K-Homology Class of the Dirac Operator

on a Compact Quantum Group

Sergey Neshveyev and Lars Tuset^{∗}

Received: March 29, 2011 Revised: September 29, 2011

Communicated by Joachim Cuntz

Abstract. By a result of Nagy, the C^{∗}-algebra of continuous functions on the q-deformation G_{q} of a simply connected semisimple compact Lie group G is KK-equivalent to C(G). We show that under this equivalence the K-homology class of the Dirac operator on G_{q}, which we constructed in an earlier paper, corresponds to that of the classical Dirac operator. Along the way we prove that for an appropriate choice of isomorphisms between completions of U_{q}𝔤 and U𝔤 a family of Drinfeld twists relating the deformed and classical coproducts can be chosen to be continuous in q.

2000 Mathematics Subject Classification: 58B34, 58B32, 46L80 Keywords and Phrases: Quantum groups, Dirac operator, Khomology

Introduction

In [10] we constructed a Dirac operator D_{q} on the q-deformation G_{q} of any simply connected semisimple compact Lie group G. The construction involved a special unitary element F^{q} in the von Neumann algebra W^{∗}(G) ¯⊗W^{∗}(G), which relates the coproducts in W^{∗}(G_{q}) and W^{∗}(G). The existence of such an element, called a unitary Drinfeld twist, is a consequence of a fundamental and highly nontrivial result in quantum group theory due to Kazhdan and Lusztig [5] (see also [11]). Since the construction of a Drinfeld twist is involved and not particularly explicit, certain properties of the operators D_{q} are not immediate. In particular, even though it is intuitively clear that D_{q} is a deformation of the classical Dirac operator and therefore should in some sense define the same index map on K-theory, it is not even obvious that the K-homology class of D_{q} is always nonzero. The goal of this note is to show that it is indeed nonzero. In fact we show that the K-homology class of D_{q} corresponds exactly to that of the classical Dirac operator via the KK-equivalence of C(G_{q}) and C(G) established by Nagy [9]. Therefore, upon identifying the K-theories of G_{q} and G, the index map defined by D_{q} does not depend on q, as expected. We should remark that the question of invariance of the index map under deformation has been studied in a recent paper by Yamashita [15] in the context of Connes-Landi θ-deformations. The paper is organized as follows. In Section 1 we show that the family of C^{∗}- algebras C(G_{q}) has a canonical continuous field structure. The result is more or less known [9], but is usually formulated in terms of standard generators of ℂ[G_{q}]. We propose a simpler approach based on a natural notion of a continuous family of isomorphisms W^{∗}(G_{q}) ≅ W^{∗}(G). In Section 2 we prove that once a continuous family of isomorphisms W^{∗}(G_{q}) ≅ W^{∗}(G) is fixed, the corresponding family of Drinfeld twists F^{q} can be chosen to be continuous in q. This result is not strictly speaking necessary for our main result on D_{q}, for which it suffices to know that D_{q} does not depend on F^{q} for a fixed isomorphism W^{∗}(G_{q}) ≅ W^{∗}(G) (see [12]), but it simplifies the arguments and is of independent interest. Both results, continuity of F^{q} and uniqueness of D_{q}, depend crucially on the fact that any two unitary Drinfeld twists differ by the coboundary of a central unitary element, a result we proved in [12]. In Section 3 we prove our main result. For this we show that the family of operators D_{q} define a Kasparov module for the algebra of continuous sections of (C(G_{q}))_{q∈[a,b]} and the algebra C[a, b]. With the preparation in the previous two sections the proof essentially boils down to observing that estimates in our paper [10] are uniform in q.

∗Supported by the Research Council of Norway.

Acknowledgement. The authors are grateful to Christian Voigt and Makoto Yamashita for stimulating discussions.

1. Continuous field of function algebras

Let G be a simply connected semisimple compact Lie group, 𝔤 its complexified Lie algebra, 𝔥 ⊂ 𝔤 the Cartan subalgebra defined by a maximal torus in G. Fix a system {α_{1}, …, α_{r}} of simple roots. Let (a_{ij})_{1≤i,j≤r} be the Cartan matrix defined by α_{1}, …, α_{r}, and d_{1}, …, d_{r} be the coprime positive integers such that (d_{i}a_{ij})_{i,j} is symmetric. For every q > 0, q ≠ 1, consider the quantized universal enveloping algebra U_{q}𝔤 with generators E_{i}^{q}, F_{i}^{q}, K_{i}^{q} (1 ≤ i ≤ r); we follow the conventions in [11, 12]. For q = 1 we let U_{1}𝔤 = U𝔤 and denote by E_{i}^{1}, F_{i}^{1}, h_{i}^{1} the standard generators of U𝔤. We will often omit various indices corresponding to q = 1. Consider the category C_{q}(𝔤) of finite dimensional admissible U_{q}𝔤-modules and denote by U(G_{q}) the endomorphism ring of the forgetful functor C_{q}(𝔤) → Vec. We think of U(G_{q}) as a completion of U_{q}𝔤. For q ≠ 1 denote by h_{i}^{q} the unique self-adjoint element of U(G_{q}) such that K_{i}^{q} = q^{d_{i}h_{i}^{q}}.

Since simple objects of C_{q}(𝔤) are classified by dominant integral weights, for every q > 0 we have a canonical identification of the centers of U(G_{q}) and U(G). It extends to a ∗-isomorphism φ^{q}: U(G_{q}) → U(G). For every dominant integral weight λ ∈ P_{+} and q > 0 fix an irreducible ∗-representation π_{λ}^{q}: U_{q}𝔤 → B(V_{λ}^{q}) with highest weight λ and a highest weight unit vector ξ_{λ}^{q} ∈ V_{λ}^{q}. Then to define φ^{q} is the same as to fix an isomorphism B(V_{λ}^{q}) ≅ B(V_{λ}) for every λ. We say that a family {φ^{q}}_{q>0} of ∗-isomorphisms extending the canonical identifications of the centers is continuous, if for every finite dimensional representation π of U𝔤 the operators π(φ^{q}(E_{i}^{q})), π(φ^{q}(F_{i}^{q})) and π(φ^{q}(h_{i}^{q})) depend continuously on q.

Lemma 1.1. A continuous family of ∗-isomorphisms φ^{q}: U(G_{q}) → U(G) always exists.

