The Modular Square for Quantum Groups

J. Kustermans J. Rognes L. Tuset

Abstract

This is mainly an expository paper with results gathered from [12], [5], [6] and [11].

We define the modular square and the modular characteristic homomorphisms for algebraic quantum groups. We recall Hopf cyclic cohomology and methods developed for computing it, which includes a Künneth exact sequence for Hopf cyclic cohomology. Focus is on the category of Hopf algebras in involution associated to compact quantum groups. We review the computations of the Hopf cyclic cohomology and the modular characteristic homomorphisms for the compact quantum group SU_{q}(2).

We define a cocyclic object which yields twisted cyclic cohomology. This cohomology is shown to have a Chern character as well, and a beautiful connection to differential calculi for compact quantum groups is recalled. As an entirely new result, we show that characteristic homomorphisms can be defined with target, twisted cyclic cohomology, without involving the modular square, and which to some degree are better behaved than the modular characteristic homomorphism.

MSC 2000 : Primary 17 B37. Secondary 19 D55, 16 W30, 20 G42. Keywords: Hopf algebras, cocyclic objects, Künneth exact sequence.

1 Introduction

We define Hopf cyclic cohomology in the sense of A. Connes and H. Moscovici. By the general machine recalled in Section 2, all that is needed to get a full-fledged cyclic cohomology theory with the long exact IBS-sequence intact, is a cocyclic object. Such an object can be canonically associated to a Hopf algebra (H, Δ) with a modular pair (δ, σ) in involution. The latter consists of a character δ and a group-like element σ satisfying certain conditions. It is shown that the corresponding Hopf Hochschild cohomology HH_{(δ,σ)}^{∗}(H) can be regarded as a derived functor in the category of bicomodules over Hopf algebras.

In the following section, we study the category of Hopf algebras with modular pairs in involution. We notice that it is closed under formation of opposites, duals and tensor products. We state a Künneth theorem for the Hopf Hochschild cohomology, and a Künneth exact sequence for the Hopf cyclic cohomology. We focus then on various examples of Hopf algebras with modular pairs in involution associated to compact quantum groups, and compute them in the case of compact quantum groups coming from certain Lie groups. We consider SU_{q}(2) and compute the Hopf cohomologies given by the Hopf algebra (U_{q}(sl_{2}), Δ) with a canonical modular pair in involution. To demonstrate the power of our Künneth theorems, we include also results for the tensor product Hopf algebra (H̃_{q}, Δ̃) of (U_{q}(sl_{2}), Δ) with its opposite.

Next we define the modular square for a Hopf algebra with a canonical modular pair in involution associated to a compact quantum group. The modular square is an intricate construction defined in order to obtain the modular characteristic homomorphism from Hopf cyclic cohomology to ordinary cyclic cohomology.

To explain what this roughly means, consider the Hopf algebra (A, Φ) of regular functions on a compact quantum group. Let (H, Δ) denote a Hopf subalgebra of its maximal dual Hopf algebra. Let {f_{z}} denote the family governing the modular properties of the Haar state h. Suppose f_{1} ∈ H, which is indeed the case when H = U_{q}(sl_{2}). Then the tensor product Hopf algebra (H̃, Δ̃) of (H, Δ) and its opposite Hopf algebra (H̃_{op}, Δ̃), is endowed with a modular pair (δ, σ) = (I ⊗ I, f_{1} ⊗ f_{1}) in involution. Next a canonical H̃-action of (H̃, Δ̃) on A is defined under which h is a δ-invariant σ-trace on A. The δ-invariance property follows from strong left invariance for h, and the σ-trace property is the KMS-condition. This assures the existence of a homomorphism γ_{∗} from HC_{(δ,σ)}^{∗}(H̃) to ordinary cyclic cohomology HC^{∗}(A) of A, which is called the modular characteristic homomorphism of (H, Δ). Thus we are dealing with a theory of characteristic classes for actions of non-unimodular Hopf algebras compatible with the modular theory of weights.

Combining the computation of the Hopf cohomologies for (U_{q}(sl_{2}), Δ) with the computations in [20] of the ordinary Hochschild and cyclic cohomology of A_{q}, gives the result that the modular characteristic homomorphism of (U_{q}(sl_{2}), Δ) is, unfortunately, ZERO.

We are then led to consider the maximal dual Hopf ∗-algebra (A_{q}^{o}, Δ) of (A_{q}, Φ). A stronger result would be to show that the modular characteristic homomorphism of (A_{q}^{o}, Δ) is zero, in which case it would be zero for any Hopf subalgebra of (A_{q}^{o}, Δ). We devote a section to this end, and show that under a certain assumption on SU_{q}(2), which we unfortunately have not been able to check, this stronger result holds. We prove a general duality theorem to the effect that HC_{(I,f_{1})}^{∗}(A^{o}) can be computed from a finitely generated free (or projective) resolution of A as an A-bimodule, provided A^{o} ⊗ A^{o} is injective as an A-bimodule. Assuming this assumption holds for SU_{q}(2), we use an explicit resolution of A_{q} given by T. Masuda, Y. Nakagami and J. Watanabe to compute HC_{(I,f_{1})}^{∗}(A_{q}^{o}). A fine study of the B-operator yields as a consequence that the modular characteristic homomorphism of (A_{q}^{o}, Δ) is again zero in Hochschild cohomology, cyclic cohomology and periodic cyclic cohomology.

We make an effort to exhibit generators for the various Hopf cohomologies, both for (U_{q}(sl_{2}), Δ) and for (A_{q}^{o}, Δ), in the respective defining complexes. Their representatives involve elements in U_{q}(sl_{2}) and, in the latter case, also an element H ∈ A_{q}^{o}, which plays the role of the Cartan element for quantum SU(2) and does not belong to U_{q}(sl_{2}) ⊂ A_{q}^{o}. We also write down the generators for their corresponding modular squares, a procedure which requires explicit formulas for the Alexander-Whitney map and the shuffle map implementing the Künneth isomorphisms.

In a separate section we define the modular square for the more general case of algebraic quantum groups, and discuss generalizations to the locally compact case as well. Loosely speaking, the modular square construction associates to a Hopf algebra A a modular pair in involution on its dual Hopf algebra A^{∘}, an action of this dual algebra A^{∘} on A itself and a linear functional on A that is ‘invariant’ with respect to this action.

The construction presented in this section was hinted at in [3] (worked out in [10]) and formally generalized in [5], thereby giving a good indication how the modular square should be build up in the general framework of locally compact quantum groups.

An intermediate step between Hopf algebras and locally compact quantum groups are the algebraic quantum groups discussed in [14] (section 6). Thanks to the algebraic nature of the latter objects (see [23]), combined with their analytic properties (see [9]), the modular square construction presented in [5] can be easily made rigorous for these algebraic quantum groups and that is precisely what we are going to do in this section. We present the Künneth theorems also in this setting, and define the modular characteristic homomorphism.

Thereafter we include an alternative approach to Hopf symmetry and the modular square construction by recalling the twisted cyclic cohomology HC^{∗}(A, θ) associated to an algebra A with an automorphism θ. It generalizes ordinary cyclic cohomology in the sense that HC^{∗}(A, ι) = HC^{∗}(A). Again this cohomology is introduced by defining a cocyclic object, which assures a satisfactory cohomology theory. We show how the Chern character can be extended to this setting, and we state a beautiful one-to-one correspondence between twisted cyclic cocycles and differential calculi with twisted graded traces [11]. In the case of a compact quantum group this correspondence preserves invariance, in that invariant cocycles correspond to covariant calculi and invariant twisted graded traces.

In Section A we prove that there exist characteristic homomorphisms

γ_{∗}^{l} : HC_{(I,f_{1})}^{∗}(A^{o}) ⟶ HC^{∗}(A, θ_{l})
γ_{∗}^{r} : HC_{(I,f_{1})}^{∗}(A^{o}) ⟶ HC^{∗}(A, θ_{r})

defined using h and canonical (left- and right- ) A^{o}-actions. Here θ_{l}(a) = (f_{−1} ⊗ ι)Φ(a) and θ_{r}(a) = (ι ⊗ f_{−1})Φ(a), for a ∈ A. These characteristic homomorphisms have certain advantages compared to the modular characteristic homomorphism.

In Section B we spell out in what sense the Hopf algebra (H̃, Δ̃) can be identified with one of Majid’s bicrossproducts known as the Mirror product H ▷◁ H_{op} [18].

2 Hopf Cyclic Cohomology

In this section we consider cyclic cohomology from the point of view of cocyclic objects. We introduce the cocyclic object that defines the cyclic cohomology HC_{(δ,σ)}^{∗}(H) of a Hopf algebra (H, Δ) with a modular pair (δ, σ) in involution. Also we include a brief reminder of the Chern character of Connes.

Let ℕ_{0} denote the set of non-negative integers. The cyclic category Λ is defined [2] [15]

[19] to be the category with objects [n] for n ∈ ℕ_{0}, and morphisms universally generated by the elements

δ_{i}^{n} : [n − 1] → [n], σ_{i}^{n} : [n + 1] → [n], τ_{n} : [n] → [n],

where i ∈ {0, 1, …, n}, satisfying the following relations:

δ_{j}^{n}δ_{i}^{n−1} = δ_{i}^{n}δ_{j−1}^{n−1}  for i < j
σ_{j}^{n}σ_{i}^{n+1} = σ_{i}^{n}σ_{j+1}^{n+1}  for i ≤ j
σ_{j}^{n}δ_{i}^{n+1} = δ_{i}^{n}σ_{j−1}^{n−1}  for i < j
σ_{j}^{n}δ_{i}^{n+1} = 1_{n}  for i = j or i = j + 1
σ_{j}^{n}δ_{i}^{n+1} = δ_{i−1}^{n}σ_{j}^{n−1}  for i > j + 1

and

τ_{n}δ_{i}^{n} = δ_{i−1}^{n}τ_{n−1}  for i ≥ 1 and τ_{n}δ_{0}^{n} = δ_{n}^{n}
τ_{n}σ_{i}^{n} = σ_{i−1}^{n}τ_{n+1}  for i ≥ 1 and τ_{n}σ_{0}^{n} = σ_{n}^{n}τ_{n+1}^{2}

and

τ_{n}^{n+1} = 1_{n}.

Here 1_{n} denotes the identity morphism from [n] to [n]. Of course, in the relations above, n has to be taken greater than zero for some of the expressions to make sense, since δ_{i}^{0} and σ_{i}^{−1} and τ_{−1} are not defined. The morphisms δ_{i}^{n} and σ_{i}^{n} are referred to as coface maps and codegeneracy maps, respectively, and generate a subcategory Δ of Λ called the cosimplicial category. The morphism τ_{n} is referred to as the cyclicity map.

One can alternatively define the category Λ in terms of its cyclic covering EΛ. The category EΛ has one object (ℤ, n), for each n ∈ ℕ_{0}, and morphisms f : (ℤ, n) → (ℤ, m) are given by non-decreasing maps f : ℤ → ℤ such that f(x + n) = f(x) + m, for x ∈ ℤ. Then Λ ≅ EΛ/ℤ with respect to the obvious action of ℤ by translation.

A cocyclic object of any category C is a covariant functor from Λ to C. By the universal property of the cyclic category Λ, a cocyclic object of a category C is equivalently described by a quadruple (C^{n}, d_{i}^{n}, s_{i}^{n}, t_{n}), where d_{i}^{n}, s_{i}^{n} and t_{n} are morphisms of C with objects C^{n} satisfying the same relations when substituted for δ_{i}^{n}, σ_{i}^{n} and τ_{n}.

Consider a cocyclic object (C^{n}, d_{i}^{n}, s_{i}^{n}, t_{n}) of an abelian category C. Define morphisms b_{n}, b′_{n} : C^{n−1} → C^{n} and λ_{n}, N^{n} : C^{n} → C^{n} by:

b_{n} = ∑_{i=0}^{n}(−1)^{i}d_{i}^{n} and b′_{n} = ∑_{i=0}^{n−1}(−1)^{i}d_{i}^{n}

λ_{n} = (−1)^{n}t_{n} and N^{n} = ∑_{i=0}^{n} λ_{n}^{i}.

It is easy to check (by using the defining relations for the morphisms δ_{i}^{n} in the category Λ) that (C, b) and (C, b′) are complexes, i.e. b_{n+1}b_{n} = b′_{n+1}b′_{n} = 0, for any n ∈ ℕ_{0}. The cohomology of (C, b) is by definition the Hochschild cohomology HH^{∗}(C) of the cosimplicial object (C^{n}, d_{i}^{n}, s_{i}^{n}). The complex (C, b′) is, however, acyclic with contracting homotopy s_{n+1} = σ_{n}^{n}τ_{n+1}. Again, using the relations between the τ’s and δ’s in Λ, one can check that (1_{n} −λ_{n})b_{n} = b′_{n}(1_{n−1} −λ_{n−1}), for any n ∈ ℕ_{0}. Therefore we get a subcomplex (C_{λ}, b) of (C, b) with objects C_{λ}^{n} = ker(1_{n} − λ_{n}) consisting of the cyclic n-cochains. The cohomology of this subcomplex is denoted by H_{λ}^{∗}(C).

Further, it can be shown (by using the relations for the δ’s and τ’s in Λ) that bN = Nb′ and (1 − λ)N = N(1 − λ) = 0. From this one obtains a bicomplex:

⋮  ⋮  ⋮  ⋮
b ↑  −b′ ↑  b ↑  −b′ ↑
C^{2}  →^{1−λ}  C^{2}  →^{N} C^{2}  →^{1−λ}  C^{2}  →^{N} ⋯
b ↑  −b′ ↑  b ↑  −b′ ↑
C^{1}  →^{1−λ}  C^{1}  →^{N} C^{1}  →^{1−λ}  C^{1}  →^{N} ⋯
b ↑  −b′ ↑  b ↑  −b′ ↑
C^{0}  →^{1−λ}  C^{0}  →^{N} C^{0}  →^{1−λ}  C^{0}  →^{N} ⋯

We denote this bicomplex by (C^{mn}, b, b′), where C^{mn} = C^{n}, for all m, n ∈ ℕ_{0}. The cohomology of the total complex of (C^{mn}, b, b′) is, by definition, the cyclic cohomology HC^{∗}(C) of the cocyclic object (C^{n}, d_{i}^{n}, s_{i}^{n}, t_{n}). The cohomology of the total complex of (C^{mn}, b, b′), where we instead let m ∈ ℤ, is by definition, the periodic cyclic cohomology HP^{∗}(C) of the cocyclic object (C^{n}, d_{i}^{n}, s_{i}^{n}, t_{n}). From the bicomplex construction the long exact IBS-sequence

⋯ ⟶ HC^{n} ⟶^{I} HH^{n} ⟶^{B} HC^{n−1} ⟶^{S} HC^{n+1} ⟶^{I} HH^{n+1} ⟶^{B} HC^{n} ⟶ ⋯

of Connes follows easily. Here we have used the abbreviations HC^{∗} for HC^{∗}(C) and HH^{∗} for HH^{∗}(C). Recall that the B-operator is given by B = Ns(1 − λ), or more explicitly

B_{n} = N^{n}σ_{n}^{n}τ_{n+1}(1_{n+1} − λ_{n+1}) : C^{n+1} → C^{n},

for n ∈ ℕ_{0}. We denote the mixed complex of C. Kassel associated to a cocyclic object (C^{n}, d_{i}^{n}, s_{i}^{n}, t_{n}) by (C, b, B) [2] [24]. It is known that H_{λ}^{∗}(C) ≅ HC^{∗}(C) whenever the ground field contains the rationals.

Now Λ is a small category, which means that the collection Mor(Λ) = {f ∈ Hom(n, m) | n, m ∈ ℕ_{0} } of all its morphisms is a set. Write C(Λ) for the vector space with basis Mor(Λ). On C(Λ) define an algebra structure with product (af)(bg) to be abfg, if the composition fg makes sense, and to be zero otherwise, where a, b ∈ ℂ and f, g ∈ Mor(Λ).

A cocyclic object (C^{n}, d_{i}^{n}, s_{i}^{n}, t_{n}) in the category of vector spaces yields in an obvious manner a C(Λ)-module C^{♮} whose underlying vector space is the direct sum ⨁_{n∈ℕ_{0}} C^{n}. We refer to C(Λ)-modules simply as Λ-modules. Associated to the Λ-module C^{♮} given by a cocyclic object, there are linear maps b = ⊕_{n∈ℕ_{0}}b_{n} and b′ = ⊕_{n∈ℕ_{0}}b′_{n} and λ = ⊕_{n∈ℕ_{0}}λ_{n} and N = ⊕_{n∈ℕ_{0}}N^{n}. We then get the following nice interpretation of the cyclic cohomology groups HC^{∗}(C) of a cocyclic object in the category of vector spaces, in terms of a derived functor over the category of Λ-modules [5], i.e.,

HC^{∗}(C) ≅ Ext_{Λ}^{∗}(ℂ^{♮}, C^{♮}).

Similarly, cyclic objects - defined as contavariant functors from Λ to any abelian category - will give rise to homology groups. And for a cyclic object in the category of vector spaces, the cyclic homology groups HC_{∗}(C) can be interpreted as a derived functor over the category of Λ-modules, i.e.,

HC_{∗}(C) ≅ Tor_{∗}^{Λ}(C^{♮}, ℂ^{♮}).

Suppose that A is a unital algebra. We recall the basic example of a cocyclic object, namely the one which gives rise to ordinary Hochschild cohomology HH^{∗}(A) and ordinary cyclic cohomology HC^{∗}(A) of A. Let n ∈ ℕ_{0}, and denote by C^{n} the vector space of ncochains, that is, the set of multilinear maps from the vector space A^{n+1} to ℂ. Let n ∈ ℕ_{0} and i ∈ {0, 1, …, n}. Define linear maps

d_{i}^{n} : C^{n−1} → C^{n}, s_{i}^{n} : C^{n+1} → C^{n}, t_{n} : C^{n} → C^{n}

as follows:

(d_{i}^{n}φ)(a^{0}, a^{1}, …, a^{n}) = φ(a^{0}, …, a^{i}a^{i+1}, …, a^{n}) for i ≤ n − 1
(d_{n}^{n}φ)(a^{0}, a^{1}, …, a^{n}) = φ(a^{n}a^{0}, a^{1}, …, a^{n−1})
(s_{i}^{n}φ)(a^{0}, a^{1}, …, a^{n}) = φ(a^{0}, …, a^{i}, I, a^{i+1}, …, a^{n})
(t_{n}φ)(a^{0}, a^{1}, …, a^{n}) = φ(a^{n}, a^{0}, a^{1}, …, a^{n−1}),

where I is the unit of A, a^{0}, …, a^{n} ∈ A and φ is a cochain. Note that C^{−1} and d_{i}^{0} are not defined, so the range of n must be restricted accordingly for some formulas to make sense. It is straightforward to check that (C^{n}, d_{i}^{n}, s_{i}^{n}, t_{n}) is a cocyclic object of the abelian category of vector spaces. In Section 8 we shall consider a twisted version of this cocyclic object.

Before we consider the cocyclic object given by a Hopf algebra, let us recall the definition of the Chern character [2] in noncommutative geometry, which manifests a profound link between ordinary cyclic cohomology and K-theory. Let φ be an n-dimensional cyclic cocycle, so φ ∈ C^{n}, λ_{n}(φ) = φ and b_{n+1}(φ) = 0, and denote by [φ] the corresponding equivalence class in HC^{n}(A) = H_{λ}^{n}(A). Let n, N ∈ ℕ_{0} and consider the standard trace Tr on M_{N}(ℂ). Denote by φ⊗Tr the (n+1)-multilinear functional on M_{N}(A) = A⊗M_{N}(ℂ) given by

(φ ⊗ Tr)(a^{0} ⊗ x^{0}, a^{1} ⊗ x^{1}, …, a^{n} ⊗ x^{n}) = φ(a^{0}, a^{1}, …, a^{n})Tr(x^{0}x^{1} ⋯ x^{n}),

for all a^{0}, …, a^{n} ∈ A and x^{0}, …, x^{n} ∈ M_{N}(ℂ).

