary tensor functor on the whole category C. Thus, we have to show that F(S ⊗ T) = F(S) ⊗ F(T) on morphisms in C. Since F is an equivalence of right (Rep G)- module categories, we already know that this is true for morphisms S in C and morphisms T in Rep G; this is also not difficult to check directly, since the formula for F(S) ⊗ ι does not involve the Yetter–Drinfeld structure, see (2.3). Therefore it remains to check that F(ι ⊗ T) = ι ⊗ F(T) for morphisms T in C.

Take T ∈ C(V, W). Let {η_{k}}_{k} be an orthonormal basis in H_{W}. We then have

F(ι_{U} ⊗ T) = ∑_{i,j,k,l} θ_{ξ_{i}⊗η_{k},ξ_{j}⊗ζ_{l}}

⊗ π(\overline{(ξ_{i} ⊗ η_{k} ⊗ \overline{(ρ^{−1/2}ξ_{j} ⊗ ρ^{−1/2}ζ_{l})})} ⊗ ((ι ⊗ T) ⊗ ι\overline{U◯_{⊤}W})R̄_{U◯_{⊤}W} ).

Similarly to the proof of Lemma 2.6, using that \overline{U ◯_{⊤} W} is equivalent to W̄ ◯_{⊤} Ū and that modulo this equivalence R̄_{U◯_{⊤}W} coincides with (ι ⊗ R̄_{W} ⊗ ι)R̄_{U}, we see that the above expression equals

∑_{i,j,k,l} θ_{ξ_{i}⊗η_{k},ξ_{j}⊗ζ_{l}} ⊗ π(\overline{(ξ_{i} ⊗ η_{k} ⊗ \overline{ρ^{−1/2}ζ_{l}} ⊗ \overline{ρ^{−1/2}ξ_{j}})} ⊗ (ι ⊗ (T ⊗ ι)R̄_{W} ⊗ ι)R̄_{U}).

The operators θ_{ξ_{i},ξ_{j}} are the matrix units m_{ij} in B(H_{U}). Recalling the definition of ⊳ we can therefore write the above expression as

∑_{i,j,k,l} m_{ij} ⊗ θ_{η_{k},ζ_{l}} ⊗ (u_{ij} ⊳ π(\overline{(η_{k} ⊗ \overline{ρ^{−1/2}ζ_{l}})} ⊗ (T ⊗ ι)R̄_{W} )),

where u_{ij} are the matrix units of U. According to (2.2) this is exactly the formula for ι_{U} ⊗ F(T).

Conversely, consider a unital braided-commutative Yetter–Drinfeld C^{∗}-al-gebra B and the corresponding pair (C_{B}, F_{B}). Let B_{C} be the Yetter–Drinfeld C^{∗}-algebra constructed from this pair. By Theorem 1.1 we know that there exists an isomorphism λ: B_{C} → B intertwining the actions of G. So all we have to do is to check that λ is also a ℂ[G]-module map. The isomorphism λ is defined by

λ(π(ζ̄ ⊗ T)) = (ζ̄ ⊗ ι)(T)  (2.13)

for ζ ∈ H_{V} and T ∈ C_{B}(𝟙, V) ⊂ B(ℂ, H_{V}) ⊗ B = H_{V} ⊗ B, see the proof of [Nes, Theorem 2.3]. As above, fix finite dimensional unitary representations U and V of G and orthonormal bases {ξ_{i}}_{i} and {ζ_{k}}_{k} in H_{U} and H_{V}, and let u_{ij} be the matrix coefficients of U. Take

T = ∑_{k} ζ_{k} ⊗ b_{k} ∈ C_{B}(𝟙, V) ⊂ H_{V} ⊗ B.

Then λ(π(ζ̄_{k_{0}} ⊗ T)) = b_{k_{0}}, and we want to check that

λ(u_{i_{0}j_{0}} ⊳ π(ζ̄_{k_{0}} ⊗ T)) = u_{i_{0}j_{0}} ⊳ b_{k_{0}}.

By definition we have

u_{i_{0}j_{0}} ⊳ π(ζ̄_{k_{0}} ⊗ T) = π((ξ̄_{i_{0}} ⊗ ξ_{j_{0}}) ˜⊳ (ζ̄_{k_{0}} ⊗ T))
= π(\overline{(ξ_{i_{0}} ⊗ ζ_{k_{0}} ⊗ \overline{ρ^{−1/2}ξ_{j_{0}}})} ⊗ (ι ⊗ T ⊗ ι)R̄_{U}).

Documenta Mathematica 19 (2014) 1105–1139

In order to compute the image of this element under λ, we need an explicit

formula for (ι ⊗ T ⊗ ι)R̄_{U} ∈ C_{B}(𝟙, U ⊗ V ⊗ Ū) ⊂ H_{U◯_{⊤}V ◯_{⊤}Ū} ⊗ B. By (2.2)

and (2.3), the element

ι_{U} ⊗ T ⊗ ι_{U} ∈ C_{B}(U ◯_{⊤} Ū, U ◯_{⊤} V ◯_{⊤} Ū) ⊂ B(H_{U}) ⊗ H_{V} ⊗ B(H̄_{U}) ⊗ B

equals ∑_{i,j,k} m_{ij} ⊗ ζ_{k} ⊗ 1 ⊗ (u_{ij} ⊳ b_{k}). It follows that

(ι ⊗ T ⊗ ι)R̄_{U} = ∑_{i,j,k}(ξ_{i} ⊗ ζ_{k} ⊗ \overline{ρ^{1/2}ξ_{j}}) ⊗ (u_{ij} ⊳ b_{k}).

