Gerstenhaber-Schack and Hochschild Cohomologies

of Hopf Algebras

Julien Bichon

Received: May 21, 2015 Revised: December 17, 2015

Communicated by Joachim Cuntz

Abstract. We show that the Gerstenhaber-Schack cohomology of a Hopf algebra determines its Hochschild cohomology, and in particular its Gerstenhaber-Schack cohomological dimension bounds its Hochschild cohomological dimension, with equality of the dimensions when the Hopf algebra is cosemisimple of Kac type. Together with some general considerations on free Yetter-Drinfeld modules over adjoint Hopf subalgebras and the monoidal invariance of Gerstenhaber-Schack cohomology, this is used to show that both Gerstenhaber-Schack and Hochschild cohomological dimensions of the coordinate algebra of the quantum permutation group are 3.

2010 Mathematics Subject Classification: 16T05, 16E40, 16E10 Keywords and Phrases: Hopf algebra, cohomology, cohomological dimension, Yetter-Drinfeld module

1 Introduction

We study homological properties of Hopf algebras by using Yetter-Drinfeld modules and tensor category techniques. We are especially interested in the following question:

Question 1.1. If A and B are Hopf algebras having equivalent tensor categories of comodules, how are their Hochschild cohomologies related? In particular do A and B have the same cohomological dimension?

We have seen in [10] that the Hochschild cohomologies of two such Hopf algebras A and B are indeed closely related, using resolutions of the trivial Yetter-Drinfeld module over A (or over B) formed by free Yetter-Drinfeld modules. In the present paper we continue this study along the same line of ideas. Our first remark in view of Question 1.1 is that there exists at least a cohomology theory for Hopf algebras that is known to be well-behaved with respect to this situation: Gerstenhaber-Schack cohomology [27, 28]. Let A be a Hopf algebra and let M be a Hopf bimodule over A: the Gerstenhaber-Schack cohomology H_{GS}^{∗}(A, M) of A with coefficients in M [28] is defined to be the homology of an explicit bicomplex whose columns are modeled on the Hochschild complex of the underlying algebra and rows are modeled on the Cartier complex of the underlying coalgebra. When M = A is the trivial Hopf bimodule, then H_{GS}^{∗}(A, A) =: H_{b}^{∗}(A) is known as the bialgebra cohomology of A. This cohomology theory, which can also be defined in terms of Yetter-Drinfeld modules, was first introduced in view of applications to deformation theory, and has been used as a key tool in the proof of several fundamental results on finite-dimensional Hopf algebras [55, 23]. If A and B are Hopf algebras having equivalent tensor categories of comodules, then there exists a tensor equivalence F : _{A}^{A}M_{A}^{A} → _{B}^{B}M_{B}^{B} between their categories of Hopf bimodules such that for any Hopf bimodule M over A, we have H_{GS}^{∗}(A, M) ≃ H_{GS}^{∗}(B, F (M)), and in particular H_{b}^{∗}(A) ≃ H_{b}^{∗}(B) and cd_{GS}(A) = cd_{GS}(B) (where cd_{GS} denotes the Gerstenhaber-Schack cohomological dimension, defined in the obvious way, see Section 5). We call these properties the monoidal invariance of Gerstenhaber-Schack cohomology. Going back to Question 1.1, the next question is to study how Hochschild and Gerstenhaber-Schack cohomologies are related. We show that the Gerstenhaber-Schack cohomology of a Hopf algebra determines its Hochschild cohomology. More precisely, we show that if A is a Hopf algebra, then there exists a functor G :_{A}M_{A} →_{A}^{A}M_{A}^{A} such that for any A-bimodule M, we have

H^{∗}(A, M) ≃ H_{GS}^{∗}(A, G(M))

In particular we have cd(A) ≤ cd_{GS}(A). Then if A and B are Hopf algebras as in Question 1.1, combining this with the monoidal invariance of Gerstenhaber-Schack cohomology, we get the existence of two functors F_{1} :_{A}M_{A} → _{B}^{B}M_{B}^{B} and F_{2} :_{B}M_{B} →_{A}^{A}M_{A}^{A} such that for any A-bimodule M and any B-bimodule N, we have

H^{∗}(A, M) ≃ H_{GS}^{∗}(B, F_{1}(M)) and H^{∗}(B, N) ≃ H_{GS}^{∗}(A, F_{2}(N))

In particular

max(cd(A), cd(B)) ≤ cd_{GS}(A) = cd_{GS}(B)

These isomorphisms and this inequality thus provide partial answers to Question 1.1. They lead to the following new question:

Question 1.2. Is it true that cd(A) = cd_{GS}(A) for any Hopf algebra A over a field of characteristic zero? Is it true at least if A is assumed to be cosemisimple?

