Braided quantum SU(2) groups

BRAIDED QUANTUM SU(2) GROUPS

arXiv:1411.3218v1 [math.OA] 12 Nov 2014

PAWEŁ KASPRZAK, RALF MEYER, SUTANU ROY, AND STANISŁAW LECH WORONOWICZ Abstract. We construct a family of q-deformations of SU(2) for complex parameters q ∈ C with 0 < |q| < 1. For real q, the deformation coincides with the compact quantum SUq (2) group introduced by S.L. Woronowicz. For q ∈ / R, SUq (2) is only a braided compact quantum group with respect to a tensor product functor ⊠q/q .

1. Introduction The q-deformations of SU(2) for real deformation parameters 0 < q < 1 discovered in [9] are among the first and most important examples of compact quantum groups. Here we construct a family of q-deformations of SU(2) for complex parameters q ∈ C, 0 < |q| < 1. For q ∈ / R, SUq (2) is not a compact quantum group, but a braided compact quantum group in a suitable tensor category. A compact quantum group G as defined in [11] is a pair G = (A, ∆) where ∆ : A → A ⊗ A is a coassociative morphism satisfying the cancellation law (1.4) below. The C∗ -algebra A is viewed as the algebra of continuous functions on a quantum group G. Any compact quantum group has a Haar state and a well-understood representation theory, described by an analogue of the Peter–Weyl Theorem and through a Hopf ∗ -algebra assigned to G, see [11]. The theory of compact quantum groups is formulated within the category C ∗ of C∗ -algebras. This category is equipped with the tensor functor ⊗, and (C ∗ , ⊗) is a monoidal category in the sense of [2]. A more general theory of compact quantum groups may be formulated within a monoidal category (D∗ , ⊠), where D∗ is a suitable category of C∗ -algebras with additional structure, and ⊠ : D∗ × D∗ → D∗ is a monoidal bifunctor on D∗ . Braided Hopf algebras may be defined in braided monoidal categories (see [4, Definition 9.4.5]); the braiding on the monoidal category is needed to axiomatise the coinverse. Since the coinverse does not enter the axioms of a compact quantum group, the theory of braided compact quantum groups may work in monoidal categories without braiding. Let T be the group of complex numbers of modulus 1 and let CT∗ the category of T-C∗ -algebras; its objects are C∗ -algebras with an action of T, arrows are T-equivariant C∗ -algebra morphisms. We consider a family of monoidal structures ⊠ζ on CT∗ parametrised by ζ ∈ T, where the monoidal bifunctor ⊠ζ : CT∗ ×CT∗ → CT∗ is introduced along the lines described in [5]. The braiding parameter ζ ∈ T and the deformation parameter q ∈ C for our q-deformation of SU(2) are related by qζ = q, and we also assume 0 < |q| < 1. The C∗ -algebra A of SUq (2) is defined as the universal unital C∗ -algebra generated by

2010 Mathematics Subject Classification. 81R50 (46L55, 46L06). Key words and phrases. braided compact quantum group; SUq (2); Uq (2). Supported by the Alexander von Humboldt-Stiftung. 1

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KASPRZAK, MEYER, ROY, AND WORONOWICZ

two elements α, γ subject to the relations  α∗ α + γ ∗ γ    2 ∗   αα + |q| γ ∗ γ (1.1) γγ ∗    αγ   αγ ∗

= I, = I, = γ ∗ γ, = qγα, = qγ ∗ α.

We shall prove that these algebras for different values of q are all isomorphic. For real q, the algebra A coincides with the algebra of continuous functions on the quantum SUq (2) group described in [9]: A = C(SUq (2)). Then there is a unique morphism ∆ : A → A ⊗ A with ∆(α) = α ⊗ α − qγ ∗ ⊗ γ, ∆(γ) = γ ⊗ α + α∗ ⊗ γ.

(1.2) It is coassociative, that is, (1.3)

(∆ ⊗ idA ) ◦ ∆ = (idA ⊗ ∆) ◦ ∆,

and has the following cancellation property: A ⊗ A = ∆(A)(A ⊗ I),

(1.4)

A ⊗ A = ∆(A)(I ⊗ A).

Here EF for two closed subspaces E and F of a C∗ -algebra denotes the norm-closed linear span of the set of products ef for e ∈ E, f ∈ F . If q is not real, then the operators on the right hand sides of (1.2) do not satisfy the relations (1.1), so there is no morphism ∆ satisfying (1.2). Instead, (1.2) defines a T-equivariant morphism A → A⊠ζ A for the monoidal functor ⊠ζ . This morphism in CT∗ satisfies appropriate analogues of the coassociative and cancellation laws (1.3) and (1.4), so we get a braided compact quantum group. For X, Y ∈ Obj(C ∗ ), X ⊗ Y contains commuting copies X ⊗ IY of X and IX ⊗ Y of Y : (x ⊗ IY )(IX ⊗ y) = (IX ⊗ y)(x ⊗ IY ) for any x ∈ X and y ∈ Y . Moreover, X ⊗ Y = (X ⊗ IY )(IX ⊗ Y ). Similarly, X ⊠ζ Y for X, Y ∈ CT∗ is a C∗ -algebra containing a copy j1 (X) of X and a copy j2 (Y ) of Y such that X ⊠ζ Y = j1 (X)j2 (Y ), where j1 ∈ Mor(X, X ⊠ζ Y ) and j2 ∈ Mor(Y, X ⊠ζ Y ) are injective morphisms. For T-homogeneous elements x ∈ Xk and y ∈ Yl (as defined in (3.4)), we have the commutation relation (1.5)