Consider first the case when n = 2m is even. It can be shown that the scalar (φ ⊗ Tr)(E, E, …, E) is invariant under homotopy for idempotents E^{2} = E ∈ M_{N}(A). Thus φ provides in this way a numerical invariant for the K-group K_{0}(A). Moreover, the scalar is independent of what representative φ in [φ] is employed. Thus we obtain a bilinear pairing

⟨·, ·⟩ : HC^{even}(A) × K_{0}(A) → ℂ

given by

⟨[φ], [E]⟩ = \frac{1}{m!}(φ ⊗ Tr)(E, E, …, E),

for all [φ] ∈ HC^{2m}(A) and projections E ∈ M_{N}(A). Here HC^{even}(A) denotes the even cyclic cohomology of A.

Consider next the case when n is odd. For u an invertible element of the unital algebra M_{N}(A) and φ a cyclic n-cocycle of A, the formula

⟨[φ], [u]⟩ = \frac{1}{\sqrt{2i}}2^{−n}Γ(\frac{n}{2} + 1)^{−1}(φ ⊗ Tr)(u^{−1} − 1, u − 1, u^{−1} − 1, …, u − 1)

defines similarly a bilinear pairing

⟨·, ·⟩ : HC^{odd}(A) × K_{1}(A) → ℂ.

Here, HC^{odd}(A) denotes the odd cyclic cohomology of A, Γ is the gamma function and [u] denotes the K-theory class of the invertible u ∈ M_{N}(A).

Let (H, Δ) be a Hopf algebra with comultiplication Δ, coinverse (antipode) S, counit ε, unit I and multiplication m : H ⊗ H → H. If U and V are vector spaces, we denote the flip from U ⊗ V to V ⊗ U by F. Our basic reference for Hopf algebras is [1].

A modular pair for (H, Δ) consists of a pair (δ, σ), where δ is a unital, multiplicative, linear functional on H and σ ∈ H satisfies Δ(σ) = σ ⊗ σ and δ(σ) = 1. Thus (ε, I) is an example of a modular pair for (H, Δ). Now δ̂: H → H given by δ̂ = (δ ⊗ ι)Δ is a unital, multiplicative, linear map with inverse δ̂^{−1} = (δS ⊗ ι)Δ. We associate to the modular pair (δ, σ) a twisted coinverse S_{δ} : H → H defined by S_{δ} = (δ ⊗ S)Δ = S ∘ δ̂, see [3] [6]. Note that if S is invertible, then S_{δ} is invertible with inverse S_{δ}^{−1} = (S^{−1} ⊗ δ)Δ. We say that the modular pair (δ, σ) is in involution if S_{δ}^{2}(x) = σxσ^{−1}, for all x ∈ H.

Let n ∈ ℕ_{0} and i ∈ {0, 1, …, n}. Define linear maps

δ_{i}^{n} : H^{⊗(n−1)} → H^{⊗n}, σ_{i}^{n} : H^{⊗(n+1)} → H^{⊗n}, τ_{n} : H^{⊗n} → H^{⊗n}

as follows:

δ_{0}^{n}(x^{1} ⊗ ... ⊗ x^{n−1}) = I ⊗ x^{1} ⊗ ... ⊗ x^{n−1}
δ_{i}^{n}(x^{1} ⊗ ... ⊗ x^{n−1}) = x^{1} ⊗ ... ⊗ Δx^{i} ⊗ ... ⊗ x^{n−1} for i ∈ {1, …, n − 1}
δ_{n}^{n}(x^{1} ⊗ ... ⊗ x^{n−1}) = x^{1} ⊗ ... ⊗ x^{n−1} ⊗ σ
σ_{i}^{n}(x^{1} ⊗ ... ⊗ x^{n+1}) = x^{1} ⊗ ... ⊗ ε(x^{i+1}) ⊗ ... ⊗ x^{n+1}
τ_{n}(x^{1} ⊗ ... ⊗ x^{n}) = (Δ^{n−1}S_{δ}(x^{1})) · (x^{2} ⊗ ... ⊗ x^{n} ⊗ σ),

where x^{1}, …, x^{n+1} ∈ H. Here H^{⊗(−1)} and δ_{i}^{0} are not defined. By H^{⊗0} we mean ℂ. We define δ_{0}^{1}, δ_{1}^{1} : ℂ → H, τ_{0} : ℂ → ℂ and σ_{0}^{0} : H → ℂ by setting δ_{0}^{1}(1) = I, δ_{1}^{1}(1) = σ, τ_{0}(1) = 1 and σ_{0}^{0} = ε. The map Δ^{n} is defined inductively by setting Δ^{n} = (ι ⊗ Δ)Δ^{n−1} for n ≥ 2. One can check directly [4] that (H^{⊗n}, δ_{i}^{n}, σ_{i}^{n}, τ_{n}) is a cocyclic object of the abelian category of vector spaces if and only if the modular pair (δ, σ) is in involution. Hereafter we assume that this is the case. Following Connes and Moscovici we denote the Hochschild cohomology and the cyclic cohomology of this cocyclic object by HH_{(δ,σ)}^{∗}(H) and HC_{(δ,σ)}^{∗}(H), respectively.

Now ℂ is an H-bicomodule with respect to α : ℂ → H⊗ℂ⊗H given by α(1) = I⊗1⊗σ, and H is an H-bicomodule with respect to α : H → H ⊗ H ⊗ H given by α = (Δ ⊗ ι)Δ. Since the face maps and degeneracy maps of our Hopf cyclic object are intimately linked to those defining the cobar resolution, we obtain the following useful result. Proposition 2.1. ([12]) Let (H, Δ) be a Hopf algebra with a modular pair (δ, σ) in involution. Then

HH_{(δ,σ)}^{∗}(H) ≅ Ext_{H−bicom}^{∗}(ℂ, H).

Alternatively, if we regard ℂ as an H-comodule by β_{1}(1) = I ⊗ 1 and denote the H-comodule ℂ with respect to β_{2}(1) = σ ⊗ 1 by ℂ_{σ}, we get the following interpretation. Proposition 2.2. ([6]) Let (H, Δ) be a Hopf algebra with a modular pair (δ, σ) in involution. Then

HH_{(δ,σ)}^{∗}(H) ≅ Cotor_{H−com}^{∗}(ℂ, ℂ_{σ}).

3 The Category of Hopf Algebras with Modular Pairs

In this section we study the category of Hopf algebras with modular pairs in involution, and recall various basic and motivating examples.

Suppose (H_{i}, Δ_{i}) are two Hopf algebras with modular pairs (δ_{i}, σ_{i}) in involution for i = 1, 2, and let φ : H_{1} → H_{2} be a morphism of Hopf algebras. We say φ is a morphism in the category of Hopf algebras with modular pairs in involution if δ_{2}φ = δ_{1} and σ_{2} = φ(σ_{1}). Consider the tensor product Hopf algebra (H_{1} ⊗ H_{2}, Δ_{1} × Δ_{2}) of (H_{1}, Δ_{1}) and (H_{2}, Δ_{2}) with comultiplication Δ_{1}×Δ_{2} = (ι⊗F ⊗ι)(Δ_{1}⊗Δ_{2}). Obviously, the pair (δ_{1}⊗δ_{2}, σ_{1}⊗σ_{2}) is a modular pair in involution for (H_{1} ⊗ H_{2}, Δ_{1} × Δ_{2}). The following Künneth theorem for this setting can now be stated. Theorem 3.1. ([12]) Consider two Hopf algebras (H_{i}, Δ_{i}) with modular pairs (δ_{i}, σ_{i}) in involution. Then

HH_{(δ_{1}⊗δ_{2},σ_{1}⊗σ_{2})}^{∗}(H_{1} ⊗ H_{2}) ≅ HH_{(δ_{1},σ_{1})}^{∗}(H_{1}) ⊗ HH_{(δ_{2},σ_{2})}^{∗}(H_{2}),

and, moreover, the isomorphism is implemented by the shuffle map sh (going from left to right) with the Alexander-Whitney map AW as its inverse.

In this context we also have the following Künneth exact sequence. Theorem 3.2. ([12]) Consider two Hopf algebras (H_{i}, Δ_{i}) with modular pairs (δ_{i}, σ_{i}) in involution. Then there is a canonical long exact sequence

⋯ ⟶^{i} HC_{(δ_{1}⊗δ_{2},σ_{1}⊗σ_{2})}^{n−1}(H_{1} ⊗ H_{2}) ⟶^{∂}

⨁_{p+q=n−2} HC_{(δ_{1},σ_{1})}^{p}(H_{1}) ⊗ HC_{(δ_{2},σ_{2})}^{q}(H_{2}) ⟶^{S⊗1−1⊗S}

⨁_{r+s=n} HC_{(δ_{1},σ_{1})}^{r}(H_{1}) ⊗ HC_{(δ_{2},σ_{2})}^{s}(H_{2}) ⟶^{i} HC_{(δ_{1}⊗δ_{2},σ_{1}⊗σ_{2})}^{n}(H_{1} ⊗ H_{2}) ⟶^{∂} ⋯

Suppose that (H, Δ) is a Hopf algebra with coinverse S. Denote by H_{op} the algebra H with the opposite multiplication mF, and denote by Δ_{op} the opposite comultiplication F Δ. Let σ ∈ H and let δ be a linear functional on H. Then the following four statements are equivalent ([12]):

1. (δ, σ) is a modular pair in involution for (H, Δ);

2. (δS, σ) is a modular pair in involution for (H_{op}, Δ);

3. (δ, σ^{−1}) is a modular pair in involution for (H, Δ_{op});

4. (δS, σ^{−1}) is a modular pair in involution for (H_{op}, Δ_{op}).

The next result shows that the category of Hopf algebras with modular pairs in involution is also closed under formation of duals. Proposition 3.3. ([12]) Suppose ⟨·, ·⟩ : H_{1} × H_{2} → ℂ is a non-degenerate dual pairing between two Hopf algebras (H_{i}, Δ_{i}) with coinverses S_{i}. Consider two elements δ ∈ H_{2} and σ ∈ H_{1}. Then (⟨·, δ⟩, σ) is a modular pair in involution for (H_{1}, Δ_{1}) if and only if (⟨S_{1}(σ), ·⟩, S_{2}(δ)) is a modular pair in involution for (H_{2}, Δ_{2}). Example 3.4. ([4][6]) Let (A, Φ) be the Hopf ∗-algebra associated to a compact quantum group in the sense of S.L. Woronowicz [25] [26]. Let {f_{z}}_{z∈ℂ} denote the family of functionals on A describing the modular properties of the Haar state, Proposition A.1, and let I denote the unit of the algebra A. Then (f_{−1}, I) is a modular pair in involution for (A, Φ). The twisted coinverse is given by κ_{δ} = (f_{−1} ⊗ κ)Φ, where κ is the coinverse of (A, Φ). If Γ is a discrete group and A is the group algebra ℂ[Γ] with Φ(g) = g ⊗ g for g ∈ Γ ⊂ A, then f_{z} = ε for all z ∈ ℂ, so (ε, I) is a modular pair in involution for (ℂ[Γ], Φ).

However, this example is not terribly interesting due to the following result, which follows easily from the existence of the Haar state for (A, Φ). Proposition 3.5. ([7]) Let (A, Φ) be the Hopf ∗-algebra associated to a compact quantum group. Then

1. HH_{(f_{−1},I)}^{n}(A) ≅ 0 if n ≠ 0, and HH_{(f_{−1},I)}^{0}(A) ≅ ℂ.

2. HC_{(f_{−1},I)}^{n}(A) ≅ ℂ if n is even, and HC_{(f_{−1},I)}^{n}(A) ≅ 0 if n is odd.

3. HP_{(f_{−1},I)}^{0}(A) ≅ ℂ and HP_{(f_{−1},I)}^{1}(A) ≅ 0.

The following proposition shows that in the absence of a Haar state, something interesting can nevertheless be obtained from the Hopf algebra of polynomial functions on a Lie group. Proposition 3.6. ([5][3]) Let (H(G), Φ) be the Hopf algebra of polynomial functions on a simply connected affine nilpotent group G, with Lie algebra 𝔤. Then

HP_{(ε,I)}^{∗}(H(G)) ≅ ⨁_{i=∗ mod 2} H^{i}(𝔤, ℂ),

where H^{∗}(𝔤, ℂ) is the Lie algebra cohomology.

For compact quantum groups the dual setting turns out to be more interesting.

Example 3.7. ([4]) Denote by (A^{o}, Δ) the maximal Hopf ∗-algebra dual to (A, Φ) [1]. It has product xy = (x ⊗ y)Φ and coproduct Δ(x)(a ⊗ b) = x(ab), for x, y ∈ A^{o} and a, b ∈ A. Thus S^{2}(x) = f_{1}xf_{−1}, where S is the coinverse of (A^{o}, Δ) given by S(x) = xκ. Note that f_{1}^{∗} = f_{1}, where x^{∗}(a) = \overline{x(κ(a)^{∗})} is the ∗-operation for (A^{o}, Δ). Suppose that (H, Δ) is a Hopf ∗-subalgebra of (A^{o}, Δ) such that f_{1} ∈ H. Regard the unit of A as the linear functional on H given by I(x) = x(I), for x ∈ H, so I is just the counit of (A^{o}, Δ). Then (I, f_{1}) is a modular pair in involution for (H, Δ). Note that S_{I} = S.

If A is the algebra of regular functions on a compact Lie group G with pointwise operations and Φ is the transpose of the group multiplication on G, then one can consider H = U(𝔤), where U(𝔤) is the (complex) universal enveloping algebra of the associated Lie algebra 𝔤. The inclusion U(𝔤) ⊂ A^{o} is given by X(a) = X(a)(e), for a ∈ A. Here e is the unit of G and X on the right hand side is regarded as a left invariant differential operator on G. Then (U(𝔤), Δ) is a Hopf ∗-subalgebra of (A^{o}, Δ) with f_{z} = ε for z ∈ ℂ, and where Δ(X) = X ⊗ ε + ε ⊗ X for X ∈ 𝔤. Hence (I, ε) is a modular pair in involution for (U(𝔤), Δ).

In view of Example 3.7 the following uniqueness result is worthwhile recalling. Let Rep B denote the category of finite-dimensional ∗-representations of the unital ∗-algebra B. We say that an element of B is positive if it can be written as a finite sum of elements of the form b^{∗}b, where b ∈ B. As usual Tr will denote the operator trace. Proposition 3.8. ([21]) Suppose (H, Δ) is a Hopf ∗-algebra with coinverse S and that Rep H separates the elements of H. Then there is at most one positive invertible f ∈ H such that S^{2}(x) = fxf^{−1}, for all x ∈ H, and Trπ(f) = Trπ(f^{−1}), for all π ∈ Rep H. Moreover, the element f is necessarily group-like, and whenever (H, Δ) is cocommutative, it equals the unit. Remark 3.9. Let ⟨·, ·⟩ : H×A → ℂ denote the dual pairing ⟨x, a⟩ = x(a) between (A, Φ) and (H, Δ) considered in Example 3.4 and Example 3.7. Then (f_{−1}, I) = (⟨S(f_{1}), ·⟩, κ(I)) is a modular pair in involution for (A, Φ) dual (in the sense of Proposition 3.3) to the modular pair (I, f_{1}) = (⟨·, I⟩, f_{1}) in involution for (H, Δ).

In Example 3.7 we saw that (I, ε) is a modular pair in involution for (U(𝔤), Δ). We can provide other types of modular pairs in involution in this case. Namely, let δ be a character of 𝔤, which means that δ : 𝔤 → ℂ is linear and δ([𝔤, 𝔤]) = 0. Such a character has a unique extension to a unital multiplicative functional on U(𝔤), which we also denote by δ. Since the twisted coinverse S_{δ} in this case is given by S_{δ}(X) = −X + δ(X), for X ∈ 𝔤, we see that (δ, ε) is a modular pair in involution for (U(𝔤), Δ) generalizing that of (I, ε). Let ℂ_{δ} denote the 𝔤-module ℂ with action induced by δ, and let H_{∗}(𝔤, ℂ_{δ}) denote the corresponding Lie algebra homology. By definition it is the homology of the Chevalley-Eilenberg complex

Λ^{0}(𝔤) ⟵^{d_{Lie}} Λ^{1}(𝔤) ⟵^{d_{Lie}} Λ^{2}(𝔤) ⟵^{d_{Lie}} Λ^{3}(𝔤) ⟵^{d_{Lie}} ⋯,

where Λ(𝔤) = ⊕_{i}Λ^{i}(𝔤) is the exterior algebra over 𝔤 and

d_{Lie}(X^{1} ∧ ... ∧ X^{n}) = ∑_{i=1}^{n}(−1)^{i+1}δ(X^{i})X^{1} ∧ ⋯ ∧ X̂^{i} ∧ ⋯ ∧ X^{n}

+ ∑_{i<j}(−1)^{i+j}[X^{i}, X^{j}] ∧ X^{1} ∧ ⋯ ∧ X̂^{i} ∧ ⋯ ∧ X̂^{j} ∧ ⋯ ∧ X^{n},

for X^{k} ∈ 𝔤. This complex can be regarded as a mixed complex (Λ^{n}(𝔤), b, B) with b = 0 and B = d_{Lie}.

It can be shown that the linear map A : Λ^{n}(𝔤) → U(𝔤)^{⊗n} given by

A(X^{1} ∧ ... ∧ X^{n}) = \frac{1}{n!} ∑_{σ} sign(σ)X^{σ(1)} ⊗ ⋯ ⊗ X^{σ(n)},

for X^{k} ∈ 𝔤, is a quasi-isomorphism from the mixed complex (Λ^{n}(𝔤), b, B) to the mixed complex associated to the Hopf algebra (U(𝔤), Δ) with the modular pair (δ, ε) in involution. Hence we get the following result. Proposition 3.10. ([4][5]) Let 𝔤 be a Lie algebra with character δ. Then

HP_{(δ,ε)}^{∗}(U(𝔤)) ≅ ⨁_{i=∗ mod 2} H_{i}(𝔤, ℂ_{δ})

and

HH_{(δ,ε)}^{n}(U(𝔤)) ≅ Λ^{n}(𝔤),

for all n ∈ ℕ_{0}.

The Hopf algebra (U(𝔤), Δ) is cocommutative. It is desirable to compute the Hopf cyclic cohomology given by a Hopf algebra of the type considered in Example 3.7 for a genuine quantum group.

Example 3.11. In this example we consider the quantum SU_{q}(2), where the parameter q ∈ ⟨0, 1⟩. The Hopf ∗-algebra associated to SU_{q}(2) is denoted by (A_{q}, Φ). The fundamental unitary corepresentation U of (A_{q}, Φ) is given by

U = (U_{ij}) = ( \begin{matrix}α & −qγ^{∗} \\ γ & α^{∗} \end{matrix} ) ,

where α and γ are the well known generators of the unital ∗-algebra A_{q} introduced by Woronowicz [14]. The multiplicative functional f_{1} is uniquely determined by

f_{1}(α) = q^{−1} , f_{1}(α^{∗}) = q , f_{1}(γ) = f_{1}(γ^{∗}) = 0 , f_{1}(I) = 1.

Consider the universal algebra U_{q}(sl_{2}) with unit ε and generators e, f, k, k^{−1} satisfying the relations:

kk^{−1} = k^{−1}k = ε , ke = qek , kf = q^{−1}fk , ef − fe = \frac{1}{q − q^{−1}} (k^{2} − k^{−2}).

It is a Hopf ∗-algebra with comultiplication Δ : U_{q}(sl_{2}) → U_{q}(sl_{2}) ⊗ U_{q}(sl_{2}) and ∗-operation uniquely determined by

Δ(k) = k ⊗ k , Δ(e) = e ⊗ k^{−1} + k ⊗ e , Δ(f) = f ⊗ k^{−1} + k ⊗ f

and

k^{∗} = k , (k^{−1})^{∗} = k^{−1} , e^{∗} = f , f^{∗} = e.

We shall regard the Hopf ∗-algebra (U_{q}(sl_{2}), Φ) as a Hopf ∗-subalgebra of (A_{q}^{o}, Δ) such that

k(U_{11}) = q^{1/2} , k(U_{22}) = q^{−1/2} , k^{−1}(U_{11}) = q^{−1/2} , k^{−1}(U_{22}) = q^{1/2} ,

k(U_{ij}) = 0 , for i ≠ j , e(U_{ij}) = δ_{i1}δ_{j2} , f(U_{ij}) = δ_{i2}δ_{j1} .

Under this identification we have k^{−2} = f_{1}.