Therefore

u_{i_{0}j_{0}} ⊳ π(ζ̄_{k_{0}} ⊗ T)

= π (\overline{(ξ_{i_{0}} ⊗ ζ_{k_{0}} ⊗ \overline{ρ^{−1/2}ξ_{j_{0}}})} ⊗ (∑_{i,j,k}(ξ_{i} ⊗ ζ_{k} ⊗ \overline{ρ^{1/2}ξ_{j}}) ⊗ (u_{ij} ⊳ b_{k}))) .

Applying λ we get the required equality λ(u_{i_{0}j_{0}} ⊳ π(ζ̄_{k_{0}} ⊗ T)) = u_{i_{0}j_{0}} ⊳ b_{k_{0}}. Since the algebra B is spanned by such elements b_{k_{0}} for different V, it follows that λ is a ℂ[G]-module map. This completes the proof of Theorem 2.1.

3 Coideals of quotient type and their generalizations

In this section we illustrate Theorem 2.1 by considering well-known examples of Yetter–Drinfeld algebras arising from quantum subgroups and Hopf–Galois extensions.

3.1 Quotient type coideals

By a closed quantum subgroup of G we mean a compact quantum group H together with a surjective homomorphism π: ℂ[G] → ℂ[H] of Hopf ∗-algebras. This is consistent with the definition used in the theory of locally compact quantum groups, but is weaker than e.g. the definition used in [Tom07]. Assuming that both G and H are reduced, the homomorphism π does not always extend to a homomorphism C(G) → C(H). Nevertheless the algebra C(G/H) of continuous functions on the quantum homogeneous space G/H is always well-defined: it is the norm closure of

ℂ[G/H] = {x ∈ ℂ[G] | (ι ⊗ π)Δ(x) = x ⊗ 1}.

The algebra C(G/H) is a braided-commutative Yetter–Drinfeld G-C^{∗}-algebra, with the left action of G defined by the restriction of Δ to C(G/H), and the action of Ĝ defined by the restriction of the right adjoint action on C(G) to C(G/H). In other words, the ℂ[G]-module structure on ℂ[G/H] is defined by

x ⊳ a = x_{(1)}aS(x_{(2)}).

It is known and is easy to see that the G-C^{∗}-algebra C(G/H) corresponds to the category Rep H with the distinguished object 𝟙, viewed as a (Rep G)- module category via the forgetful tensor functor Rep G → Rep H. Namely, in the notation of Section 2.3, by identifying Hom_{H}(ℂ, H_{U}) with a subspace of H_{U}, we can view the algebra B̃ corresponding to the functor Rep G → Rep H as a subalgebra of { \atop ℂ[G]} = ⨁_{U}(H̄_{U} ⊗ H_{U}). Then the map π_{G}: { \atop ℂ[G]} → ℂ[G] induces a G-equivariant isomorphism B ≅ ℂ[G/H]. We claim that the ℂ[G]-module structure on ℂ[G/H] defined by the tensor functor Rep G → Rep H is exactly the adjoint action. In order to show this it is enough to consider the case of trivial H, since it corresponds to the inclusion Rep G ↪ Hilb_{f}, while the general case corresponds to the intermediate inclusion Rep G ↪ Rep H. As in the proof of Lemma 2.8, fix unitary representations U and V and orthonormal bases {ξ_{i}}_{i} in H_{U} and {ζ_{k}}_{k} in H_{V} such that ρξ_{i} = ρ_{i}ξ_{i}. Denote matrix coefficients of U, V and Ū by u_{ij}, v_{kl}, ū_{ij}. Recall that by (2.10) we have ū_{ij} = ρ_{i}^{1/2}ρ_{j}^{−1/2}S(u_{ji}). Then

(ξ̄_{i} ⊗ ξ_{j}) ˜⊳ (ζ̄_{k} ⊗ ζ_{l}) = ∑_{m} \overline{(ξ_{i} ⊗ ζ_{k} ⊗ ρ_{j}^{−1/2}ξ̄_{j})} ⊗ (ρ_{m}^{1/2}ξ_{m} ⊗ ζ_{l} ⊗ ξ̄_{m})

It follows that

u_{ij} ⊳ v_{kl} = ∑_{m} ρ_{j}^{−1/2}ρ_{m}^{1/2}u_{im}v_{kl}ū_{jm} = ∑_{m} u_{im}v_{kl}S(u_{mj}),

which is exactly the formula for the adjoint action. As a simple application of Theorem 2.1 we now get the following result, which under slightly stronger assumptions has been already established in [Tom07] and [Sal11].

Theorem 3.1. Let G be a reduced compact quantum group. Then any unital left G- and right Ĝ-invariant C^{∗}-subalgebra of C(G) has the form C(G/H) for a unique closed quantum subgroup H of G.

Proof. Let B ⊂ C(G) be a unital left G- and right Ĝ-invariant C^{∗}-algebra. Consider the corresponding pair (D_{B}, E_{B}) = T (B) ∈ Tens(Rep G). By the ergodicity of the G-action on B, the unit object in D_{B} is simple. Since D_{B} is generated by the image of Rep G and the category Rep G is rigid, the C^{∗}-tensor category D_{B} is rigid as well. The inclusion B ↪ C(G) defines a morphism

(D_{B}, E_{B}) → (D_{C(G)}, E_{C(G)}) ≅ (Hilb_{f}, F),

where F: Rep G → Hilb_{f} is the forgetful fiber functor. This means that D_{B} has a unitary fiber functor E: D_{B} → Hilb_{f} such that F = EE_{B}. By Woronowicz’s Tannaka–Krein duality theorem, the pair (D_{B}, E) defines a compact quantum group H. Then the functor E_{B} defines a functor Rep G → Rep H such that the forgetful fiber functor F on Rep G factors through that on Rep H. It follows that H can be regarded as a quantum subgroup of G.

