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Idempotent states on compact quantum groups and their classification on U_q(2), SU_q(2), and SO_q(3)

Unlike for locally compact groups, idempotent states on locally compact quantum groups do not necessarily arise as Haar states of compact quantum subgroups. We give a simple characterisation of those idempotent states on compact quantum groups which do arise as Haar states on quantum subgroups. We also show that all idempotent states on the quantum groups U_q(2), SU_q(2), and SO_q(3) (q in (-1,0) \cup (0,1]) arise in this manner and list the idempotent states on the compact quantum semigroups U_0(2), SU_0(2), and SO_0(3). In the Appendix we provide a short new proof of coamenability of the deformations of classical compact Lie groups based on their representation theory.


Introduction
It is well known that if X is a locally compact topological semigroup, then the space of regular probability measures on X possesses a natural convolution product. Analogously if A is a compact quantum semigroup, i.e. a unital C *algebra together with a coproduct (i.e., coassociative unital * -homomorphism) ∆ : A → A ⊗ A then one can consider a natural associative convolution product on the state space of A, λ ⋆ µ = (λ ⊗ µ) • ∆, λ, µ ∈ S(A).
It is natural to ask whether one can characterise the states which satisfy the idempotent property µ ⋆ µ = µ.
A particular and most important example of an idempotent state is the Haar state on a given compact quantum group in the sense of Woronowicz [Wor98]. More general idempotent states arise naturally in considerations of Césaro limits of convolution operators on compact quantum groups, cf. [FS08a]. They are also an important ingredient in the construction of quantum hypergroups [CV99] and occur as initial value ϕ 0 of convolution semigroups (ϕ t ) t≥0 of states on quantum groups, if one relaxes the initial condition ϕ 0 = ε, cf. [FSc00].
For classical compact groups, Kawada and Itô have proven that all idempotent measures are induced by Haar measures of compact subgroups, see [KI40,theorem 3]. Later this result was extended to arbitrary locally compact topological groups, see [Hey77] and references therein. In [Pal96] Pal showed that this characterisation does not extend to quantum groups by giving an example of an idempotent state on the Kac-Paljutkin quantum group which cannot arise as the Haar state on a quantum subgroup. In [FS08b] the first two authors began a systematic study of idempotent states on compact quantum groups. In particular we exhibited further examples of idempotent states on quantum groups that are not induced by Haar states of quantum subgroups and gave a characterisation of idempotent states on finite quantum groups in terms of sub quantum hypergroups.
In this work we continue the analysis began in [FS08b] and show a simple characterisation of those idempotent states which arise as Haar states on quantum subgroups, so-called Haar idempotents. The main result of the present paper is the classification of all idempotent states on the compact quantum groups U q (2), SU q (2), and SO q (3) for q ∈ (−1, 1] \ {0}. It turns out that they are all induced by quantum subgroups, cf. Theorems 4.5, 5.1, and 5.2. As a byproduct we obtain the classification of quantum subgroups of the afore-mentioned quantum groups, giving a new proof of the known result of Podleś [Pod95] for SU q (2) and SO q (3).
For the value q = 0, the quantum cancellation properties fail and U 0 (2), SU 0 (2), and SO 0 (3) are no longer compact quantum groups. But they can still be considered as compact quantum semigroups so that as explained above their state spaces have natural convolution products. Using this we determine all idempotent states on U 0 (2), SU 0 (2), and SO 0 (3), see Theorems 6.3 and 6.5 and 6.6. It turns out that in the case q = 0 there exist additional families of idempotent states, which do not appear when q = 0.
The detailed plan of the paper is as follows: in Section 2 we list the background results and definitions we need in the rest of the paper. In particular we recall the definitions of the quantum groups being the subject of the paper, discuss their corepresentation theory and present explicit formulas for their Haar states. Section 3 introduces idempotent states on compact quantum groups, provides the characterisation of those idempotent states which arise as Haar states on quantum subgroups (extending the results for finite quantum groups given in [FS08b]) and briefly discusses the commutative and cocommutative situation under the coamenability assumption. Section 4 contains main technical arguments of the paper and ends with the characterisation of all idempotent states on U q (2) for q ∈ (−1, 1] \ {0}. In Section 5 we show how one can deduce the corresponding statements for SU q (2) and SO q (3). Section 6 contains the classification of the idempotent states on compact quantum semigroups U 0 (2), SU 0 (2), and SO 0 (3). Finally in Section 7, we use the result of [FS08b] showing that idempotent states on compact quantum groups are group-like projections in the dual quantum group, giving rise to algebraic quantum hypergroups by the construction given in [LvD07], and discuss the quantum hypergroups associated to these group-like projections for the case of SU q (2). The appendix contains a short direct proof of coamenability of deformations of classical compact Lie groups based on the representation theory developed in [KS98].