Recall that the coboundary operator b_{n} : (A_{q}^{o})^{⊗(n−1)} → (A_{q}^{o})^{⊗n} is given by b_{n} = ∑_{i=0}^{n}(−1)^{i}δ_{i}^{n}, for n ∈ ℕ_{0}. Thus b_{1}(1) = ε − f_{1} ≠ 0, so HH_{(I,f_{1})}^{0}(A_{q}^{o}) = Ker(b_{1}) ≅ 0 and therefore also HH_{(I,f_{1})}^{0}(U_{q}(sl_{2})) ≅ 0. Next, we get b_{2}(x) = ε ⊗ x − Δ(x) + x ⊗ f_{1}, for all x ∈ A_{q}^{o}, so

Ker(b_{2}) = {x ∈ A_{q}^{o} | Δ(x) = x ⊗ f_{1} + ε ⊗ x}.

Clearly, the elements {f_{1} −ε, ek^{−1}, fk^{−1}} are linearly independent and belong to Ker(b_{2}). It is not hard to show, for instance by using methods from differential calculi for quantum groups [12], that these elements generate the vector space Ker(b_{2}). Since f_{1} − ε is a generator for Im(b_{1}), we thus see that the two elements {[ek^{−1}], [fk^{−1}]} are generators for HH_{(I,f_{1})}^{1}(A_{q}^{o}).

To compute HC_{(I,f_{1})}^{1}(A_{q}^{o}), first note that λ_{1} : A_{q}^{o} → A_{q}^{o} is given by λ_{1}(x) = −τ_{1}(x) = −S_{I}(x)f_{1} = −S(x)f_{1}, for all x ∈ A_{q}^{o}. Thus for any x ∈ Ker(b_{2}), we have λ_{1}(x) = x if and only if x(I) = 0. Since (ek^{−1})(I) = (fk^{−1})(I) = 0, we therefore see that the two elements {[ek^{−1}⊕0], [fk^{−1}⊕0]} are generators in the total complex for HC_{(I,f_{1})}^{1}(A_{q}^{o}) (with ek^{−1}⊕0 and fk^{−1} ⊕ 0 in the total complex of the bicomplex defining cyclic cohomology).

Since these generators also belong to U_{q}(sl_{2}), the same is true for the groups HH_{(I,f_{1})}^{1}(U_{q}(sl_{2})) and HC_{(I,f_{1})}^{1}(U_{q}(sl_{2})). We have shown part of the following result due to M. Crainic [6]. The computation of the full Hopf Hochschild cohomology uses essentially Proposition 2.2 and Proposition 2.1. The result for Hopf cyclic cohomology then follows from the long exact IBS-sequence, whereas the Hopf periodic cyclic cohomology is obtained as the limit of Hopf cyclic cohomology. Proposition 3.12. ([6]) Consider the Hopf algebra (U_{q}(sl_{2}), Δ) with the modular pair (I, f_{1}) in involution. Then:

1. HH_{(I,f_{1})}^{1}(U_{q}(sl_{2})) ≅ ℂ^{2} with generators {[ek^{−1}], [fk^{−1}]}.

2. HH_{(I,f_{1})}^{n}(U_{q}(sl_{2})) ≅ 0, for n ≠ 1.

3. HC_{(I,f_{1})}^{2n}(U_{q}(sl_{2})) ≅ 0, for n ∈ ℕ_{0}.

4. HC_{(I,f_{1})}^{2n+1}(U_{q}(sl_{2})) ≅ ℂ^{2} with generators {S^{n}[ek^{−1} ⊕ 0], S^{n}[fk^{−1} ⊕ 0]}, for n ∈ ℕ_{0}.

5. HP_{(I,f_{1})}^{0}(U_{q}(sl_{2})) ≅ 0.

6. HP_{(I,f_{1})}^{1}(U_{q}(sl_{2})) ≅ ℂ^{2}.

Remark 3.13. This should be compared with the classical case (q → 1), where by Proposition 3.10 we have:

1. HH_{(I,ε)}^{3}(U(sl_{2})) ≅ HH_{(I,ε)}^{0}(U(sl_{2})) ≅ Λ^{0}(sl_{2}) ≅ ℂ.

2. HH_{(I,ε)}^{2}(U(sl_{2})) ≅ HH_{(I,ε)}^{1}(U(sl_{2})) ≅ Λ^{1}(sl_{2}) ≅ ℂ^{3}.

3. HH_{(I,ε)}^{n}(U(sl_{2})) ≅ 0, for n ≥ 4.

Example 3.14. Here we combine the results in Example 3.4 with the general discussion preceding it, to construct Hopf ∗-algebras with modular pairs in involution which will play an important role in the next section. Let (H, Δ) be a Hopf subalgebra of (A^{o}, Δ) such that f_{1} ∈ H. Let H̃ = H ⊗ H_{op} and Δ̃ = Δ × Δ. Then (H̃, Δ̃) is a Hopf algebra with a modular pair (δ, σ) = (I ⊗ I, f_{1} ⊗ f_{1}) in involution. Note that its coinverse is S ⊗ S^{−1}, its counit is I ⊗ I and its unit is ε ⊗ ε, where ε is the counit of (A, Φ).

To compute the cohomologies of such tensor products, we state a useful corollary of Theorem 3.1 and Theorem 3.2. Corollary 3.15. ([12]) Let (H, Δ) be a Hopf subalgebra of (A^{o}, Δ) such that f_{1} ∈ H. Consider the Hopf algebra (H̃, Δ̃) with a modular pair (δ, σ) = (I⊗I, f_{1}⊗f_{1}) in involution. Then

HH_{(δ,σ)}^{n}(H̃) ≅ ⨁_{i+j=n} HH_{(I,f_{1})}^{i}(H) ⊗ HH_{(I,f_{1})}^{j}(H),

for n ∈ ℕ_{0}. Moreover, the isomorphism is implemented by the shuffle map sh (going from left to right) with the Alexander-Whitney map AW as its inverse. Concerning Hopf cyclic cohomology, there exists a canonical long exact sequence

⋯ ⟶^{i} HC_{(δ,σ)}^{n−1}(H̃) ⟶^{∂} ⨁_{p+q=n−2} HC_{(I,f_{1})}^{p}(H) ⊗ HC_{(I,f_{1})}^{q}(H) ⟶^{S⊗1−1⊗S}

⨁_{r+s=n} HC_{(I,f_{1})}^{r}(H) ⊗ HC_{(I,f_{1})}^{s}(H) ⟶^{i} HC_{(δ,σ)}^{n}(H̃) ⟶^{∂} ⋯

Combining Theorem 3.12 and Corollary 3.15 we immediately get the following result. Theorem 3.16. ([12]) Let (H_{q}, Δ) = (U_{q}(sl_{2}), Δ), so (H̃_{q}, Δ̃) is a Hopf ∗-algebra with a modular pair (δ, σ) = (I ⊗ I, f_{1} ⊗ f_{1}) in involution. Then:

1. HH_{(δ,σ)}^{2}(H̃_{q}) ≅ ℂ^{4}.

2. HH_{(δ,σ)}^{n}(H̃_{q}) ≅ 0, for n ≠ 2.

3. HC_{(δ,σ)}^{0}(H̃_{q}) ≅ 0.

4. HC_{(δ,σ)}^{2n}(H̃_{q}) ≅ ℂ^{4}, for n ∈ ℕ.

5. HC_{(δ,σ)}^{2n+1}(H̃_{q}) ≅ 0, for n ∈ ℕ_{0}.

6. HP_{(δ,σ)}^{0}(H̃_{q}) ≅ ℂ^{4}.

7. HP_{(δ,σ)}^{1}(H̃_{q}) ≅ 0.

To find representatives for the generators of HH_{(δ,σ)}^{2}(H̃_{q}) we use the Alexander-Whitney map

AW_{i,j} : HH_{(I,f_{1})}^{i}(U_{q}(sl_{2})) ⊗ HH_{(I,f_{1})}^{j}(U_{q}(sl_{2})) ⟶ HH_{(δ,σ)}^{n}(H̃_{q}),

where i + j = n ∈ ℕ_{0}. We can thus exhibit the following generators (see Section 6) of HH_{(δ,σ)}^{2}(H̃_{q}) in (H̃_{q}^{⊗n}, b):

[ek^{−1} ⊗ ε ⊗ f_{1} ⊗ ek^{−1}], [ek^{−1} ⊗ ε ⊗ f_{1} ⊗ fk^{−1}],

[fk^{−1} ⊗ ε ⊗ f_{1} ⊗ ek^{−1}], [fk^{−1} ⊗ ε ⊗ f_{1} ⊗ fk^{−1}].

Proposition 3.17. ([12]) Let n ∈ ℕ. The following four elements

1. S^{n−1}[(ek^{−1} ⊗ ε ⊗ f_{1} ⊗ ek^{−1}) ⊕ (−ek^{−1} ⊗ ek^{−1}) ⊕ 0] ,

2. S^{n−1}[(ek^{−1} ⊗ ε ⊗ f_{1} ⊗ fk^{−1}) ⊕ (−ek^{−1} ⊗ fk^{−1}) ⊕ 0] ,

3. S^{n−1}[(fk^{−1} ⊗ ε ⊗ f_{1} ⊗ ek^{−1}) ⊕ (−fk^{−1} ⊗ ek^{−1}) ⊕ 0] ,

4. S^{n−1}[(fk^{−1} ⊗ ε ⊗ f_{1} ⊗ fk^{−1}) ⊕ (−fk^{−1} ⊗ fk^{−1}) ⊕ 0] ,

are generators of HC_{(δ,σ)}^{2n}(H̃_{q}).

4 The Modular Square for Compact Quantum Groups

Let (H, Δ) be a Hopf algebra with a modular pair (δ, σ) in involution and let A be an arbitrary algebra. We use the Sweedler notation, so Δ(x) = ∑ x_{(1)} ⊗ x_{(2)}, for x ∈ H. Recall that by an H-action of (H, Δ) on A we mean a linear map ▷ : H ⊗ A → A such that y ▷ (x ▷ a) = (yx) ▷ a and I ▷ a = a and

x ▷ (ab) = ∑(x_{(1)} ▷ a)(x_{(2)} ▷ b),

for a, b ∈ A and x, y ∈ H (where we have used the convention ▷(x ⊗ a) = x ▷ a).

Now suppose ▷ : H ⊗ A → A is an H-action of (H, Δ) on A. A linear functional τ on A is said to be a σ-trace under ▷ if τ(ab) = τ(b(σ ▷ a)) holds for all a, b ∈ A. A linear functional τ on A is said to be δ-invariant under ▷ if τ((x ▷ a)b) = τ(a(S_{δ}(x) ▷ b)) holds for all a, b ∈ A and x ∈ H.

The proof of the following lemma is quite similar to that of Theorem 8.2. Lemma 4.1. ([5]) Suppose τ is a δ-invariant σ-trace for an H-action ▷ : H ⊗ A → A of a Hopf algebra (H, Δ) with a modular pair (δ, σ) in involution. Consider the ordinary cocyclic object (C^{n}, d_{i}^{n}, s_{i}^{n}, t_{n}) associated to the algebra A, and the cocyclic object (H^{⊗n}, δ_{i}^{n}, σ_{i}^{n}, τ_{n}) associated to the Hopf algebra (H, Δ) with the modular pair (δ, σ) in involution. Let n ∈ ℕ. Define a linear map

γ_{n} : H^{⊗n} → C^{n} ; x^{1} ⊗ ⋯ ⊗ x^{n} ↦ γ_{n}(x^{1} ⊗ ⋯ ⊗ x^{n})

by

γ_{n}(x^{1} ⊗ ⋯ ⊗ x^{n})(a^{0}, …, a^{n}) = τ(a^{0}(x^{1} ▷ a^{1}) ⋯ (x^{n} ▷ a^{n})),

for a^{0}, …, a^{n} ∈ A and x^{1}, …, x^{n} ∈ H. Define γ_{0} : H^{⊗0} = ℂ → C^{0} = A′ by γ_{0}(1) = τ, for 1 ∈ ℂ. Then

γ_{n}δ_{i}^{n} = d_{i}^{n}γ_{n−1}, γ_{n}σ_{i}^{n} = s_{i}^{n}γ_{n+1}, γ_{n}τ_{n} = t_{n}γ_{n},

for any n ∈ ℕ_{0} and i ∈ {0, …, n}.

The significance of Lemma 4.1 is that γ_{n} induces a homomorphism from HH_{(δ,σ)}^{n}(H) to HH^{n}(A) and from HC_{(δ,σ)}^{n}(H) to HC^{n}(A) and from HP_{(δ,σ)}^{n}(H) to HP^{n}(A), for n ∈ ℕ_{0}. We denote also these maps by γ_{n}.

In terms of Λ-modules, the lemma says that the linear map

γ : ⨁_{n∈ℕ_{0}} H^{⊗n} ⟶ ⨁_{n∈ℕ_{0}} C^{n}

defined by γ = ⊕_{n∈ℕ_{0}}γ_{n}, is a Λ-module morphism from H_{(δ,σ)}^{♮} to C^{♮}.

Let (K, Δ) be any Hopf subalgebra of (H, Δ) such that σ ∈ K, and let γ′ : K_{(δ,σ)}^{♮} → C^{♮} denote the Λ-module morphism given by Lemma 4.1. Clearly, the inclusion j : K → H induces an inclusion j of associated Λ-modules such that γ′ = γj, and thus also a homomorphism

j : HH_{(δ,σ)}^{∗}(K) → HH_{(δ,σ)}^{∗}(H)

on the level of Hopf Hochschild cohomology such that γ′ = γj. The same is true on the level of Hopf cyclic cohomology and Hopf periodic cyclic cohomology.

Consider now a Hopf ∗-subalgebra (H, Δ) of the maximal Hopf ∗-algebra (A^{o}, Δ) dual to the Hopf ∗-algebra (A, Φ) associated to a compact quantum group with Haar state h. Assume that f_{1} ∈ H. Define x ∗ a and a ∗ x in A, for x ∈ A^{o} and a ∈ A, by x ∗ a = (ι ⊗ x)Φ(a) and a ∗ x = (x ⊗ ι)Φ(a). Then x ⊗ a ↦ x ∗ a, for a ∈ A and x ∈ H, is an H-action of (H, Δ) on A. Similarly, the map x ⊗ a ↦ a ∗ x, for a ∈ A and x ∈ H_{op}, is an H_{op}-action of (H_{op}, Δ) on A. Now consider the Hopf algebra (H̃, Δ̃) from Example 3.14. Define a linear map ▷ : H̃ ⊗ A → A by (x ⊗ y) ▷ a = x ∗ a ∗ y, for a ∈ A, x ∈ H and y ∈ H_{op}. It is easily seen to be an H̃-action of (H̃, Δ̃) on A. Theorem 4.2. ([5][12]) Let notation be as in the preceeding paragraph. Then the Haar state h of (A, Φ) is a δ-invariant σ-trace for the H̃-action ▷ on A of the Hopf algebra (H̃, Δ̃) with modular pair (δ, σ) = (I ⊗ I, f_{1} ⊗ f_{1}) in involution.

A detailed proof of this theorem will be given in a later section, in the more general setting of algebraic quantum groups. The data used in the statement of Theorem 4.2 is referred to as the modular square associated to the Hopf algebra (H, Δ). By Lemma 4.1, it immediately yields the following main result. Theorem 4.3. ([5][12]) Consider a Hopf ∗-subalgebra (H, Δ) of the maximal Hopf ∗- algebra (A^{o}, Δ) dual to the Hopf ∗-algebra (A, Φ) associated to a compact quantum group with Haar state h. Assume that f_{1} ∈ H. Let H̃_{(δ,σ)}^{♮} denote the Λ-module canonically associated to the modular square of (H, Δ). Then the formulas γ_{0}(1) = h and

γ_{n}((x^{1} ⊗ y^{1}) ⊗ ⋯ ⊗ (x^{n} ⊗ y^{n}))(a^{0}, …, a^{n}) = h(a^{0}(x^{1} ∗ a^{1} ∗ y^{1}) ⋯ (x^{n} ∗ a^{n} ∗ y^{n})),

for a^{0}, …, a^{n} ∈ A and x^{1}, y^{1}, …, x^{n}, y^{n} ∈ H and n ∈ ℕ, yield a Λ-module morphism γ : H̃_{(δ,σ)}^{♮} → C^{♮}, which in turn induces characteristic homomorphisms in cyclic cohomology

γ_{∗} : HC_{(δ,σ)}^{∗}(H̃) → HC^{∗}(A),

where (δ, σ) = (I⊗I, f_{1}⊗f_{1}) is a modular pair in involution for (H̃, Δ̃) = (H⊗H_{op}, Δ×Δ). Characteristic homomorphisms are similarly induced on the level of Hochschild cohomology and periodic cyclic cohomology.

The map γ from Theorem 4.3 is called the modular characteristic homomorphism and in general it is neither injective nor surjective. In fact, in the case of SU_{q}(2), we shall presently see that it is zero.

5 The Modular Characteristic Homomorphism for SU_{q}(2)

First recall [20] that HH^{n}(A_{q}) is infinite dimensional for n ∈ {0, 1}, and zero otherwise. Since

γ_{∗} : HH_{(δ,σ)}^{∗}(H̃_{q}) → HH^{∗}(A_{q}),

it is by Theorem 3.16 already evident that γ_{∗} is zero on the level of Hochschild cohomology.

Next recall [20] the following result for ordinary cyclic cohomology. Proposition 5.1. Consider the algebra A_{q} of regular functions associated to SU_{q}(2). Then:

1. HC^{0}(A_{q}) ≅ ℂ[τ_{even}] ⊕ KerS, where

KerS ≅ (⊕_{l>0}ℂ[τ_{x}^{l}]) ⊕ (⊕_{l>0}ℂ[τ_{y}^{l}]) ⊕ (⊕_{l>0}ℂ[τ_{u}^{l}]) ⊕ (⊕_{l>0}ℂ[τ_{v}^{l}]).

2. HC^{2n+1}(A_{q}) ≅ ℂS^{n}[τ_{odd}], for n ∈ ℕ_{0}.

3. HC^{2n}(A_{q}) ≅ ℂS^{n}[τ_{even}], for n ∈ ℕ.

In [20] the generators τ_{x}^{l}, τ_{y}^{l}, τ_{u}^{l}, τ_{v}^{l}, τ_{even} and τ_{odd} are written down explicitly in terms of the Poincarè-Birkhoff-Witt-type linear basis

{α^{r}γ^{m}(γ^{∗})^{n} | m, n, r ∈ ℕ_{0} } ∪ {(α^{∗})^{s}γ^{m}(γ^{∗})^{n} | m, n ∈ ℕ_{0} , s ∈ ℕ }

for A_{q}. In that paper α = y, γ = u, α^{∗} = x, γ^{∗} = −q^{−1}v and q = μ. Also recall that τ_{even} is the Chern character of the 1-summable even Fredholm module which generates the K-homology group K^{0}(A_{q}) ≅ ℤ, where A_{q} is the C^{∗}- algebra envelope of A_{q}. And τ_{odd} is the Chern character of the 1-summable odd Fredholm module which generates the K-homology group K^{1}(A_{q}) ≅ ℤ. It would be desirable to hit at least these two generators with the modular characteristic homomorphism. A direct check could in principle be carried out using the explicit expressions for the generators involved together with a formula for the Haar functional in terms of the Poincarè-Birkhoff-Witt-type basis. However, this can be avoided by introducing the following element.

There exist a functional H ∈ A_{q}^{o} such that

Δ(H) = H ⊗ ε + ε ⊗ H,

which is uniquely determined by H(U_{ij}) = 0, for i ≠ j, and by H(U_{11}) = −1, H(U_{22}) = 1 and H(I) = 0. It is easily checked that e^{hH} = k^{2}, where h = −ln(q) > 0, so H ∉ U_{q}(sl_{2}). Thus we have the following familiar identities

Hk = kH , Hk^{−1} = k^{−1}H , He − eH = −2e , Hf − fH = 2f.