j1 (x)j2 (y) = ζ kl j2 (y)j1 (x)

The following theorem contains the main result of this paper: Theorem 1.1. Assume that qζ = q.

(1.6)

Then (1) there is a unique T-equivariant morphism ∆ ∈ Mor(A, A ⊠ζ A) with ∆(α) = j1 (α)j2 (α) − qj1 (γ)∗ j2 (γ), (1.7)

∆(γ) = j1 (γ)j2 (α) + j1 (α)∗ j2 (γ),

(2) ∆ is coassociative, that is, (∆ ⊠ζ idA ) ◦ ∆ = (idA ⊠ζ ∆) ◦ ∆,


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(3) ∆ obeys the cancellation law j1 (A)∆(A) = ∆(A)j2 (A) = A ⊠ζ A. Braided Hopf algebras that deform SL(2, C) are already described in [3]. We could not find a precise relationship between Majid’s braided Hopf algebra BSL(2) and our braided quantum group SUq (2). A few words about the notation used later. Usually, z will denote the variable running over T. It may also be treated as a coordinate function on T and viewed as z ∈ C(T). It will always be clear from the context which of these two meanings is used. The comultiplication δ ∈ Mor(C(T), C(T) ⊗ C(T)) on C(T) is δz = z ⊗ z. ∗

Let A and B be C -algebras. The multiplier algebra of B is denoted by M(B). A morphism π ∈ Mor(A, B) is a ∗ -homomorphism π : A → M(B) with π(A)B = B. Since A and A⊠A are unital, we mostly need morphisms between unital C∗ -algebras. These are simply unital ∗ -homomorphisms. 2. The algebra of SUq (2) The following elementary observation explains the meaning of the relations (1.1): Lemma 2.1. Two elements α and γ of a C∗ -algebra satisfy the relations (1.1) if and only if the matrix α , −qγ ∗ γ , α∗ is unitary. There are many ways to introduce a C∗ -algebra with given generators and relations. One may consider the algebra A of all non-commutative polynomials in the generators and their adjoints and take the largest C∗ -seminorm on A vanishing on the given relations. The set N of elements with vanishing seminorm is an ideal in A. The seminorm becomes a norm on A/N. Completing A/N with respect to this norm gives the desired C∗ -algebra A. Another way is to consider the operator domain consisting of all families of operators satisfying the relations. Then A is the algebra of all continuous operator functions on that domain (see [1]). Applying one of these procedures to the relations (1.1) gives a C∗ -algebra A with two distinguished elements α, γ ∈ A that is universal in the following sense: e be a C∗ -algebra with two elements α e satisfying Theorem 2.2. Let A e, e γ∈A  ∗ ∗ α ˜ α ˜ + γ˜ γ˜ = I,    2  α ˜α ˜ ∗ + |q| γ˜ ∗ γ˜ = I,  (2.1) γ˜ γ˜ ∗ = γ˜ ∗ γ˜ ,   γα ˜,  α˜ ˜ γ = q˜   α ˜ γ˜ ∗ = q˜ γ∗α ˜.

e with ρ(α) = α Then there is a unique morphism ρ ∈ Mor(A, A) e and ρ(γ) = γ e.

The elements α e = IC(T) ⊗ α and e γ = z ⊗ γ of C(T) ⊗ A satisfy relations (2.1). Hence Theorem 2.2 gives a unique morphism ρA ∈ Mor(A, C(T) ⊗ A) with (2.2)

ρ(α) = IC(T) ⊗ α, ρ(γ) = z ⊗ γ.

This is a continuous T-action, so we may view (A, ρA ) as an object in the category CT∗ described in detail in the next section. Theorem 2.3. The C∗ -algebra A does not depend on q, that is, the algebras A for q ∈ C with 0 < |q| < 1 are all isomorphic.