Preliminaries
The symbol ⊗ will denote the spatial tensor product of C * -algebras, ⊙ will be reserved for the purely algebraic tensor product.
2.1. Compact quantum groups. The notion of compact quantum groups has been introduced in [Wor87a]. Here we adopt the definition from [Wor98] (Definition 2.1 of that paper).
If the quantum cancellation properties are satisfied, then the pair (A, ∆) is called a compact quantum group.
In quantum group theory it is quite common to write A = C(G), to emphasize that A is considered as the algebra of functions on a compact quantum group G -but note that the symbol G itself has no meaning.
The map ∆ is called the coproduct of A, it induces the convolution product It is said to be irreducible, if the only matrices T ∈ M n (C) with T U = UT are multiples of the identity matrix.
Possibly the most important feature of compact quantum groups is the existence of the dense * -subalgebra A (the algebra of matrix coefficients of irreducible unitary corepresentations of A), which is in fact a Hopf * -algebra -so for example ∆ : A → A ⊙ A. This * -Hopf algebra is also denoted by A = Pol G, and treated as the analog of polynomial functions of G.
Another fact of the crucial importance is given in the following result, Theorem 2.3 of [Wor98].
Proposition 2.2. Let A be a compact quantum group. There exists a unique state h ∈ A * (called the Haar state of A) such that for all a ∈ A A compact quantum group is said to be in reduced form if the Haar state h is faithful. If it is not the case we can always quotient out the kernel of h. This procedure in particular does not influence the underlying Hopf * -algebra A; in fact the reduced object may be viewed as the natural completion of A in the GNS representation with respect to h (as opposed for example to the universal completion of A, for details see [BMT01]). In general the reduced and universal object need not coincide. This leads to certain technical complications which are not of essential importance in our context (for example if a discrete group Γ is not amenable, the reduced C * -algebra of Γ is a proper quantum subgroup of the universal C * -algebra of Γ, even though they have 'identical' Haar states). To avoid such difficulties we focus on the class of coamenable compact quantum groups ( [BMT01], see the Appendix to this paper for more information), for which the reduced and universal C * -algebraic completions of A are naturally isomorphic. All deformations of classical compact Lie groups, so in particular quantum groups U q (2), SU q (2) and SO q (3) we consider in Sections 4 and 5 are known to be coamenable ( [Ban99]); we give a simple proof of this fact in the Appendix.
The following definition was introduced by Podleś in the context of compact matrix pseudogroups (Definition 1.3 of [Pod95]). Definition 2.3. A compact quantum group B is said to be a quantum subgroup of a compact quantum group A if there exists a surjective compact quantum group morphism j : A → B, i.e. a surjective unital * -homomorphism j : A → B such that Strictly speaking, one should consider the pairs (B, j), since A can contain several copies of B with different morphisms. We will not distinguish between (B, j) and (B ′ , j ′ ) if there exists an isomorphism of quantum groups Θ : B → B ′ such that Θ•j = j ′ . Note that such isomorphic pairs induce the same idempotent Any compact quantum group contains itself and the trivial compact quantum group C as quantum subgroups; further these two quantum subgroups will be called trivial. If G is a compact group and a compact quantum group A contains G as a quantum subgroup, via a morphism j : A → C(G), we will sometimes simply say that G is a subgroup of A.
2.2. q-Numbers. Let q = 1. We will use the following notation for q-numbers, for n ∈ Z + , 0 ≤ k ≤ n, x ∈ R.
2.3.1. Representation theory of C(SU q (2)). [Wor87b,VS88] The C * -algebra C(SU q (2)) has two families of irreducible representations. The first family consists of the one-dimensional representations ρ θ , 0 ≤ θ < 2π, given by The other family consist of infinite-dimensional representations π θ , 0 ≤ θ < 2π, acting on a separable Hilbert space h by where {e n ; n ∈ Z + } is an orthonormal basis for h. This list is complete, i.e. any irreducible representation of C(SU q (2)) is unitarily equivalent to a representation in one of the two families above (Theorem 3.2 of [VS88]). It is known that the C * -algebra C(SU q (2)) is of type I, therefore any representation can be written as a direct integral over the irreducible representations given above.