Consider the cocyclic object ((A_{q}^{o})^{⊗n}, δ_{i}^{n}, σ_{i}^{n}, τ_{n}) associated to the Hopf ∗-algebra (A_{q}^{o}, Δ) with the modular pair (I, f_{1}) in involution. Recall that

B_{1} = N^{1}σ_{1}^{1}τ_{2}(1_{2} − λ_{2}) = N^{1}(σ_{1}^{1}τ_{2} − τ_{1}σ_{0}^{1}),

so

B_{1}(x ⊗ y) = S(x)y − S(y)f_{1}x − x(I)S(y)f_{1} + x(I)y ,

for all x, y ∈ A_{q}^{o}. Recall also that the coboundary operator b_{3} : A_{q}^{o} ⊗ A_{q}^{o} → A_{q}^{o} ⊗ A_{q}^{o} ⊗ A_{q}^{o} is given by

b_{3}(x ⊗ y) = ε ⊗ x ⊗ y − Δ(x) ⊗ y + x ⊗ Δ(y) − x ⊗ y ⊗ f_{1},

for all x, y ∈ A_{q}^{o}. The following lemma follows by direct verification. Lemma 5.2. ([12]) Let notation be as in the above paragraph and consider B_{1} : A_{q}^{o}⊗A_{q}^{o} → A_{q}^{o}. Then:

1. b_{3}(H ⊗ ek^{−1}) = b_{3}(H ⊗ fk^{−1}) = 0.

2. B_{1}(H ⊗ ek^{−1}) = 2ek^{−1}.

3. B_{1}(H ⊗ fk^{−1}) = −2fk^{−1}.

This lemma tells us that the induced map

IB_{1} : HH_{(I,f_{1})}^{2}(A_{q}^{o}) → HH_{(I,f_{1})}^{1}(A_{q}^{o})

from the long exact IBS-sequence is surjective. Moreover, the equivalence classes {[H ⊗ ek^{−1}], [H ⊗ fk^{−1}]} are linearly independent in HH_{(I,f_{1})}^{2}(A_{q}^{o}), because B_{1}([H ⊗ ek^{−1}]) = 2[ek^{−1} ⊕ 0] and B_{1}([H ⊗ fk^{−1}]) = −2[fk^{−1} ⊕ 0].

A somewhat complicated diagram chase involving the homomorphism

j : HC_{(I,f_{1})}^{∗}(H̃_{q}) → HC_{(I,f_{1})}^{∗}(Ã_{q}^{o})

such that γj = γ′ (from the discussion following Lemma 4.1), Theorem 3.16 and Proposition 5.1, now yields the following surprising result. Theorem 5.3. ([12]) The modular characteristic homomorphisms in Hochschild cohomology, cyclic cohomology and periodic cyclic cohomology of the Hopf algebra (U_{q}(sl_{2}), Δ) with the modular pair (I, f_{1}) in involution, are all ZERO.

6 Additional Considerations

In this section we expand on results for SU_{q}(2) briefly presented in [12]. Due to the presence of the element H ∈ A_{q}^{o} it is evident that HH_{(I,f_{1})}^{∗}(A_{q}^{o}) ≅ H̸H_{(I,f_{1})}^{∗}(H_{q}), so in principle the modular characteristic homomorphism γ of the Hopf ∗-algebra (A_{q}^{o}, Δ) could be non-zero on the level of cyclic cohomology and periodic cyclic cohomology, although we have already seen that the restricted modular characteristic homomorphism γ′ of the Hopf ∗-algebra (U_{q}(sl_{2}), Δ) is zero. In this section we prove that provided a certain plausible assumption (to be explained below in Theorem 6.1) on A_{q}^{o} holds, which we unfortunately have not been able to check, then γ = 0 and therefore γ′ = γj = 0. By the same argument, the modular characteristic homomorphism of any other Hopf subalgebra of (A_{q}^{o}, Δ) containing f_{1} will then also be zero. Of course, γ = 0 in Hochschild cohomology, so we need only focus on the case of cyclic cohomology.

We need a theorem showing how HH_{(I,f_{1})}^{∗}(A^{o}) can be computed from a resolution of A. Throughout this discussion (A, Φ) denotes a compact quantum group with maximal dual Hopf algebra (A^{o}, Δ) and modular element f_{1} ∈ A^{o}, and B denotes the algebra A ⊗ A_{op}.

Proposition 2.1 tells us that HH_{(I,f_{1})}^{∗}(A^{o}) can be computed from any injective resolution of A^{o} as an A^{o}-bicomodule. Using this we shall give a formula for computing HH_{(I,f_{1})}^{∗}(A^{o}) from a given finitely generated free resolution

⋯ ⟶ M_{2} ⟶^{d_{2}} M_{1} ⟶^{d_{1}} M_{0} ⟶^{d_{0}} A ⟶ 0

of A as a left B-module (or as an A-bimodule). We assume that M_{0} = B and that d_{0} : M_{0} → A is the algebra multiplication.

Suppose M_{nL} is a finite dimensional vector space, that M_{n} ≅ B ⊗ M_{nL} as left Bmodules, so that d_{n} : M_{n} → M_{n−1} needs only be specified on elements of the type 1_{B} ⊗z, where 1_{B} = I ⊗I is the unit of B and z ∈ M_{nL}. Let {e_{k}^{n}} be a linear basis of M_{nL}, and let {e_{k}^{n∗}} be a basis for the dual vector space M_{nL}^{o} of M_{nL} such that e_{k}^{n∗}(e_{l}^{n}) = δ_{kl}, for all k, l. We shall regard A^{o} ⊗A^{o} as an A-bimodule with respect to the action a(ξ ⊗η)b = aξ ⊗ηb, where ξ, η ∈ A^{o} and a, b ∈ A and where by aξ, ηb ∈ A^{o} we mean aξ(c) = ξ(ca) and ηb(c) = η(bc) for c ∈ A. Theorem 6.1. ([12]) Let notation be as in the preceeding paragraph. Assume that (M_{∗}, d) is a finitely generated free resolution of A as a left B-module. Suppose the A-bimodule A^{o} ⊗ A^{o} defined above is injective. Then the linear map δ_{n} : M_{(n−1)L}^{o} → M_{nL}^{o} given by

δ_{n}(y) = ∑_{k}(ε ⊗ f_{1} ⊗ y)d_{n}(1_{B} ⊗ e_{k}^{n}) e_{k}^{n∗},

for all y ∈ M_{(n−1)L}^{o} and n ∈ ℕ, defines a complex (M_{∗L}^{o}, δ) such that its cohomology coincides with HH_{(I,f_{1})}^{∗}(A^{o}). Hence

dim(HH_{(I,f_{1})}^{n}(A^{o})) = dim(M_{nL}^{o}) − dim(Imδ_{n}) − dim(Imδ_{n+1}),

for all n ∈ ℕ_{0}. Remark 6.2. Since A^{o} ⊗ A^{o} ≅ (A ⊗ A)^{o} where the latter is defined with respect to (A⊗A, Φ×Φ), our assumption reduces to requiring that the left A-module A^{o} is injective. Also A^{o} is a left A-submodule of the injective left A-module A′. However, injectivity is unfortunately not necessarily inherited to submodules.

Let us now return to the case SU_{q}(2). In the rest of this section we assume that the A_{q}-bimodule A_{q}^{o} ⊗ A_{q}^{o} defined above is injective. In [20] an explicit finitely generated free resolution

⋯ ⟶ M_{2} ⟶^{d_{2}} M_{1} ⟶^{d_{1}} M_{0} ⟶^{m} A_{q} ⟶ 0

of A_{q} as a left B-module (here M_{0} = B and m is the multiplication) was constructed by hand, to compute HH^{∗}(A_{q}). We will now use this resolution together with Theorem 6.1 to write down the complex (M_{∗L}^{o}, δ), which we will then use to compute HH_{(I,f_{1})}^{∗}(A_{q}^{o}).

Below we follow closely the notation and conventions in [20] which was used for specifying the left B-modules M_{n} and the differentials d_{n} : M_{n} → M_{n−1}, for n ∈ ℕ. Define the vector spaces M_{nL} as the linear spans of the B-bases {e_{k}^{n}} for M_{n}, given by e_{0}^{0} = I and

e_{1}^{1} = e_{x}, e_{2}^{1} = e_{y}, e_{3}^{1} = e_{u}, e_{4}^{1} = e_{v}

and

e_{1}^{2} = ϑ_{T}^{(1)},  e_{2}^{2} = ϑ_{S}^{(1)},  e_{3}^{2} = e_{u} ∧ e_{x},  e_{4}^{2} = e_{u} ∧ e_{y}
e_{5}^{2} = e_{v} ∧ e_{x},  e_{6}^{2} = e_{v} ∧ e_{y},  e_{7}^{2} = e_{v} ∧ e_{u}

and

e_{1}^{3} = e_{x} ∧ ϑ_{S}^{(1)},  e_{2}^{3} = e_{y} ∧ ϑ_{T}^{(1)},  e_{3}^{3} = e_{u} ∧ ϑ_{T}^{(1)},  e_{4}^{3} = e_{u} ∧ ϑ_{S}^{(1)},
e_{5}^{3} = e_{v} ∧ ϑ_{T}^{(1)},  e_{6}^{3} = e_{v} ∧ ϑ_{S}^{(1)},  e_{7}^{3} = e_{v} ∧ e_{u} ∧ e_{x},  e_{8}^{3} = e_{v} ∧ e_{u} ∧ e_{y},

whereas {e_{k}^{2p+4}} is listed as follows

e_{1}^{2p+4} = ϑ_{T}^{(p+2)},  e_{2}^{2p+4} = ϑ_{S}^{(p+2)},  e_{3}^{2p+4} = e_{u} ∧ e_{x} ∧ ϑ_{S}^{(p+1)},
e_{4}^{2p+4} = e_{u} ∧ e_{y} ∧ ϑ_{T}^{(p+1)}, e_{5}^{2p+4} = e_{v} ∧ e_{x} ∧ ϑ_{S}^{(p+1)}, e_{6}^{2p+4} = e_{v} ∧ e_{y} ∧ ϑ_{T}^{(p+1)},
e_{7}^{2p+4} = e_{v} ∧ e_{u} ∧ ϑ_{T}^{(p+1)}, e_{8}^{2p+4} = e_{v} ∧ e_{u} ∧ ϑ_{S}^{(p+1)},

for p ≥ 0, and finally {e_{k}^{2p+3}} is listed as follows

e_{1}^{2p+3} = e_{x} ∧ ϑ_{S}^{(p+1)},  e_{2}^{2p+3} = e_{y} ∧ ϑ_{T}^{(p+1)},  e_{3}^{2p+3} = e_{u} ∧ ϑ_{T}^{(p+1)},
e_{4}^{2p+3} = e_{u} ∧ ϑ_{S}^{(p+1)},  e_{5}^{2p+3} = e_{v} ∧ ϑ_{T}^{(p+1)},  e_{6}^{2p+3} = e_{v} ∧ ϑ_{S}^{(p+1)},
e_{7}^{2p+3} = e_{v} ∧ e_{u} ∧ e_{x} ∧ ϑ_{S}^{(p)}, e_{8}^{2p+3} = e_{v} ∧ e_{u} ∧ e_{y} ∧ ϑ_{T}^{(p)},

for p > 0.

Denote the elements e_{k}^{n∗} of the dual basis {e_{k}^{n∗}} ⊂ M_{nL}^{o} of {e_{k}^{n}} by v_{k}^{n}. Using the formula

δ_{n}(v_{i}^{n−1}) = ∑_{k}(ε ⊗ f_{1} ⊗ v_{i}^{n−1})d_{n}(1_{B} ⊗ e_{k}^{n})v_{k}^{n},

for n ∈ ℕ from Theorem 6.1 for the maps δ_{n} : M_{(n−1)L}^{o} → M_{nL}^{o}, and the formulas for d_{n} : M_{n} → M_{n−1} given on pp. 160-163 in [20], we can then obtain the explicit expressions for δ_{n} written below.

To show how they come about, consider for example the linear map δ_{1} : M_{0L}^{o} → M_{1L}^{o}, which can be calculated as follows

δ_{1}(v_{0}^{0}) = ∑_{k=1}^{4}(ε ⊗ f_{1} ⊗ v_{0}^{0})d_{1}(1_{B} ⊗ e_{k}^{1})v_{k}^{1}
= (ε ⊗ f_{1})d_{1}(1_{B} ⊗ e_{x})v_{1}^{1} + (ε ⊗ f_{1})d_{1}(1_{B} ⊗ e_{y})v_{2}^{1}
+(ε ⊗ f_{1})d_{1}(1_{B} ⊗ e_{u})v_{3}^{1} + (ε ⊗ f_{1})d_{1}(1_{B} ⊗ e_{v})v_{4}^{1}
= (ε ⊗ f_{1})(x ⊗ I − I ⊗ x)v_{1}^{1} + (ε ⊗ f_{1})(y ⊗ I − I ⊗ y)v_{2}^{1}
+ (ε ⊗ f_{1})(u ⊗ I − I ⊗ u)v_{3}^{1} + (ε ⊗ f_{1})(v ⊗ I − I ⊗ v)v_{4}^{1}
= (1 − q)v_{1}^{1} + (1 − q^{−1})v_{2}^{1}.

Similarly, consider δ_{2} : M_{1L}^{o} → M_{2L}^{o}, which can be calculated, for any number i ∈ {1, …, 4}, as follows

δ_{2}(v_{i}^{1}) = ∑_{k=1}^{7}(ε ⊗ f_{1} ⊗ v_{i}^{1})d_{2}(1_{B} ⊗ e_{k}^{2})v_{k}^{2}
= (ε ⊗ f_{1} ⊗ v_{i}^{1})(I ⊗ y ⊗ e_{1}^{1} + x ⊗ I ⊗ e_{2}^{1} − qu ⊗ I ⊗ e_{4}^{1} − I ⊗ q^{−1}v ⊗ e_{3}^{1})v_{1}^{2}

+(ε ⊗ f_{1} ⊗ v_{i}^{1})(y ⊗ I ⊗ e_{1}^{1} + I ⊗ x ⊗ e_{2}^{1} − qu ⊗ I ⊗ e_{4}^{1} − I ⊗ qv ⊗ e_{3}^{1})v_{2}^{2} +(ε ⊗ f_{1} ⊗ v_{i}^{1})((u ⊗ I − I ⊗ qu) ⊗ e_{1}^{1} + (qx ⊗ I − I ⊗ x) ⊗ e_{3}^{1}))v_{3}^{2} +(ε ⊗ f_{1} ⊗ v_{i}^{1})((qu ⊗ I − I ⊗ u) ⊗ e_{2}^{1} + (y ⊗ I − I ⊗ qy) ⊗ e_{3}^{1}))v_{4}^{2} +(ε ⊗ f_{1} ⊗ v_{i}^{1})((v ⊗ I − I ⊗ qv) ⊗ e_{1}^{1} + (qx ⊗ I − I ⊗ x) ⊗ e_{4}^{1}))v_{5}^{2} +(ε ⊗ f_{1} ⊗ v_{i}^{1})((qv ⊗ I − I ⊗ v) ⊗ e_{2}^{1} + (y ⊗ I − I ⊗ qy) ⊗ e_{3}^{1}))v_{6}^{2} +(ε ⊗ f_{1} ⊗ v_{i}^{1})((v ⊗ I − I ⊗ v) ⊗ e_{3}^{1} + (u ⊗ I − I ⊗ u) ⊗ e_{4}^{1}))v_{7}^{2},

so for instance,

δ_{2}(v_{1}^{1}) = (ε ⊗ f_{1})(I ⊗ y)v_{1}^{2} + (ε ⊗ f_{1})(y ⊗ I)v_{2}^{2}
+(ε ⊗ f_{1})(u ⊗ I − I ⊗ qu)v_{3}^{2} + (ε ⊗ f_{1})(v ⊗ I − I ⊗ qv)v_{5}^{2}
= q^{−1}v_{1}^{2} + v_{2}^{2}.

We trust that the reader can now easily verify the following result:

δ_{1}(v_{0}^{0}) = (1 − q)v_{1}^{1} + (1 − q^{−1})v_{2}^{1}

and

δ_{2}(v_{1}^{1}) = q^{−1}v_{1}^{2} + v_{2}^{2},  δ_{2}(v_{2}^{1}) = v_{1}^{2} + qv_{2}^{2}
δ_{2}(v_{3}^{1}) = 0,  δ_{2}(v_{4}^{1}) = 0

and

δ_{3}(v_{1}^{2}) = −qv_{1}^{3} + v_{2}^{3},  δ_{3}(v_{2}^{2}) = v_{1}^{3} − q^{−1}v_{2}^{3}
δ_{3}(v_{3}^{2}) = −q^{−1}v_{3}^{3} − q^{−1}v_{4}^{3},  δ_{3}(v_{4}^{2}) = −v_{3}^{3} − v_{4}^{3}
δ_{3}(v_{5}^{2}) = −q^{−1}v_{5}^{3} − q^{−1}v_{6}^{3},  δ_{3}(v_{6}^{2}) = −v_{5}^{3} − v_{6}^{3}
δ_{3}(v_{7}^{2}) = (q^{2} − q)v_{7}^{3} + (1 − q)v_{8}^{3}

and

δ_{4}(v_{1}^{3}) = q^{−1}v_{1}^{4} + v_{2}^{4},  δ_{4}(v_{2}^{3}) = v_{1}^{4} + qv_{2}^{4}
δ_{4}(v_{3}^{3}) = qv_{3}^{4} − v_{4}^{4},  δ_{4}(v_{4}^{3}) = −qv_{3}^{4} + v_{4}^{4}
δ_{4}(v_{5}^{3}) = qv_{5}^{4} − v_{6}^{4},  δ_{4}(v_{6}^{3}) = −qv_{5}^{4} + v_{6}^{4}
δ_{4}(v_{7}^{3}) = q^{−1}v_{7}^{4} + q^{−2}v_{8}^{4},  δ_{4}(v_{8}^{3}) = v_{7}^{4} + q^{−1}v_{8}^{4}

and for p ∈ ℕ, we get

δ_{2p+3}(v_{1}^{2p+2}) = −qv_{1}^{2p+3} + v_{2}^{2p+3},  δ_{2p+3}(v_{2}^{2p+2}) = v_{1}^{2p+3} − q^{−1}v_{2}^{2p+3}
δ_{2p+3}(v_{3}^{2p+2}) = −q^{−1}v_{3}^{2p+3} − q^{−1}v_{4}^{2p+3}, δ_{2p+3}(v_{4}^{2p+2}) = −v_{3}^{2p+3} − v_{4}^{2p+3}
δ_{2p+3}(v_{5}^{2p+2}) = −q^{−1}v_{5}^{2p+3} − q^{−1}v_{6}^{2p+3}, δ_{2p+3}(v_{6}^{2p+2}) = −v_{5}^{2p+3} − v_{6}^{2p+3}
δ_{2p+3}(v_{7}^{2p+2}) = −qv_{7}^{2p+3} + v_{8}^{2p+3},  δ_{2p+3}(v_{8}^{2p+2}) = q^{2}v_{7}^{2p+3} − qv_{8}^{2p+3}

and finally

δ_{2p+4}(v_{1}^{2p+3}) = q^{−1}v_{1}^{2p+4} + v_{2}^{2p+4},  δ_{2p+4}(v_{2}^{2p+3}) = v_{1}^{2p+4} + qv_{2}^{2p+4}
δ_{2p+4}(v_{3}^{2p+3}) = qv_{3}^{2p+4} − v_{4}^{2p+4},  δ_{2p+4}(v_{4}^{2p+3}) = −qv_{3}^{2p+4} + v_{4}^{2p+4}
δ_{2p+4}(v_{5}^{2p+3}) = qv_{5}^{2p+4} − v_{6}^{2p+4},  δ_{2p+4}(v_{6}^{2p+3}) = −qv_{5}^{2p+4} + v_{6}^{2p+4}
δ_{2p+4}(v_{7}^{2p+3}) = q^{−1}v_{7}^{2p+4} + q^{−2}v_{8}^{2p+4}, δ_{2p+4}(v_{8}^{2p+3}) = v_{7}^{2p+4} + q^{−1}v_{8}^{2p+4}.