Proof. It suffices to show that for every λ ∈ P_{+} there exist unitaries u_{λ}^{q}: V_{λ}^{q} → V_{λ} such that the operators u_{λ}^{q}π_{λ}^{q}(X^{q})u_{λ}^{q}^{∗} depend continuously on q for X^{q} = E_{i}^{q}, F_{i}^{q}, h_{i}^{q}. For this, in turn, it is enough to show that such unitaries exist locally. Therefore fix λ and q_{0} > 0. For every multi-index I = (i_{1}, …, i_{k}) (1 ≤ i_{j} ≤ r) define

e_{I}^{q} = π_{λ}^{q}(F_{i_{1}}^{q} … F_{i_{k}}^{q})ξ_{λ}^{q} ∈ V_{λ}^{q}.

We can choose multi-indices I_{1}, …, I_{n} such that the vectors e_{I_{1}}^{q_{0}}, …, e_{I_{n}}^{q_{0}} form a basis in V_{λ}^{q_{0}}. The quantum Serre relations, together with the identities F_{i}^{q}^{∗} = (K_{i}^{q})^{−1}E_{i}^{q}, imply that the scalar products (e_{I}^{q}, e_{J}^{q}) depend continuously on q. Hence, for some ε > 0, the vectors e_{I_{1}}^{q}, …, e_{I_{n}}^{q} form a basis in V_{λ}^{q} for all q ∈ (q_{0} − ε, q_{0} + ε). Applying the Gram-Schmidt orthogonalization we get an orthonormal basis ζ_{1}^{q}, …, ζ_{n}^{q} in V_{λ}^{q}. Let v^{q}: V_{λ}^{q} → V_{λ}^{q_{0}} be the unitary mapping ζ_{i}^{q} into ζ_{i}^{q_{0}}. By construction, for every multi-index I, the coefficients of e_{I}^{q} in the basis ζ_{1}^{q}, …, ζ_{n}^{q} depend continuously on q, hence the matrix coefficients of π_{λ}^{q}(F_{i}^{q}) in this basis also depend continuously on q. It follows that the operators v^{q}π_{λ}^{q}(F_{i}^{q})v^{q}^{∗} depend continuously on q. The same is clearly true for h_{i}^{q} in place of F_{i}^{q} (in fact, we even have v^{q}π_{λ}^{q}(h_{i}^{q})v^{q}^{∗} = h_{i}^{q_{0}}), hence also for E_{i}^{q} = q^{d_{i}h_{i}^{q}}F_{i}^{q}^{∗}. Now take an arbitrary unitary u: V_{λ}^{q_{0}} → V_{λ}. Then the unitaries u_{λ}^{q} = uv^{q}, q ∈ (q_{0} − ε, q_{0} + ε), have the required properties. □

¿From now on we will fix a continuous family of ∗-isomorphisms φ^{q}: U(G_{q}) → U(G) such that φ^{1} = ι. For every q > 0 denote by ℂ[G_{q}] ⊂ U(G_{q})^{∗} the Hopf ∗-algebra of matrix coefficients of finite dimensional admissible U_{q}𝔤-modules, and by C(G_{q}) its C^{∗}-completion.

Theorem 1.2. The family of C^{∗}-algebras C(G_{q}), q > 0, has a unique structure of a continuous field of C^{∗}-algebras such that for every a ∈ ℂ[G] the section q ↦ aφ^{q} ∈ C(G_{q}) is continuous. This structure does not depend on the choice of a continuous family of ∗-isomorphisms φ^{q}.

It is known that the C^{∗}-algebras C(SU_{q}(N)) have a continuous field structure such that the matrix coefficients of the fundamental representation form continuous sections. For N = 2 this was proved by Bauval [2] and for all N ≥ 2 by Nagy [9]. As mentioned in [9] the same proof as for SU_{q}(N) works for all other classical simple compact Lie groups. In principle the same result is also true for exceptional groups once explicit generators of ℂ[G_{q}] have been found. The point of the above theorem is that there is actually no need to do this, it is enough to know that there exists a ‘continuous’ choice of generators. Although we will not need this here, we note that the theorem and its proof also imply that the families of function algebras on q-deformations of homogeneous spaces of G can be given a continuous field structure without working out explicit generators and relations in those algebras. This, as well as the relation of the above result to Rieffel’s notion of strict deformation quantization, will be discussed in a subsequent paper [13].

Proof of Theorem 1.2. First consider the dependence of the continuous field structure on the isomorphisms φ^{q}. Assume we have another continuous family of isomorphisms ψ^{q}. For every λ ∈ P_{+} denote by γ_{λ}^{q} the unique automorphism of B(V_{λ}) such that π_{λ}ψ^{q} = γ_{λ}^{q}π_{λ}φ^{q}. Then the map q ↦ γ_{λ}^{q} ∈ Aut(B(V_{λ})) is continuous. It follows that for any linear functional ω on B(V_{λ}) the elements ωγ^{q}π_{λ} ∈ ℂ[G] decompose into finite linear combinations of elements νπ_{λ} with continuous coefficients, so that the sections q ↦ ωπ_{λ}ψ^{q} are finite linear combinations of sections q ↦ νπ_{λ}φ^{q} with continuous coefficients. Therefore if the latter sections are continuous, the former are continuous as well.

Since ℂ[G_{q}] is dense in C(G_{q}), it is also clear that the continuous field structure is unique if it exists.

To prove existence, first consider the case G = SU(2). As usual identify the weight lattice P with the half-integers. For every s ∈ \frac{1}{2}ℤ_{+} consider the orthonormal basis in V_{s}^{q} consisting of the vectors ∥(F^{q})^{k}ξ_{s}^{q}∥^{−1}(F^{q})^{k}ξ_{s}^{q}, k = 0, …, 2s. Let u_{ij}^{s;q} ∈ ℂ[SU_{q}(2)] be the matrix coefficients of π_{s}^{q} in this basis. We also use these bases to construct the isomorphisms φ^{q}, so that u_{ij}^{s;q} = u_{ij}^{s}φ^{q}. By [2], see also [3], there exists a continuous field structure on the C^{∗}-algebras C(SU_{q}(2)) such that the sections q ↦ u_{ij}^{1/2;q} are continuous. To prove that the sections q ↦ u_{ij}^{s;q} are continuous in this structure we just have to show that u_{ij}^{s;q} can be expressed as polynomials of the elements u_{ij}^{1/2;q} with continuous coefficients. That this is indeed possible is easy to see using the unique embedding of V_{s}^{q} into (V_{1/2}^{q})^{⊗2s} mapping ξ_{s}^{q} into (ξ_{1/2}^{q})^{⊗2s}; in fact an explicit expression for these polynomials is known [14].