The matrix elements u (s) kℓ , s ∈ 1 2 Z + , −s ≤ k, ℓ ≤ s, span Pol SU q (2) and they are linearly dense in C(SU q (2)), therefore will be sufficient for our calculations. We have and the matrix elements of the higher-dimensional corepresentations are of the form kℓ is a polynomial in γ * γ. In particular, for s an integer, u If we define a Z-grading on Pol SU q (2) by deg α = deg α * = 0, deg γ = 1, deg γ * = −1, With this grading, it is straight-forward to verify the following formula for the square of the antipode on homogeneous elements, S 2 (a) = q 2 deg a a.
2.4. The compact quantum group SO q (3). A compact quantum group B is called a quotient group of (A, ∆), if there exists an injective morphism of quantum groups j : B → A. The compact quantum group SO q (3) can be defined as the quotient of SU q (2) by the quantum subgroup Z 2 , cf. [Pod95]. Pol SO q (3) is the subalgebra of Pol SU q (2) spanned by the matrix elements of the unitary irreducible corepresentations with integer label, and C(SO q (3)) its norm closure. The Haar state on SO q (3) is simply the restriction of the Haar state on SU q (2).

2.4.2.
The conditional expectation E : C(SU q (2)) → C(SO q (3)). Looking at the defining relations of SU q (2), it is clear that ϑ : α → −α, γ → −γ extends to a unique * -algebra automorphism of C(SU q (2)). Therefore E = 1 2 (id + ϑ) defines a completely positive unital map from C(SU q (2)) to itself. If φ 2 = h Z 2 • j denotes the idempotent state on SU q (2) induced by the Haar measure of Z 2 (with j : SU q (2) → C(Z 2 ) the corresponding surjective morphism), then we can write E also as (2) and its closed subgroups as quantum subgroups. Podleś [Pod95] has shown that these are the only non-trivial quantum subgroups of SO q (3). Again this can be deduced from the results of Section 5.
Another quantum subgroup of U q (2) is the two-dimensional torus. Denote by Pol T 2 the * -Hopf algebra generated by two commuting unitaries, i.e. by w 1 , w 2 with the relations and C(T 2 ) the compact quantum group obtained as its C * -enveloping algebra. Then the mapping a → w 1 , c → 0, v → w 2 extends to a unique surjective compact quantum group morphism C(U q (2)) → C(T 2 ).
We will see that the twisted products SU q (2) ⋉ σ Z n , n ∈ N, the torus T 2 , and its closed subgroups are the only non-trivial quantum subgroups of U q (2), cf. Corollary 4.7.
2.5.2. Corepresentations of U q (2). Unitary irreducible corepresentations of U q (2) can be obtained as tensor product of unitary irreducible corepresentations of SU q (2) with corepresentations of U(1), cf. [Wys04]. In this way one obtains the following family of unitary irreducible corepresentations of U q (2), for p ∈ Z, s ∈ 1 2 Z + . The matrix elements of these corepresentations clearly span Pol U q (2). Therefore they are dense in C(U q (2)) and will be sufficient for the calculations in this paper.
Assume q = 0. We want to compute the action of the square of the antipode on the matrix elements of the unitary irreducible corepresentations defined above. Since we have S 2 (a) = a, S 2 (c) = q 2 c, and S 2 (v) = v, we get Then we have for all a ∈ D T and b ∈ A.

Idempotent states on compact quantum groups
In this section we formally introduce the notion of idempotent states on a C * -bialgebra, provide a characterisation of those idempotent staets on compact quantum hroups which arise as Haar states on quantum subgroups and discuss commutative and cocommutative cases.
i.e. if it is idempotent for the convolution product.
Assume now that (A, ∆) is a compact quantum group. It is easy to check that each Haar state on a quantum subgroup of A is idempotent. It follows from the example of Pal in [Pal96] and our work in [FS08b] that not every idempotent state is a Haar idempotent. We have the following simple characterisation, extending Theorem 4.5 of [FS08b].
Proof. It is an easy consequence of the Cauchy-Schwarz inequality that N φ is a left ideal; thus it is a two-sided ideal if and only if it is selfadjoint.
Suppose first that φ is a Haar idempotent, i.e. there exists a compact quantum group B and a surjective compact quantum group morphism j : A → B such that φ = h B • j. Recall that we assumed h B to be faithful, so that N φ = {a ∈ A : j(a * a) = 0} = {a ∈ A : j(a) = 0}, which is obviously self-adjoint.