The result of this is the following result. By ‘our assumption’ we refer to the one in Theorem 6.1 Theorem 6.3. ([12]) Provided our assumption holds for SU_{q}(2), we get:

1. HH_{(I,f_{1})}^{1}(A_{q}^{o}) ≅ HH_{(I,f_{1})}^{2}(A_{q}^{o}) ≅ ℂ^{2}

2. HH_{(I,f_{1})}^{n}(A_{q}^{o}) ≅ 0, for all n ∉ {1, 2}.

Proof. By Theorem 6.1 we have

dim(HH_{(I,f_{1})}^{n}(A_{q}^{o})) = dim(M_{nL}^{o}) − dim(Imδ_{n}) − dim(Imδ_{n+1}),

for all n ∈ ℕ_{0}. Now the dimensions dim(M_{nL}^{o}) are

dim(M_{0L}^{o}) = 0, dim(M_{1L}^{o}) = 4, dim(M_{2L}^{o}) = 7, dim(M_{nL}^{o}) = 8,

for n ≥ 3. Whereas by looking at the formulas for δ_{n} listed above, we see that the dimensions dim(Imδ_{n}) are

dim(Imδ_{0}) = 0, dim(Imδ_{1}) = 1, dim(Imδ_{2}) = 1, dim(Imδ_{n}) = 4,

for n ≥ 3. Combining these facts yields the result for HH_{(I,f_{1})}^{∗}(A_{q}^{o}). ⊔ ⊓

Thus by Lemma 5.2 the generators for HH_{(I,f_{1})}^{1}(A_{q}^{o}) are {[ek^{−1}], [fk^{−1}]}, whereas those of HH_{(I,f_{1})}^{2}(A_{q}^{o}) are {[H ⊗ ek^{−1}], [H ⊗ fk^{−1}]}.

Remark 6.4. In the classical limit q → 1, we easily see that the numbers dim(Imδ_{n}) are

dim(Imδ_{0}) = 0, dim(Imδ_{1}) = 0, dim(Imδ_{2}) = 1,
dim(Imδ_{3}) = 3, dim(Imδ_{n}) = 4,

for n ≥ 4. Hence, when q → 1, we get the following results, which should be compared with the remark following Proposition 3.12:

1. HH_{(I,f_{1})}^{0}(A_{q}^{o}) ≅ HH_{(I,f_{1})}^{3}(A_{q}^{o}) ≅ ℂ

2. HH_{(I,f_{1})}^{1}(A_{q}^{o}) ≅ HH_{(I,f_{1})}^{2}(A_{q}^{o}) ≅ ℂ^{3}

3. HH_{(I,f_{1})}^{n}(A_{q}^{o}) ≅ 0, for all n ≥ 4.

Remark 6.5. Before we move on let us briefly explain how the oracle could possibly come up with the two generators {[H ⊗ ek^{−1}] , [H ⊗ fk^{−1}]} in Lemma 5.2. We know that both the bar resolution B(A_{q})_{∗} ⟶^{≃} A_{q} and resolution M_{∗} ⟶^{≃} A_{q} in [20] are free resolutions of A_{q}. We can then built up inductively a chain map h_{∗} from M_{∗} ⟶^{≃} A_{q} with boundary d to B(A_{q})_{∗} ⟶^{≃} A_{q} with boundary b, which by the fundamental theorem in Homological algebra automatically will be a quasi-isomorphism.

We start with h_{0} : M_{0} = A_{q} ⊗ A_{q} → A_{q} ⊗ A_{q} = B(A_{q})_{0}, which is the identity. The map h_{1} : M_{1} = A_{q} ⊗ M_{1L} ⊗ A_{q} → A_{q} ⊗ A_{q} ⊗ A_{q} has to satisfy b_{1}h_{1} = h_{0}d_{1} = d_{1}, and since b_{1}(a_{0} ⊗ a_{1} ⊗ a_{2}) = a_{0}a_{1} ⊗ a_{2} − a_{0} ⊗ a_{1}a_{2}, for a_{i} ∈ A_{q}, a possible choice for h_{1} sends (e_{1}^{1}, …, e_{4}^{1}) to

(I ⊗ x ⊗ I, I ⊗ y ⊗ I, I ⊗ u ⊗ I, I ⊗ v ⊗ I).

Similarly, the map h_{2} : M_{2} = A_{q} ⊗ M_{2L} ⊗ A_{q} → A_{q} ⊗ A_{q} ⊗ A_{q} ⊗ A_{q} has to satisfy b_{2}h_{2} = h_{1}d_{2}, and since

b_{2}(a_{0} ⊗ a_{1} ⊗ a_{2} ⊗ a_{3}) = a_{0}a_{1} ⊗ a_{2} ⊗ a_{3} − a_{0} ⊗ a_{1}a_{2} ⊗ a_{3} + a_{0} ⊗ a_{1} ⊗ a_{2}a_{3} ,

for a_{i} ∈ A_{q}, a possible choice for h_{2} is:

h_{2}(e_{1}^{2}) = I ⊗ x ⊗ y ⊗ I − I ⊗ q^{−1}u ⊗ v ⊗ I − I ⊗ I ⊗ I ⊗ I,
h_{2}(e_{2}^{2}) = I ⊗ y ⊗ x ⊗ I − I ⊗ qu ⊗ v ⊗ I − I ⊗ I ⊗ I ⊗ I,
h_{2}(e_{3}^{2}) = I ⊗ u ⊗ x ⊗ I − I ⊗ qx ⊗ u ⊗ I,
h_{2}(e_{4}^{2}) = I ⊗ qu ⊗ y ⊗ I − I ⊗ y ⊗ u ⊗ I,
h_{2}(e_{5}^{2}) = I ⊗ v ⊗ x ⊗ I − I ⊗ qx ⊗ v ⊗ I,
h_{2}(e_{6}^{2}) = I ⊗ qv ⊗ y ⊗ I − I ⊗ y ⊗ v ⊗ I,
h_{2}(e_{7}^{2}) = I ⊗ v ⊗ u ⊗ I − I ⊗ u ⊗ v ⊗ I,

where we have invoked the relations of the generators u, v, x, y ∈ A_{q}.

Tracing this chain map through the proof of Theorem 6.1, our assumption on SU_{q}(2) thus gives us a quasi-isomorphism h_{∗}^{o} from (M_{∗L}^{o}, δ) to the chain complex for (A_{q}^{o})_{(I,f_{1})}^{♮}. Now two convenient generators for HH_{(I,f_{1})}^{2}(A_{q}^{o}) ≅ ℂ^{2} in (M_{∗L}^{o}, δ) are {[qv_{3}^{2} − v_{4}^{2}], [qv_{5}^{2} − v_{6}^{2}]}. It is easy to see that h_{2}^{o}(qv_{3}^{2} − v_{4}^{2}) ∈ A_{q}^{o} ⊗ A_{q}^{o} is then a linear map such that

x ⊗ y − q^{−1}u ⊗ v − I ⊗ I ↦ 0,
y ⊗ x − qu ⊗ v − I ⊗ I ↦ 0,
u ⊗ x − qx ⊗ u ↦ q,
qu ⊗ y − y ⊗ u ↦ −1,
v ⊗ x − qx ⊗ v ↦ 0,
qv ⊗ y − y ⊗ v ↦ 0,
v ⊗ u − u ⊗ v ↦ 0.

Staring at these formulas suggests the (admittingly still ad hoc) candidate −q^{1/2}H ⊗ fk^{−1} as the obvious linear map satisfying these evaluation requirements. By its nature it belongs to Ker(b_{3}). Similar considerations connect the two elements qv_{5}^{2} − v_{6}^{2} and −q^{−1/2}H ⊗ ek^{−1}.

The method of inductively establishing a chain map also gives a hint to how the (seemingly ad hoc) free resolution of A_{q} in [20] was constructed.

Using Theorem 6.3, the IBS-sequence and Lemma 5.2, one can now check that HC_{(I,f_{1})}^{1}(A_{q}^{o}) ≅ ℂ^{2} is the only non-vanishing component in cyclic cohomology, with generators {[ek^{−1} ⊕ 0], [fk^{−1} ⊕ 0]}, so HP_{(I,f_{1})}^{∗}(A_{q}^{o}) ≅ 0. The non-zero B-operator IB_{1} : HH_{(I,f_{1})}^{2}(A_{q}^{o}) → HH_{(I,f_{1})}^{1}(A_{q}^{o}) kills the periodic cyclic cohomology classes for U_{q}(sl_{2}) found in Proposition 3.12.

Together with Corollary 3.15, we therefore see that HH_{(δ,σ)}^{n}(Ã_{q}^{o}) is ℂ^{4} for n ∈ {2, 4} and ℂ^{8} for n = 3 and otherwise zero. Using the Alexander-Whitney map and the shuffle map, we can further find explicit generators for these cohomologies in the respective original complexes. Lemma 6.6. ([12]) Let n ∈ ℕ_{0}. The Alexander-Whitney map AW_{n} = ⊕_{i+j=n}AW_{ij}, which induces the isomorphism

AW_{n} : ⨁_{i+j=n} HH_{(I,f_{1})}^{i}(A_{q}^{o}) ⊗ HH_{(I,f_{1})}^{j}(A_{q}^{o}) ⟶ HH_{(δ,σ)}^{n}(Ã_{q}^{o}),

is in low degrees given by:

1. AW_{11}(x_{0} ⊗ y_{0}) = x_{0} ⊗ ε ⊗ f_{1} ⊗ y_{0}.

2. AW_{12}(x_{0} ⊗ y_{0} ⊗ y_{1}) = x_{0} ⊗ ε ⊗ f_{1} ⊗ y_{0} ⊗ f_{1} ⊗ y_{1}.

3. AW_{21}(x_{0} ⊗ x_{1} ⊗ y_{0}) = x_{0} ⊗ ε ⊗ x_{1} ⊗ ε ⊗ f_{1} ⊗ y_{0}.

4. AW_{22}(x_{0} ⊗ x_{1} ⊗ y_{0} ⊗ y_{1}) = x_{0} ⊗ ε ⊗ x_{1} ⊗ ε ⊗ f_{1} ⊗ y_{0} ⊗ f_{1} ⊗ y_{1},

for x_{i}, y_{i} ∈ A_{q}^{o}.

The shuffle map sh_{n} = ⊕_{i+j=n}sh_{ij}, which induces an isomorphism

sh_{n} : HH_{(δ,σ)}^{n}(Ã_{q}^{o}) ⟶ ⨁_{i+j=n} HH_{(I,f_{1})}^{i}(A_{q}^{o}) ⊗ HH_{(I,f_{1})}^{j}(A_{q}^{o})

is in low degrees given by:

1. sh_{11}(x_{0} ⊗ y_{0} ⊗ x_{1} ⊗ y_{1}) = x_{0}I(x_{1}) ⊗ I(y_{0})y_{1} − I(x_{0})x_{1} ⊗ y_{0}I(y_{1}),

2. sh_{12}(x_{0} ⊗ y_{0} ⊗ x_{1} ⊗ y_{1} ⊗ x_{2} ⊗ y_{2}) = x_{0}I(x_{1})I(x_{2}) ⊗ I(y_{0})y_{1} ⊗ y_{2}

−I(x_{0})x_{1}I(x_{2}) ⊗ y_{0}I(y_{1}) ⊗ y_{2} + I(x_{0})I(x_{1})x_{2} ⊗ y_{0} ⊗ y_{1}I(y_{2}),

3. sh_{21}(x_{0} ⊗ y_{0} ⊗ x_{1} ⊗ y_{1} ⊗ x_{2} ⊗ y_{2}) = x_{0} ⊗ x_{1}I(x_{2}) ⊗ I(y_{0})I(y_{1}) ⊗ y_{2}

−x_{0}I(x_{1}) ⊗ x_{2} ⊗ I(y_{0})y_{1}I(y_{2}) + I(x_{0})x_{1} ⊗ x_{2} ⊗ y_{0}I(y_{1})I(y_{2}),

for all x_{i}, y_{i} ∈ A_{q}^{o}.

Proposition 6.7. ([12]) We have the following generators of HH_{(δ,σ)}^{n}(Ã_{q}^{o}):

1. n = 2: [ek^{−1} ⊗ ε ⊗ f_{1} ⊗ ek^{−1}], [ek^{−1} ⊗ ε ⊗ f_{1} ⊗ fk^{−1}],

[fk^{−1} ⊗ ε ⊗ f_{1} ⊗ ek^{−1}], [fk^{−1} ⊗ ε ⊗ f_{1} ⊗ fk^{−1}].

2. n = 3: [ek^{−1} ⊗ ε ⊗ f_{1} ⊗ H ⊗ f_{1} ⊗ ek^{−1}], [ek^{−1} ⊗ ε ⊗ f_{1} ⊗ H ⊗ f_{1} ⊗ fk^{−1}], [fk^{−1} ⊗ ε ⊗ f_{1} ⊗ H ⊗ f_{1} ⊗ ek^{−1}], [fk^{−1} ⊗ ε ⊗ f_{1} ⊗ H ⊗ f_{1} ⊗ fk^{−1}],

[H ⊗ ε ⊗ ek^{−1} ⊗ ε ⊗ f_{1} ⊗ ek^{−1}], [H ⊗ ε ⊗ ek^{−1} ⊗ ε ⊗ f_{1} ⊗ fk^{−1}],
[H ⊗ ε ⊗ fk^{−1} ⊗ ε ⊗ f_{1} ⊗ ek^{−1}], [H ⊗ ε ⊗ fk^{−1} ⊗ ε ⊗ f_{1} ⊗ fk^{−1}],

3. n = 4: [H ⊗ ε ⊗ ek^{−1} ⊗ ε ⊗ f_{1} ⊗ H ⊗ f_{1} ⊗ ek^{−1}],

[H ⊗ ε ⊗ ek^{−1} ⊗ ε ⊗ f_{1} ⊗ H ⊗ f_{1} ⊗ fk^{−1}], [H ⊗ ε ⊗ fk^{−1} ⊗ ε ⊗ f_{1} ⊗ H ⊗ f_{1} ⊗ ek^{−1}],

[H ⊗ ε ⊗ fk^{−1} ⊗ ε ⊗ f_{1} ⊗ H ⊗ f_{1} ⊗ fk^{−1}].

Again by Corollary 3.15 we see that HC_{(δ,σ)}^{n}(Ã_{q}^{o}) is ℂ^{4} for n ∈ {2, 3} and otherwise zero, so HP_{(δ,σ)}^{∗}(Ã_{q}^{o}) ≅ 0. To illustrate the role of the Leibniz rule for the B-operator in the Künneth exact sequence in Corollary 3.15 it is perhaps more instructive to present the generators for HC_{(δ,σ)}^{∗}(Ã_{q}^{o}) as below, rather than applying the Alexander-Whitney map. Remark 6.8. Let B denote the linear map IB_{1} : HH_{(I,f_{1})}^{2}(A_{q}) → HH_{(I,f_{1})}^{1}(A_{q}). Consider ξ = [H ⊗ ek^{−1}] and η = [H ⊗ fk^{−1}] in HH_{(I,f_{1})}^{2}(A_{q}). Then by Corollary 3.15, the vector space

HH_{(δ,σ)}^{n}(Ã_{q}^{o}) ≅ ⨁_{n=i+j} HH_{(I,f_{1})}^{i}(A_{q}) ⊗ HH_{(I,f_{1})}^{j}(A_{q})

has the following generators:

1. {ξ ⊗ ξ, ξ ⊗ η, η ⊗ ξ, η ⊗ η}, for n = 4.

2. {B(ξ) ⊗ ξ, B(ξ) ⊗ η, B(η) ⊗ ξ, B(η) ⊗ η, ξ ⊗ B(ξ), ξ ⊗ B(η), η ⊗ B(ξ), η ⊗ B(η)}, for n = 3.

3. {B(ξ) ⊗ B(ξ), B(ξ) ⊗ B(η), B(η) ⊗ B(ξ), B(η) ⊗ B(η)}, for n = 2.

This is a consequence of the following Leibniz rule

IB_{n} = ⨁_{n+1=i+j}(IB_{i−1} ⊗ 1_{j} ⨁(−1)^{i}1_{i} ⊗ IB_{j−1})

for IB_{n} : HH_{(δ,σ)}^{n+1}(Ã_{q}^{o}) → HH_{(δ,σ)}^{n}(Ã_{q}^{o}), where n ∈ ℕ_{0}, with respect to the corresponding operators for HH_{(I,f_{1})}^{∗}(A_{q}).

We conclude this discussion with the following results. Theorem 6.9. ([12]) Provided our assumption on SU_{q}(2) holds, we have:

1. HH_{(δ,σ)}^{n}(Ã_{q}^{o}) ≅ ℂ^{4}, for n ∈ {2, 4}.

2. HH_{(δ,σ)}^{3}(Ã_{q}^{o}) ≅ ℂ^{8}.

3. HH_{(δ,σ)}^{n}(Ã_{q}^{o}) ≅ 0, for n ∉ {2, 3, 4}.

4. HC_{(δ,σ)}^{n}(Ã_{q}^{o}) ≅ ℂ^{4}, for n ∈ {2, 3}.

5. HC_{(δ,σ)}^{n}(Ã_{q}^{o}) ≅ 0, for n ∉ {2, 3}.

6. HP_{(δ,σ)}^{∗}(Ã_{q}^{o}) ≅ 0.

Moreover, explicit representatives of the generators in the respectively defining complexes can be written down in each case (see the results prior to this). Theorem 6.10. ([12]) Provided our assumption on SU_{q}(2) holds, the modular characteristic homomorphisms in Hochschild cohomology, cyclic cohomology and periodic cyclic cohomology of the Hopf algebra (A_{q}^{o}, Δ) with the modular pair (I, f_{1}) in involution, are all ZERO.

Proof. By Proposition 5.1 the operator S : HC^{n}(A_{q}) → HC^{n+2}(A_{q}) is an isomorphism for n ≥ 1. By Theorem 6.9 we see that for any [x] ∈ HC_{(δ,σ)}^{n}(Ã_{q}^{o}), with n ≥ 2, we have

Sγ_{n}([x]) = γ_{n+2}S([x]) = γ_{n+2}(0) = 0.

By injectivity of S : HC^{n}(A_{q}) → HC^{n+2}(A_{q}), for n ≥ 2, we therefore have γ_{n}([x]) = 0. Since clearly γ_{n} = 0, for n ∈ {0, 1}, we thus conclude that γ = 0 in the case of cyclic cohomology. ⊔ ⊓

7 The Modular Square for Algebraic Quantum Groups

Fix an algebraic quantum group (A, Δ) in the sense of definition 6.13 in [14]. First we recall the most important objects naturally associated to (A, Δ):

(i) The counit and antipode of (A, Δ) will be denoted by ε and S respectively. We will denote the unitary antipode of (A, Δ) by R, its scaling group by τ (see [14] Propositions 6.16 and 6.17). Recall that both these objects provide the polar decomposition of the antipode:

S = R τ_{− \frac{i}{2}} = τ_{− \frac{i}{2}} R .

(ii) Let us also fix a non-zero, positive left invariant linear functional φ on A and define a non-zero, positive right invariant linear functional ψ on A by ψ = φR.

Denote the modular automorphism groups of φ by σ (see [14] Proposition 6.15). Recall that

φ(ab) = φ(b σ_{−i}(a))  and  φσ_{z} = φ

for all a, b ∈ A and z ∈ ℂ. (iii) The modular element of (A, Δ) will be denoted by δ (see [14] Prop. 6.8). Remember that δ belongs to M(A) and that φ(S(a)) = φ(aδ) for all a ∈ A. As we mentioned after Prop. 6.17, we can define any complex power δ^{z} ∈ M(A) (z ∈ ℂ). Then ψ(a) = φ(δ^{\frac{1}{2}} aδ^{\frac{1}{2}} ) for all a ∈ A. (iv) The scaling constant ν > 0 of (A, Δ) was defined in [14], Prop. 6.16. Thus, φτ_{z} = ν^{−z} φ for all z ∈ ℂ. Let us also mention that σ_{z}(δ^{r}) = ν^{rz} δ^{r} for all z ∈ ℂ and r ∈ ℝ.

As in [14] Prop. 6.10 and Thm. 6.11, we let (Â, Δ̂) denote the dual algebraic quantum group of (A, Δ). Remember that M(Â) can be realized as a subspace of the dual A′ (see the discussion before Thm. 6.11 ). Hence, the counit ε is the unit of M(Â).

Denoting the modular element of (Â, Δ̂) by δ̂, one can show that δ̂^{z} = ε σ_{iz} for all z ∈ ℂ (see Prop. 5.14 of [9]).

In order to perform the modular square construction, we need a Hopf algebra that is paired with (A, Δ). But in general, (M(Â), Δ̂) is not a Hopf algebra since it can happen that Δ̂(M(Â)) ⊈ M(Â) ⊗ M(Â). (where we used the canonical extension of Δ̂ to M(Â)).

Therefore we suppose that we have a unital subalgebra Ĥ of M(Â) at our disposal such that

1. Δ̂(Ĥ) ⊆ Ĥ ⊗ Ĥ,

2. Ŝ(Ĥ) = Ĥ (where we use the canonical extension of the antipode Ŝ of (Â, Δ̂) ),

3. δ̂^{\frac{1}{2}} , δ̂^{− \frac{1}{2}} belong to Ĥ.