A positive answer would give the monoidal invariance of cohomological dimension and fully answer the last part of Question 1.1, and would also be a natural infinite-dimensional generalization of a famous result by Larson-Radford [38], which states that, in characteristic zero, a finite-dimensional cosemisimple Hopf algebra is semisimple. See Remark 5.8. We provide (Corollary 5.10) a partial positive answer to Question 1.2 in the case where A is cosemisimple of Kac type (the square of the antipode is the identity), and in turn this gives a partial positive answer to Question 1.1 (see Corollary 5.11). We then apply our general considerations to quantum symmetry Hopf algebras, which were the first motivation for this work. Let (R, φ) semisimple measured algebra of dimension ≥ 4, and let A_{aut}(R, φ) be its quantum symmetry Hopf algebra [61, 8]. We compute, in the cosemisimple case, the bialgebra cohomology of A_{aut}(R, φ), and we show that cd(A_{aut}(R, φ)) ≤ cd_{GS}(A_{aut}(R, φ)) = 3, with equality if φ is a trace. These results include in particular the coordinate algebra of Wang’s quantum permutation group S_{n}^{+} [61]. As a last comment to further motivate the use of Gerstenhaber-Schack cohomology as an appropriate cohomology theory for Hopf algebras (apart from its use to get information on Hochschild cohomology itself), we would like to point out that, in the examples computed so far, it also has the merit to avoid the “dimension drop” phenomenon usually encountered for quantum algebras (see [31, 32]): the canonical choice of coefficients (the trivial Hopf bimodule) is the good one to get the cohomological dimension. It would be interesting to know if this can be further generalized. The paper is organized as follows. Section 2 consists of preliminaries. In Section 3 we discuss the cohomological dimension of a Hopf subalgebra and the sub-additivity of the cohomological dimension under extensions. Section 4 is devoted to Yetter-Drinfeld modules: we recall the concept of free (resp. cofree) Yetter Drinfeld module and we introduce the notion of relative projective (resp. injective) Yetter-Drinfeld module, which corresponds, via the tensor equivalence between Yetter-Drinfeld modules and Hopf bimodules [49], to the notion of relative projective (resp. injective) Hopf bimodule considered in [53]. We show that relative projective (resp. injective) Yetter-Drinfeld modules are precisely the direct summands of free (resp. co-free) Yetter-Drinfeld modules. This section also contains some considerations on free Yetter-Drinfeld modules over adjoint Hopf subalgebras. In Section 5, after having recalled some basic facts on Gerstenhaber-Schack cohomology, we provide an explicit complex that computes the Gerstenhaber-Schack cohomology H_{GS}^{∗}(A, V), if A is cosemisimple or if the Yetter-Drinfeld module V is relative injective, using results from [53] in this last case (see Proposition 5.3). We then show that Gerstenhaber-Schack cohomology determines Hochschild cohomology, and show that Question 1.2 has a positive answer in the case of cosemisimple Hopf algebras of Kac type. In Section 6 we study the examples mentioned earlier in the introduction. In Section 7 we discuss the Gerstenhaber-Schack cohomological dimension in the setting of Hopf algebras having a projection.

2 Preliminaries

In this preliminary section we fix some notation, we recall some basic definitions and facts on the Hochschild cohomology of a Hopf algebra, and we discuss exact sequences of Hopf algebras.

2.1 Notations and conventions

We work over ℂ (or over any algebraically closed field of characteristic zero). This assumption does not affect any of the theoretical results in the paper, but is important for the examples we consider. We assume that the reader is familiar with the theory of Hopf algebras and their tensor categories of comodules, as e.g. in [34, 35, 42]. If A is a Hopf algebra, as usual, Δ, ε and S stand respectively for the comultiplication, counit and antipode of A. We use Sweedler’s notations in the standard way. The category of right A-comodules is denoted M^{A}, the category of right A-modules is denoted M_{A}, etc... The trivial (right) A-module is denoted ℂ_{ε}. The set of A-module morphisms (resp. A-comodule morphisms) between two A-modules (resp. two A-comodules) V and W is denoted Hom_{A}(V, W) (resp. Hom^{A}(V, W)).

2.2 Hochschild cohomology of a Hopf algebra

If A is an algebra and M is an A-bimodule, then H^{∗}(A, M) denotes, as usual, the Hochschild cohomology of A with coefficients in M. See e.g. [62].

Definition 2.1. The Hochschild cohomological dimension of an algebra A is defined to be

cd(A) = sup{n : H^{n}(A, M) ≠ 0 for some A − bimodule M} ∈ ℕ ∪ {∞}

As noted by several authors (see [25], [29], [31], [13], [17], [10]), the Hochschild cohomology of a Hopf algebra can be described by using a suitable Ext functor on the category of left or right A-modules. Indeed, if A is a Hopf algebra and M is an A-bimodule, we have

H^{∗}(A, M) ≃ Ext_{A}^{∗}(ℂ_{ε}, M′)

where the above Ext is in the category of right A-modules and M′ is M equipped with the right A-module structure given by x ← a = S(a_{(1)}) · x · a_{(2)}. This leads to the following description of the cohomological dimension of a Hopf algebra.