Suppose then that N φ is a two-sided selfadjoint ideal. Let B := A/N φ and let π φ : A → B denote the canonical quotient map. We want to define the coproduct on B by the formula We need to check that it is well-defined -to this end we employ a slightly modified idea from the proof of Theorem 2.1 of [BMT01]. A standard use of Cauchy-Schwarz inequality implies that φ| N φ = 0, so that there exists a faithful state ψ ∈ B * such that ψ • π φ = φ. Faithfulness of ψ implies that also the map id B ⊗ ψ : B ⊗ B → B is faithful (note that here faithfulness of a positive map T is understood in the usual sense, namely T a = 0 and a ≥ 0 imply a = 0) and thus also ψ ⊗ ψ ∈ (B ⊗ B) * is faithful. Suppose then that a ∈ N φ . We have then so also (π φ ⊗ π φ )∆(a * a) = 0. The last statement implies that (π φ ⊗ π φ )∆(a) = 0 and validity of the definition given in the formula (3.1) is established. The fact that ∆ B is a coassociative unital * -homomorphism follows immediately from the analogous properties of ∆; similarly the cancellation properties of B follow from obvious equalities of the type and the cancellation properties of A. Thus (B, ∆ B ) is a compact quantum group and it remains to check that ψ defined above is actually the invariant state on B. This is however an immediate consequence of the following observation: so that ψ is an idempotent state and, as it is faithful, it has to coincide with the Haar state of B ( [Wor98]).
Note that the first implication remains valid without the assumption of faithfulness of h B ; alternatively one could exploit the modular properties of Haar states on not-necessarily-coamenable compact quantum groups implying that their null spaces are always selfadjoint.
The following proposition will be useful for the classification of idempotent states on in the next two sections, cf. [FS08b, Section 3]. 3.1. Idempotent states on cocommutative compact quantum groups. Suppose now that A is cocommutative, i.e. ∆ = τ • ∆, where τ : A ⊗ A → A ⊗ A denoted the usual tensor flip. It is easy to deduce from the general theory of duality for quantum groups ( [KV00]) that A is isomorphic to the dual of C 0 (Γ), where Γ is a (classical) discrete group. For the reasons mentioned in the previous section in general we need to distinguish between the reduced and the universal dual; thus we restrict our attention to amenable Γ. The following generalises Theorem 6.2 of [FS08b] to the infinite-dimensional context.
where f is as above. So-defined j is bounded: note that it is a restriction of the transpose of the map T : B * → A * given by The map T is well defined as it maps positive definite functions into positive definite functions; these generate the relevant Fourier-Stieltjes algebras. Further the closed graph theorem allows to prove that T is bounded; therefore so is T * : A * * → B * * and j = T * | F (Γ) . It is now easy to check that the extension of j to A is a surjective unital * -homomorphism (onto B). As the Haar state on B is given by so that h B • j corresponds via the identification of A * to B(Γ) exactly to the characteristic function of S.
The other direction follows exactly as in [FS08b]; we reproduce the argument for the sake of completeness. Suppose that S is a subgroup of Γ which is not normal and let γ 0 ∈ Γ, s 0 ∈ S be such that γ 0 s 0 γ −1 0 / ∈ S. Denote by φ S the state on A corresponding to the indicator function of S. Define f ∈ A by f = λ γ 0 s 0 −λ γ 0 . Then , so that Ker φ S is not selfadjoint and φ S cannot be a Haar idempotent.
Corollary 3.6. Let A be a coamenable cocommutative compact quantum group. The following are equivalent: (1) all idempotent states on A are Haar idempotents; (2) A ∼ = C * (Γ) for an amenable hamiltonian (i.e. containing no non-normal subgroups) discrete group Γ.
4. Idempotent states on U q (2) (q ∈ (−1, 0) ∪ (0, 1]) For q = 1, C(U q (2)) is equal to the C * -algebra of continuous functions on the unitary group U(2), and by Kawada and Itô's classical theorem all idempotent states on C(U(2)) come from Haar measures of compact subgroups of U(2). In this section we shall classify the idempotent states on C(U q (2)) for −1 < q < 1, q = 0. It turns out that they all correspond to Haar states of quantum subgroups of U q (2).
We begin with some preparatory lemmas.
Lemma 4.1. Let φ : Pol U q (2) → C be an idempotent state. Then we have 1}, for all s ∈ 1 2 Z + , r ∈ Z, −s ≤ k, ℓ ≤ s. Proof. By Proposition 3.4, we have φ•S = φ on Pol U q (2). Therefore, by Equation for all s ∈ 1 2 Z + , p ∈ Z. As we have already seen that these matrices are diagonal, it follows that the diagonal entries can take only the values 0 and 1.