So (Ĥ, Δ̂) is a Hopf algebra consisting of linear functionals on A (in the classical case, Ĥ could be the universal enveloping algebra of the Lie algebra associated to the Lie group of which A is the appropriate function algebra).

In order to obtain a good modular pair in action, it turns out that not Ĥ is the Hopf algebra we should focus on, but rather the Hopf algebra (H̃, Δ̃) defined by

H̃ = Ĥ ⊗ Ĥ^{op}  and  Δ̃ = (ι ⊗ χ ⊗ ι)(Δ ⊗ Δ) ,

where χ denotes the flip map. Hence, the antipode S̃ of (H̃, Δ̃) is given by S̃ = Ŝ ⊗ Ŝ^{−1}. Next we introduce the modular pair for (H̃, Δ̃).

, (1) Set α̃ = δ̂^{− \frac{1}{2}} ⊗ δ̂^{− \frac{1}{2}} ∈ H̃. We know that δ̂^{− \frac{1}{2}} is group-like in M(Â). Hence, α̃ is group-like in H̃. , (2) Since Δ(δ^{\frac{1}{2}} ) = δ^{\frac{1}{2}} ⊗ δ^{\frac{1}{2}} , we can define non-zero characters θ, θ^{−1} : Â → ℂ such that

θ(ω) = ω(δ^{\frac{1}{2}} )  and  θ^{−1}(ω) = ω(δ^{− \frac{1}{2}} )

for all ω ∈ Â. The characters θ and θ^{−1} extend uniquely to characters on M(Â). For instance, θ(ωa) = ω(aδ^{\frac{1}{2}} ) for all a ∈ A and ω ∈ A′. These extensions restrict to characters on H̃ which we denote by the same symbols. Define the character θ̃: H̃ → ℂ as θ̃ = θ⊗θ^{−1}.

Let S̃θ̃ be the twisted antipode on (H̃, Δ̃), i.e. S̃θ̃ = (θ̃⊗ S̃)Δ̃. In terms of the twisted antipodes Ŝ_{θ} = (θ ⊗ Ŝ)Δ̂ on (Ĥ, Δ̂) and Ŝ_{θ^{−1}}^{−1} = (θ^{−1} ⊗ Ŝ^{−1})Δ̂ on (Ĥ^{op}, Δ̂), the twisted antipode S̃θ̃ is given by

(7.1)  S̃θ̃ = Ŝ_{θ} ⊗ Ŝ_{θ^{−1}}^{−1} .

and an easy calculation reveals that

(7.2)  Ŝ_{θ}(ω)(a) = ω(δ^{\frac{1}{2}} S(a))  and  Ŝ_{θ^{−1}}^{−1}(ω)(a) = ω(δ^{− \frac{1}{2}} S^{−1}(a))

for all ω ∈ Ĥ and a ∈ A.

It is not so difficult to check that Proposition 7.1. The couple (θ̃, α̃) is a modular pair in involution for (H̃, Δ̃).

Proof.

(1) We have that

θ̃(α̃) = θ(δ̂^{− \frac{1}{2}} ) θ^{−1}(δ̂^{− \frac{1}{2}} ) = δ̂^{− \frac{1}{2}} (δ^{\frac{1}{2}} ) δ̂^{− \frac{1}{2}} (δ^{− \frac{1}{2}} ) = (ε σ_{− \frac{i}{2}} )(δ^{\frac{1}{2}} ) (ε σ_{− \frac{i}{2}} )(δ^{− \frac{1}{2}} ) ,

which by the fact that δ^{\frac{1}{2}} and δ^{− \frac{1}{2}} are group-like, implies that

θ̃(α̃) = (ν^{− \frac{i}{4}} ε(δ^{\frac{1}{2}} )) (ν^{\frac{i}{4}} ε(δ^{− \frac{1}{2}} )) = 1 .

(2) Choose ω ∈ Ĥ, η ∈ Ĥ^{op} and a, b ∈ A. Then Eqs. (7.1) and (7.2) imply that

S̃2 ; θ̃(ω ⊗ η)(a ⊗ b) = Ŝ_{θ}^{2}(ω)(a) (Ŝ_{θ^{−1}}^{−1})^{2}(η)(b)

= ω(δ^{\frac{1}{2}} S^{2}(a)δ^{− \frac{1}{2}} ) η(δ^{− \frac{1}{2}} S^{−2}(b)δ^{\frac{1}{2}} ) = ω(δ^{\frac{1}{2}} τ_{−i}(a)δ^{− \frac{1}{2}} ) η(δ^{− \frac{1}{2}} τ_{i}(b)δ^{\frac{1}{2}} ) .

Hence, Lem. 5.15 of [9] guarantees that

S̃2 ; θ̃(ω ⊗ η)(a ⊗ b) = ω( (εσ_{− \frac{i}{2}} ⊗ ι ⊗ εσ_{\frac{i}{2}} )Δ^{(2)}(a) ) η( (εσ_{\frac{i}{2}} ⊗ ι ⊗ εσ_{− \frac{i}{2}} )Δ^{(2)}(b) )
= ω( (δ̂^{− \frac{1}{2}} ⊗ ι ⊗ δ̂^{\frac{1}{2}} )Δ^{(2)}(a) ) η( (δ̂^{\frac{1}{2}} ⊗ ι ⊗ δ̂^{− \frac{1}{2}} )Δ^{(2)}(b) )
= (δ̂^{− \frac{1}{2}} ω δ̂^{\frac{1}{2}} )(a) (δ̂^{− \frac{1}{2}} ω δ̂^{\frac{1}{2}} )(b) ,

(remember that we use the opposite product on Ĥ^{op}). It follows that S̃2 ; θ̃(ω ⊗ η) = α̃ (ω ⊗ η) α̃^{−1}.

⊔ ⊓

In the next step, we let H̃ act on A. Since Ĥ ⊆ M(Â), the discussion before Thm. 6.11 in [14] guarantees that (ω ⊗ ι)Δ(a) and (ι ⊗ ω)Δ(a) belong to A for all a ∈ A and ω ∈ H̃. Thus, we can define left Hopf actions

(7.3)  Ĥ × A → A : (ω, a) ↦ ω ∗ a = (ι ⊗ ω)Δ(a) and
Ĥ^{op} × A → A : (ω, a) ↦ a ∗ ω = (ω ⊗ ι)Δ(a) .

We leave it to the reader to check that these are really left Hopf actions and that they commute. For ω ∈ Ĥ, η ∈ Ĥ^{op} and a ∈ A, we set

ω ∗ a ∗ η = (ω ∗ a) ∗ η = ω ∗ (a ∗ η) .

So we can define a left Hopf action ▷ : H̃ × A → A so that

(ω ⊗ η) ▷ a = ω ∗ a ∗ η

for all ω ∈ Ĥ, η ∈ Ĥ^{op} and a ∈ A.

In order to define the γ-map that connects the Hopf cyclic cohomology of the Hopf algebra Ĥ and the ordinary cyclic cohomology of the algebra A, one needs a θ̃-invariant α̃-trace. To this end, Connes and Moscovici introduced in [5] the mid-weight ρ : A → ℂ defined by

ρ(a) = φ(δ^{\frac{1}{4}} aδ^{\frac{1}{4}} ) = ψ(δ^{− \frac{1}{4}} aδ^{− \frac{1}{4}} )

for all a ∈ A (recall that ψ = φ(δ^{\frac{1}{2}} . δ^{\frac{1}{2}} ).

Proposition 7.2. The linear functional ρ is a θ̃-invariant α̃-trace.

Proof.

(1) ρ is θ̃-invariant:

Choose η ∈ Ĥ^{op} and a, b ∈ A. Then, since σ_{−i}(δ^{\frac{1}{4}} ) = ν^{− \frac{i}{4}} δ^{\frac{1}{4}} ,

ρ((a ∗ η) b) = φ(δ^{\frac{1}{4}} (a ∗ η) b δ^{\frac{1}{4}} )
= ν^{− \frac{i}{4}} φ((η ⊗ ι)(Δ(a)) b δ^{\frac{1}{2}} ) = ν^{− \frac{i}{4}} η( (ι ⊗ φ)(Δ(a)(1 ⊗ b δ^{\frac{1}{2}} )) ) .

The proof of Prop. 3.11 of [23] shows that

S( (ι ⊗ φ)(Δ(p)(1 ⊗ q)) ) = (ι ⊗ φ)((1 ⊗ p)Δ(q))

for all p, q ∈ A. Thus,

ρ((a ∗ η) b) = ν^{− \frac{i}{4}} η( S^{−1}( (ι ⊗ φ)((1 ⊗ a)Δ(bδ^{\frac{1}{2}} )) ) )
= ν^{− \frac{i}{4}} η( S^{−1}( (ι ⊗ φ)((1 ⊗ a)Δ(b)(1 ⊗ δ^{\frac{1}{2}} )) δ^{\frac{1}{2}} ) )
= η( δ^{− \frac{1}{2}} S^{−1}( (ι ⊗ φ)((1 ⊗ δ^{\frac{1}{4}} )(1 ⊗ a)Δ(b)(1 ⊗ δ^{\frac{1}{4}} )) ) )
= η( δ^{− \frac{1}{2}} S^{−1}( (ι ⊗ ρ)((1 ⊗ a)Δ(b)) ) ) = Ŝ_{θ^{−1}}^{−1}(η)( (ι ⊗ ρ)((1 ⊗ a)Δ(b)) )
= ρ(a (b ∗ Ŝ_{θ^{−1}}^{−1}(η))) .

Using the fact that ρ = ψ(δ^{− \frac{1}{4}} . δ^{− \frac{1}{4}} ), one proves in a similar fashion that ρ((ω ∗ a) b) = ρ(a (Ŝ_{θ}(ω) ∗ b)) for all a, b ∈ A and ω ∈ Ĥ. Combining these two results and Eq. (7.1), one sees that ρ is θ̃-invariant, i.e.

ρ((φ ▷ a) b) = ρ(a (S̃θ̃(φ) ▷ b))

for all a, b ∈ A and φ ∈ H̃.

(2) ρ is an α̃-trace:

Choose a, b ∈ A. Then,

ρ(ab) = φ(δ^{\frac{1}{4}} ab δ^{\frac{1}{4}} ) = φ(bδ^{\frac{1}{4}} σ_{−i}(δ^{\frac{1}{4}} a)) = ν^{− \frac{1}{4}} φ(b δ^{\frac{1}{2}} σ_{−i}(a))
= ν^{− \frac{1}{4}} φ(b δ^{\frac{1}{2}} σ_{−i}(a)δ^{− \frac{1}{2}} δ^{\frac{1}{4}} δ^{\frac{1}{4}} ) = φ(δ^{\frac{1}{4}} b δ^{\frac{1}{2}} σ_{−i}(a)δ^{− \frac{1}{2}} δ^{\frac{1}{4}} )
= ρ(b δ^{\frac{1}{2}} σ_{−i}(a)δ^{− \frac{1}{2}} )

According to Lem. 5.15 of [9], we have that

δ^{\frac{1}{2}} σ_{−i}(a)δ^{− \frac{1}{2}} = (εσ_{− \frac{i}{2}} ⊗ι⊗εσ_{− \frac{i}{2}} )Δ^{(2)}(a) = (δ̂^{− \frac{1}{2}} ⊗ι⊗δ̂^{− \frac{1}{2}} )Δ^{(2)}(a) = (δ̂^{− \frac{1}{2}} ⊗δ̂^{− \frac{1}{2}} )▷a = α̃▷a .

Hence, ρ(ab) = ρ(b (α̃ ▷ a)), meaning that ρ is an α̃-trace.

⊔ ⊓

We end this section by commenting on the case of locally compact quantum groups in a formal way. In this case we start of with a locally compact quantum group (A, Δ) as in [14] Def. 8.1, which we assume to act on the GNS-space H_{φ} of its left Haar weight φ. Let W ∈ B(H_{φ} ⊗ H_{φ}) be the multiplicative unitary as defined before Thm. 8.3 in [14] and recall from Prop. 8.4 in [14] that A is the norm closure of the algebra B := { (ι ⊗ ω)(W) | ω ∈ B(H_{φ})_{∗} }.

Also, the dual quantum group (Â, Δ̂) of (A, Δ) is defined such that Â is the norm closure of the algebra { (ω ⊗ ι)(W) | ω ∈ B(H_{φ})_{∗} } and

(7.4)  Δ̂(x) = W (x ⊗ 1)W^{∗} for all x ∈ Â .

So we see that Â can be identified with a subspace of the dual B′ such that

(7.5)  ⟨x, (ι ⊗ ω)(W)⟩ = ω(x)

for x ∈ Â and ω ∈ B(H_{φ})_{∗}. In a formal sense, one obtains a dual pairing between Â and B this way.

Regarding Â as sitting inside B′ this way, referring to Eq. (7.3) and using the fact that (Δ ⊗ ι)(W) = W_{13} W_{23} (see [14] Thm. 8.3 and Prop. 8.4), we should define left actions of Â and Â^{op} on B in the following way:

Â × B → B : (x, (ι ⊗ ω)(W)) ↦ x ∗ (ι ⊗ ω)(W) = (ι ⊗ ω)(W (1 ⊗ x))

and

(7.6)  Â^{op} × B → B : (x, (ι ⊗ ω)(W)) ↦ (ι ⊗ ω)(W) ∗ x = (ι ⊗ ω)((1 ⊗ x)W) .

In a lot of specific examples it happens to be the case that there is a Hopf algebra Ĥ around that can play the role of the Hopf algebra in the construction of the modular square (as far as we know, the possible presence and the desirable properties of such a Hopf algebra has not been seriously studied yet in a more general theoretical framework). However, the precise relation between this Hopf algebra Ĥ and the actual C^{∗}-algebra Â or von Neumann Â′′ can be cumbersome to describe. Let us discuss this a little bit further.

In a lot of cases, there is some dense subspace D of H_{φ} present together with a number of linear operators T_{1}, …, T_{n} on D (in order not to cloud the discussion, we ignore the important ^{∗}-operation) such that Ĥ is the algebra of linear operators on D that is generated by T_{1}, …, T_{n}.

However, the linear operators T_{1}, …, T_{n} that generate this Hopf algebra Ĥ are not really the ones that are of fundamental importance to Â. The important ones are most of the time (possibly unbounded) closed linear operators T̄_{1}, …, T̄_{n} in H_{φ} that are extensions of T_{1}, …, T_{n} respectively and are affiliated to the C^{∗}-algebra Â or to the von Neumann algebra Â′′. It also can happen that the proper extension T̄_{i} is not merely the closure of T_{i} but that a more refined description of the extension T̄_{i} is needed (comparable to the theory of self-adjoint extensions of linear operators that are not essentially self-adjoint).

In accordance with Eq. (7.4), the comultiplication on Â and Ĥ should be related through the fact that Δ(T_{i}) is a restriction of W (T̄_{i} ⊗ 1)W^{∗}.

It can happen that the C^{∗}-algebra Â is generated by T̄_{1}, …, T̄_{n} in the sense of Def. 1.24 in [14] (e.g. quantum E(2), quantum az+b). But there also instances (e.g. quantum ax+b, quantum S̃U(1, 1)) where one needs to add an extra bounded linear operator S on D so that Â is generated by T̄_{1}, …, T̄_{n} and S̄ (where S̄ is the unique bounded linear operator on H_{φ} extending S) and for which there does not exist an element X in the algebraic tensor product Ĥ_{∗}⊗Ĥ_{∗} (where Ĥ_{∗} denotes the algebra of linear operators on D generated by Ĥ and S) so that W (1 ⊗ S̄)W^{∗} is an extension of X. In short, we can not extend the Hopf algebra structure on Ĥ to an appropriate Hopf algebra structure on Ĥ_{∗}. We refer to [14] section ?? (ref: part of S.L. Woronowicz concerning examples) for more details on this subject.

Let us end this discussion by saying that a reason for not working with the ‘Hopf algebra generated by’ the extensions T̄_{1}, …, T̄_{n} lies in the fact that for such unbounded operators, addition and multiplication are tricky and sometimes ill-defined operations.

But let us ignore this important issue for the rest of this section and focus on Ĥ. Recall that in the case of algebraic quantum groups we used the inherent pairing between A and Ĥ to define actions of Ĥ on A (see Eq. (7.3) ). So we would like to use the formula in Eq. (7.5) to define a pairing between Ĥ and B. Due to the unboundedness of the elements in Ĥ, the expression in Eq. (7.5) does not make sense for every ω ∈ B(H_{φ})_{∗} and x ∈ Ĥ. But if ω is the vector functional ω_{v,w} for v ∈ D and w ∈ H_{φ} (see the comments before Def. 1.29), we can interpret ω(x) to be equal to ⟨x(v), w⟩. So we should look for an appropriate subspace E ⊆ B(H_{φ})_{∗} for which ω(x) can be well defined, a technical issue we will not further go into. But the vector functionals introduced above indicate that E can be chosen to be dense in B(H_{φ})_{∗}.

This way, Eq. (7.5) will provide us with a pairing between Ĥ and B_{0} := { (ι⊗ω)(W) | ω ∈ E } ⊆ B. The above discussion is only intended to give a first impression of the technical subtleties involved. It turns out that further restrictions on E (and hence on B_{0}) are needed in order to

1. well define the left actions of Ĥ and Ĥ^{op} on B_{0} inspired by formula (7.6),

2. be able to apply the mid-weight on elements of B_{0} and products thereof.

The data used in the statement of Proposition 7.1 and Proposition 7.2 is referred to as the modular square associated to the Hopf algebra (Ĥ, Δ̂). By Lemma 4.1, these propositions immediately yield the following important result. Theorem 7.3. ([5]) Consider a Hopf algebra (Ĥ, Δ̂) with Ĥ ⊂ M(Â) as a unital algebra, with coproduct given by restricting Δ̂ : Â → M(Â × Â) to Ĥ, and such that δ̂^{\frac{1}{2}} , δ̂^{− \frac{1}{2}} ∈ Ĥ. Let H̃_{(θ̃,α̃)}^{♮} denote the Λ-module canonically associated to the modular square of (Ĥ, Δ̂). Then the formulas γ_{0}(1) = ρ and

γ_{n}((x^{1} ⊗ y^{1}) ⊗ ⋯ ⊗ (x^{n} ⊗ y^{n}))(a^{0}, …, a^{n}) = ρ(a^{0}(x^{1} ∗ a^{1} ∗ y^{1}) ⋯ (x^{n} ∗ a^{n} ∗ y^{n})),

for a^{0}, …, a^{n} ∈ A and x^{1}, y^{1}, …, x^{n}, y^{n} ∈ Ĥ and n ∈ ℕ, yield a Λ-module morphism γ : H̃_{(θ̃,α̃)}^{♮} → C^{♮}, which in turn induces characteristic homomorphisms in cyclic cohomology

γ_{∗} : HC_{(θ̃,α̃)}^{∗}(H̃) → HC^{∗}(A).

Characteristic homomorphisms are similarly induced on the level of Hochschild cohomology and periodic cyclic cohomology.

The map γ from Theorem 7.3 is called the modular characteristic homomorphism associated to (Ĥ, Δ̂).