Turning to the general case, for every simple root α_{i} consider the ∗- homomorphism σ_{i}^{q}: C(G_{q}) → C(SU_{q^{d_{i}}} (2)) which is dual to the embedding ρ_{i}^{q}: U(SU_{q^{d_{i}}} (2)) ↪ U(G_{q}) defined by E^{q^{d_{i}}} ↦ E_{i}^{q}, F^{q^{d_{i}}} ↦ F_{i}^{q} and h^{q^{d_{i}}} ↦ h_{i}^{q}. Let w = s_{i_{1}} … s_{i_{n}} be the longest element in the Weyl group of G written in reduced form. Put A_{q} = C(SU_{q^{d_{i_{1}}}} (2)) ⊗ ⋯ ⊗ C(SU_{q^{d_{in}}} (2)). Since C(SU_{q}(2)) is a C^{∗}-algebra of type I, the C^{∗}-algebra of continuous sections of the field (C(SU_{q}(2)))_{q>0} vanishing at infinity is of type I as well, hence exact. By [7, Theorem 4.6] it follows that the field (A_{q})_{q>0} has a continuous field structure such that the tensor product of continuous sections is a continuous section. Define a ∗-homomorphism

σ^{q}: C(G_{q}) → A_{q} by σ^{q}(a) = (σ_{i_{1}}^{q} ⊗ ⋯ ⊗ σ_{i_{n}}^{q})Δ_{q}^{(n−1)}(a).

It follows from the description of irreducible representations of C(G_{q}), see e.g. [8, Theorem 6.2.7], that σ^{q} is injective for every q. Therefore the field (C(G_{q}))_{q>0} embeds into (A_{q})_{q>0}, so to prove existence of the required continuous field structure on (C(G_{q}))_{q>0} it suffices to show that for every a ∈ ℂ[G] the section q ↦ σ^{q}(aφ^{q}) ∈ A_{q} is continuous. Since φ^{q} is an algebra homomorphism, the dual map ℂ[G] → ℂ[G_{q}], a ↦ aφ^{q}, is a coalgebra homomorphism. Therefore, using Sweedler’s sumless notation, Δ_{q}^{(n−1)}(aφ^{q}) = a_{(0)}φ^{q} ⊗ ⋯ ⊗ a_{(n−1)}φ^{q}. Hence to prove that the sections q ↦ σ^{q}(aφ^{q}) ∈ A_{q} are continuous for all a ∈ ℂ[G] it suffices to show that the sections q ↦ σ_{i}^{q}(aφ^{q}) of the field (C(SU_{q^{d_{i}}} (2)))_{q>0} are continuous for all a ∈ ℂ[G] and 1 ≤ i ≤ r. Fix i and a continuous family of ∗-isomorphisms θ^{q}: U(SU_{q}(2)) → U(SU(2)) such that θ^{1} = ι. To simplify the notation assume d_{i} = 1, so σ_{i}^{q} is defined by an embedding ρ_{i}^{q}: U_{q}𝔰𝔩_{2} → U_{q}𝔤. To finish the proof it suffices to show that there exists a continuous family of ∗-automorphisms γ^{q} of U(G) such that γ^{q}φ^{q}ρ_{i}^{q} = ρ_{i}θ^{q}. Indeed, then exactly as in the first part of the proof, a section q ↦ σ_{i}^{q}(aφ^{q}) is continuous if and only if q ↦ σ_{i}^{q}(aγ^{q}φ^{q}) is continuous. Since

σ_{i}^{q}(aγ^{q}φ^{q}) = aγ^{q}φ^{q}ρ_{i}^{q} = aρ_{i}θ^{q},

the latter section is indeed continuous by definition of the continuous field structure on the C^{∗}-algebras C(SU_{q}(2)). The automorphisms γ^{q} will be defined by a family of automorphisms γ_{λ}^{q} of B(V_{λ}). Fix λ ∈ P_{+}. Let N ∈ ℕ be such that ω(h_{i}) ≤ N for every ω ∈ P such that V_{λ}(ω) ≠ 0. For every q > 0 consider the direct sum ⊕_{s≤N/2}V_{s}^{q} of U_{q}𝔰𝔩_{2}-modules and the corresponding surjective homomorphism α^{q} from U_{q}𝔰𝔩_{2} into the algebra B_{q} = ⊕_{s≤N/2}B(V_{s}^{q}). Then the representation π_{λ}^{q}ρ_{i}^{q}: U_{q}𝔰𝔩_{2} → B(V_{λ}^{q}) factors through B_{q}, so π_{λ}^{q}ρ_{i}^{q} = β^{q}α^{q} for a unique β^{q}: B_{q} → B(V_{λ}^{q}). We summarize all the maps involved in the following diagram, which is commutative along solid lines:

(1.1)  \xymatrix{U(G_{q}) \ar[r]^{π_{λ}^{q}} & B(V_{λ}^{q}) \ar@{-->}[r]^{φ_{λ}^{q}} & B(V_{λ}) & U(G) \ar[l]_{π_{λ}} \\  & B_{q} \ar[r]^{θ̃^{q}} \ar[u]^{β^{q}} & B \ar[u]_{β} \\ U(SU_{q}(2)) \ar[uu]^{ρ_{i}^{q}} \ar[rrr]^{θ^{q}} \ar[ur]^{α^{q}} &  &  & U(SU(2)) \ar[uu]_{ρ_{i}} \ar[ul]_{α}} ,