Lemma 4.2. If φ : Pol U q (2) → C is an idempotent state with φ(u (1) 00 ) = 1, then there exists an idempotent stateφ : Pol T 2 → C such that Proof. By the previous lemma φ(c) = φ(c * ) = 0 We have u (1) 00 = 1 − (1 + q 2 )c * c, therefore φ(u (s) 00 ) = 1 is equivalent to φ(c * c) = 0. Then by Lemma 2.4, c, c * ∈ D φ , and φ vanishes on expressions of the form uc, cu, uc * , c * u with u ∈ Pol U q (2). But since vc = cv and u (s) kℓ for s ∈ 1 2 Z + , −s ≤ k, ℓ ≤ s, we can deduce that φ vanishes on the ideal I c = {u 1 cu 2 , u 1 c * u 2 ; u 1 , u 2 ∈ Pol U q (2)} generated by c and c * . It follows that we can divide out I c , i.e. there exists a unique stateφ on Pol U q (2)/I c such that the diagram w w n n n n n n n n n n n n n C commutes.
By the representation theory of C(SU q (2)), c * c is positive self-adjoint contraction, with the spectrum σ(c * c) ⊆ {q 2n ; n ∈ Z + } ∪ {0}, and therefore the product (1 − q 2 c * c) · · · (1 − q 2s c * c) defines a strictly positive operator. Therefore φ (u (s) kℓ v r = 0 for all r ∈ Z, s ∈ 1 2 Z + , s > 0, and −s ≤ k, ℓ ≤ s. Proof. By Lemma 4.1, we already know that φ(u kk v r ) = 1. We will show that this is impossible, if φ agrees with the Haar state h on the subalgebra generated by c * c.
By the Cauchy-Schwarz inequality, we have Applying φ to We can now give a description of all idempotent states on U q (2). It turns out that they are all induced by Haar states of quantum subgroups of U q (2).

(3) The states induced by the Haar states of the compact quantum subgroups
The case of n = 1 corresponds to the Haar state on SU q (2) viewed as a quantum subgroup of U q (2).
(1) The compact quantum group C(T 2 ) is commutative, by Kawada and Itô's theorem all idempotent states on C(T 2 ) are induced by Haar measures of compact subgroups of the two-dimensional torus T 2 .
Proof. (of Theorem 4.5) Let φ : C(U q (2)) → C be an idempotent state on U q (2). Clearly φ is uniquely determined by its restriction to Pol U q (2).
We distinguish two cases. Case (i) φ(u (1) 00 ) = 1. In this case Lemma 4.2 shows that φ is induced by an idempotent state on the quantum subgroup T 2 of U q (2), i.e. φ =φ • π T 2 for some idempotent stateφ : C(T 2 ) → C. This case includes the counit ε of U q (2), it corresponds to the trivial subgroup {1} of T 2 . Case (ii) φ(u kℓ v r ) = 0 for all s ∈ 1 2 Z + , s > 0, −s ≤ k, ℓ ≤ s, and r ∈ Z. It remains to determine φ on the * -subalgebra alg{v, v * } generated by v, since Pol U q (2) = A 0 ⊕ alg{v, v * } as a vector space. But this subalgebra is isomorphic to the * -Hopf algebra Pol U(1) of polynomials on the unit circle, and therefore φ| alg{v,v * } has to be induced by the Haar measure of a compact subgroup of U(1). We have the following possibilities.
(2) φ| alg{v,v * } = h U (1) , i.e. the restriction of φ to alg{v, v * } is equal to the Haar state of Pol U(1) . In this case we have φ(u We see that in this case φ is the Haar state h of U q (2).
(3) φ| alg{v,v * } is the idempotent state on U(1) induced by the Haar measure of the subgroup Z n ⊆ U(1) for some n ∈ N, n ≥ 2. In this case we have It follows that φ is induced by the Haar state of the quantum subgroup Conversely, all the states we have found are induced by Haar states on quantum subgroups of U q (2), therefore they are clearly idempotent. It can be also checked directly.
We see that all idempotent states on U q (2) are induced from Haar states of quantum subgroups of U q (2). We can also deduce the complete list of quantum subgroups of U q (2): Corollary 4.7. Let q ∈ (−1, 0) ∪ (0, 1). Then the following is a complete list of the non-trivial quantum subgroups of U q (2).
(1) The two-dimensional torus and its closed subgroups.
(2) The compact quantum groups of the form SU q (2) ⋉ σ Z n , with n ∈ N (here the twisting is identical to that appearing in the identification U q (2) ∼ = SU q (2) ⋉ σ T).
5. Idempotent states on compact quantum groups SU q (2) and Let us first discuss the case q = 1. C(SU 1 (2)) and C (SO 1 (3)) are the algebras of continuous functions on the groups SU(2) and SO(3). All idempotent states correspond to Haar measures on compact subgroups. The list of these subgroups can be found, e.g., in [Pod95].