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Proof. The equations (1.1) show that γ is normal with kγk ≤ 1. So we may use the functional calculus for continuous functions on the unit disc D1 = {λ ∈ C : |λ| ≤ 1}. We claim that (2.3)

αf (γ) = f (qγ)α 1

for all f ∈ C(D ). Indeed, the set B ⊂ C(D1 ) of functions satisfying (2.3) is a norm-closed, unital subalgebra of C(D1 ). The last two equations in (1.1) say that B contains the functions f (λ) = λ and f ∗ (λ) = λ. Since these separate the points of D1 , the Stone–Weierstrass Theorem gives B = C(D1 ). Let q = eiθ |q| be the polar decomposition of q. For λ ∈ D1 , let ( λeiθ log|q| |λ| for λ 6= 0, g(λ) = 0 for λ = 0. Then g is a homeomorphism of D1 (we get g −1 if we replace θ by −θ), and g(qλ) = |q| g(λ). Let γ ′ = g(γ), then γ and γ ′ generate the same C∗ -algebra. Inserting f = g and f = g in (2.3) gives aγ ′ = |q| γ ′ α,

a(γ ′ )∗ = |q| (γ ′ )∗ α.

Moreover, |g(λ)| = |λ|. Therefore, |γ ′ | = |γ| and we may replace γ by γ ′ in the first three equations of (1.1). As a result, α and γ ′ satisfy the relations (1.1) with |q| instead of q. Since g is a homeomorphism, it follows that we get the same algebra A if we replace q by |q|. Now [9, Theorem A2.2, page 180] finishes the proof. 3. Monoidal structure on the category of T-C∗ -algebras Let ζ ∈ T. In this section, we describe the monoidal category (CT∗ , ⊠ζ ) that will be the framework for our construction of braided quantum groups. See [2] for the definition of a monoidal category. Let CT∗ be the category of T-C∗ -algebras. An object of CT∗ is, by definition, a pair (X, ρX ) where X is a C∗ -algebra and ρX ∈ Mor(X, C(T) ⊗ X) is such that the diagram ρX

X (3.1)

/ C(T) ⊗ X

ρX

δ⊗id

C(T) ⊗ X

idC(T) ⊗ρX

/ C(T) ⊗ C(T) ⊗ X

is commutative and ρX satisfies the Podleś condition (3.2)

ρX (X)(C(T) ⊗ IX ) = C(T) ⊗ X.

Such a morphism ρX is equivalent to a continuous T-action on X by [8, Proposition 2.3]. Let X, Y be T-C∗ -algebras. The set of morphisms from X to Y in CT∗ is the set MorT (X, Y ) of T-equivariant morphisms X → Y . By definition, ϕ ∈ Mor(X, Y ) is T-equivariant if and only if the following diagram commutes: X (3.3)

ρX

idC(T) ⊗ϕ

ϕ

Y

/ C(T) ⊗ X

ρY

/ C(T) ⊗ Y

Let X ∈ CT∗ . An element x ∈ X is homogeneous of degree n ∈ Z if (3.4)

ρX (x) = z n ⊗ x.


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The degree of a homogeneous element x will be denoted by deg(x). Let Xn be the set of homogeneous elements of X of degree n. This is a closed linear subspaces of X, and Xn Xm ⊂ Xn+m and Xn∗ = X−n for n, m ∈ Z. Moreover, finite sums of homogeneous elements are dense in X. Let ζ ∈ T. The monoidal functor ⊠ζ : CT∗ × CT∗ → CT∗ is introduced as in [5]. We describe X ⊠ζ Y using quantum tori. By definition, the C∗ -algebra C(T2ζ ) of the quantum torus is the C∗ -algebra generated by two unitary elements U, V subject to the relation U V = ζ V U . There are unique injective morphisms ι1 , ι2 ∈ Mor(C(T), C(T2ζ )) with ι1 (z) = U and ι2 (z) = V . We define j1 ∈ Mor(X, C(T2ζ ) ⊗ X ⊗ Y ) and j2 ∈ Mor(Y, C(T2ζ ) ⊗ X ⊗ Y ) by j1 (x) = (ι1 ⊗ idX ) ◦ ρX (x) Y

j2 (y) = (ι2 ⊗ idY ) ◦ ρ (y)

for all x ∈ X, for all y ∈ Y.

Let x ∈ Xk and y ∈ Yl . Then j1 (x) = U k ⊗ x ⊗ 1 and j2 (y) = V l ⊗ 1 ⊗ y, so that we get the commutation relation (1.5). This implies j1 (X)j2 (Y ) = j2 (Y )j1 (X), so that j1 (X)j2 (Y ) is a C∗ -algebra. We define X ⊠ζ Y = j1 (X)j2 (Y ). This construction agrees with the one in [5] because C(T2ζ ) ∼ = C(T) ⊠ζ C(T), see also the end of [5, Section 5.2]. There is a unique continuous T-action ρX⊠ζ Y on X ⊠ζ Y for which j1 and j2 are T-equivariant, that is, j1 ∈ MorT (X, X ⊠ζ Y ) and j2 ∈ MorT (Y, X ⊠ζ Y ). This action is constructed in a more general context in [6]. We always equip X ⊠ζ Y with this T-action and thus view it as an object of CT∗ . The construction ⊠ζ is a bifunctor; that is, T-equivariant morphisms π1 ∈ MorT (X1 , Y1 ) and π2 ∈ MorT (X2 , Y2 ) induce a unique T-equivariant morphism π1 ⊠ζ π2 ∈ MorT (X1 ⊠ζ X2 , Y1 ⊠ζ Y2 ) with (3.5)