To compute the cyclic cohomology of the modular square, we include a convenient corollary of Theorem 3.1 and Theorem 3.2. Corollary 7.4. Consider a Hopf ∗-algebra (Ĥ, Δ̂) with Ĥ ⊂ M(Â) as a unital ∗-algebra, with coproduct given by restricting Δ̂ : Â → M(Â × Â) to Ĥ, and such that δ̂^{\frac{1}{2}} , δ̂^{− \frac{1}{2}} ∈ Ĥ. Let (H̃, Δ̃) denote its modular square with modular pair (θ̃, α̃) = (θ ⊗ θ^{−1}, δ̂^{− \frac{1}{2}} ⊗ δ̂^{− \frac{1}{2}} ) in involution. Then

HH_{(θ̃,α̃)}^{n}(H̃) ≅ ⨁_{i+j=n} HH_{(θ,δ̂^{− \frac{1}{2}} )}^{i}(Ĥ) ⊗ HH_{(θ,δ̂^{− \frac{1}{2}} )}^{j}(Ĥ),

for n ∈ ℕ_{0}. Moreover, the isomorphism is implemented by the shuffle map sh (going from left to right) with the Alexander-Whitney map AW as its inverse. There exists a canonical long exact sequence

⋯ ⟶^{i} HC_{(θ̃,α̃)}^{n−1}(H̃) ⟶^{∂} ⨁_{p+q=n−2} HC_{(θ,δ̂^{− \frac{1}{2}} )}^{p}(Ĥ) ⊗ HC_{(θ,δ̂^{− \frac{1}{2}} )}^{q}(Ĥ) ⟶^{S⊗1−1⊗S}

⨁_{r+s=n} HC_{(θ,δ̂^{− \frac{1}{2}} )}^{r}(Ĥ) ⊗ HC_{(θ,δ̂^{− \frac{1}{2}} )}^{s}(Ĥ) ⟶^{i} HC_{(θ̃,α̃)}^{n}(H̃) ⟶^{∂} ⋯

Notation. We recover Theorem 4.3 and Corollary 3.15 in the compact case from Theorem 7.3 and Corollary 7.4, respectively, by first observing that (A, Δ) = (A, Φ) and M(Â) ≅ A′, so (Ĥ, Δ̂) is a Hopf subalgebra of (A^{o}, Δ). Now since (A, Φ) is a compact quantum group, we have ν = 1, δ = I and δ̂ = f_{−2}, so ρ = h and (θ̃, α̃) = (I ⊗ I, f_{1} ⊗ f_{1}).

8 Twisted Cyclic Cohomology

Suppose A is a unital algebra and let θ : A → A be an automorphism of A. We now define a new example of a cocyclic object, namely the one which gives rise to twisted (see [11] and [7]) cyclic cohomology HC^{∗}(A, θ) of A. Fix n ∈ ℕ_{0}. Let C^{n} denote (as in the case of ordinary cyclic cohomology) the vector space of n-cochains, that is, the set of multilinear maps from the vector space A^{n+1} to ℂ. Let i ∈ {0, 1, …, n}. Define linear maps

d_{i}^{n} : C^{n−1} → C^{n}, s_{i}^{n} : C^{n+1} → C^{n}, t_{n} : C^{n} → C^{n}

as follows:

(d_{i}^{n}φ)(a^{0}, a^{1}, …, a^{n}) = φ(a^{0}, …, a^{i}a^{i+1}, …, a^{n}) for i ≤ n − 1
(d_{n}^{n}φ)(a^{0}, a^{1}, …, a^{n}) = φ(θ(a^{n})a^{0}, a^{1}, …, a^{n−1})
(s_{i}^{n}φ)(a^{0}, a^{1}, …, a^{n}) = φ(a^{0}, …, a^{i}, I, a^{i+1}, …, a^{n})
(t_{n}φ)(a^{0}, a^{1}, …, a^{n}) = φ(θ(a^{n}), a^{0}, a^{1}, …, a^{n−1}),

where I is the unit of A, a^{0}, …, a^{n} ∈ A and φ is a cochain. As before C^{−1} and d_{i}^{0} are not defined, so the range of n must be restricted accordingly for some formulas to make sense. Note that (t_{n}^{n+1}φ)(a^{0}, …, a^{n}) = φ(θ(a^{0}), …, θ(a^{n})), for any φ ∈ C^{n} and a^{i} ∈ A. Define a subspace C^{n}(θ) of C^{n} by

C^{n}(θ) = {φ ∈ C^{n} | t_{n}^{n+1}φ = φ}.

It is straightforward to check that d_{i}^{n}(C^{n−1}(θ)) ⊂ C^{n}(θ), s_{i}^{n}(C^{n+1}(θ)) ⊂ C^{n}(θ) and t_{n}(C^{n}(θ)) ⊂ C^{n}(θ). Thus (C^{n}(θ), d_{i}^{n}, s_{i}^{n}, t_{n}) is a cocyclic object of the abelian category of vector spaces. We denote the Hochschild cohomology, the cyclic cohomology and the periodic cyclic cohomology canonically associated to this cocyclic object by HH^{∗}(A, θ), HC^{∗}(A, θ) and HP^{∗}(A, θ), respectively.

Dually, define the invariant subspace C_{n}(θ) of C_{n} = A^{⊗(n+1)} by

C_{n}(θ) = {x ∈ A^{⊗(n+1)} | θ^{⊗(n+1)}(x) = x }.

Let i ∈ {0, 1, …, n}, and define linear maps

d_{n}^{i} : C_{n−1}(θ) → C_{n}(θ), s_{n}^{i} : C_{n+1}(θ) → C_{n}(θ), t^{n} : C_{n}(θ) → C_{n}(θ)

as follows:

d_{n}^{i}(a^{0} ⊗ a^{1} ⊗ ... ⊗ a^{n}) = a^{0} ⊗ ... ⊗ a^{i}a^{i+1} ⊗ ... ⊗ a^{n} for i ≤ n − 1
d_{n}^{n}(a^{0} ⊗ a^{1} ⊗ ... ⊗ a^{n}) = θ(a^{n})a^{0} ⊗ a^{1} ⊗ ... ⊗ a^{n−1}
s_{n}^{i}(a^{0} ⊗ a^{1} ⊗ ... ⊗ a^{n}) = a^{0} ⊗ ... ⊗ a^{i} ⊗ I ⊗ a^{i+1} ⊗ ... ⊗ a^{n}
t^{n}(a^{0} ⊗ a^{1} ⊗ ... ⊗ a^{n}) = θ(a^{n}) ⊗ a^{0} ⊗ a^{1} ⊗ ... ⊗ a^{n−1},

where I is the unit of A and a^{0}, …, a^{n} ∈ A. Note that C_{−1}(θ) and d_{0}^{i} are not defined, so the range of n must be restricted accordingly for some formulas to make sense. Then (C_{n}(θ), d_{i}^{n}, s_{i}^{n}, t_{n}) is a cyclic object of the abelian category of vector spaces. Let C_{θ}^{♮} denote the corresponding Λ-module. We denote the Hochschild homology, the cyclic homology and the periodic cyclic homology canonically associated to C_{θ}^{♮} by HH_{∗}(A, θ), HC_{∗}(A, θ) and HP_{∗}(A, θ), respectively.

Of course, these definitions extend to yield cyclic and cocyclic objects over arbitrary fields, rings or modules. Note that when θ = ι, then the twisted cohomology (homology) groups coincide with the ordinary cohomology (homology) groups of A.

Notice that C^{n}(θ) is isomorphic (as a vector space) to the annihilator

(Im(θ^{⊗(n+1)} − ι^{⊗(n+1)}))^{⊥}

in A^{⊗(n+1)} of the image of the linear map θ^{⊗(n+1)} − ι^{⊗(n+1)}, and, moreover, that

Im(θ^{⊗(n+1)} − ι^{⊗(n+1)}) = A^{⊗(n+1)}/C_{n}(θ),

for all n ∈ ℕ_{0}. When θ = ι, then C_{n}(θ) = A^{⊗(n+1)} and C^{n}(θ) = C_{n}(θ)′.

We also have a Chern character in the twisted case generalizing the ordinary one. Recall that C_{0}(θ) is the unital fix point algebra A_{θ} in A of θ, which obviously coincides with A if and only if θ = ι. Let C^{♮}(A_{θ}) denote the ordinary Λ-module of the algebra A_{θ} (to distinguish it from the Λ-modules C_{ι}^{♮} = C^{♮} and C_{θ}^{♮} associated to (A, ι) and (A, θ), respectively).

It is readily seen that the map

p : (A_{θ}, ι) ↦ (A, θ)

of algebras with automorphisms induces a Λ-module morphism p : C^{♮}(A_{θ}) → C_{θ}^{♮}, and thus group homomorphisms

p_{∗} : HC_{∗}(A_{θ}) → HC_{∗}(A, θ)
p^{∗} : HC^{∗}(A, θ) → HC^{∗}(A_{θ})

in cyclic homology and cyclic cohomology. Of course the same holds for Hochschild (co-) homology and periodic cyclic (co-) homology.

Let

⟨·, ·⟩ : HC^{∗}(A_{θ}) × K_{∗}(A_{θ}) → ℂ

denote the ordinary bilinear pairing between ordinary (even or odd) cyclic cohomology and K-theory associated to the algebra A_{θ}. Then we have the following bilinear pairing

HC^{∗}(A, θ) × K_{∗}(A_{θ}) ⟶^{p^{∗}×ι} HC^{∗}(A_{θ}) × K_{∗}(A_{θ}) ⟶^{⟨·,·⟩} ℂ.

Similarly, using the Chern character

ch_{∗} : K_{∗}(A_{θ}) → HC_{∗}(A_{θ})

from K-theory to (even or odd) cyclic homology associated to the algebra A_{θ}, we get the following Chern character

K_{∗}(A_{θ}) ⟶^{ch_{∗}} HC_{∗}(A_{θ}) ⟶^{p_{∗}} HC_{∗}(A, θ)

for twisted cyclic homology.

There exists a beautiful relation between twisted cyclic cocycles and differential calculi of quantum groups [27][8], which we shall briefly sketch (see also the proceeding of Gerard Murphy). Henceforth let (A, Φ) denote the Hopf ∗-algebra associated to a compact quantum group. Theorem 8.1. ([11]) Let notation be as above and let Z_{λ}^{n}(A, θ) denote the vector space of θ-twisted n-cyclic cocycles of A. For any φ ∈ Z_{λ}^{n}(A, θ) there exists a canonical triple (Ω, d, ∫), where (Ω, d) is an n-dimensional differential calculus and ∫ is a closed faithful θ-twisted graded trace on (Ω, d) (in the sense of [11]). The assignment

φ ∈ Z_{λ}^{n}(A, θ) ⟶ (Ω, d, ∫ )

is bijective. Moreover, with respect to the natural notions of invariance for φ and ∫, we have for this correspondence that:

1. φ is left invariant if and only if (Ω, d) is left covariant.

2. φ is right invariant if and only if (Ω, d) is right covariant.

3. φ is bi-invariant if and only if (Ω, d) is bicovariant.

Finite dimensional covariant differential calculi come with a unique (up to a scalar) closed faithful twisted graded trace exactly when there is only one (up to a scalar) top invariant volume form [11]. In the bicovariant case the twist is then usually given by θ_{D}(a) = f_{−1} ∗ a ∗ f_{−1}, for a ∈ A, which can be shown to be the case for the 4D_{+}-calculus of Woronowicz associated to SU_{q}(2).

In the twisted world we can define characteristic homomorphisms without involving the modular square. Theorem 8.2. Let θ_{l} and θ_{r} denote the automorphisms of A defined for a ∈ A, by θ_{l}(a) = a ∗ f_{−1} and θ_{r}(a) = f_{−1} ∗ a, respectively, so θ_{l}θ_{r} = θ_{r}θ_{l} = θ_{D}. Denote the corresponding Λ-modules by C_{l}^{♮} and C_{r}^{♮}. Consider a Hopf subalgebra (H, Δ) of (A^{o}, Δ) with the modular pair (I, f_{1}) in involution. Define linear maps γ_{n}^{l,r} : H^{⊗n} → C^{n} by

γ_{n}^{l}(x^{1} ⊗ ⋯ ⊗ x^{n})(a^{0}, …, a^{n}) = h(a^{0}(x^{1} ∗ a^{1}) ⋯ (x^{n} ∗ a^{n}))

and

γ_{n}^{r}(x^{1} ⊗ ⋯ ⊗ x^{n})(a^{0}, …, a^{n}) = h(a^{0}(a^{1} ∗ x^{1}) ⋯ (a^{n} ∗ x^{n})),

for a^{0}, …, a^{n} ∈ A, x^{1}, …, x^{n} ∈ H and n ∈ ℕ. For n = 0, define linear maps

γ_{n}^{l,r} : H^{⊗0} = ℂ → C^{0} = A′

by γ_{0}^{l}(1) = γ_{0}^{r}(1) = h, where h is the Haar state of (A, Φ). Then γ_{n}^{l,r}(H^{⊗n}) ⊂ C^{n}(θ_{l,r}), for n ∈ ℕ_{0}. Moreover, the linear maps

γ^{l,r} = ⊕_{n=0}^{∞}γ_{n}^{l,r} : H_{(I,f_{1})}^{♮} → C_{l,r}^{♮}

are injective Λ-module morphisms. Hence γ^{l} and γ^{r} induce characteristic homomorphisms

γ^{l,r} : HC_{(I,f_{1})}^{∗}(H) → HC^{∗}(A, θ_{l,r})

of the corresponding cyclic cohomology groups. The same holds for Hochschild cohomology and periodic cyclic cohomology. Remark 8.3. The map γ^{l} does not in general map into the Λ-module underlying the ordinary cyclic cohomology HC^{∗}(A) because h is not a δ-invariant σ-trace with respect to the modular pair (δ, σ) = (I, f_{1}) in involution and the H-action ▷_{l} : x ⊗ a → x ∗ a, where x ∈ H and a ∈ A. As we have seen, γ^{l} maps instead into the Λ-module C_{l}^{♮} underlying the twisted cyclic cohomology HC^{∗}(A, θ_{l}) of A, and in general, there exists no Λ-module morphism between C^{♮} and C_{l}^{♮}, nor any interesting homomorphism between their corresponding cohomology groups. Similar remarks hold for γ^{r} with the H_{op}-action ▷_{r} : x ⊗ a → a ∗ x, where x ∈ H_{op} and a ∈ A.

We have seen that to get a characteristic homomorphism into HC^{∗}(A), one needs to involve both ▷_{l} and ▷_{r} and form the modular square (H̃, Δ̃) of (H, Δ). The H̃-action on A is given by

x ⊗ y ↦ x ▷_{l} (y ▷_{r} a) = y ▷_{r} (x ▷_{l} a),

for x ∈ H, y ∈ H_{op} and a ∈ A. However, as we have seen, the γ-map obtained this way is not injective on the level of Λ-modules, whereas both γ^{l,r} are.

Also there is another important feature of the maps γ^{l,r}. The image γ^{l}(H_{(I,f_{1})}^{♮}) consists

entirely of left-invariant cochains. Similarly, the image γ^{r}(H_{(I,f_{1})}^{♮}) consists entirely of right-invariant cochains. The characteristic homomorphism associated to the modular square, on the other hand, has an image which consists of cochains that are neither leftnor right-invariant.

Finally, it follows from [11], that a left-invariant θ_{l}-twisted cyclic cocycle will necessary belong to γ^{l}(H_{(I,f_{1})}^{♮}). Similarly, a right-invariant θ_{r}-twisted cyclic cocycle will

necessary belong to γ^{r}(H_{(I,f_{1})}^{♮}). This suggests that a characterization of the images of γ^{l,r}(HC_{(I,f_{1})}^{∗}(H)) in HC^{∗}(A, θ_{l,r}) can be made. There is work in progress discussing these matters, where the case of SU_{q}(2) is considered in great detail.

A Proof of Theorem 8.2

In this section we include a detailed proof of Theorem 8.2, which has not been written down elsewhere. To this end we need the following result [25][26]. Proposition A.1. Let (A, Φ) denote the Hopf ∗-algebra associated to a compact quantum group. Then there exists a canonical family {f_{z}}_{z∈ℂ} of unital multiplicative linear functionals on A such that:

1. f_{0} = ε and f_{z}f_{w} = f_{z+w} for all z, w ∈ ℂ.

2. f_{z}(κ(a)) = f_{−z}(a) and \overline{f_{z}(a)} = f_{−\overline{z}}(a^{∗}) for all a ∈ A and z ∈ ℂ.

3. κ^{2}(a) = f_{−1} ∗ a ∗ f_{1} for all a ∈ A.

4. h(ab) = h(bf_{1} ∗ a ∗ f_{1}) for all a, b ∈ A.

We also need the following result [12], which follows from Proposition A.1. Proposition A.2. ([12]) Let x, y ∈ H ⊂ A^{o} and a, b ∈ A. Then:

1. κ((h ⊗ ι)((a ⊗ I)Φ(b))) = (h ⊗ ι)(Φ(a)(b ⊗ I)).

2. κ((ι ⊗ h)(Φ(a)(I ⊗ b))) = (ι ⊗ h)((I ⊗ a)Φ(b)).

3. h((y ∗ a)b) = h(a(S(y) ∗ b)).

4. h((a ∗ y)b) = h(a(b ∗ S^{−1}(y))).

Proof of Theorem 8.2 We consider first the map γ^{l} corresponding to the left action ▷_{l}. Let n ∈ ℕ_{0}. Now γ_{n}^{l}(H^{⊗n}) ⊂ C^{n}(θ_{l}) if and only if t_{n}^{n+1}γ_{n}^{l} = γ_{n}^{l}, which we may express as

h(θ_{l}(a^{0})(x^{1} ∗ θ_{l}(a^{1})) ⋯ (x^{n} ∗ θ_{l}(a^{n}))) = h(a^{0}(x^{1} ∗ a^{1}) ⋯ (x^{n} ∗ a^{n})),

for a^{0}, …, a^{n} ∈ A and x^{1}, …, x^{n} ∈ H. But

h(θ_{l}(a^{0})(x^{1} ∗ θ_{l}(a^{1})) ⋯ (x^{n} ∗ θ_{l}(a^{n}))) = h(a^{0}(x^{1} ∗ a^{1}) ⋯ (x^{n} ∗ a^{n}))

holds for any twist given by a → a ∗ ξ, where a ∈ A, as long as ξ ∈ A′ satisfies ξ(I) = 1.

To see that γ^{l} is injective, consider first n ∈ ℕ and pick a typical element ∑_{i} x_{i}^{1} ⊗ ⋯ ⊗ x_{i}^{n} of H^{⊗n} such that γ_{n}^{l}(∑_{i} x_{i}^{1} ⊗ ⋯ ⊗ x_{i}^{n}) = 0. Then by faithfulness of h on A, we conclude that

∑_{i}(x_{i}^{1} ∗ a^{1}) ⋯ (x_{i}^{n} ∗ a^{n}) = 0,

for any a^{1}, …, a^{n} ∈ A. By applying the counit ε of (A, Φ) to this expression, we get

(∑_{i} x_{i}^{1} ⊗ ⋯ ⊗ x_{i}^{n})(a^{1}, …, a^{n}) = ∑_{i} x_{i}^{1}(a^{1}) ⋯ x_{i}^{n}(a^{n}) = 0,

for any a^{1}, …, a^{n} ∈ A. Thus ∑_{i} x_{i}^{1} ⊗ ⋯ ⊗ x_{i}^{n} = 0 and γ_{n}^{l} is injective. Since γ_{0}^{l}(c) = ch, for any c ∈ ℂ, obviously γ_{0}^{l} is injective. Thus γ^{l} is injective.

To show that γ^{l} is a Λ-module morphism we need to check the conditions

γ_{n}^{l}δ_{i}^{n} = d_{i}^{n}γ_{n−1}^{l}, γ_{n}^{l}σ_{i}^{n} = s_{i}^{n}γ_{n+1}^{l}, γ_{n}^{l}τ_{n} = t_{n}γ_{n}^{l},

for all n ∈ ℕ_{0} and i ∈ {0, …, n}. Recall that d_{i}^{0} and δ_{i}^{0} are not defined.

Consider first the condition γ_{n}^{l}δ_{i}^{n} = d_{i}^{n}γ_{n−1}^{l}, where we restrict to n ≥ 2 and i ∈ {1, …, n − 1}. Let a^{0}, …, a^{n} ∈ A and x^{1}, …, x^{n−1} ∈ H. Then

(d_{i}^{n}γ_{n−1}^{l})(x^{1} ⊗ ⋯ ⊗ x^{n−1})(a^{0}, …, a^{n})
= γ_{n−1}^{l}(x^{1} ⊗ ⋯ ⊗ x^{n−1})(a^{0}, …, a^{i}a^{i+1}, …, a^{n})
= h(a^{0}(x^{1} ∗ a^{1}) ⋯ (x^{i−1} ∗ a^{i−1})(x^{i} ∗ (a^{i}a^{i+1}))(x^{i+1} ∗ a^{i+2}) ⋯ (x^{n−1} ∗ a^{n}))

= h(a^{0}(x^{1} ∗ a^{1}) ⋯ (x^{i−1} ∗ a^{i−1})(∑(x_{(1)}^{i} ∗ a^{i})(x_{(2)}^{i} ∗ a^{i+1}))(x^{i+1} ∗ a^{i+2}) ⋯ (x^{n−1} ∗ a^{n})).