Proposition 2.2. Let A be a Hopf algebra. We have

cd(A) = sup{n : Ext_{A}^{n}(ℂ_{ε}, M) ≠ 0 for some A − module M}

= inf{n : Ext_{A}^{i}(ℂ_{ε}, −) = 0 for i > n}

= inf{n : ℂ_{ε} admits a projective resolution of length n}

Proof. The previous isomorphism ensures that

cd(A) ≤ sup{n : Ext_{A}^{n}(ℂ_{ε}, M) ≠ 0 for some A − module M}

If V is a right A-module, let _{ε}V be the A-bimodule whose right structure is that of V and whose left structure is trivial, i.e. given by ε. Then (_{ε}V)′ = V, hence the converse inequality holds, and the first equality in the statement is proved, as well as the second one. The last one is shown similarly as in the case of group cohomology, see e.g. [14, Chapter VIII, Lemma 2.1].

Examples 2.3. 1. If G is a linear algebraic group, with coordinate algebra O(G), it is well-known that cd(O(G)) = dim G.

2. If Γ is a (discrete) group, then cd(ℂΓ) = cd_{ℂ}(Γ), the cohomological dimension of Γ with coefficients ℂ. We have cd(ℂΓ) = 0 if and only if Γ is finite, and if Γ is finitely generated, then cd(ℂΓ) = 1 if and only if Γ contains a free normal subgroup of finite index, see [22].

3. If A is a finite-dimensional Hopf algebra, then either cd(A) = 0 (when A is semisimple) or cd(A) = ∞, a finite-dimensional Hopf algebra being Frobenius and hence self-injective.

2.3 Exact sequences of Hopf algebras

A sequence of Hopf algebra maps

ℂ → B →^{i} A →^{p} L → ℂ

is said to be exact [3] if the following conditions hold:

1. i is injective and p is surjective,

2. ker p = Ai(B)^{+} = i(B)^{+}A, where i(B)^{+} = i(B) ∩ Ker(ε),

3. i(B) = A^{co L} = {a ∈ A : (id ⊗ p)Δ(a) = a ⊗ 1} = ^{co L}A = {a ∈ A : (p ⊗ id)Δ(a) = 1 ⊗ a}.

Note that condition (2) implies pi = ε1.

Proposition 2.4. Let

ℂ → B →^{i} A →^{p} L → ℂ

be a sequence of Hopf algebra maps where i is injective, p is surjective and pi = ε1. Assume that the antipode of A is bijective. Consider the following three assertions.

1. A is faithfully flat as a right B-module and Ker(p) = Ai(B)^{+} = i(B)^{+}A.

2. ^{co L}A = A^{co L} = i(B) and p is left or right faithfully coflat.

3. The sequence is exact.

Then we have (1) ⇒ (3) and (2) ⇒ (3), and if (3) holds, then we have (1) ⟺ (2).

An exact sequence satisfying (1) and (2) is called strict [52]. Note that p is automatically faithfully coflat if L is cosemisimple. That (1) ⇒ (3) holds is well-known (see [3, Proposition 1.2.4], [51, Lemma 1.3], [42, Proposition 3.4.3], or more generally [59, Theorem 1]). Also that (1) ⟺ (2) if (3) holds is known, see [51, Corollary 1.8]. I wish to thank the referee for pointing out that (2) ⇒ (3) follows from [59, Theorem 2], combined with [44, Remark 1.3].

3 Cohomological dimension of a Hopf subalgebra

In this section we discuss the behavior of cohomological dimension when passing to a Hopf subalgebra, which, under mild assumptions, is similar to the group cohomology case.

Proposition 3.1. Let B ⊂ A be a Hopf subalgebra. Assume that one of the following conditions holds.

1. A is projective as a right B-module.

2. The antipode of A is bijective and A is faithfully flat as a right B-module.

3. A is cosemisimple.

4. The exists a Hopf algebra map π : A → B such that π_{|B} = id_{B}.

5. The antipode of A is bijective and B is commutative.

Then cd(B) ≤ cd(A).

Proof. If A is projective as a right B-module, any projective right A-module is projective as a right B-module, thus a resolution of length n of ℂ_{ε} in M_{A} yields a resolution of length n in M_{B}, and thus Proposition 2.2 ensures that cd(B) ≤ cd(A). Assuming (2), Corollary 1.8 in [51] yields that A is projective as a right B-module, and we conclude by (1). If we assume that A is cosemisimple, then its antipode is bijective and by [16] A is faithfully flat as a right B-module, and we conclude by (2). If we assume (4), then A is free as a right B-module, see [47] (we will come back to this situation in Section 7), thus we conclude by (1). If B is commutative, then A is faithfully flat over B by Proposition 3.12 in [4], and again we conclude by (2).