(π1 ⊠ζ π2 )(jX1 (x1 )jX2 (x2 )) = jY1 (π1 (x1 ))jY2 (π2 (x2 ))

for all x1 ∈ X1 and x2 ∈ X2 . Proposition 3.1. Let x ∈ X and y ∈ Y be homogeneous elements. Then j1 (x)j2 (Y ) = j2 (Y )j1 (x), j1 (X)j2 (y) = j2 (y)j1 (X). Proof. Equation (1.5) shows that j1 (x)j2 (y) = j2 (y)j1 (ρX ζ deg(y) (x)) for any x ∈ X and any homogeneous y ∈ Y . Since ρX is an automorphism of X, ζ deg(y) this implies j1 (X)j2 (y) = j2 (y)j1 (X). Similarly, j1 (x)j2 (y) = j2 (ρYζdeg(x) (y))j1 (x) for homogeneous x ∈ X and any y ∈ Y implies j1 (x)j2 (Y ) = j2 (Y )j1 (x). 4. Proof of Theorem 1.1 Let α and γ be the distinguished elements of A. Let α e and e γ be the elements of A ⊠ζ A appearing on the right hand side of (1.7): (4.1)

α e = j1 (α)j2 (α) − qj1 (γ)∗ j2 (γ), γ = j1 (γ)j2 (α) + j1 (α)∗ j2 (γ). e

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We have deg(α) = deg(α∗ ) = 0, deg(γ) = 1 and deg(γ ∗ ) = −1 by (2.2). Assume that ζ and q are related by (1.6). Using (1.5) we may rewrite (4.1) in the following form: α e = j2 (α)j1 (α) − qj2 (γ)j1 (γ)∗ , Therefore, (4.2)

γ = j2 (α)j1 (γ) + j2 (γ)j1 (α)∗ . e

α e∗ = j1 (α)∗ j2 (α)∗ − qj1 (γ)j2 (γ)∗ , γ e∗ = j1 (γ)∗ j2 (α)∗ + j1 (α)j2 (γ)∗ .

The four equations (4.1) and (4.2) together are equivalent to the single matrix equation α e , −qe γ∗ j1 (α) , −qj1 (γ)∗ j2 (α) , −qj2 (γ)∗ (4.3) = . γ e , α e∗ j1 (γ) , j1 (α)∗ j2 (γ) , j2 (α)∗ Lemma 2.1 shows that the matrix α , −qγ ∗ (4.4) u= ∈ M2×2 (A) γ , α∗

is unitary. Therefore, the matrix j1 (u)j2 (u) on the right hand side of (4.3) is also unitary. Using Lemma 2.1 in the other direction implies that α e, γ e ∈ A ⊠ζ A satisfy (2.1). So the universal property of A in Theorem 2.2 gives a unique morphism ∆ with ∆(α) = α e and ∆(γ) = γ e. The elements α and γ are homogeneous of degrees 0 and 1, respectively, by (2.2). Hence α e and e γ are homogeneous of degree 0 and 1 as well. Since α and γ generate A, it follows that ∆ is T-equivariant. This proves statement (1) in Theorem 1.1. Here A⊠ A A we use the action ρA⊠ζ A of T with ρz ζ (j1 (a1 )j2 (a2 )) = j1 (ρA z (a1 ))j2 (ρz (a2 )). Since ∆ is T-equivariant, we may form ∆ ⊠ζ id and id ⊠ ∆ζ . Now (4.3) may be rewritten as ∆(α) , −q∆(γ)∗ j1 (α) , −qj1 (γ)∗ j2 (α) , −qj2 (γ)∗ = , ∆(γ) , ∆(α)∗ j1 (γ) , j1 (α)∗ j2 (γ) , j2 (α)∗ Identifying M2×2 (A) with M2×2 (C) ⊗ A, we may further rewrite this as

(4.5)

(id ⊗ ∆)(u) = (id ⊗ j1 )(u) (id ⊗ j2 )(u),

where id is the identity map on M2×2 (C). Now we prove statement (2) in Theorem 1.1. Let j1 , j2 , j3 be the natural embeddings of A into A ⊠ζ A ⊠ζ A. Applying id ⊗ (∆ ⊠ζ idA ) and id ⊗ (idA ⊠ζ ∆) to the right hand side of (4.5) gives the same result: (id ⊗ (∆ ⊠ζ idA ) ◦ ∆) (u) = (id ⊗ j1 )(u) (id ⊗ j2 )(u) (id ⊗ j3 )(u), (id ⊗ (idA ⊠ζ ∆) ◦ ∆) (u) = (id ⊗ j1 )(u) (id ⊗ j2 )(u) (id ⊗ j3 )(u). Thus (∆ ⊠ζ idA ) ◦ ∆ and (idA ⊠ζ ∆) ◦ ∆ coincide on α, γ, α∗ , γ ∗ . Since the latter generate A, this proves statement (2) of Theorem 1.1. Now we prove statement (3). Let S = {x ∈ A : j1 (x) ∈ ∆(A)j2 (A)} . This is a closed subspace of A. We may also rewrite (4.5) as ∗ j1 (α) , −qj1 (γ)∗ ∆(α) , −q∆(γ)∗ j2 (α) , −qj2 (γ)∗ (4.6) = . j1 (γ) , j1 (α)∗ ∆(γ) , ∆(α)∗ j2 (γ) , j2 (α)∗