Documenta Mathematica 16 (2011) 767–780

where φ_{λ}^{q}: B(V_{λ}^{q}) → B(V_{λ}) and θ̃^{q}: B_{q} → B are the isomorphisms defined by the isomorphisms φ^{q}: U(G_{q}) → U(G) and θ^{q}: U(SU_{q}(2)) → U(SU(2)), respectively. Consider the family of homomorphisms φ_{λ}^{q}β^{q}(θ̃^{q})^{−1} : B → B(V_{λ}). Since the families of homomorphisms θ̃^{q}α^{q} = αθ^{q}: U_{q}𝔰𝔩_{2} → B and φ_{λ}^{q}β^{q}α^{q} = φ_{λ}^{q}π_{λ}^{q}ρ_{i}^{q} = π_{λ}φ^{q}ρ_{i}^{q}: U_{q}𝔰𝔩_{2} → B(V_{λ}) are continuous in the sense defined earlier, and the homomorphisms in the first family are surjective, it follows that the homomorphisms φ_{λ}^{q}β^{q}(θ̃^{q})^{−1} : B → B(V_{λ}) depend continuously on q. Furthermore, for q = 1 we get the homomorphism β. Hence, by a standard result on homomorphisms of finite dimensional C^{∗}-algebras, we can choose a continuous family of ∗-automorphisms γ_{λ}^{q} of B(V_{λ}) such that γ_{λ}^{q}φ_{λ}^{q}β^{q}(θ̃^{q})^{−1} = β for all q > 0. In other words, if we replace φ_{λ}^{q} by γ_{λ}^{q}φ_{λ}^{q} in (1.1), we get a commutative diagram, which is what we need. □

Remark 1.3. The space of sections of the form q ↦ aφ^{q}, a ∈ ℂ[G], is not an algebra and is not closed under involution. But the space of finite sums of sections of the form q ↦ f(q)aφ^{q}, where a ∈ ℂ[G] and f is a continuous function, is a ∗-algebra. Indeed, assume a is a matrix coefficient of a finite dimensional representation of G and {a_{i}}_{i} is a basis in the space spanned by the matrix coefficients of the contragradient representation. Then (aφ^{q})^{∗} = ∑_{i} f_{i}(q)a_{i}φ^{q} for uniquely defined functions f_{i}. Since q ↦ (aφ^{q})^{∗} is a continuous section, the functions f_{i} must be continuous. Therefore the space is closed under involution. Similarly we check that the space is closed under multiplication. This of course can also be checked without relying on the above theorem. Note also that this space does not depend on the choice of φ^{q}.

For b > a > 0 denote by C(G_{[a,b]}) the C^{∗}-algebra of continuous sections of the field (C(G_{q}))_{q∈[a,b]}.

Theorem 1.4 ([9]). For any b > a > 0 and q ∈ [a, b] the evaluation map ev_{q}: C(G_{[a,b]}) → C(G_{q}) is a KK-equivalence.

Since this is not exactly how the result is formulated in [9, Corollaries 3.8 and 3.11], some comments are in order. What is proved in [9], is that C(SU_{q}(N)) is KK-equivalent to C(SU(N)) for q ∈ (0, 1), and it is mentioned that the same can be proved for the other classical simple compact Lie groups. The proof, however, shows that the above more precise result holds for b ≤ 1, and once the family of C^{∗}-algebras C(G_{q}) is given a continuous field structure as described above, the general case is essentially identical to G = SU(N), see [13, Section 6] for details. To deal with the case b > 1 we can argue as follows. It is easy to see that it suffices to prove the theorem for the evaluations at the end points, see the proof of [13, Lemma 6.3]. We therefore have to show that the C^{∗}-algebras I_{a,b} (resp. J_{a,b}) of continuous sections of (C(G_{q}))_{q∈[a,b]} vanishing at a (resp., at b) are KK-contractible, knowing already that this is true when b ≤ 1. If a ≥ 1, this follows from the canonical isomorphisms G_{q} ≅ G_{q^{−1}}. If a < 1 < b then the KK-contractibility of I_{a,b} and J_{a,b} follows from the exact sequences 0 → I_{1,b} → I_{a,b} → I_{a,1} → 0 and 0 → J_{a,1} → J_{a,b} → J_{1,b} → 0.

2. Continuous family of Drinfeld twists

As in the previous section, fix a continuous family of ∗-isomorphisms φ^{q}: U(G_{q}) → U(G) with φ^{1} = ι. For q > 0, let ℏ_{q} ∈ iℝ be such that q = e^{πiℏ_{q}}. Denote by t ∈ 𝔤 ⊗ 𝔤 the 𝔤-invariant element defined by the ad-invariant symmetric form on 𝔤 such that the induced form on 𝔥^{∗} satisfies (α_{i}, α_{j}) = d_{i}a_{ij}. By a result of Kazhdan and Lusztig [5], see [11] for details, for every q > 0 there exists a unitary element F^{q} ∈ U(G × G), which we call a unitary Drinfeld twist, such that

(i) (φ^{q} ⊗ φ^{q})Δ̂_{q} = F^{q}Δ̂φ^{q}(·)F^{q}^{∗}; (ii) (ε̂⊗ ι)(F^{q}) = (ι ⊗ ε̂)(F^{q}) = 1; (iii) (φ^{q} ⊗ φ^{q})(R_{q}) = F_{21}^{q}q^{t}F^{q}^{∗}, where R_{q} ∈ U(G_{q} × G_{q}) is the universal Rmatrix;

(iv)(ι⊗Δ̂)(F^{q})^{∗}(1⊗F^{q})^{∗}(F^{q} ⊗1)(Δ̂⊗ι)(F^{q}) = Φ_{KZ}(ℏ_{q}t_{12}, ℏ_{q}t_{23}), where Φ_{KZ} is Drinfeld’s KZ-associator. Such an element is not unique, but by [12] any other unitary Drinfeld twist (for the same isomorphism φ^{q}) has the form (c ⊗ c)F^{q}Δ̂(c)^{∗} for a unitary element c in the center of U(G). We say that a family {F^{q}}_{q>0} of unitary Drinfeld twists is continuous, if the map

q ↦ F^{q} ∈ W^{∗}(G × G)

is continuous in the strong operator topology on the von Neumann algebra W^{∗}(G × G) ⊂ U(G × G) of G × G. In other words, the map q ↦ (π_{λ} ⊗ π_{ν})(F^{q}) is continuous for all λ, ν ∈ P_{+}.

Theorem 2.1. There exists a continuous family of unitary Drinfeld twists F^{q} such that F^{1} = 1. Furthermore, if {ψ^{q}: U(G_{q}) → U(G)}_{q>0} is another continuous family of ∗-isomorphisms such that ψ^{1} = ι, and {E^{q}}_{q>0} is a corresponding continuous family of unitary Drinfeld twists with E^{1} = 1, then there exists a unique continuous family of unitary elements u^{q} ∈ U(G) such that

u^{1} = 1, and ψ^{q} = u^{q}φ^{q}(·)u^{q}^{∗} and E^{q} = (u^{q} ⊗ u^{q})F^{q}Δ̂(u^{q})^{∗} for all q > 0.