The following result is the generalization of the sub-additivity of cohomological dimension under extensions (see e.g. Proposition 2.4 in [14]) with essentially the same proof, using Stefan’s spectral sequence [54] as the natural generalization of the Hochschild-Serre spectral sequence.

Proposition 3.2. Let

ℂ ⟶ B ⟶^{i} A ⟶^{p} L ⟶ ℂ

be a strict exact sequence of Hopf algebras , and assume that the antipode of A is bijective. Then we have cd(B) ≤ cd(A) ≤ cd(B) + cd(L). If moreover L is semisimple, then cd(B) = cd(A).

Proof. By [51, Lemma 1.3], (or more generally [59, Theorem 1], see also [42, Proposition 3.4.3]), the canonical map

A ⊗_{B} A ⟶ A ⊗ L
a ⊗_{B} a′ ⟼ aa′_{(1)} ⊗ p(a′_{(2)})

is bijective. Thus B ⊂ A is an L-Galois extension, and A is faithfully flat both as a left and right B-module (the antipode of A is bijective). Thus for any A-A-bimodule M there exists a spectral sequence [54]

E_{2}^{pq} = H^{p}(L; H^{q}(B, M)) ⇒ H^{p+q}(A, M)

The spectral sequence is concentrated in the rectangle 0 ≤ p ≤ cd(L), 0 ≤ q ≤ cd(B), and it follows that for i > cd(L) + cd(B), we have H^{i}(A, M) = 0, and this proves the inequality. If L is semisimple, then cd(L) = 0, and hence cd(B) = cd(A).

4 Yetter-Drinfeld modules

Let A be a Hopf algebra. Recall that a (right-right) Yetter-Drinfeld module over A is a right A-comodule and right A-module V satisfying the condition, ∀v ∈ V, ∀a ∈ A,

(v ← a)_{(0)} ⊗ (v ← a)_{(1)} = v_{(0)} ← a_{(2)} ⊗ S(a_{(1)})v_{(1)}a_{(3)}

The category of Yetter-Drinfeld modules over A is denoted YD_{A}^{A}: the morphisms are the A-linear A-colinear maps. Endowed with the usual tensor product of modules and comodules, it is a tensor category, with unit the trivial Yetter-Drinfeld module, denoted ℂ. An important example of Yetter-Drinfeld module is the right coadjoint Yetter-Drinfeld module A_{coad}: as a right A-module A_{coad} = A and the right Acomodule structure is defined by

ad_{r}(a) = a_{(2)} ⊗ S(a_{(1)})a_{(3)}, ∀a ∈ A

The coadjoint Yetter-Drinfeld module has a natural generalization, discussed in the next subsection.

4.1 Free and co-free Yetter-Drinfeld modules

We now discuss some important constructions of Yetter-Drinfeld modules (leftright versions of these constructions were first given in [15], see [53] as well, in the context of Hopf bimodules). Let V be a right A-comodule. The Yetter-Drinfeld module V ⊠ A is defined as follows [10]. As a vector space V ⊠ A = V ⊗ A, the right module structure is given by multiplication on the right, and the right coaction α_{V ⊠A} is defined by

α_{V ⊠A}(v ⊗ a) = v_{(0)} ⊗ a_{(2)} ⊗ S(a_{(1)})v_{(1)}a_{(3)}

Note that A_{coad} = ℂ ⊠ A. A Yetter-Drinfeld module is said to be free if it is isomorphic to V ⊠A for some comodule V. The construction of the free Yetter-Drinfeld module on a comodule yields a functor L = − ⊠ A : M^{A} ⟶ YD_{A}^{A} which is left adjoint to the forgetful functor R : YD_{A}^{A} ⟶ M^{A} . Indeed we have natural isomorphisms

Hom^{A}(V, R(X)) ⟶ Hom_{YD}_{A}^{A}(V ⊠ A, X)
f ⟼ f̃, f̃(v ⊗ a) = f(v) ← a

for any A-comodule V and any Yetter-Drinfeld module X. Now let M be a right A-module. The Yetter-drinfeld module M#A is defined as follows: the underlying vector space is M ⊗A, the right coaction is id_{M} ⊗Δ, while the right action is given by

(x ⊗ a) ← b = x · b_{(2)} ⊗ S(b_{(1)})ab_{(3)}

The Yetter-Drinfeld module ℂ#A is the adjoint Yetter-Drinfeld module, denoted A_{ad}. A Yetter-Drinfeld module will be said to be co-free if it is isomorphic to M#A for some module M. The construction of the co-free Yetter-Drinfeld module on a module yields a functor L = −#A : M_{A} ⟶ YD_{A}^{A} which is right adjoint to the forgetful functor L : YD_{A}^{A} ⟶ M_{A}. Indeed we have natural isomorphisms

Hom_{YD}_{A}^{A}(X, M#A) ⟶ Hom_{A}(L(X), M)
f ⟼ (id_{M} ⊗ ε)f,

for any A-module M and any Yetter-Drinfeld module X.