Thus α, γ, α∗ , γ ∗ ∈ S. Let x, y ∈ S and with homogeneous y. Proposition 3.1 gives j1 (xy) = j1 (x)j1 (y) ∈ ∆(A)j2 (A)j1 (y) = ∆(A)j1 (y)j2 (A) ⊂ ∆(A)∆(A)j2 (A)j2 (A) = ∆(A)j2 (A).


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That is, xy ∈ S. Therefore, all monomials in α, γ, α∗ , γ ∗ belong to S, so that S = A. Hence j1 (A) ⊂ ∆(A)j2 (A). Now A ⊠ζ A = j1 (A)j2 (A) ⊂ ∆(A)j2 (A)j2 (A) = ∆(A)j2 (A), which is one of the Podleś conditions. Similarly, let R = {x ∈ A : j2 (x) ∈ j1 (A)∆(A)} . Then R is a closed subspace of A. We may also rewrite (4.5) as ∗ j2 (α) , −qj2 (γ)∗ j1 (α) , −qj1 (γ)∗ ∆(α) , −q∆(γ)∗ (4.7) = . j2 (γ) , j2 (α)∗ j1 (γ) , j1 (α)∗ ∆(γ) , ∆(α)∗ Thus α, γ, α∗ , γ ∗ ∈ R. Let x, y ∈ R with homogeneous x. Proposition 3.1 gives j2 (xy) = j2 (x)j2 (y) ∈ j2 (x)j1 (A)∆(A) = j1 (A)j2 (x)∆(A) ⊂ j1 (A)j1 (A)∆(A)∆(A) = j1 (A)∆(A). Thus xy ∈ R. Therefore, all monomials in α, γ, α∗ , γ ∗ belong to R, so that R = A, that is, j2 (A) ⊂ j1 (A)∆(A). This implies the other Podleś condition: A ⊠ζ A = j1 (A)j2 (A) ⊂ j1 (A)j1 (A)∆(A) = j1 (A)∆(A). This finishes the proof of Theorem 1.1. 5. Towards the representation theory of SUq Let H be a T-Hilbert space, that is, a Hilbert space with a unitary representation U : T → U(H). For z ∈ T and x ∈ K(H) we define ρzK(H) (x) = Uz xUz∗ . Thus (K(H), ρK(H) ) is a T-C∗ -algebra. Let (X, ρX ) ∈ Obj(CT∗ ). Since ρK(H) is inner, the braided tensor product K(H) ⊠ζ X may (and will) be identified with K(H) ⊗ X – see [5, Corollary 5.18] and [5, Example 5.19]. Definition 5.1. Let H be a T-Hilbert space and let v ∈ M(K(H) ⊗ A) be a K(H) ⊗ ρX unitary element which is T-invariant, that is, (ρz z )(v) = v. We call v a representation of SUq (2) on H if (idH ⊗ ∆)(v) = (idH ⊗ j1 )(v) (idH ⊗ j2 )(v). If v1 ∈ M(K(H1 ) ⊗ A) and v2 ∈ M(K(H2 ) ⊗ A) are representations of SUq (2), then so is their direct sum v = v1 ⊕ v2 ∈ M(K(H1 ⊕ H2 ) ⊗ A). This leads to a notion of irreducibility (or indecomposability). Theorem 6.1 below will show that representations of SUq (2) are equivalent to representations of the compact quantum group Uq (2). Since this is an ordinary compact quantum group, this allows us to carry over all the usual structural results about representations of compact quantum groups to those of the braided compact quantum group SUq (2). In particular, there should be a tensor product of representations. We describe it directly, using the following result: Proposition 5.2. Let X, Y, U, T be T-C∗ -algebras. Let v ∈ X ⊗ T and w ∈ Y ⊗ U be homogeneous elements of degree 0. Denote the natural embeddings by i1 : X → X ⊠ζ Y,

i2 : Y → X ⊠ζ Y,

j1 : U → U ⊠ζ T,

j2 : T → U ⊠ζ T.

Then (i1 ⊗ j2 )(v) and (i2 ⊗ j1 )(w) commute in (X ⊠ζ Y ) ⊗ (U ⊠ζ T ).