Since by the discussion preceding the theorem the factorization of the fiber functor F: Rep G → Hilb_{f} through Rep G → Rep H corresponds to the inclusion C(G/H) ↪ C(G), we have therefore shown that there exists a closed quantum subgroup H ⊂ G and an isomorphism (D_{B}, E_{B}) ≅ (D_{C(G/H)}, E_{C(G/H)}) such that the morphism (D_{B}, E_{B}) → (D_{C(G)}, E_{C(G)}) under this isomorphism becomes the morphism (D_{C(G/H)}, E_{C(G/H)}) → (D_{C(G)}, E_{C(G)}) defined by the inclusion C(G/H) ↪ C(G). Since T is an equivalence of categories, this implies that B = C(G/H). It remains to prove the uniqueness. In other words, we want to show that C(G/H) ⊂ C(G) determines the kernel of the restriction map ℂ[G] → ℂ[H]. Since ℂ[G/H] is spanned by the matrix coefficients a_{ξ,ζ} = ((· ζ, ξ) ⊗ ι)(U) such that ζ is an H-invariant vector, we can recover Hom_{H}(𝟙, U) ⊂ H_{U} for any representation U of G from C(G/H). Using the duality morphisms in Rep G we can then recover Hom_{H}(V, U) ⊂ B(H_{V} , H_{U}) for all V and U. Finally, observe that a finite combination ∑_{i} a_{ξ_{i},ζ_{i}} of matrix coefficients in ℂ[G], with ξ_{i}, ζ_{i} ∈ H_{U}, is in the kernel of the restriction map ℂ[G] → ℂ[H] if and only if ∑_{i}(· ζ_{i}, ξ_{i}) vanishes on the commutant of End_{H}(H_{U}) in B(H_{U}).

3.2 Invariant subalgebras of linking algebras

The considerations of the previous subsection can be generalized to the linking algebras defined by monoidal equivalences. Let F be the forgetful functor Rep G → Hilb_{f}, and F′: Rep G → Hilb_{f}, U ↦ H′_{U}, be another unitary fiber functor. We denote the compact quantum group corresponding to F′ by G′. Then it is not difficult to check that the linking algebra between G and G′, introduced in the C^{∗}-algebraic setting in [BDRV06] and in the purely algebraic setting earlier in [Sch96], is exactly the C^{∗}-algebra B(F, F′) corresponding to the pair (Hilb_{f}, F′) by our construction. In addition to the left action of G it carries also a commuting right action of G′, which is easy to see using that the regular subalgebra of B(F, F′) is B(F, F′) = ⨁_{s}(H̄_{s} ⊗ H′_{s}). The G-C^{∗}-algebras B of the form B(F, F′) can be abstractly characterized by saying that the regular subalgebra B ⊂ B is a Hopf–Galois extension of ℂ over ℂ[G], which is a well-studied notion in the algebraic approach to quantum groups, see [Bic10]. By definition, this means that the Galois map

Γ: B ⊗ B → ℂ[G] ⊗ B, x ⊗ y ↦ x_{(1)} ⊗ x_{(2)}y,

is bijective. Analogously to the case of ℂ[G] (which is the linking algebra B(F, F)), there is a standard structure of a braided-commutative Yetter– Drinfeld algebra over G on B. Namely, the action of ℂ[G] on B is the so called Miyashita–Ulbrich action, defined by

x ⊳ a = Γ^{−1}(x ⊗ 1)_{1}aΓ^{−1}(x ⊗ 1)_{2}.

We claim that this action is the same as the one induced by the pair (Hilb_{f}, F′) by our construction. In order to show this, replace (Hilb_{f}, F′) by an isomorphic pair consisting of a strict C^{∗}-tensor category C containing Rep G and the embedding functor Rep G → C, as explained in Section 2.1. What is now special about C, is that the unit object is simple and the maps C(𝟙, U) ⊗ C(𝟙, V) → C(𝟙, U ◯_{⊤} V) are bijective. As in the previous subsection, fix unitary representations U and V of G and an orthonormal basis {ξ_{i}}_{i} in H_{U} such that ρξ_{i} = ρ_{i}ξ_{i}. We can find elements T_{l} ∈ C(𝟙, U) and S_{l} ∈ C(𝟙, Ū) such that

R̄_{U} = ∑_{l} T_{l} ⊗ S_{l} in C.

Then, for any P ∈ C(𝟙, V) and ζ ∈ H_{V}, we have

(ξ̄_{i} ⊗ ξ_{j}) ˜⊳(ζ̄ ⊗ P) = ρ_{j}^{−1/2}\overline{(ξ_{i} ⊗ ζ ⊗ ξ̄_{j})} ⊗ (ι ⊗ P ⊗ ι)R̄_{U}

= ρ_{j}^{−1/2} ∑_{l}(ξ̄_{i} ⊗ T_{l}) · (ζ̄ ⊗ P) · (¯ξ̄_{j} ⊗ S_{l}).

Therefore in order to prove the claim it suffices to check that

∑_{l} Γ(π(ξ̄_{i} ⊗ T_{l}) ⊗ π(¯ξ̄_{j} ⊗ S_{l})) = ρ_{j}^{1/2}u_{ij} ⊗ 1.