Consider now the generic case q ∈ (−1, 0) ∪ (0, 1). Every idempotent state on SU q (2) induces an idempotent state on U q (2), since SU q (2) is a quantum subgroup of U q (2). This observation allows us to deduce all idempotent states on SU q (2) from Theorem 4.5. We omit the details and just state the result.
Since the morphism j : C(SU q (2)) → C(U(1)) gives the diagonal matrices Consider now the idempotent states on SO q (3). Since C(SO q (3)) is a subalgebra of C(SU q (2)) and since the inclusion map is a quantum group morphism, every idempotent state on SU q (2) gives an idempotent state on SO q (3) by restriction. We will show that all idempotent states on SO q (3) arise in this way.
It follows that all idempotent states on SO q (3) are induced from Haar states of quantum subgroups.
Furthermore, the Haar state, the counit, and the states induced from the Haar states on the quantum subgroups U(1) ∼ = SO(2) and its closed subgroups are the only idempotent states on SO q (3).
Let now φ be an idempotent state on SO q (3). Denote by E the conditional expectation from C(SU q (2)) onto C(SO q (3)) introduced in Paragraph 2.4.2. Then φ = φ • E defines an idempotent state on SU q (2) such that φ =φ| C (SOq(3)) . With this observation Theorem 5.2 follows immediately from Theorem 5.1.
Remark 5.3. This method applies to quotient quantum groups in general. Let A be a compact quantum group. A quantum subgroup (B, j) is called normal, if the images of the conditional expectations As a corollary to Theorems 5.1 and 5.2, we recover Podleś' classification [Pod95] of the quantum subgroups of SU q (2) and SO q (3).
Note that π T 2 : C(U q (2)) → C(T 2 ) is a well-defined * -algebra and coalgebra morphism also for q = 0, so T 2 and its compact subgroups induce idempotent states on U 0 (2).
As a is a coisometry, we have a decreasing family of orthogonal projections (a * ) n a n , n ∈ N , which are group-like, i.e. ∆((a * ) n a n ) = (a * ) n a n ⊗ (a * ) n a n , and c * c = 1 − a * a is also an orthogonal projection.
For k ∈ Z and n ∈ Z + , define functions Θ k n : M → C by The span of these functions is dense in C(M), and they satisfy Θ k n Θ ℓ m = Θ k+ℓ max(n,m) , (Θ k n ) * = Θ −k n , and ε(Θ k n ) = Θ k n (e M ) = 1. For their coproduct, we have ∆Θ k n (z 1 , m 1 ), (z 2 , m 2 ) = Θ k n z 1 z 2 , min(m 1 , m 2 ) i.e. ∆Θ k n = Θ k n ⊗ Θ k n . Proposition 6.2. The semigroup M is a quantum quotient semigroup of U 0 (2), in the sense that there exists an injective * -algebra homomorphism j from Pol M := span{Θ k n ; n ∈ Z + , k ∈ Z} to Pol U 0 (2) such that Proof. For n ∈ Z + and k ∈ Z, define E k n = (α * ) n α n v k ∈ Pol U q (2). From the defining relations of U 0 (2), one can check that the E k n satisfy the same *algebraic and coalgebraic relations as the Θ k n , i.e. j(Θ k n ) = E k n defines a * -bialgebra homomorphism j : Pol M → Pol U 0 (2).
Let us show that j is injective. Assume there exists a non-zero function f = k,n λ k,n Θ k n such that j(f ) = 0. Let n 0 be the smallest integer for which there exists a k ∈ Z such that λ k,n = 0. Take the representation π = π 0 ⊗ id L 2 (T) of C(U 0 (2)) ∼ = C(SU 0 (2)) ⊗ C(U(1)) (recall that π 0 was defined in Section 2.3.1).
We can now give a description of all idempotent states on U 0 (2).
Theorem 6.3. The following gives a complete list of the idempotents states on U 0 (2).
(1) The idempotent states induced by the Haar measures on the two-dimensional torus T 2 and its closed subgroups. If ρ denotes the Haar measure of T 2 or one of its closed subgroups, then the corresponding idempotent φ ρ state is given by for n, m ∈ Z + , k, ℓ ∈ Z. This includes the counit of U 0 (2), for the trivial subgroup {1} of T 2 .