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Proof. We may assume that v = x ⊗ t and w = y ⊗ u for homogeneous elements x ∈ X, t ∈ T , y ∈ Y and u ∈ U . Since deg(v) = deg(w) = 0 we get deg(x) = − deg(t) and deg(y) = − deg(u). We compute (i1 ⊗ j2 )(v) (i2 ⊗ j1 )(w) = i1 (x) ⊗ j2 (t) i2 (y) ⊗ j1 (u) = i1 (x)i2 (y) ⊗ j2 (t)j1 (u)

= ζ deg(x) deg(y)−deg(t) deg(u) i2 (y)i1 (y) ⊗ j1 (u)j2 (t) = ζ deg(x) deg(y)−deg(t) deg(u) i2 (y) ⊗ j1 (u) i1 (x) ⊗ j2 (t)

= ζ deg(x) deg(y)−deg(t) deg(u) (i2 ⊗ j1 )(w) (i1 ⊗ j2 )(v) = (i2 ⊗ j1 )(w) (i1 ⊗ j2 )(v).

Proposition 5.3. Let H1 and H2 be T-Hilbert spaces and let v1 ∈ M(K(H1 ) ⊗ A) and v2 ∈ M(K(H2 ) ⊗ A) be representations of SUq (2) on H1 and H2 , respectively. Define v = (ι1 ⊗ idA )(v1 )(ι2 ⊗ idA )(v2 ) ∈ M(K(H1 ) ⊠ζ K(H2 ) ⊗ A) ∼ K(H1 ⊗ H2 ). Then v ∈ M(K(H1 ⊗ H2 ) ⊗ A) is a and identify K(H1 ) ⊠ζ K(H2 ) = ⊤ v2 and called the tensor representation of SUq (2) on H1 ⊗ H2 . It is denoted v1 product of v1 and v2 . Proof. The proof boils down to the following computation, where the third equality uses Proposition 5.2: (idH1 ⊗H2 ⊗ ∆)(v) = (idH1 ⊗H2 ⊗ ∆)((ι1 ⊗ idA )(v1 )(ι2 ⊗ idA )(v2 )) = (ι1 ⊗ j1 )(v1 ) (ι1 ⊗ j2 )(v1 ) (ι2 ⊗ j1 )(v2 ) (ι2 ⊗ j2 )(v2 ) = (ι1 ⊗ j1 )(v1 ) (ι2 ⊗ j1 )(v2 ) (ι1 ⊗ j2 )(v1 ) (ι2 ⊗ j2 )(v2 ) = (idH1 ⊗H2 ⊗ j1 )(v) (idH1 ⊗H2 ⊗ j2 )(v).

Now consider the Hilbert space C2 , let {e1 , e2 } be its canonical orthonormal basis. We equip it with the representation U : T → U(C2 ) defined by Uz e1 = ze1 and Uz e2 = e2 . Let ρM2×2 (C) be the action implemented by U : a11 , za12 a11 , a12 M2 (C) , = ρz za21 , a22 a21 , a22 where aij ∈ C. Then u=

α , −qγ ∗ γ , α∗

∈ M2×2 (C) ⊗ A

is a representation of SUq (2) on C2 . ⊤ u ∈ M2×2 (C) ⊗ M2×2 (C) ⊗ A. Consider We may form the representation v = u the vector ξ = e1 ⊗ e2 − qe2 ⊗ e1 ∈ C2 ⊗ C2 and let H1 = Cξ, H2 = H1⊥ ⊂ C2 ⊗ C2 . Then the orthogonal projections P1 , P2 ∈ B(C2 ⊗ C2 ) onto H1 and H2 are T-invariant, that is, Uz P1 Uz∗ = P1 Uz P2 Uz∗ = P2 . A computation analogous to the one for q ∈ R but taking the braiding into account ⊤u ∼ shows that u = u1 ⊕u2 with irreducible representations u1 ∈ M(K(H1 )⊗A) and u2 ∈ M(K(H2 ) ⊗ A); namely, u1 = (P1 ⊗ 1)u(P1 ⊗ 1) and u2 = (P2 ⊗ 1)u(P2 ⊗ 1).


9

6. Quantum U(2) groups The braided quantum group SUq (2) described above may be incorporated into a genuine compact quantum group Uq (2). The passage from SUq (2) to Uq (2) may be viewed as a quantum counterpart of the semidirect product construction. The details of this general construction are given in [6]. In our case, the C∗ -algebra C(Uq (2)) is the universal C∗ -algebra with three generators α, γ, z, with the SUq (2)relations for α and γ and the relations zαz ∗ = α, zγz ∗ = ζ −1 γ, zz ∗ = z ∗ z = I, The comultiplication on Uq (2) is given by ∆Uq (2) (z) = z ⊗ z, ∆Uq (2) (α) = α ⊗ α − qγ ∗ z ⊗ γ, ∆Uq (2) (γ) = γ ⊗ α + α∗ z ⊗ γ. The compact quantum group Uq (2) is already described in [12]. For real q, its relationship to the compact quantum group SUq (2) is also discussed in [12]. There are two embeddings ι1 , ι2 : C(SUq (2)) ⇒ C(Uq (2)) ⊗ C(Uq (2)) defined by ι1 (α) = α ⊗ I

ι2 (α) = I ⊗ α,

ι1 (γ) = γ ⊗ I

ι2 (γ) = z ⊗ γ.