But this is true by the following simple computation:

∑_{l} Γ(π(ξ̄_{i} ⊗ T_{l}) ⊗ π(¯ξ̄_{j} ⊗ S_{l})) = ∑_{k,l} u_{ik} ⊗ π(ξ̄_{k} ⊗ T_{l})π(¯ξ̄_{j} ⊗ S_{l})

= ∑_{k} u_{ik} ⊗ π(\overline{(ξ_{k} ⊗ ξ̄_{j})} ⊗ R̄_{U}) = ∑_{k} u_{ik} ⊗ \overline{R̄_{U}^{∗}(ξ_{k} ⊗ ξ̄_{j})} = ρ_{j}^{1/2}u_{ij} ⊗ 1.

If H′ is a closed quantum subgroup of G′, then, similarly to the C^{∗}-algebras C(G/H) ⊂ C(G), we may define C^{∗}-algebras B(F, F′)^{H′} ⊂ B(F, F′). Then by a completely analogous argument to that in the proof of Theorem 3.1 we obtain the following result.

Theorem 3.2. Let G be a reduced compact quantum group and B = B(F, F′) be the linking C^{∗}-algebra defined by the forgetful fiber functor F: Rep G → Hilb_{f} and a unitary fiber functor F′: Rep G → Hilb_{f}. Let G′ be the compact quantum group defined by F′. Then any unital left G- and right Ĝ-invariant C^{∗}- subalgebra of B(F, F′) has the form B(F, F′)^{H′} for a unique closed quantum subgroup H′ ⊂ G′.

Let us finally say a few words about the differences between our approach to reconstructing the tensor functor F′ from B(F, F′) and that in [BDRV06]. Assume B is a unital G-C^{∗}-algebra such that B^{G} = ℂ1. We can define a weak unitary tensor functor E: Rep G → Hilb_{f}, called the spectral functor, by letting

E(U) = D_{B}(B, B×U) and E_{2;U,V} : E(U)⊗E(V) → E(U ◯_{⊤}V), T ⊗S ↦ (T ⊗ι)S.

The scalar product on E(U) is defined by S^{∗}T = (T, S)1, which makes sense by the ergodicity assumption. In general the maps E_{2;U,V} are not unitary but only isometric. When they are unitary, so that (E, E_{2}) becomes a unitary tensor functor, then the ergodic action of G on B is said to be of full quantum multiplicity. In this case, if G is reduced, then B ≅ B(F, E) as G-C^{∗}-algebras [BDRV06] (see also [Nes], where a more general result is proved in the notation consistent with the present work). In particular, another way of formulating the Hopf–Galois condition is to say that the action of G is of full quantum multiplicity. The spectral functor is constructed in a simple way using only the action of G, while in order to construct a tensor functor in our approach we also have to use the Miyashita–Ulbrich action. The reason why the two constructions give isomorphic functors is basically the following observation. Given a unitary fiber functor F′: Rep G → Hilb_{f}, we can define a new unitary fiber functor E: Rep G → Hilb_{f} by letting E(U) = Hom(ℂ, F′(U)) and

E_{2;U,V} : E(U) ⊗ E(V) → E(U ◯_{⊤} V), T ⊗ S ↦ F′_{2;U,V} (T ⊗ ι)S.

But it is clear that under the identification of Hom(ℂ, H) with H, the tensor functor E becomes identical to F′.

4 Noncommutative Poisson boundaries

In this section we show that Theorem 2.1 provides a link between Izumi’s theory of Poisson boundaries of discrete quantum groups [Izu02] and categorical Poisson boundaries introduced in [NY14a]. We start by giving a categorical description of discrete duals.

4.1 Discrete dual

Consider the algebra ℓ^{∞}(Ĝ) ⊂ U(G) = ℂ[G]^{∗} of bounded functions on Ĝ. We have a left adjoint action α of G on

ℓ^{∞}(Ĝ) ≅ ℓ^{∞}- ⨁_{s} B(H_{s})

defined by

B(H_{s}) ∋ T ↦ (U_{s})_{21}^{∗}(1 ⊗ T)(U_{s})_{21}.  (4.1)

This action is continuous only in the von Neumann algebraic sense, so in order to stay within the class of G-C^{∗}-algebras, instead of ℓ^{∞}(Ĝ) we should rather consider the norm closure B(Ĝ) of the regular subalgebra ℓ_{alg}^{∞}(Ĝ) ⊂ ℓ^{∞}(Ĝ). Then the right action Δ̂ of Ĝ on ℓ^{∞}(Ĝ) makes this algebra into a unital braidedcommutative Yetter–Drinfeld C^{∗}-algebra. In other words, the left ℂ[G]-module structure on ℓ_{alg}^{∞}(Ĝ) is defined by

x ⊳ a = (ι ⊗ x)Δ̂(a).  (4.2)

In the subsequent computations we will use the notation Δ̂(a) = a^{(1)} ⊗ a^{(2)}. Literally this does not make sense, but the expressions like a^{(1)} ⊗ π_{U}(a^{(2)}) are still meaningful, since (ι⊗π_{U})Δ̂(a) is an element of the algebraic tensor product ℓ^{∞}(Ĝ) ⊗ B(H_{U}).

We want to describe the corresponding C^{∗}-tensor category C = C_{B(Ĝ)} and the unitary tensor functor F = F_{B(Ĝ)} : Rep G → C. By definition, the category C is the idempotent completion of the category with the same objects as in Rep G, but with the morphism sets C(U, V) ⊂ B(H_{U}, H_{V}) ⊗ ℓ_{alg}^{∞}(Ĝ). In fact, for the reasons that will become apparent in a moment, it is more convenient to consider C(U, V) as a subset of ℓ_{alg}^{∞}(Ĝ) ⊗ B(H_{U}, H_{V}). Thus, we define C(U, V) as the set of elements T ∈ ℓ_{alg}^{∞}(Ĝ) ⊗ B(H_{U}, H_{V}) such that

V_{31}^{∗}(α ⊗ ι)(T)U_{31} = 1 ⊗ T.