4.2 Relative projective and relative injective Yetter-Drinfeld modules

We will use the following notions.

Definition 4.1. Let V be a Yetter-Drinfeld module over A. Then V is said to be relative projective if the functor Hom_{YD}_{A}^{A}(V, −) transforms exact sequences of Yetter-Drinfeld modules that split as sequences of comodules to exact sequences of vector spaces. Similarly V is said to be relative injective if the functor Hom_{YD}_{A}^{A}(−, V) transforms exact sequences of Yetter-Drinfeld modules that split as sequences of modules to exact sequences of vector spaces.

Relative projective Yetter-Drinfeld modules have the following characterization.

Proposition 4.2. Let P be a Yetter-Drinfeld module over A. The following assertions are equivalent.

1. P is relative projective.

2. Any surjective morphism of Yetter-Drinfeld modules f : M → P that admits a section which is a map of comodules admits a section which is a morphism of Yetter-Drinfeld modules.

3. P is a direct summand of a free Yetter-Drinfeld module.

If A is cosemisimple, these conditions are equivalent to P being a projective object of YD_{A}^{A}.

Proof. The proof of (1)⇒(2) is similar to the usual one for modules. Assume (2), and consider the surjective Yetter-Drinfeld module morphism R(P) ⊠ A → P, x ⊗ a ↦ x ← a. The map P → R(P) ⊠ A, x ↦ x ⊗ 1 is an A-colinear section, so by (2) P is indeed, as a Yetter-Drinfeld module, a direct summand of R(P) ⊠ A. Assume now that P is free, i.e. P = V ⊠ A for some comodule V, and let

0 → M →^{i} N →^{p} Q → 0

be an exact sequence of Yetter-Drinfeld modules that splits as a sequence of comodules. The sequence

0 → Hom_{YD}_{A}^{A}(P, M) ⟶^{i−} Hom_{YD}_{A}^{A}(P, N) ⟶^{p−} Hom_{YD}_{A}^{A}(P, Q)

is exact and we have to show the surjectivity of the map on the right. Let s : Q → N be a morphism of comodules such that ps = id_{Q}. Let φ ∈ Hom_{YD}_{A}^{A}(V ⊠A, Q), and let φ_{0} : V → Q be defined by φ_{0}(v) = φ(v⊗1): φ_{0} is a map of comodules, and so is sφ_{0}. Considering now s̃φ_{0} ∈ Hom_{YD}_{A}^{A}(V ⊠ A, N), we have ps̃φ_{0} = φ, which gives the expected surjectivity result. Now if V ⊠ A ≃ P ⊕ M as Yetter-Drinfeld modules, then Hom_{YD}_{A}^{A}(V ⊠ A, −) ≃ Hom_{YD}_{A}^{A}(P, −) ⊕ Hom_{YD}_{A}^{A}(M, −), and the usual argument for projective modules work to conclude that P is relative projective. It is clear that a projective Yetter-Drinfeld module is relative projective, and if A is cosemisimple, a free Yetter-Drinfeld module is a projective object in YD_{A}^{A} (Proposition 3.3 in [10]), hence a direct summand of a free Yetter-Drinfeld module is projective, and so is a relative projective Yetter-Drinfeld module.

Similarly, relative injective Yetter-Drinfeld modules are characterized as follows. The proof is similar to the one of the previous result, and is left to the reader.

Proposition 4.3. Let I be a Yetter-Drinfeld module over A. The following assertions are equivalent.

1. I is relative injective.

2. Any injective morphism of Yetter-Drinfeld modules f : I → M that admits a section which is a map of modules admits a section which is a morphism of Yetter-Drinfeld modules.

3. P is a direct summand of a co-free Yetter-Drinfeld module.

4.3 Yetter-Drinfeld modules and Hopf bimodules

In this subsection we briefly recall the category equivalence between Yetter-Drinfeld modules and Hopf bimodules [49], and check that the notion of relative projective objects (resp. relative injective objects) for Yetter-Drinfeld modules corresponds to that for Hopf bimodules considered in [53]. First recall that a Hopf bimodule over A is an A-bimodule and A-bicomodule M whose respective left and right coactions λ : M → A ⊗ M and ρ : M → M ⊗ A are A-bimodule maps. The category of Hopf bimodules over A, whose morphisms are the bimodule and bicomodule maps, is denoted _{A}^{A}M_{A}^{A}. If M is Hopf bimodule over A, then ^{coA}M = {x ∈ M | λ(x) = 1 ⊗ x} is a right subcomodule of M, and inherits a right A-module structure given by x ← a = S(a_{(1)}).x.a_{(2)}, making it into a Yetter-Drinfeld module over A. This defines a functor