Homogeneous elements x, y ∈ C(SUq (2)) satisfy (6.1)

ι1 (x)ι2 (y) = ζ deg(x) deg(y) ι2 (y)ι2 (x).

Thus we may rewrite the comultiplication formulas as ∆Uq (2) (z) = z ⊗ z, ∆Uq (2) (α) = ι1 (α)ι2 (α) − qι1 (γ)∗ ι2 (γ), ∆Uq (2) (γ) = ι1 (γ)ι2 (α) + ι1 (α)∗ ι2 (γ). In particular, ∆Uq (2) respects the commutation relations for (α, γ, z), so it is a welldefined ∗ -homomorphism C(Uq (2)) → C(Uq (2)) ⊗ C(Uq (2)). It is routine to check the cancellation conditions (1.4) for C(Uq (2)), so (C(Uq (2)), ∆Uq (2) ) is a compact quantum group. This is a compact quantum group with a projection as in [7]. In this case, the projection π : C(Uq (2)) → C(Uq (2)) is the unique ∗ -homomorphism with π(α) = 1C(Uq (2)) , π(γ) = 0 and π(z) = z; this is an idempotent bialgebra morphism. Its “image” is the copy of C(T) generated by z, its “kernel” is the copy of C(SUq (2)) generated by α and γ. Theorem 6.1. Let H be a Hilbert space with a unitary representation U ∈ M(K(H)⊗ C(T)) of T. There is a bijection between representations of SUq (2) and Uq (2) on H. Proof. Let v ∈ M(K(H) ⊗ C(SUq (2))) be a unitary representation of SUq (2) on H. Since C(Uq (2)) contains copies of C(SUq (2)) and C(T), we may view u = vU ∗ as an element of M(K(H) ⊗ C(Uq (2))). The T-invariance of v, ∗ (id ⊗ ρA )(v) = U12 v13 U12

and the formula for ι2 (which is basically given by the action ρA ) show that ∗ U12 (id ⊗ ι2 )(v)U12 = v13 .

10


Using (id ⊗ ι2 )(v) = v12 , we conclude that u is a unitary representation of Uq (2): ∗ ∗ ∗ ∗ (id ⊗ ∆Uq (2) )(u) = v12 (id ⊗ ι2 )(v)U12 U13 = v12 U12 v13 U13 = u12 u13 .

Going back and forth between u and v is the desired bijection.

7. Haar states on SUq (2) Let (CT∗ , ⊠ζ ) be the braided monoidal category introduced in Section (3) and let A = C(SUq (2)). It is shown in [5, Section 5.2] that ⊠ζ is functorial for T-equivariant completely positive maps. Let h : A → C be a T-invariant state on A, that is, h is a state with (idC(T) ⊗ h)ρA (x) = h(x)1 for all x ∈ A. Equipping C with a trivial action of T, we may view h : A → C as a T-equivariant, completely positive, unital map and form T-equivariant, completely positive, unital maps id ⊠ζ h : A ⊠ζ A → A, h ⊠ζ id : A ⊠ζ A → A. Definition 7.1. A T-invariant state h : A → C is a Haar state on SUq (2) if, for any x ∈ A, (id ⊠ζ h)(∆(x)) = h(x)I, (7.1) (h ⊠ζ id)(∆(x)) = h(x)I. Adapting the proof of [11, Theorem 1.3] by replacing the bifunctor ⊗ with ⊠ζ , we may show that there is a Haar state h : A → C on SUq (2). In a separate paper, we shall prove this fact in the context of braided compact quantum groups over an arbitrary compact quantum group G instead of T. Let h1 , h2 : A → C be states on A. If h1 is right-invariant and h2 is left-invariant, then h1 = h2 : (7.2)

h1 (a) = h1 (a)h2 (1A ) = (h1 ⊠ h2 )(∆(a)) = h1 (1A )h2 (a) = h2 (a).