From the definition of the adjoint action α we see that an element T ∈ ℓ_{alg}^{∞}(Ĝ)⊗ B(H_{U}, H_{V}) lies in C(U, V) if and only if it defines a G-equivariant map H_{s} ⊗ H_{U} → H_{s}⊗H_{V} for all s. It follows that C(U, V) can be identified with the space Nat_{b}(ι⊗U, ι⊗V) of bounded natural transformations between the functors ι⊗U and ι ⊗ V on Rep G. Using this picture we get a natural tensor structure on C: the tensor product of objects is defined as in Rep G, while the tensor product of natural transformations ν: ι ⊗ U → ι ⊗ V and η: ι ⊗ W → ι ⊗ Z is defined by

ν ⊗ η = (ν ⊗ ι_{Z})(ι_{U} ⊗ η) = (ι_{V} ⊗ η)(ν ⊗ ι_{W}),

where ν ⊗ ι_{Z} is defined by (ν ⊗ ι_{Z})_{X} = ν_{X} ⊗ ι_{Z}, while ι_{U} ⊗ η is defined by

(ι_{U} ⊗ η)_{X} = η_{X◯_{⊤}U}. Explicitly, if ν = ∑_{i} a_{i} ⊗ T_{i} ∈ ℓ_{alg}^{∞}(Ĝ) ⊗ B(H_{U}, H_{V}) and

η = ∑_{j} b_{j} ⊗ S_{j} ∈ ℓ_{alg}^{∞}(Ĝ) ⊗ B(H_{W} , H_{Z}), then

ν ⊗η = ∑_{i,j} a_{i}b_{j}^{(1)}⊗(T_{i}π_{U}(b_{j}^{(2)})⊗S_{j}) ∈ ℓ_{alg}^{∞}(Ĝ)⊗B(H_{U} ⊗H_{W} , H_{V} ⊗H_{Z}). (4.3)

The functor F: Rep G → C is now the strict tensor functor such that F(U) = U on objects and F(T) = 1 ⊗ T on morphisms. It remains to show that the tensor structure on C defines the same ℂ[G]-module structure on ℓ_{alg}^{∞}(Ĝ) as (4.2). Consider an element ζ̄ ⊗ T ∈ H̄_{V} ⊗ C(𝟙, V). Identifying B(ℂ, H_{V}) with H_{V} we can write T = ∑_{k} a_{k} ⊗ ζ_{k} for some a_{k} ∈ ℓ_{alg}^{∞}(Ĝ) and ζ_{k} ∈ H_{V}. Then, identifying the algebra B constructed from the pair (C, F) with ℓ_{alg}^{∞}(Ĝ), the element a = π(ζ̄ ⊗ T) ∈ B = ℓ_{alg}^{∞}(Ĝ) equals ∑_{k}(ζ_{k}, ζ)a_{k}, see equation (2.13). Choose a unitary representations U and an orthonormal basis {ξ_{i}}_{i} in H_{U} consisting of eigenvectors of ρ, so ρξ_{i} = ρ_{i}ξ_{i}. By (4.3) the morphism

(ι ⊗ T ⊗ ι)R̄_{U} ∈ C(𝟙, U ◯_{⊤} V ◯_{⊤} Ū) ⊂ ℓ^{∞}(Ĝ) ⊗ (H_{U} ⊗ H_{V} ⊗ H̄_{U})

is represented by the element

∑_{k,l} a_{k}^{(1)} ⊗ (ρ_{l}^{1/2}a_{k}^{(2)}ξ_{l} ⊗ ζ_{k} ⊗ ξ̄_{l}).

Then by definition (2.4) of the map ˜⊳ we get

(ξ̄_{i} ⊗ ξ_{j}) ˜⊳ (ζ̄ ⊗ (∑_{k} a_{k} ⊗ ζ_{k}))

= \overline{(ξ_{i} ⊗ ζ ⊗ ρ_{j}^{−1/2}ξ̄_{j})} ⊗ (∑_{k,l} a_{k}^{(1)} ⊗ (ρ_{l}^{1/2}a_{k}^{(2)}ξ_{l} ⊗ ζ_{k} ⊗ ξ̄_{l})) ,

whence

u_{ij} ⊳ a = ∑_{k}(a_{k}^{(2)}ξ_{j}, ξ_{i})(ζ_{k}, ζ)a_{k}^{(1)} = (a^{(2)}ξ_{j}, ξ_{i})a^{(1)}.

But this is exactly how the action (4.2) is defined.

4.2 Poisson boundaries

Let us briefly overview the theory of noncommutative Poisson boundaries developed by Izumi [Izu02]. For a finite dimensional unitary representation U of G, consider the state ϕ_{U} on B(H_{U}) defined by

ϕ_{U}(T) = \frac{Tr(T π_{U}(ρ)^{−1})}{dim_{q} U} for T ∈ B(H).  (4.4)

If U is irreducible, it can be characterized as the unique state satisfying

(ι ⊗ ϕ_{U})(U_{21}^{∗}(1 ⊗ T)U_{21}) = ϕ_{U}(T).

For our fixed representatives of irreducible representations {U_{s}}_{s} of G, we write ϕ_{s} instead of ϕ_{U_{s}}. When ϕ is a normal state on ℓ^{∞}(Ĝ), we define a completely positive map P_{ϕ} on ℓ^{∞}(Ĝ) by

P_{ϕ}(a) = (ϕ ⊗ ι)Δ̂(a).