Proof. To prove existence, consider the set Ω of pairs (q, F), where q > 0 and F is a unitary Drinfeld twist for φ^{q}. It is a closed subset of the direct product of ℝ_{+}^{∗} and the unitary group of W^{∗}(G × G) (this is used already in the proof of [11, Lemma 3.2]), so it is a locally compact space. Let p: Ω → ℝ_{+}^{∗} be the projection onto the first coordinate. The compact abelian group of elements of the form (c ⊗ c)Δ̂(c)^{∗}, where c is a unitary element in the center of U(G), acts freely by multiplication on the right on Ω, and by [12, Theorem 5.2] this action is transitive on each fiber of the map p. Therefore if this group were a compact Lie group, then by a theorem of Gleason [4], p: Ω → ℝ_{+}^{∗} would be a fiber bundle, hence p would have a continuous section. Since the group of elements of the form (c⊗c)Δ̂(c)^{∗} is not a Lie group, we cannot apply Gleason’s theorem directly and will proceed as follows.

Choose an increasing sequence of finite subsets P_{n} ⊂ P_{+} such that P_{1} = {0} and ∪_{n}P_{n} = P_{+}. For every q > 0 we will construct a sequence of unitary Drinfeld twists F_{n}^{q} such that F_{n}^{1} = 1, the map q ↦ (π_{λ} ⊗ π_{ν})(F_{n}^{q}) is continuous for all λ, ν ∈ P_{n} and n ≥ 1, and (π_{λ} ⊗ π_{ν})(F_{n+1}^{q}) = (π_{λ} ⊗ π_{ν})(F_{n}^{q}) for all λ, ν ∈ P_{n} and n ≥ 1. Then, for every q > 0, the sequence {F_{n}^{q}}_{n} converges to a unitary Drinfeld twist F^{q} with the required properties. For n = 1 and q ≠ 1 we take F_{1}^{q} to be any unitary Drinfeld twist, and we take F_{1}^{1} = 1. Assume the Drinfeld twists F_{n}^{q} are already constructed for some n. Denote by Ω_{n+1} the set of pairs (q, U), where U = (U_{λ,ν})_{λ,ν} is a unitary element in ∏_{(λ,ν)∈P_{n+1}×P_{n+1}\P_{n}×P_{n}} B(V_{λ}⊗V_{ν}) such that there exists a unitary Drinfeld twist F for φ^{q} satisfying

(π_{λ} ⊗ π_{ν})(F) = U_{λ,ν} for all (λ, ν) ∈ P_{n+1} × P_{n+1} \ P_{n} × P_{n},
(π_{λ} ⊗ π_{ν})(F) = (π_{λ} ⊗ π_{ν})(F_{n}^{q}) for all λ, ν ∈ P_{n}.

Let p_{n+1} : Ω_{n+1} → ℝ_{+}^{∗} be the projection onto the first coordinate. The set Ω_{n+1} is a closed subset of the direct product of ℝ_{+}^{∗} and the unitary group of ∏_{(λ,ν)∈P_{n+1}×P_{n+1}\P_{n}×P_{n}} B(V_{λ} ⊗ V_{ν}). For every q > 0 the fiber p_{n+1}^{−1}(q) is nonempty, as it contains the element ((π_{λ} ⊗ π_{ν})(F_{n}^{q}))_{λ,ν}. Let S_{n+1} be the set of weights λ ∈ P_{+} such that either λ ∈ P_{n+1}, or V_{λ} is equivalent to a subrepresentation of V_{ν} ⊗ V_{η} for some ν, η ∈ P_{n+1}, in which case we write V_{λ} ≺ V_{ν} ⊗ V_{η}. Let K_{n+1} = ∏_{λ∈S_{n+1}} 𝕋. We have a homomorphism ρ_{n+1} from K_{n+1} into the unitary group of ∏_{(λ,ν)∈P_{n+1}×P_{n+1}\P_{n}×P_{n}} B(V_{λ} ⊗ V_{ν}): ρ_{n+1}(c) acts on the isotypic component of V_{ν} ⊗ V_{η} of type V_{λ} as the multiplication by c_{ν}c_{η}c̄_{λ}. We also have a similar homomorphism θ_{n+1} from K_{n+1} into the unitary group of ∏_{λ,ν∈P_{n}} B(V_{λ} ⊗ V_{ν}). The group ker θ_{n+1} acts on Ω_{n+1} by multiplication by ρ_{n+1}(c) on the right. On every fiber of p_{n+1} this action is transitive, and the stabilizer of every point is ker ρ_{n+1} ∩ ker θ_{n+1}. Since ker θ_{n+1} is a compact Lie group, by Gleason’s theorem we conclude that p_{n+1} : Ω_{n+1} → ℝ_{+}^{∗} is a fiber bundle, hence it is a trivial bundle. Choosing a continuous section of this bundle, by definition of Ω_{n+1} we conclude that there exist unitary Drinfeld twists E^{q} such that the map q ↦ (π_{λ} ⊗ π_{ν})(E^{q}) is continuous for all λ, ν ∈ P_{n+1} and (π_{λ} ⊗ π_{ν})(E^{q}) = (π_{λ}⊗π_{ν})(F_{n}^{q}) for all λ, ν ∈ P_{n}. There exists a unitary central element c in U(G) such that E^{1} = (c^{∗} ⊗ c^{∗})Δ̂(c). We can then set F_{n+1}^{q} = E^{q}(c ⊗ c)Δ̂(c)^{∗}. This finishes the proof of existence.

Assume now that {ψ^{q}: U(G_{q}) → U(G)}_{q>0} is another continuous family of ∗-isomorphisms such that ψ^{1} = ι, and {E^{q}}_{q>0} is a corresponding continuous family of unitary Drinfeld twists with E^{1} = 1. For every λ ∈ P_{+}, let φ_{λ}^{q}, ψ_{λ}^{q}: B(V_{λ}^{q}) → B(V_{λ}) be the isomorphisms defined by φ^{q} and ψ^{q}. The set of unitaries v ∈ B(V_{λ}) such that ψ_{λ}^{q} = vφ_{λ}^{q}(·)v^{∗} forms a circle bundle over ℝ_{+}^{∗}, so it has a continuous section v_{λ}^{q}. Since ψ_{λ}^{1} = φ_{λ}^{1} = ι, we may assume that v_{λ}^{1} = 1. The unitaries v_{λ}^{q} define a continuous family of unitaries v^{q} ∈ U(G).