Therefore

(d_{i}^{n}γ_{n−1}^{l})(x^{1} ⊗ ⋯ ⊗ x^{n−1})(a^{0}, …, a^{n})
= γ_{n}^{l}(x^{1} ⊗ ⋯ ⊗ x^{i−1} ⊗ Δ(x^{i}) ⊗ x^{i+1} ⊗ ⋯ ⊗ x^{n−1})(a^{0}, …, a^{n})

and thus

(d_{i}^{n}γ_{n−1}^{l})(x^{1} ⊗ ⋯ ⊗ x^{n−1})
= γ_{n}^{l}(x^{1} ⊗ ⋯ ⊗ x^{i−1} ⊗ Δ(x^{i}) ⊗ x^{i+1} ⊗ ⋯ ⊗ x^{n−1}) = (γ_{n}^{l}δ_{i}^{n})(x^{1} ⊗ ⋯ ⊗ x^{n−1}),

so γ_{n}^{l}δ_{i}^{n} = d_{i}^{n}γ_{n−1}^{l}, for n ≥ 2 and i ∈ {1, …, n − 1}.

Next consider the condition γ_{n}^{l}δ_{n}^{n} = d_{n}^{n}γ_{n−1}^{l}. We have

(d_{n}^{n}γ_{n−1}^{l})(x^{1} ⊗ ⋯ ⊗ x^{n−1})(a^{0}, a^{1}, …, a^{n})
= γ_{n−1}^{l}(x^{1} ⊗ ⋯ ⊗ x^{n−1})(θ_{l}(a^{n})a^{0}, a^{1}, …, a^{n−1})
= h((a^{n} ∗ f_{−1})a^{0}(x^{1} ∗ a^{1}) ⋯ (x^{n−1} ∗ a^{n−1})).

Using Proposition A.1, we thus get

(d_{n}^{n}γ_{n−1}^{l})(x^{1} ⊗ ⋯ ⊗ x^{n−1})(a^{0}, a^{1}, …, a^{n})
= h(a^{0}(x^{1} ∗ a^{1}) ⋯ (x^{n−1} ∗ a^{n−1})(f_{1} ∗ a^{n}))
= γ_{n}^{l}(x^{1} ⊗ ⋯ ⊗ x^{n−1} ⊗ f_{1})(a^{0}, …, a^{n})
= γ_{n}^{l}(δ_{n}^{n}(x^{1} ⊗ ⋯ ⊗ x^{n−1}))(a^{0}, …, a^{n}),

for all x^{1}, …, x^{n−1} ∈ H and a^{0}, …, a^{n} ∈ A, so γ_{n}^{l}δ_{n}^{n} = d_{n}^{n}γ_{n−1}^{l}. Next consider the condition γ_{n}^{l}δ_{0}^{n} = d_{0}^{n}γ_{n−1}^{l}. We have

(d_{0}^{n}γ_{n−1}^{l})(x^{1} ⊗ ⋯ ⊗ x^{n−1})(a^{0}, …, a^{n})
= γ_{n−1}^{l}(x^{1} ⊗ ⋯ ⊗ x^{n−1})(a^{0}a^{1}, a^{2}, …, a^{n})
= h(a^{0}a^{1}(x^{1} ∗ a^{2}) ⋯ (x^{n−1} ∗ a^{n})) = h(a^{0}(ε ∗ a^{1})(x^{1} ∗ a^{2}) ⋯ (x^{n−1} ∗ a^{n}))

= γ_{n}^{l}(ε ⊗ x^{1} ⊗ ⋯ ⊗ x^{n−1})(a^{0}, …, a^{n}) = γ_{n}^{l}(δ_{0}^{n}(x^{1} ⊗ ⋯ ⊗ x^{n−1}))(a^{0}, …, a^{n}),

for all x^{1}, …, x^{n−1}H and a^{0}, …, a^{n}A, so γ_{n}^{l}δ_{0}^{n} = d_{0}^{n}γ_{n−1}^{l}.

We now look at the case when n = 1. To see that γ_{1}^{l}δ_{i}^{1} = d_{i}^{1}γ_{0}^{l}, for i = 0 and i = 1, observe that

γ_{1}^{l}(δ_{i}^{1}(1))(a^{0}, a^{1}) = h(a^{0}δ_{i}^{1}(1) ∗ a^{1}),

whereas

d_{i}^{1}(γ_{0}^{l}(1))(a^{0}, a^{1}) = d_{i}^{1}(h)(a^{0}, a^{1}),

for all a^{0}, a^{1} ∈ A. Since δ_{0}^{1}(1) = ε, we therefore get

γ_{1}^{l}(δ_{0}^{1}(1))(a^{0}, a^{1}) = h(a^{0}ε ∗ a^{1}) = h(a^{0}a^{1}) = d_{0}^{1}(h)(a^{0}, a^{1}),

for all a^{0}, a^{1} ∈ A, as wanted. Since δ_{1}^{1}(1) = f_{1}, we get

γ_{1}^{l}(δ_{1}^{1}(1))(a^{0}, a^{1}) = h(a^{0}(f_{1} ∗ a^{1})) = h((a^{1} ∗ f_{−1})a^{0})
= h(θ_{l}(a^{1})a^{0}) = d_{1}^{1}(h)(a^{0}, a^{1}),

for all a^{0}, a^{1} ∈ A, also as wanted.

Consider now the condition γ_{n}^{l}σ_{i}^{n} = s_{i}^{n}γ_{n+1}^{l}, for n ∈ ℕ_{0} and i ∈ {0, …, n}. We have

(s_{i}^{n}γ_{n+1}^{l})(x^{1} ⊗ ⋯ ⊗ x^{n+1})(a^{0}, …, a^{n})
= γ_{n+1}^{l}(x^{1} ⊗ ⋯ ⊗ x^{n+1})(a^{0}, …, a^{i}, I, a^{i+1}, …, a^{n})
= h(a^{0}(x^{1} ∗ a^{1}) ⋯ (x^{i} ∗ a^{i})(x^{i+1} ∗ I)(x^{i+2} ∗ a^{i+1}) ⋯ (x^{n+1} ∗ a^{n}))
= h(a^{0}(x^{1} ∗ a^{1}) ⋯ (x^{i} ∗ a^{i})I(x^{i+1})(x^{i+2} ∗ a^{i+1}) ⋯ (x^{n+1} ∗ a^{n}))
= γ_{n}^{l}(x^{1} ⊗ ⋯ ⊗ x^{i} ⊗ I(x^{i+1}) ⊗ x^{i+2} ⊗ ⋯ ⊗ x^{n+1})(a^{0}, …, a^{n}),
= γ_{n}^{l}(σ_{i}^{n}(x^{1} ⊗ ⋯ ⊗ x^{n+1}))(a^{0}, …, a^{n}),

for all x^{1}, …, x^{n+1} ∈ H and a^{0}, …, a^{n} ∈ A, so γ_{n}^{l}σ_{i}^{n} = s_{i}^{n}γ_{n+1}^{l}.

Consider the condition γ_{n}^{l}τ_{n} = t_{n}γ_{n}^{l}, for n ≥ 2. We have

(t_{n}γ_{n}^{l})(x^{1} ⊗ ⋯ ⊗ x^{n})(a^{0}, …, a^{n}) = γ_{n}^{l}(x^{1} ⊗ ⋯ ⊗ x^{n})(θ_{l}(a^{n}), a^{0}, …, a^{n−1})

= h((a^{n} ∗ f_{−1})(x^{1} ∗ a^{0}) ⋯ (x^{n} ∗ a^{n−1})) = h((x^{1} ∗ a^{0}) ⋯ (x^{n} ∗ a^{n−1})(f_{1} ∗ a^{n}))

= h(a^{0}S(x^{1}) ∗ ((x^{2} ∗ a^{1}) ⋯ (x^{n} ∗ a^{n−1})(f_{1} ∗ a^{n}))),

where we in the last step have used Proposition A.2. Say

Δ^{n−1}(S(x^{1})) = ∑_{i} y_{i}^{2} ⊗ ⋯ ⊗ y_{i}^{n} ⊗ z_{i}.

Then

(t_{n}γ_{n}^{l})(x^{1} ⊗ ⋯ ⊗ x^{n})(a^{0}, …, a^{n})

= ∑_{i} h(a^{0}(y_{i}^{2} ∗ (x^{2} ∗ a^{1})) ⋯ (y_{i}^{n} ∗ (x^{n} ∗ a^{n−1}))(z_{i} ∗ (f_{1} ∗ a^{n})))

= ∑_{i} h(a^{0}((y_{i}^{2}x^{2}) ∗ a^{1}) ⋯ ((y_{i}^{n}x^{n}) ∗ a^{n−1})((z_{i}f_{1}) ∗ a^{n}))

= γ_{n}^{l}(∑_{i} y_{i}^{2}x^{2} ⊗ ⋯ ⊗ y_{i}^{n}x^{n} ⊗ z_{i}f_{1})(a^{0}, …, a^{n}),

for all a^{0}, …, a^{n} ∈ A, so

(t_{n}γ_{n}^{l})(x^{1} ⊗ ⋯ ⊗ x^{n}) = γ_{n}^{l}(∑_{i} y_{i}^{2}x^{2} ⊗ ⋯ ⊗ y_{i}^{n}x^{n} ⊗ z_{i}f_{1})

= γ_{n}^{l}((∑_{i} y_{i}^{2} ⊗ ⋯ ⊗ y_{i}^{n} ⊗ z_{i})(x^{2} ⊗ ⋯ ⊗ x^{n} ⊗ f_{1}))

= γ_{n}^{l}(Δ^{n−1}(S(x^{1})) · (x^{2} ⊗ ⋯ ⊗ x^{n} ⊗ f_{1})) = γ_{n}^{l}(τ_{n}(x^{1} ⊗ ⋯ ⊗ x^{n})),

for all x^{1}, …, x^{n} ∈ H. Therefore γ_{n}^{l}τ_{n} = t_{n}γ_{n}^{l}, for n ≥ 2.

Also, we have γ_{1}^{l}τ_{1} = t_{1}γ_{1}^{l} because

(t_{1}γ_{1}^{l})(x^{1})(a^{0}, a^{1}) = γ_{1}^{l}(x^{1})((a^{1} ∗ f_{−1})a^{0}) = h((a^{1} ∗ f_{−1})(x^{1} ∗ a^{0}))
= h((x^{1} ∗ a^{0})(f_{1} ∗ a^{1})) = h(a^{0}S(x^{1}) ∗ (f_{1} ∗ a^{1})) = h(a^{0}(S(x^{1})f_{1}) ∗ a^{1})
= γ_{1}^{l}(S(x^{1})f_{1})(a^{0}, a^{1}) = γ_{1}^{l}(τ_{1}(x^{1}))(a^{0}, a^{1}),

for all x^{1} ∈ H and a^{0}, a^{1} ∈ A.

The condition γ_{0}^{l}τ_{0} = t_{0}γ_{0}^{l} is true because (γ_{0}^{l}τ_{0})(1) = γ_{0}^{l}(1) = h and

(t_{0}γ_{0}^{l})(1)(a^{0}) = t_{0}(h)(a^{0}) = hθ_{l}(a^{0}) = h(a^{0}),

for all a^{0} ∈ A. This finally shows that all the conditions involving the basic operations for the Λ-modules are satisfied, and thus γ^{l} is a Λ-module map. The statements regarding γ^{r} corresponding to the H_{op}-action ▷_{r} are proved in a similar fashion. In order to check the condition γ_{n}^{r}τ_{n} = t_{n}γ_{n}^{r}, one needs to use the formula S^{2}(x) = f_{1}xf_{−1}, for all x ∈ H, in conjunction with Proposition A.2 involving ▷_{r}. This completes the proof.

B The Bicrossproduct Hopf Algebra H ▷◁ H_{op}

The tensor product Hopf algebra (H̃, Δ̃), from Theorem 4.2, is a special case of a left-right bicrossproduct H ▷◁ H_{op} [17]. In this section we define what is mean by this. We also discuss the Hopf algebra H(G) appearing in the calculations of the transverse index theorem of Connes and Moscovici [3].

Let (H, Δ) be a Hopf algebra with counit ε, coinverse S, multiplication m : H⊗H → H and unit I. Let ad : H_{op} ⊗H → H be the left action of (H_{op}, Δ) on the algebra H defined by ad(x ⊗ y) = ∑ S(x_{(1)})yx_{(2)}, where x, y ∈ H. Note that ad(x ⊗ I) = ε(x)I, for x ∈ H. Hence H is a left H_{op}-module algebra with respect to the action ad [17].

According to Proposition 1.6.6 in [17], one may thus form a left cross product algebra H ▷◁_{left} H_{op} built on the vector space H ⊗ H_{op} with product m_{l} given by

m_{l}((x ⊗ y) ⊗ (x′ ⊗ z)) = ∑ x ad(y_{(1)} ⊗ x′) ⊗ y_{(2)}z,

where x, x′ ∈ H and y, z ∈ H_{op}. The unit element I_{l} of the algebra H ▷◁_{left} H_{op} is I ⊗ I. Define a linear map ad_{c} : H_{op} → H_{op} ⊗ H by

ad_{c} = (ι ⊗ m)(ι ⊗ S ⊗ ι)(F ⊗ ι)(Δ ⊗ ι)Δ.

This is a coaction of (H, Δ) on H_{op}, that is, the identities (ι ⊗ Δ)ad_{c} = (ad_{c} ⊗ ι)ad_{c} and (ι ⊗ ε)ad_{c} = ι hold. Also H_{op} is a right H-comodule coalgebra under the coaction ad_{c}, that is, the additional identities (Δ ⊗ ι)ad_{c} = (ι ⊗ ι ⊗ m)(ι ⊗ F ⊗ ι)(ad_{c} ⊗ ad_{c})Δ and (ε ⊗ ι)ad_{c} = ι hold.

According to Proposition 1.6.16 in [17], one may thus form a right cross coproduct coalgebra H ▷◁_{right} H_{op} built on the vector space H ⊗ H_{op} with coproduct Δ_{r} given by

Δ_{r}(x ⊗ y) = ∑ x_{(1)} ⊗ y_{(1)}^{(1)} ⊗ x_{(2)}y_{(1)}^{(2)} ⊗ y_{(2)},

where x ∈ H and y ∈ H_{op} with ad_{c}(y_{(1)}) = ∑ y_{(1)}^{(1)} ⊗ y_{(1)}^{2)}. Note also that the counit ε_{r} of H ▷◁_{right} H_{op} is given by ε_{r}(x ⊗ y) = ε(x)ε(y).

The maps ad and ad_{c} satisfy the compatibility conditions of Theorem 6.2.2 in [17], and therefore the product m_{l} from H ▷◁_{left} H_{op} and the coproduct Δ_{r} from H ▷◁_{right} H_{op} turn the vector space H ⊗ H_{op} into a Hopf algebra H ▷◁ H_{op} called a left-right bicrossproduct Hopf algebra. The coinverse S_{r} of H ▷◁ H_{op} is given by

S_{r}(x ⊗ y) = ∑(I ⊗ S^{−1}(y^{(1)}))(S(xy^{(2)}) ⊗ I),

where x ∈ H and y ∈ H_{op} with ad_{c}(y) = ∑ y^{(1)} ⊗ y^{(2)}. Thus, the coinverse S_{r} may alternatively be written S_{r} = (S ⊗ S^{−1})(m ⊗ ι)(ι ⊗ F)(ι ⊗ ad_{c}).

For the sake of convenience, we spell out the product m_{l} and coproduct Δ_{r} of H ▷◁ H_{op}. Recall that as a vector space H ▷◁ H_{op} is just H ⊗ H_{op}. Suppose x, z ∈ H and y, w ∈ H_{op}. Then

m_{l}((x ⊗ y) ⊗ (z ⊗ w)) = ∑ xS(y_{(1)})zy_{(2)} ⊗ y_{(3)}w,
Δ_{r}(x ⊗ y) = ∑ x_{(1)} ⊗ y_{(2)} ⊗ x_{(2)}S(y_{(1)})y_{(3)} ⊗ y_{(4)}

and

S_{r}(x ⊗ y) = ∑ S(xS(y_{(1)})y_{(3)}) ⊗ S^{−1}(y_{(2)}).

The Hopf algebra H ▷◁ H_{op} has a simpler description [17], namely, the map E : H ⊗ H_{op} → H ▷◁ H_{op} defined by E(x ⊗ y) = (x ⊗ I)Δ(y), where x ∈ H and y ∈ H_{op}, is a Hopf algebra isomorphism from the tensor product Hopf algebra (H̃, Δ̃) to H ▷◁ H_{op} with inverse E^{−1}(x ⊗ y) = (x ⊗ I)(S ⊗ ι)Δ(y), and it restricts to embeddings of (H, Δ) and (H_{op}, Δ) into H ▷◁ H_{op}.

Suppose, for i = 1, 2, G_{i} are finite subgroups of a group G such that any g ∈ G admits a decomposition as g = ka for unique k ∈ G_{1} and a ∈ G_{2}. Let π_{i} : G → G_{i} be the canonical epimorphisms. Define a right action α : G_{2} × G_{1} → G_{2} of G_{1} on G_{2} by α(a, k) = α_{k}(a) = π_{2}(ak) for a ∈ G_{2} and k ∈ G_{1}. Similarly define a left action β : G_{2} × G_{1} → G_{1} of G_{2} on G_{1} by β(a, k) = β_{a}(k) = π_{1}(ak) for a ∈ G_{2} and k ∈ G_{1}.

With respect to the actions α and β, the groups (G_{2}, G_{1}) form a right-left matched pair [17], that is, the actions α and β satisfy the following compatibility conditions:

α_{k}(1) = 1, α_{k}(a_{1}a_{2}) = α_{β_{a_{2}} (k)}(a_{1})α_{k}(a_{2}),

and

β_{a}(1) = 1, β_{a}(k_{1}k_{2}) = β_{a}(k_{1})β_{α_{k_{1}} (a)}(k_{2}),

where k, k_{i} ∈ G_{1} and a, a_{i} ∈ G_{2}.

According to Example 6.2.11 in [17], the induced action α̃ of ℂ[G_{2}] on C(G_{1}) and the induced coaction β̃ of C(G_{1}) on ℂ[G_{2}], give a left-right bicrossproduct Hopf algebra C(G_{1}) ▷◁ ℂ[G_{2}] as described in Theorem 6.2.2 in [17]. Let δ_{s} ∈ C(G_{1}) and ε_{u} ∈ ℂ[G_{2}], where s ∈ G_{1} and u ∈ G_{2}, be given by δ_{s}(t) = 1, if t ∈ G_{1} equals s and is zero otherwise, whereas ε_{u}(v) = 1, if v ∈ G_{2} equals u and is zero otherwise. Thus {δ_{s} ⊗ ε_{u} | s ∈ G_{1}, u ∈ G_{2} } is a basis of the vector space C(G_{1}) ⊗ ℂ[G_{2}]. The product m of C(G_{1}) ▷◁ ℂ[G_{2}] is now explicitly given by

m((δ_{s} ⊗ ε_{u}) ⊗ (δ_{t} ⊗ ε_{v})) = δ_{α_{u}(s),t}(δ_{s} ⊗ ε_{uv})

and the coproduct Δ of C(G_{1}) ▷◁ ℂ[G_{2}] is given by

Δ(δ_{s} ⊗ ε_{u}) = ∑_{ab=s} δ_{a} ⊗ ε_{β_{b}(u)} ⊗ δ_{b} ⊗ ε_{u},

where s, t ∈ G_{1} and u, v ∈ G_{2}. Here δ_{s,t} is the number which is one if s = t and is zero otherwise.

The left-right bicrossproduct Hopf algebra C(G_{1}) ▷◁ ℂ[G_{2}] defined above is isomorphic to the Hopf algebra H(G) defined in Section 5 in [3]. In the latter paper, the notation a · k = α_{k}(a) and a(k) = β_{a}(k) and X_{k} = ε_{k} and ε_{a} = δ_{a} for k ∈ G_{1} and a ∈ G_{2}, is used, and the same product but the opposite coproduct is defined.

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Authors: Johan Kustermans, Department of Mathematics, K.U. Leuven, Belgium, E-mail: Johan.Kustermanswis.kuleuven.ac.be John Rognes, Department of Mathematics, University of Oslo, Oslo, Norway, E-mail: rognesmath.uio.no Lars Tuset, Faculty of Engineering, Oslo University College, Norway, E-mail: Lars.Tusetiu.hio.no