Besides the appropriate halves of the invariance condition (7.1) for h1 and h2 , the computation also used the normalisation conditions h1 (1A ) = h2 (1A ) = 1 for states. In particular, (7.2) shows that there is only one Haar state on a braided compact quantum group. Proposition 7.2. The restriction of the Haar state on C(Uq (2)) to C(SUq (2)) is the Haar state on SUq (2). Proof. Equation (6.1) implies that ι1 and ι2 generate a homomorphism ψ

C(SUq (2)) ⊠ C(SUq (2)) − → C(Uq (2)) ⊗ C(Uq (2)). An argument similar to the proof of [6, Proposition 6.6] shows that this representation is faithful. Comparing the comultiplication formulas for SUq (2) and Uq (2), we see that ∆SUq (2) composed with ψ is the restriction of ∆Uq (2) to C(SUq (2)). Hence (id ⊗ h)ψ(∆SUq (2) (x)) = h(x)1 for all x ∈ C(SUq (2)) if h : C(Uq (2)) → C is the Haar state on Uq (2). Since h is Uq (2)-invariant, it is also invariant under T. Let x, y ∈ C(SUq (2)) be homogeneous elements of degree n. Since h(y) = 0 if n 6= 0, we get (id ⊗ h)(ι1 (x)ι2 (y)) = (id ⊗ h)(xz n ⊗ y) = xz n h(y) = xh(y) = (id ⊠ h)(ι1 (x)ι2 (y)). In particular, (id ⊠ h)(∆SUq (2) (x)) = h(x)1, so the restriction of h to C(SUq (2)) is a left invariant state on SUq (2). Taking the existence of a Haar state for granted, (7.2) now shows that h is the Haar state because it is a left-invariant state.


11

We give an explicit formula for the Haar state on SUq (2). Let H be the Hilbert space with an orthonormal basis (Ψnk | n = 0, 1, 2, . . . , k ∈ Z) and define the representation π : A → B(H) by q 2 π(α)Ψnk = 1 − |q| Ψn−1,k , π(γ)Ψnk = q n Ψn,k+1 .

The Haar state h on SUq (2) is given by (7.3)

h(a) = N −1

∞ X

2n

|q|

(Ψn,0 | π(a)Ψn,0 )

n=0

P 2n for all a ∈ A, with the normalisation factor N = ∞ = (1 − |q|2 )−1 . The n=0 |q| formula (7.3) for real q was given in [10, Appendix A1]. For q ∈ C with 0 < |q| < 1, it follows from the explicit formula for the Haar state on Uq (2) in [12, Theorem 4.1] and Proposition 7.2. References [1] Paweł Kruszyński and Stanisław Lech Woronowicz, A noncommutative Gelfand– Na˘ımark theorem, J. Operator Theory 8 (1982), no. 2, 361–389, available at http://www.theta.ro/jot/archive/1982-008-002/1982-008-002-009.pdf. MR 677419 [2] Saunders MacLane, Categories for the working mathematician, Springer-Verlag, New York, 1971. Graduate Texts in Mathematics, Vol. 5. MR 0354798 [3] Shahn Majid, Examples of braided groups and braided matrices, J. Math. Phys. 32 (1991), no. 12, 3246–3253, doi: 10.1063/1.529485. MR1137374 (93i:17019) [4] , Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995., doi: 10.1017/CBO9780511613104 MR 1381692 [5] Ralf Meyer, Sutanu Roy, and Stanisław Lech Woronowicz, Quantum group-twisted tensor products of C∗ -algebras, Internat. J. Math. 25 (2014), no. 2, 1450019, 37, doi: 10.1142/S0129167X14500190. MR 3189775 [6] , Quantum group-twisted tensor products of C∗ -algebras II (2014), in preparation. [7] Sutanu Roy, C∗ -Quantum groups with projection, Ph.D. Thesis, Georg-August Universität Göttingen, 2013, http://hdl.handle.net/11858/00-1735-0000-0022-5EF9-0. [8] Piotr Mikołaj Sołtan, Examples of non-compact quantum group actions, J. Math. Anal. Appl. 372 (2010), no. 1, 224–236, doi: 10.1016/j.jmaa.2010.06.045. MR 2672521 [9] Stanisław Lech Woronowicz, Twisted SU(2) group. An example of a noncommutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987), no. 1, 117–181, doi: 10.2977/prims/1195176848. MR 890482 [10] , Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), no. 4, 613–665, available at http://projecteuclid.org/euclid.cmp/1104159726. MR 901157 , Compact quantum groups, Symétries quantiques (Les Houches, 1995), North-Holland, [11] Amsterdam, 1998, pp. 845–884. MR 1616348 [12] Xiao Xia Zhang and Ervin Yunwei Zhao, The compact quantum group Uq (2). I, Linear Algebra Appl. 408 (2005), 244–258, doi: 10.1016/j.laa.2005.06.004. MR 2166867 E-mail address: [email protected] Katedra Metod Matematycznych Fizyki, Wydział Fizyki, Uniwersytet Warszawski, Hoża 74, 00-682 Warszawa, Poland E-mail address: [email protected] Mathematisches Institut, Georg-August Universität Göttingen, Bunsenstraße 3–5, 37073 Göttingen, Germany E-mail address: [email protected] Indian Statistical Institute, 203, B.T. Road, Kolkata 700108, India E-mail address: [email protected] Katedra Metod Matematycznych Fizyki, Wydział Fizyki, Uniwersytet Warszawski, Hoża 74, 00-682 Warszawa, Poland