_{A}^{A}M_{A}^{A} ⟶ YD_{A}^{A}
M ⟼ ^{coA}M

Conversely, starting from a Yetter-Drinfeld module V, one defines a Hopf bimodule structure on A ⊗ V as follows. The bimodule structure is given by

a.(b ⊗ v).c = abc_{(1)} ⊗ (v ← c_{(2)})

and the bicomodule structure is given by the following left and right coactions

λ : A ⊗ V ⟶ A ⊗ A ⊗ V  ρ : A ⊗ V ⟶ A ⊗ V ⊗ A
a ⊗ v ⟼ a_{(1)} ⊗ a_{(2)} ⊗ v  a ⊗ v ⟼ a_{(1)} ⊗ v_{(0)} ⊗ a_{(2)}v_{(1)}

If f : V ⟶ W is a morphism of Yetter-Drinfeld module, then id_{A}⊗f : A⊗V → A ⊗ W is a morphism of Hopf bimodules, and hence we get a functor

YD_{A}^{A} ⟶_{A}^{A}M_{A}^{A}
V ⟼ A ⊗ V

The two functors just defined are quasi-inverse equivalences, see [49].

Lemma 4.4. Relative projective (resp. relative injective) objects in YD_{A}^{A} correspond, via the category equivalence YD_{A}^{A} ≃ _{A}^{A}M_{A}^{A}, to relative projective (resp. relative injective) objects of _{A}^{A}M_{A}^{A} in the sense of [53].

Proof. Let M be a Hopf bimodule over A. That M is relatively projective means that the functor Hom_{_{A}^{A}M}_{A}^{A}(M, −) transforms exact sequences of Hopf bimodules that split as sequences of bicomodules to exact sequences of vector spaces. The proof of the lemma easily reduces to the following statement. Let f : V → W be a surjective morphism of Yetter-Drinfeld modules, inducing a surjective morphism of Hopf bimodules id_{A} ⊗f : A⊗V → A⊗W. Then there exists an A-comodule section to f if and only if there exists an A-bicomodule section to id_{A} ⊗ f. Indeed, if s : W → V is A-colinear with fs = id_{W}, then id_{A}⊗s : A⊗W → A⊗V is A-bicolinear and is a section to id_{A} ⊗ f. Conversely starting with an Abicolinear map T : A ⊗ W → A ⊗ V with (id_{A} ⊗ f)T = id_{A⊗W}, then the map s : W → V defined by s(w) = ε ⊗ id_{V} (T (1 ⊗ w)) is A-colinear, and satisfies fs = id_{W}. In a similar manner, that M is relatively injective means that the functor Hom_{_{A}^{A}M}_{A}^{A}(−, M,) transforms exact sequences of Hopf bimodules that split as sequences of bimodules to exact sequences of vector spaces. The proof that this corresponds to the notion of relative injective Yetter-Drinfeld module is left to the reader.

4.4 Adjoint Hopf subalgebras

We now discuss the way to restrict certain free Yetter-Drinfeld to adjoint Hopf subalgebras.

Proposition 4.5. Let B ⊂ A be a Hopf subalgebra. The following assertions are equivalent.

1. For any a ∈ A and b ∈ B, we have

a_{(2)} ⊗ S(a_{(1)})ba_{(3)} ∈ A ⊗ B

2. For any B-comodule W, we have α_{V ⊠A}(W ⊠ A) ⊂ (W ⊠ A) ⊗ B so that W ⊠ A is an object of YD_{B}^{B}.

Proof. (1) ⇒ (2) follows from the definition of α_{V ⊠A}. Conversely, assuming that (2) holds, take W = B the regular B-comodule. Then for any a ∈ A and b ∈ B, we have

b_{(1)} ⊗ a_{(2)} ⊗ S(a_{(1)})b_{(2)}a_{(3)} ∈ A ⊗ A ⊗ B

and hence

a_{(2)} ⊗ S(a_{(1)})ba_{(3)} = a_{(2)} ⊗ S(a_{1})ε(b_{(1)})b_{(2)}a_{(3)} ∈ A ⊗ B

Thus (1) holds.

Definition 4.6. A Hopf subalgebra B ⊂ A is said to be adjoint if it satisfies the equivalent conditions of Proposition 4.5.

Very often adjoint Hopf subalgebras are obtained in the following way. Recall that a Hopf algebra map f : A → L is said to be cocentral if f(a_{(1)}) ⊗ a_{(2)} = f(a_{(2)}) ⊗ a_{(1)} for any a ∈ A.

Proposition 4.7. Let B ⊂ A be a Hopf subalgebra. Assume that there exists a cocentral and surjective Hopf algebra map p : A → L such that B = A^{coL}. Then B ⊂ A is an adjoint Hopf subalgebra. Conversely if B ⊂ A is an adjoint Hopf subalgebra, if A and B have bijective antipodes and if A is faithfully flat as a right B-module, then there exists a cocentral surjective Hopf algebra map p : A → L such that B = A^{coL}.