(2) The family Ψ n,m = ψ n ⊗ φ m , n ∈ Z + , m ∈ N ∪ {∞}. Here φ m is an idempotent state on C(U(1)), namely the Haar measure for m = ∞ and the idempotent state induced by the Haar measure of Z m for m ∈ N. And ψ n , n ∈ Z + is the idempotent state on C(SU 0 (2)) defined by ψ n (α * ) r γ k α s = 1 if r = s ≤ n and k = 0 0 else. Remark 6.4. The state Ψ 0,∞ can be considered as the Haar state on U 0 (2) since it is invariant, i.e.
for any f ∈ C(U 0 (2)) * . But Ψ 0,∞ is not faithful. Its null space is a left ideal, but not self-adjoint or two-sided. It is a subcoalgebra, i.e. we have but it is not a coideal, since the counit does not vanish on N Ψ 0,∞ .
Proof. (of Theorem 6.3) Let φ : C(U 0 (2)) → C be an idempotent state on U 0 (2). Then φ induces an idempotent state φ • j on C(M). By Lemma 6.1, φ • j is integration against a probability measure of the form ρ ⊗ δ n with n ∈ Z + ∪ {∞} and ρ an idempotent measure on U(1). This determines φ on the subalgebra generated by v and (a * ) r a r , r ∈ N, we have for k ∈ Z, r ∈ Z + . Case (i): n = ∞. For k > 0 and any r, s ∈ Z + , ℓ ∈ Z, we have and therefore φ vanishes on the * -ideal I c generated by c. As in the proof of Lemma 4.2, it follows that φ is induced by an idempotent state on Pol U 0 (2)/I c ∼ = Pol T 2 , i.e. φ is of the form given in (1). Case (ii): n = 0. Using again the Cauchy-Schwarz inequality, we get φ (a * ) r a s v ℓ 2 ≤ φ ((a * ) r a r ) φ ((a * ) s a s ) , (6.3) for k, r, s ∈ Z + , ℓ ∈ Z. This shows that φ (a * ) r c k a r v ℓ vanishes, unless r = s = 0. different from 0. But then φ = φ ⋆ φ implies for k > 0, s ∈ Z. By hermitianity φ (c * ) k v ℓ = 0 for k > 0, and φ has the form given in (2) with n = 0. Case (iii): n ∈ N. We use once more the Cauchy-Schwarz inequality. For k = 0, (6.1) and (6.2) imply that φ (a * ) r c k a s v ℓ vanishes unless r = s = n. But then we can show that φ = φ ⋆ φ implies φ (a * ) n c k a n v ℓ = 0 in the same way as in the previous case.
For k = 0, we see from (6.3) that φ (a * ) r a s v ℓ vanishes unless r, s ≤ n. The elements (a * ) r a s v ℓ are group-like, therefore φ (a * ) r a s v ℓ ∈ {0, 1}. If we can show φ (a * ) r a s v ℓ = 1 for r = s, we are done, since then φ (a * ) r c k a s v ℓ is non-zero only if r = s ≤ n. We get φ (a * ) r a s v ℓ = δ rs ρ(v ℓ ) for r, s ≤ n, i.e. φ has the form given in (2).
We show φ (a * ) r a s v ℓ = 1 for r = s by contradiction. Assume there exists a triple (r 0 , s 0 , ℓ 0 ) such that φ (a * ) r a s v ℓ = 1 and choose such a triple with Maximality of r 0 implies r 0 > s 0 and which is clearly a contradiction to the positivity of φ. Conversely, using the formulas for r, s, k ∈ Z + , ℓ ∈ Z, for the coproduct in Pol U 0 (2), one can check that all states given in the theorem are indeed idempotent.
The complete description of the idempotent states on SU 0 (2) follows now directly from Theorem 6.3 and the comments before Theorem 4.5.
Theorem 6.5. The following gives a complete list of the idempotents states on SU 0 (2).
(1) The family φ n , n ∈ N ∪ {∞} where φ 1 is the counit, φ ∞ the idempotent induced by the Haar state of quantum subgroup U(1), and φ n , 2 ≤ n < ∞, denotes the idempotent state induced by the Haar state on the quantum subgroup Z n .
(2) The family ψ n , n ∈ Z + defined by Similarly using the conditional expectation introduced in Paragraph 2.4.2, we can derive a complete classification of the idempotent states on SO 0 (3). The proof is identical to the proof of Theorem 5.2 and therefore omitted.
7. The idempotent states on SU q (2) as elements of the dual and associated quantum hypergroups Let A be a compact quantum group and A the corresponding Hopf * -algebra dense in A. Then A is an algebraic quantum group in the sense of van Daele, and so is its dualÂ, given by the functionals of the form h(· a) with a ∈ A. By [FS08b, Lemma 3.1], an idempotent state φ on A defines a group-like projection p φ in the multiplier algebra M(Â) of the dual, and therefore, by [LvD07, Theorem 2.7] and [FS08b, Theorem 2.4] an algebraic quantum hypergroupÂ p φ . As an algebra, A p φ = p φÂ p φ , and the coproduct ofÂ p φ is given bŷ for a ∈Â p φ , where∆ denotes the coproduct ofÂ.