For every q > 0, the element (v^{q} ⊗ v^{q})F^{q}Δ̂(v^{q})^{∗} is a unitary Drinfeld twist for ψ^{q}. Hence, for every q, there exists a unitary central element c ∈ U(G) such that

(2.1)  E^{q} = (v^{q} ⊗ v^{q})F^{q}Δ̂(v^{q})^{∗}(c ⊗ c)Δ̂(c)^{∗}.

Furthermore, the element c is defined up to a group-like unitary element in the center of U(G), that is, up to an element of the center Z(G) of G. Therefore, applying once again Gleason’s theorem (which in this case is quite obvious as Z(G) is finite), we see that the set of pairs (q, c) with c satisfying (2.1) is a principal Z(G)-bundle over ℝ_{+}^{∗}, hence it has a continuous section q ↦ (q, c^{q}). The element c^{1} is group-like, so replacing c^{q} by c^{q}c^{1}^{∗} we may assume that c^{1} = 1. Letting u^{q} = c^{q}v^{q}, we get the required continuous family of unitary elements. Finally, if ũ^{q} is another continuous family of unitary elements with the same properties, then c^{q} = ũ^{q}u^{q}^{∗} is a unitary central group-like element in U(G), hence c^{q} ∈ Z(G). Since c^{q} depends continuously on q, Z(G) is finite and c^{1} = 1, we conclude that c^{q} = 1 for all q. □

Another way of formulating the above result is to say that the set of triples (q, φ, F) such that q > 0, φ: U(G_{q}) → U(G) is a ∗-isomorphism extending the canonical identification of the centers and F is a unitary Drinfeld twist for φ, has a structure of a principal U(W^{∗}(G))/Z(G)-bundle over ℝ_{+}^{∗}, where U(W^{∗}(G)) is the unitary group of W^{∗}(G). Note also that by analyzing the proof of Kazhdan and Lusztig [6] one could hope to prove a stronger result: the family of unitary Drinfeld twists F^{q} can be chosen to be real-analytic for an appropriate choice of φ^{q}.

3. Family of Dirac operators

We continue by fixing a continuous family of ∗-isomorphisms φ^{q}: U(G_{q}) → U(G) with φ^{1} = ι and a continuous family of unitary Drinfeld twists F^{q} for φ^{q} with F^{1} = 1. For every q > 0 we have a Dirac operator D_{q} on G_{q} defined in [10] as follows. Consider a basis {x_{i}}_{i} of 𝔤 such that (x_{i}, x_{j}) = −δ_{ij}, and let γ: 𝔤 → Cl(𝔤) denote the inclusion of 𝔤 into the complex Clifford algebra with the convention that γ(x_{i})^{2} = −1. Identifying 𝔰𝔬(𝔤) with 𝔰𝔭𝔦𝔫(𝔤), the adjoint action is defined by the representation ãd: 𝔤 → 𝔰𝔭𝔦𝔫(𝔤) ⊂ Cl(𝔤) given by

ãd(x) = \frac{1}{4} ∑_{i} γ(x_{i})γ([x, x_{i}]).

We denote by the same symbol ãd the corresponding homomorphism U(G) → Cl(𝔤). Let s: Cl(𝔤) → End(𝕊) be an irreducible representation. Denote by ∂ the representation of U𝔤 by left-invariant differential operators. Identifying the sections Γ(S) of the spin bundle S over G with C^{∞}(G) ⊗ 𝕊, the Dirac operator D: C^{∞}(G) ⊗ 𝕊 → C^{∞}(G) ⊗ 𝕊 defined using the Levi-Civita connection, can be written as D = (∂ ⊗ s)(D), where D ∈ U𝔤 ⊗ Cl(𝔤) is given by the formula

D = ∑_{i}(x_{i} ⊗ γ(x_{i}) + \frac{1}{2} ⊗ γ(x_{i})ãd(x_{i})).

Let (L^{2}(G_{q}), π_{q,r}, ξ_{h}^{q}) be the GNS-triple defined by the Haar state on C(G_{q}). The right regular representation of the von Neumann algebra W^{∗}(G_{q}) ⊂ U(G_{q}) of G_{q} on L^{2}(G_{q}), denoted by ∂_{q}, is defined by

∂_{q}(ω)π_{r,q}(a)ξ_{h}^{q} = (π_{r,q} ⊗ ω)Δ_{q}(a)ξ_{h}^{q} = a_{(1)}(ω)π_{r,q}(a_{(0)})ξ_{h}^{q}.

The Dirac operator D_{q} on G_{q} is the unbounded operator on L^{2}(G_{q})⊗𝕊 defined by

D_{q} = (∂_{q} ⊗ s)(D_{q}),

where D_{q} ∈ U(G_{q}) ⊗ Cl(𝔤) is given by

D_{q} = (φ^{q} ⊗ ι)^{−1}((ι ⊗ ãd)(F^{q})D(ι ⊗ ãd)(F^{q})^{∗}).

Our goal is to show that the family (D_{q})_{q} is continuous in the sense that it defines a Kasparov (C(G_{[a,b]}), C[a, b])-module.

Lemma 3.1. The family (L^{2}(G_{q}))_{q>0} has a unique structure of a continuous field of Hilbert spaces such that the vector field q ↦ π_{r,q}(a^{q})ξ_{h}^{q} is continuous for every continuous section q ↦ a^{q} of the field (C(G_{q}))_{q>0}.