Proof. Let a ∈ A and b ∈ B. Since p(b) = ε(b)1, we have, using the cocentrality of p,

id_{A} ⊗ id_{B} ⊗ p (a_{(2)} ⊗ (S(a_{(1)})ba_{(3)})_{(1)} ⊗ (S(a_{(1)})ba_{(3)})_{(2)})
= id_{A} ⊗ id_{B} ⊗ p (a_{(3)} ⊗ S(a_{(2)})b_{(1)}a_{(4)} ⊗ S(a_{(1)})b_{(2)}a_{(5)})
= a_{(3)} ⊗ S(a_{(2)})b_{(1)}a_{(4)} ⊗ pS(a_{(1)})p(b_{(2)})p(a_{(5)})
= a_{(3)} ⊗ S(a_{(2)})ba_{(4)} ⊗ pS(a_{(1)})p(a_{(5)})
= a_{(2)} ⊗ S(a_{(1)})ba_{(3)} ⊗ 1

Hence a_{(2)} ⊗ S(a_{(1)})ba_{(3)} ∈ A ⊗ A^{coL} = A ⊗ B: this shows that B ⊂ A is adjoint. Conversely, assume that B ⊂ A is adjoint, that A and B have bijective antipodes and that A is faithfully flat as a right B-module. Then for any a ∈ A and b ∈ B, we have

S(a_{(1)})ba_{(2)} = ε(a_{(2)})ε(b_{(1)})S(a_{(1)})b_{(2)}a_{(3)} ∈ B

It is well-known that this implies B^{+}A ⊂ AB^{+}, and hence AB^{+} ⊂ B^{+}A by the bijectivity of the antipodes. It follows that B^{+}A is a Hopf ideal in A, and we denote by p : A → A/B^{+}A = L the canonical Hopf algebra surjection. By construction we have B ⊂ A^{coL}, and for b ∈ B we have p(b) = ε(b). Hence we have for any a ∈ A, a ⊗ 1 = a_{(2)} ⊗ p(S(a_{(1)})a_{(3)}), hence

a_{(2)} ⊗ p(a_{(1)}) = (1 ⊗ p(a_{(1)}))(a_{(2)} ⊗ 1)
= (1 ⊗ p(a_{(1)})(a_{(3)} ⊗ p(S(a_{(2)})a_{(4)})) = a_{(1)} ⊗ p(a_{(2)})

and this shows that p is cocentral. Finally we have B = A^{coL} by Corollary 1.8 in [51].

We now discuss a condition that ensures that the restriction of a free Yetter-Drinfeld module to an adjoint Hopf subalgebra as in Proposition 4.5 remains a relative projective Yetter-Drinfeld module.

Proposition 4.8. Let B ⊂ A be a Hopf subalgebra with B = A^{coL} for some cocentral and surjective Hopf algebra map p : A → L. Assume that there exists a linear map σ : L → A such that

1. pσ = id_{L};

2. σ(x)_{(1)} ⊗ p(σ(x)_{(2)}) = σ(x_{(1)}) ⊗ x_{(2)}, for any x ∈ L;

3. σ(x)_{(1)}S(σ(x)_{(3)}) ⊗ σ(x)_{(2)} = 1_{B} ⊗ σ(x), for any x ∈ L.

Then for any B-comodule W, the object W ⊠ A ∈ YD_{B}^{B} is relative projective. Such a map σ exists if A is cosemisimple.

Proof. We first claim that for any a ∈ A, we have

σp(a_{(1)})_{(1)} ⊗ S(σp(a_{(1)})_{(2)})a_{(2)} ∈ A ⊗ B

For any x ∈ L, we have, by (2)

σ(x)_{(1)} ⊗ σ(x)_{(2)} ⊗ p(σ(x)_{(3)}) = σ(x_{(1)})_{(1)} ⊗ σ(x_{(1)})_{(2)} ⊗ x_{(2)}

and hence for any a ∈ A

σp(a)_{(1)} ⊗ σp(a)_{(2)} ⊗ p(σp(a)_{(3)}) = σp(a_{(1)})_{(1)} ⊗ σp(a_{(1)})_{(2)} ⊗ p(a_{(2)})

We have

(id_{A} ⊗ p ⊗ id_{A})(id_{A} ⊗ Δ) (σp(a_{(1)})_{(1)} ⊗ S(σp(a_{(1)})_{(2)})a_{(2)})
= σp(a_{(1)})_{(1)} ⊗ Sp(σp(a_{(1)})_{(3)})p(a_{(2)}) ⊗ S(σp(a_{(1)})_{(2)})a_{(3)}
= σp(a_{(1)})_{(1)} ⊗ Sp(a_{(2)})p(a_{(3)}) ⊗ S(σp(a_{(1)})_{(2)})a_{(4)}
= σp(a_{(1)})_{(1)} ⊗ 1 ⊗ S(σp(a_{(1)})_{(2)})a_{(2)}

and this proves our claim. We thus get for any B-comodule W, a linear map

ι : W ⊠ A ⟶ (W ⊠ A) ⊠ B
w ⊗ a ⟼ w ⊗ σp(a_{(1)})_{(1)} ⊗ S(σp(a_{(1)})_{(2)})a_{(2)}

that we claim to be a morphism of Yetter-Drinfeld modules over B. That ι is a left B-module map is easily checked. Denoting by β the B-coaction on (W ⊠ A) ⊠ B, we have