Let q ∈ (−1, 0) ∪ (0, 1). In this section we will consider the case of the compact quantum group SU q (2) and describe the algebraic quantum hypergroups associated to its idempotent states. Note that in this case the dense Hopf * -algebra is A = Pol SU q (2) = span {u (s) kℓ : s ∈ 1 2 Z + , −s ≤ k, ℓ ≤ s}. We will use the basis e (s) kℓ ) * · forÂ = Pol SU q (2). Using the orthogonality relation Eq. (5.12)], we can check that this basis is dual to the basis {u (s) kℓ : s ∈ 1 2 Z + , −s ≤ k, ℓ ≤ s} of Pol SU q (2). The algebraic quantum groupÂ = Pol SU q (2) is of discrete type and equal to the algebraic direct sumÂ The e i.e. p φ 2 is the sum of the identity matrices from the odd-dimensional matrix algebras M 2s+1 , s ∈ Z + . This projection is in the center of M(Â), thereforê A 2 = p φ 2Â p φ 2 will be an algebraic quantum group. We get i.e.Â p φ 2 is discrete algebraic quantum group dual to SO q (3). This is to be expected as Z 2 is the only nontrivial normal quantum subgroup of SU q (2) and SO q (3) is the corresponding quotient quantum group. and this projection is not central. We get which is a commutative algebraic quantum hypergroup of discrete type. This is the dual of the hypergroup introduced in [Koo91, Section 7].
Note that as the quantum subgroups consider in the last two paragraphs are not normal, the objects we obtain have only the quantum hypergroup structure (and can be informally thought of as duals of quantum hypergroups obtained via the double coset construction, [CV99]).

Appendix
The goal of the appendix is to provide a short proof of coamenability of the deformations of classical compact Lie groups. To facilitate the discussion we use here the symbol G to denote a compact quantum group, G to denote the dual of G, C(G) red and C(G) to denote respectively the reduced and universal C * -algebras associated with G and L ∞ (G) to denote the corresponding von Neumann algebra (we refer for example to [Tom07] for precise definitions). Note that contrary to the main body of the paper we do not assume that the Haar state on G is faithful, so that G need not be in the reduced form. We adopt the following definition ( [BMT02], [Tom06]).
Definition A.1. A compact quantum group G is said to be coamenable if the dual quantum group G is amenable, that is, L ∞ ( G) has an invariant mean.
The following result gives a useful criterion to check coamenability: The second equivalence is fairly easy to show, in the first the forward implication was established in [BMT02] and the backward implication in [Tom06].
Let G be a classical compact Lie group and G q the q-deformation with the parameter −1 < q < 1, q = 0 (see [KS98]). The function algebra C(G q ) is the universal C * -algebra generated by certain polynomial elements. The Haar state is denoted by h.
The following theorem was proved by T. Banica ([Ban99, Corollary 6.2]). We present another direct proof using L. I. Korogodski-Y. S. Soibelman's results on the representation theory of C(G q ).
Theorem A.3. The quantum group G q is coamenable.
Proof. Let us introduce the left ideal N h := {a ∈ C(G q ) | h(a * a) = 0}, which is in fact an ideal of C(G q ). The reduced compact quantum group C(G q ) red is defined as the quotient C(G q )/N h . By Theorem A.2 to show that G q is coamenable it suffices to show that the C * -algebra C(G q ) red has a character.
Consider an irreducible representation π : C(G q ) red → B(H π ). Composing this map with the canonical surjection ρ : C(G q ) → C(G q ) red , we get an irreducible representation π • ρ of C(G q ). Thanks to [KS98, Theorem 6.2.7 (3), §3], we may assume that π • ρ is of the following form: π • ρ = (π s i 1 ⊗ · · · ⊗ π s i k ⊗ π t ) • δ (k) or π • ρ = π t , where s i 1 · · · s i k is the reduced decomposition in the Weyl group of G, and t ∈ T , the maximal torus of G. In the latter case π is a one-dimensional representation. In the former case, we remark that the counit of C(G q ) factors through Im π s i for every i, that is, there exists η i : Im π s i → C such that η i •π s i = ε (See the argument in [Tom07, p. 294]).
Thus we have proved in each case the existence of a one-dimensional representation of the C * -algebra C(G) red , and G q is co-amenable.