If μ is a probability measure on the set Irr(G) of isomorphism classes of irreducible representations of G, we define a normal unital completely positive map P_{μ} on ℓ^{∞}(Ĝ) by P_{μ} = ∑_{s} μ(s)P_{ϕ_{s}}. The space

H^{∞}(Ĝ, μ) = {x ∈ ℓ^{∞}(Ĝ) | x = P_{μ}(x)}

of P_{μ}-harmonic elements is called the noncommutative Poisson boundary of Ĝ with respect to μ. This is an operator subspace of ℓ^{∞}(Ĝ) closed under the left adjoint action α of G defined by (4.1) and the right action Δ̂ of Ĝ on itself by translations. It has a new product structure

x · y = lim_{n→∞} P_{μ}^{n}(xy),

where the limit is taken in the strong^{∗} operator topology. With this product H^{∞}(Ĝ, μ) becomes a von Neumann algebra (with the original operator space structure), and the actions of G and Ĝ on ℓ^{∞}(Ĝ) define continuous, in the von Neumann algebraic sense, actions on H^{∞}(Ĝ, μ). Consider the regular subalgebra H_{alg}^{∞}(Ĝ, μ) = H^{∞}(Ĝ, μ) ∩ ℓ_{alg}^{∞}(Ĝ) of H^{∞}(Ĝ, μ) and denote by B(Ĝ, μ) its norm closure. In other words, in the notation of Section 4.1, B(Ĝ, μ) = B(Ĝ) ∩ H^{∞}(Ĝ, μ). We will show in Theorem 4.1 that the action of Ĝ on H^{∞}(Ĝ, μ) restricts to a continuous action on B(Ĝ, μ) and that B(Ĝ, μ) becomes a braided-commutative Yetter–Drinfeld G-C^{∗}-algebra.

Let us now recall the construction of the Poisson boundary of (Rep G, μ) defined in [NY14a]. The image of Hom_{G}(U ◯_{⊤} V, U ◯_{⊤} W) under the map

ϕ_{U} ⊗ ι: B(H_{U}) ⊗ B(H_{V} , H_{W}) → B(H_{V} , H_{W})

is contained in Hom_{G}(V, W), and the maps

ϕ_{U} ⊗ ι: Hom_{G}(U ◯_{⊤} V, U ◯_{⊤} W) → Hom_{G}(V, W)

we thus get, are what we called the partial categorical traces on Rep G in [NY14a]. They allow us to define an operator P_{U} on the space of natural transformations Nat(ι ⊗ V, ι ⊗ W) by

P_{U}(η)_{X} = (ϕ_{U} ⊗ ι)(η_{U⊗X}).

It is easy to see that this operation preserves the subspace Nat_{b}(ι ⊗ V, ι ⊗ W) of bounded natural transformations. Given a probability measure μ on Irr(G), we define an operator P_{μ} acting on Nat_{b}(ι ⊗ V, ι ⊗ W) by P_{μ} = ∑_{s} μ(s)P_{U_{s}}. A bounded natural transformation η: ι ⊗ V → ι ⊗ W is called P_{μ}-harmonic if P_{μ}(η) = η. Any morphism T: V → W defines a bounded natural transformation (ι_{X} ⊗ T)_{X}, which is obviously P_{μ}-harmonic for every μ. The categorical Poisson boundary (P, Π) of (Rep G, μ) consists of the C^{∗}-tensor category P and the strict unitary tensor functor Π: C → P defined as follows. The category P is the idempotent completion of Rep G with the new morphism sets

P(U, V) = {η ∈ Nat_{b}(ι ⊗ U, ι ⊗ V) | P_{μ}(η) = η},

endowed with the composition law

(η · ν)_{X} = lim_{n→∞} P_{μ}^{n}(ην)_{X}.

On objects in Rep G the tensor product in P is the same as in Rep G, while on morphisms it is given by

η ⊗ ν = (η ⊗ ι) · (ι ⊗ ν) = (ι ⊗ ν) · (η ⊗ ι),

where η ⊗ ι and ι ⊗ ν are defined as in Section 4.1. The functor Π: C → P is defined by letting Π(U) = U on objects and Π(T) = (ι_{X} ⊗ T)_{X} on morphisms. We usually omit Π and consider Rep G as a subcategory of P.

Theorem 4.1. Let G be a compact quantum group and μ be a probability measure on Irr(G). Then the dense C^{∗}-subalgebra B(Ĝ, μ) ⊂ H^{∞}(Ĝ, μ) is a unital braided-commutative Yetter–Drinfeld G-C^{∗}-algebra and the pair (D_{B(Ĝ,μ)}, E_{B(Ĝ,μ)}), consisting of the C^{∗}-tensor category D_{B(Ĝ,μ)} of Gequivariant finitely generated Hilbert B(Ĝ, μ)-modules and the unitary tensor functor E_{B(Ĝ,μ)} : Rep G → D_{B(Ĝ,μ)}, is isomorphic to the categorical Poisson boundary of (Rep G, μ).

Proof. When μ = δ_{e}, in which case H^{∞}(Ĝ, μ) = ℓ^{∞}(Ĝ), this theorem is the contents of Section 4.1. The general case easily follows from this. Indeed, denote by B̃_{μ} and B_{μ} the algebras constructed from the Poisson boundary (P, Π) of (Rep G, μ) as described in Section 2.3. If μ = δ_{e}, we simply write B̃ and B. Thus,

B̃ = ⨁_{U}(H̄_{U} ⊗ Nat_{b}(ι, ι ⊗ U)).