Proof. It suffices to show that the function q ↦ (π_{r,q}(a^{q})ξ_{h}^{q}, ξ_{h}^{q}) is continuous. But this is clear, since if a^{q} = aφ^{q}, where a is a matrix coefficient of a nontrivial irreducible representation of G, then the function is zero by the orthogonality relations. □

It is easy to see that the continuous field (L^{2}(G_{q}))_{q>0} is trivial. To formulate a more precise result, recall the exact form of the orthogonality relations. First let us introduce some notation. For a weight β = ∑_{i} c_{i}α_{i} (c_{i} ∈ ℂ) put h_{β}^{q} = ∑_{i} c_{i}d_{i}h_{i}^{q} ∈ U(G_{q}) and K_{β}^{q} = q^{h_{β}^{q}}; note that for q = 1 the element h_{β} is characterized by λ(h_{β}) = (λ, β) for any weight λ, and for q ≠ 1 we have K_{α_{i}}^{q} = K_{i}^{q}. Let ρ be half the sum of positive roots. Then K_{2ρ}^{q} is the Woronowicz character f_{−1} for G_{q}; in particular, for the square of the antipode Ŝ_{q} on U(G_{q}) we have Ŝ_{q}^{2}(ω) = K_{−2ρ}^{q}ωK_{2ρ}^{q}. Put dim_{q}(V_{λ}^{q}) = Tr(π_{λ}^{q}(K_{2ρ}^{q})) = Tr(π_{λ}^{q}(K_{−2ρ}^{q})). For ξ, ζ ∈ V_{λ}^{q} define a_{ξ,ζ}^{λ;q} ∈ ℂ[G_{q}] by

a_{ξ,ζ}^{λ;q}(ω) = (π_{λ}^{q}(ω)ζ, ξ) for ω ∈ U(G_{q}).

Then the orthogonality relations state that the vectors π_{r,q}(a_{ξ,ζ}^{λ;q})ξ_{h}^{q} are mutually orthogonal for different λ, and

(π_{r,q}(a_{ξ,ζ}^{λ;q})ξ_{h}^{q}, π_{r,q}(a_{ξ′,ζ′}^{λ;q})ξ_{h}^{q}) = \frac{1}{dim_{q}(V_{λ}^{q})}\overline{(π_{λ}^{q}(K_{2ρ}^{q})ξ, ξ′)}(ζ, ζ′).

Documenta Mathematica 16 (2011) 767–780

Let d^{q} ∈ U(G_{q}) be the element such that

π_{λ}^{q}(d^{q}) = \frac{dim(V_{λ})^{1/2}}{dim_{q}(V_{λ}^{q})^{1/2}} π_{λ}^{q}(K_{ρ}^{q}) for all λ ∈ P_{+}.

Lemma 3.2. The linear operator W_{q}: π_{r,q}(ℂ[G_{q}])ξ_{h}^{q} → π_{r}(ℂ[G])ξ_{h} defined by

W_{q}π_{r,q}(aφ^{q})ξ_{h}^{q} = a_{(0)}(φ^{q}(d^{q}))π_{r}(a_{(1)})ξ_{h} for a ∈ ℂ[G]

is unitary. It has the property

W_{q}∂_{q}(ω) = ∂(φ^{q}(ω))W_{q} for all ω ∈ U(G_{q}).

Proof. By the orthogonality relations we have a decomposition L^{2}(G_{q}) = ⊕_{λ∈P_{+}} \overline{V_{λ}^{q}} ⊗ V_{λ}^{q}, with π_{r,q}(a_{ξ,ζ}^{λ;q})ξ_{h}^{q} ∈ L^{2}(G_{q}) corresponding to dim_{q}(V_{λ}^{q})^{−1/2}\overline{π_{λ}^{q}(K_{ρ}^{q})ξ} ⊗ ζ ∈ \overline{V_{λ}^{q}} ⊗ V_{λ}^{q}. The isomorphism φ^{q}: U(G_{q}) → U(G) is implemented by unitaries u_{λ}^{q}: V_{λ}^{q} → V_{λ}. Then the unitaries

\overline{u_{λ}^{q}} ⊗ u_{λ}^{q}: \overline{V_{λ}^{q}} ⊗ V_{λ}^{q} → \overline{V_{λ}} ⊗ V_{λ}

define a unitary L^{2}(G_{q}) → L^{2}(G). This is exactly the unitary W_{q}, since if a = a_{ξ,ζ}^{λ} ∈ ℂ[G] then aφ^{q} = a_{u_{λ}^{q}^{∗}ξ,u_{λ}^{q}^{∗}ζ}^{λ;q} and hence the vector π_{r,q}(aφ^{q})ξ_{h}^{q} is mapped onto

\frac{dim(V_{λ})^{1/2}}{dim_{q}(V_{λ}^{q})^{1/2}} π_{r}(a_{u_{λ}^{q}π_{λ}^{q}(K_{ρ}^{q})u_{λ}^{q}^{∗}ξ,ζ}^{λ})ξ_{h},

which gives the formula in the formulation, as

a_{u_{λ}^{q}π_{λ}^{q}(K_{ρ}^{q})u_{λ}^{q}^{∗}ξ,ζ}^{λ} = (π_{λ}(·)ζ, π_{λ}(φ^{q}(K_{ρ}^{q}))ξ) = a(φ^{q}(K_{ρ}^{q}) ·) = a_{(0)}(φ^{q}(K_{ρ}^{q}))a_{(1)}.

The last statement in the formulation follows either by a direct computation or by observing that ∂_{q}(ω) acts on \overline{V_{λ}^{q}} ⊗ V_{λ}^{q} ⊂ L^{2}(G_{q}) as 1 ⊗ π_{λ}^{q}(ω). □

The extension of W_{q} to L^{2}(G_{q}) we continue to denote by the same symbol. Therefore the unitaries W_{q} define an isomorphism of the continuous field (L^{2}(G_{q}))_{q>0} onto the trivial field with fiber L^{2}(G). For b > a > 0 denote by L^{2}(G_{[a,b]}) the right Hilbert C[a, b]-module of continuous sections of (L^{2}(G_{q}))_{q∈[a,b]}. The C^{∗}-algebra C(G_{[a,b]}) acts on L^{2}(G_{[a,b]}) via GNS-representations π_{r,q}; we denote by π the corresponding homomorphism from C(G_{[a,b]}) into the algebra of adjointable operators on L^{2}(G_{[a,b]}). The operators D_{q} define an unbounded operator D_{[a,b]} on L^{2}(G_{[a,b]})⊗𝕊 with domain of definition consisting of continuous vector fields ξ such that ξ(q) ∈ Dom(D_{q}) for all q and the vector field q ↦ D_{q}ξ(q) is continuous. When G is even-dimensional we also define a grading on L^{2}(G_{[a,b]})⊗𝕊 using the chirality element χ ∈ Cl(𝔤).