As we showed in Section 4.1, the Yetter–Drinfeld algebra B can be identified with ℓ_{alg}^{∞}(Ĝ), and then the homomorphism π: B̃ → B = ℓ_{alg}^{∞}(Ĝ) is given by

π (ξ̄⊗ (∑_{k} a_{k} ⊗ ζ_{k})) = ∑_{k}(ζ_{k}, ξ)a_{k},

if we view Nat_{b}(ι, ι ⊗ U) as a subspace of ℓ^{∞}- ⨁_{s}(B(H_{s}) ⊗ H_{U}). The Markov operators P_{μ} on Nat_{b}(ι, ι ⊗ U) define an operator ι ⊗ P_{μ} on B̃. Then by definition, the algebra B̃_{μ} is the subspace of (ι⊗P_{μ})-invariant elements in B̃. Furthermore, by construction we have π(ι⊗P_{μ}) = P_{μ}π, where on the right hand side by P_{μ} we mean the operator on ℓ^{∞}(Ĝ) used to define the Poisson boundary of Ĝ. This already implies that the restriction of π to B̃_{μ} defines a surjective homomorphism B̃_{μ} → H_{alg}^{∞}(Ĝ, μ). Recalling how B_{μ} is obtained from B̃_{μ}, we then conclude that this restriction factors through B_{μ} and defines a G-equivariant ∗-isomorphism B_{μ} ≅ H_{alg}^{∞}(Ĝ, μ). It remains to compare the ℂ[G]-module structures. For this part the computation is in fact exactly the same as for μ = δ_{e}. The point is that, in the formula (2.4) for the ℂ[G]-action, one only needs to compute the compositions of the form (ι ⊗ T ⊗ ι)R̄_{U} for U, V ∈ Rep G and T ∈ P(𝟙, V). In general, if η ∈ P(U, V) and S ∈ W → U is a morphism in Rep G, the composition η · S is represented by the family (η_{X}(ι_{X} ⊗ S))_{X}, which is independent of μ. Thus, the ℂ[G]-module structure on H_{alg}^{∞}(Ĝ, μ) induced by the tensor category structure of P via the isomorphism B_{μ} ≅ H_{alg}^{∞}(Ĝ, μ), is the restriction of that on ℓ_{alg}^{∞}(Ĝ). But this is exactly how the original ℂ[G]-module structure was defined on H_{alg}^{∞}(Ĝ, μ).

Recall that a probability measure μ on Irr(G) is called ergodic, if the only P_{μ}- harmonic functions on Irr(G) are the constant functions, that is, H^{∞}(Ĝ, μ)^{G} reduces to ℂ1. Such a measure exists if and only if Irr(G) is at most countable and Rep G is weakly amenable, see [NY14a, Sections 2 and 7.1]. From the above theorem and our results on categorical Poisson boundaries in [NY14a] we then get the following theorem, originally proved by Tomatsu [Tom07, Theorem 4.8]. (To be more precise, Tomatsu formulates the result in a more restricted form, but his proof shows that a stronger result formulated below is true.)

Theorem 4.2. Let G be a coamenable compact quantum group, and μ be an ergodic probability measure on Irr(G). Then the Poisson boundary H^{∞}(Ĝ, μ) is G- and Ĝ-equivariantly isomorphic to L^{∞}(G/K), where K is the maximal Kac quantum subgroup of G.

Proof. The results of [NY14a, Section 4] imply that the Poisson boundary of (Rep G, μ) is isomorphic to the forgetful functor F: Rep G → Rep K, see [NY14b, Section 2] for details. From Theorem 4.1 and the discussion in Section 3.1, where we showed that we have an isomorphism (D_{C(G/K)}, E_{C(G/K)}) ≅ (Rep K, F), we conclude that there exists a G- and Ĝ-equivariant isomorphism B(Ĝ, μ) ≅ C(G/K). Since H^{∞}(Ĝ, μ) and L^{∞}(G/K) are the von Neumann algebras generated by B(Ĝ, μ) and C(G/K), respectively, in the GNSrepresentations defined by the unique G-invariant states, we conclude that H^{∞}(Ĝ, μ) ≅ L^{∞}(G/K).

Of course, conversely, the argument of Tomatsu could be used to show that the Poisson boundary of (Rep G, μ) is F: Rep G → Rep K without relying on [NY14a, Section 4]. Note that in order to prove Theorem 4.2 we do not need the full strength of Theorem 2.1, it suffices to understand how B(Ĝ, μ) and C(G/K) are reconstructed from the functors Π: Rep G → P and F: Rep G → Rep K. It is also worth noting that independently of which approach to Theorem 4.2 one prefers, all the results of this type have so far relied in a crucial, but every time different, way on the so called Izumi’s Poisson integral [Izu02, INT06, Tom07, NY14a]. We finish the paper by proving a converse to Theorem 4.2.

Proposition 4.3. Let G be a compact quantum group and μ be a probability measure on Irr(G). Assume that the Poisson boundary H^{∞}(Ĝ, μ) is G- and Ĝequivariantly isomorphic to L^{∞}(G/H) for a closed quantum subgroup H of G. Then G is coamenable, and hence H is the maximal Kac quantum subgroup of G.

1138 S. Neshveyev and M. Yamashita

Proof. Theorem 4.1 and the assumptions of the proposition imply that the Poisson boundary of (Rep G, μ) is isomorphic to the forgetful functor F: Rep G → Rep H. On the other hand, since the action of G on L^{∞}(G/H) is ergodic, the measure μ is ergodic, and therefore by [NY14a, Theorem 5.1] the Poisson boundary of (Rep G, μ) defines the amenable dimension function on Rep G. It follows that the classical dimension function on Rep G is amenable, which exactly means that G is coamenable.

References

[Bic10] J. Bichon, Hopf–Galois objects and cogroupoids. lecture notes (2010), available

at arXiv:1006.3014 [math.QA].

[BDRV06] J. Bichon, A. De Rijdt, and S. Vaes, Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups, Comm. Math. Phys. 262 (2006), no. 3, 703–728, DOI 10.1007/s00220-005-1442-2, available at

arXiv:math/0502018 [math.OA]. MR2202309 (2007a:46072)

[BN11] A. Bruguières and S. Natale, Exact sequences of tensor categories, Int. Math. Res. Not. IMRN 24 (2011), 5644–5705, DOI 10.1093/imrn/rnq294, available at

arXiv:1006.0569 [math.QA]. MR2863377

[CVOZ94] S. Caenepeel, F. Van Oystaeyen, and Y. H. Zhang, Quantum Yang-Baxter module algebras, Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), 1994, pp. 231–255, DOI 10.1007/BF00960863. MR1291020 (95e:16031)

[CW94] M. Cohen and S. Westreich, From supersymmetry to quantum commutativity, J. Algebra 168 (1994), no. 1, 1–27, DOI 10.1006/jabr.1994.1217. MR1289088 (95e:16033)

[DMNO13] A. Davydov, M. Müger, D. Nikshych, and V. Ostrik, The Witt group of nondegenerate braided fusion categories, J. Reine Angew. Math. 677 (2013), 135– 177, available at arXiv:1009.2117 [math.QA]. MR3039775

[DCY13] K. De Commer and M. Yamashita, Tannaka-Kreĭn duality for compact quantum homogeneous spaces. I. General theory, Theory Appl. Categ. 28 (2013), No. 31, 1099–1138, available at arXiv:1211.6552 [math.OA]. MR3121622

[Izu02] M. Izumi, Non-commutative Poisson boundaries and compact quantum group actions, Adv. Math. 169 (2002), no. 1, 1–57, DOI 10.1006/aima.2001.2053. MR1916370 (2003j:46105)

[INT06] M. Izumi, S. Neshveyev, and L. Tuset, Poisson boundary of the dual of SU_{q}(n), Comm. Math. Phys. 262 (2006), no. 2, 505–531, DOI 10.1007/s00220-005-1439-x, available at arXiv:math/0402074 [math.OA]. MR2200270 (2007f:58012)

[Lan92] M. B. Landstad, Ergodic actions of nonabelian compact groups, Ideas and methods in mathematical analysis, stochastics, and applications (Oslo, 1988), 1992, pp. 365–388. MR1190512 (93j:46072)

[Nes] S. Neshveyev, Duality theory for nonergodic actions, Münster J. Math., available

at arXiv:1303.6207 [math.OA]. to appear in print.

[NT13] S. Neshveyev and L. Tuset, Compact quantum groups and their representation categories, Cours Spécialisés [Specialized Courses], vol. 20, Société Mathématique de France, Paris, 2013. preliminary version available at

http://folk.uio.no/sergeyn/papers/CQGRC.pdf.

Duality for Yetter–Drinfeld Algebras 1139

[NY14a] S. Neshveyev and M. Yamashita, Poisson boundaries of monoidal categories. preprint (2014), available at arXiv:1405.6572[math.OA].

[NY14b] S. Neshveyev and M. Yamashita, Classification of non-Kac compact quantum groups of SU(n) type. preprint (2014), available at arXiv:1405.6574[math.QA].

[NV10] R. Nest and C. Voigt, Equivariant Poincaré duality for quantum group actions, J. Funct. Anal. 258 (2010), no. 5, 1466–1503, DOI 10.1016/j.jfa.2009.10.015,

available at arXiv:0902.3987 [math.KT]. MR2566309

[Sal11] P. Salmi, Compact quantum subgroups and left invariant C^{∗}-subalgebras of locally compact quantum groups, J. Funct. Anal. 261 (2011), no. 1, 1–24, DOI 10.1016/j.jfa.2011.03.003, available at arXiv:1004.4161 [math.OA]. MR2785890 (2012f:46159)

[Sch96] P. Schauenburg, Hopf bi-Galois extensions, Comm. Algebra 24 (1996), no. 12, 3797–3825, DOI 10.1080/00927879608825788. MR1408508 (97f:16064)

[Tom07] R. Tomatsu, A characterization of right coideals of quotient type and its application to classification of Poisson boundaries, Comm. Math. Phys. 275 (2007), no. 1, 271–296, DOI 10.1007/s00220-007-0267-6, available at

arXiv:math/0611327 [math.OA]. MR2335776 (2008j:46058)

[Ulb89] K.-H. Ulbrich, Fibre functors of finite-dimensional comodules, Manuscripta Math. 65 (1989), no. 1, 39–46, DOI 10.1007/BF01168365. MR1006625 (90e:16014)

[Was88] A. Wassermann, Ergodic actions of compact groups on operator algebras. II. Classification of full multiplicity ergodic actions, Canad. J. Math. 40 (1988), no. 6, 1482–1527. MR990110 (92d:46168)

S. Neshveyev Department of Mathematics University of Oslo P.O. Box 1053 Blindern NO-0316 Oslo, Norway sergeyn@math.uio.no M. Yamashita Department of Mathematics Ochanomizu University Otsuka 2-1-1 192-0361 Tokyo, Japan yamashita.makoto@ocha.ac.jp 1140

Documenta Mathematica 19 (2014)