Theorem 3.3. For any b > a > 0 the triple (L^{2}(G_{[a,b]}) ⊗ 𝕊, π(·) ⊗ 1, D_{[a,b]}) is an unbounded Kasparov (C(G_{[a,b]}), C[a, b])-module of the same parity as the dimension of G.

Proof. By definition of an unbounded Kasparov module [1] it suffices to check that (i) D_{[a,b]} is a regular self-adjoint operator such that (1+D_{[a,b]}^{2})^{−1} is generalized compact (recall that regularity means that the operator 1+D_{[a,b]}^{2} is surjective);

(ii) there exists a dense ∗-subalgebra A of C(G_{[a,b]}) and an A-invariant core of D_{[a,b]} such that the commutators [D_{[a,b]}, π(c) ⊗ 1] are bounded on this core for all c ∈ A.

By definition of D_{q} and Lemma 3.2 we have

(3.1)  (∂ ⊗ s ãd)(F^{q})^{∗}(W_{q} ⊗ 1)D_{q} = D(∂ ⊗ s ãd)(F^{q})^{∗}(W_{q} ⊗ 1).

Therefore the unitaries (∂ ⊗ s ãd)(F^{q})^{∗}(W_{q} ⊗ 1) define an isomorphism of the continuous field (L^{2}(G_{q}) ⊗ 𝕊)_{q} onto the constant field with fiber L^{2}(G) ⊗ 𝕊, which maps D_{[a,b]} onto the operator which acts as D on every fiber. This immediately gives (i). To prove (ii) consider the space A ⊂ C(G_{[a,b]}) of finite sums of sections of the form q ↦ f(q)c φ^{q} with f a continuous function and c ∈ ℂ[G]. By Remark 1.3 this is a ∗-algebra. It is dense in C(G_{[a,b]}) since it is dense in every fiber and is closed under multiplication by continuous functions. The linear span of vectors fields of the form q ↦ π_{r,q}(c^{q})ξ_{h}^{q} ⊗ ζ, where c ∈ A and ζ ∈ 𝕊, is an A-invariant core for D_{[a,b]}. This follows e.g. from (3.1) and the fact that vectors of the form π_{r}(c)ξ^{h} ⊗ ζ, where c ∈ ℂ[G] and ζ ∈ 𝕊, span a core for D which is a union of D-invariant finite dimensional subspaces. It remains to check boundedness of commutators. For this it is enough to show that for every c ∈ ℂ[G] the commutators [D_{q}, π_{r,q}(c φ^{q}) ⊗ 1] are uniformly bounded on [a, b]. By [10, Proposition 3.1] we have

[D_{q}, π_{r,q}(c φ^{q}) ⊗ 1] = −(π_{r,q}(c_{(0)}φ^{q}) ⊗ 1)(∂_{q} ∘ (φ^{q})^{−1} ⊗ s)(c_{(1)} ⊗ ι ⊗ ι)(U_{q}T_{q}U_{q}^{∗}),

where

U_{q} = (ι ⊗ ι ⊗ ãd)((F^{q} ⊗ 1)(Δ̂ ⊗ ι)(F^{q})) ∈ W^{∗}(G) ¯⊗W^{∗}(G) ⊗ Cl(𝔤)

and T_{q} ∈ U(G × G) ⊗ Cl(𝔤) is defined by

T_{q} = (ι⊗ι⊗γ)(t_{13})+(ι⊗ι⊗γ)(t_{23})−(ι⊗ι⊗ãd)(Φ^{q})^{∗}(ι⊗ι⊗γ)(t_{23})(ι⊗ι⊗ãd)(Φ^{q}),

with Φ^{q} = Φ_{KZ}(ℏ_{q}t_{12}, ℏ_{q}t_{23}). It is therefore enough to prove that the operators

(c ⊗ ι ⊗ ι)(U_{q}T_{q}U_{q}^{∗}) ∈ U(G) ⊗ Cl(𝔤)

are uniformly bounded on [a, b] for all c ∈ ℂ[G]. Equivalently, the operators (π ⊗ ι ⊗ ι)(T_{q}) are uniformly bounded for any finite dimensional unitary representation π of G, which, in turn, is the same as uniform boundedness of

[(π ⊗ ι ⊗ γ)(t_{23}), (π ⊗ ι ⊗ ãd)(Φ_{KZ}(ℏ_{q}t_{12}, ℏ_{q}t_{23}))].

By [10, Proposition 3.6] the latter property indeed holds: the norm of the above commutator is bounded by 6∥(π ⊗ γ)(t)∥ independently of q > 0. □

Recall that by Theorem 1.4 the evaluation map ev_{q}: C(G_{[a,b]}) → C(G_{q}) is a KK-equivalence for any q ∈ [a, b]. For q, q′ ∈ [a, b] define an invertible element γ_{q,q′} in KK(C(G_{q}), C(G_{q′})) by γ_{q,q′} = [ev_{q}]^{−1}[ev_{q′}]. It does not depend on the segment [a, b] containing q and q′. Denote also by [D_{q}] the element of KK_{i}(C(G_{q}), ℂ) defined by D_{q}, where i = dim G (mod 2).

Corollary 3.4. For any q, q′ > 0 we have γ_{q,q′}[D_{q′}] = [D_{q}].

Proof. Denote by [D_{[a,b]}] ∈ KK_{i}(C(G_{[a,b]}), C[a, b]) the class of the Kasparov module

(L^{2}(G_{[a,b]}) ⊗ 𝕊, π(·) ⊗ 1, D_{[a,b]}).

If ẽv_{q}: C[a, b] → ℂ is the evaluation at q then clearly [D_{[a,b]}] [ẽv_{q}] = [ev_{q}] [D_{q}]. Since [ẽv_{q}] does not depend on q, we thus see that the class [ev_{q}] [D_{q}] ∈ KK_{i}(C(G_{[a,b]}), ℂ) does not depend on q ∈ [a, b] either, which is what we need. □

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Sergey Neshveyev Department of Mathematics University of Oslo P.O. Box 1053 Blindern NO-0316 Oslo Norway sergeyn@math.uio.no Lars Tuset Faculty of Engineering Oslo University College P.O. Box 4 St. Olavs plass NO-0130 Oslo Norway Lars.Tuset@iu.hio.no