βι(w ⊗ a) = w_{(0)} ⊗ σp(a_{(1)})_{(2)} ⊗ S(σp(a_{(1)})_{(5)})a_{(3)}⊗

S (S(σp(a_{(1)})_{(6)}a_{(2)}) S(σp(a_{(1)})_{(1)})w_{(1)}σp(a_{(1)})_{(3)}S(σp(a_{(1)})_{(4)})a_{(4)}

=w_{(0)} ⊗ σp(a_{(1)})_{(2)} ⊗ S(σp(a_{(1)})_{(3)})a_{(3)}⊗
S (S(σp(a_{(1)})_{(4)})a_{(2)}) S(σp(a_{(1)})_{(1)})w_{(1)}a_{(4)}
=w_{(0)} ⊗ σp(a_{(1)})_{(2)} ⊗ S(σp(a_{(1)})_{(3)})a_{(3)}⊗
S(a_{(2)})S (σp(a_{(1)})_{1}S(σp(a_{(1)})_{(4)})) w_{(1)}a_{(4)}

By (3), for x ∈ L, we have

σ(x)_{(2)} ⊗ σ(x)_{(1)}S(σ(x)_{(3)}) = σ(x) ⊗ 1_{B}

and hence

σ(x)_{(2)} ⊗ S(σ(x)_{(3)}) ⊗ σ(x)_{(1)}S(σ(x)_{(4)}) = σ(x)_{(1)} ⊗ S(σ(x)_{(2)}) ⊗ 1_{B}

Thus

βι(w ⊗ a) = w_{(0)} ⊗ σp(a_{(1)})_{(1)} ⊗ S(σp(a_{(1)})_{(2)})a_{(3)} ⊗ S(a_{(2)})w_{(1)}a_{(4)}

Now let γ be the B-coaction on W ⊠ A. We have

(ι ⊗ id_{B})γ(w ⊗ a) = ι ⊗ id_{B}(w_{(0)} ⊗ a_{(2)} ⊗ S(a_{(1)})w_{(1)}a_{(3)})
= w_{(0)} ⊗ σp(a_{(2)})_{(1)} ⊗ S(σp(a_{(2)})_{(2)})a_{(3)} ⊗ S(a_{(1)})w_{(1)}a_{(4)}

= w_{(0)} ⊗ σp(a_{(1)})_{(1)} ⊗ S(σp(a_{(1)})_{(2)})a_{(3)} ⊗ S(a_{(2)})w_{(1)}a_{(4)} = βι(w ⊗ a)

where we have used the cocentrality of p. It follows that ι is B-colinear, and hence is a morphism of Yetter-Drinfeld modules over B. Consider now

μ : (W ⊠ A) ⊠ B ⟶ W ⊠ A
w ⊗ a ⊗ b ⟼ w ⊗ ab

It is straightforward to check that μ is a morphism of Yetter-Drinfeld modules over B, with μι = id_{W ⊠A} and hence we conclude from Proposition 4.2 that W ⊠ A is a relative projective Yetter-Drinfeld module over B. For the last assertion, note that L and A both admit right B^{cop} ⊗ L-comodule structures given by

L ⟶ L ⊗ (B^{cop} ⊗ L),  A ⟶ A ⊗ (B^{cop} ⊗ L)
x ⟼ x_{(1)} ⊗ 1 ⊗ x_{(2)},  a ⟼ a_{(2)} ⊗ a_{(1)}S(a_{(3)}) ⊗ p(a_{(4)})

and that p is B^{cop} ⊗ L-colinear. If A is cosemisimple then so is B and so is L (since p is cocentral), hence B^{cop} ⊗ L is cosemisimple. Thus there exists a B^{cop} ⊗ L-colinear section to p, which satisfies our 3 conditions.

There are also situations where the Hopf algebra in the proposition is not cosemisimple and the map σ still exists, see Section 6.

5 Gerstenhaber-Schack cohomology.

5.1 Generalities.

Let A be a Hopf algebra and let V be a Yetter-Drinfeld module over A. The Gerstenhaber-Schack cohomology of A with coefficients in V, that we denote H_{GS}^{∗}(A, V), was introduced in [27, 28] by using an explicit bicomplex. In fact Gerstenhaber-Schack used Hopf bimodules instead of Yetter-Drinfeld modules to define their cohomology, but in view of the equivalence between Hopf bimodules and Yetter-Drinfeld modules, we shall work with the simpler framework of Yetter-Drinfeld modules (a Yetter-Drinfeld version of the Gerstenhaber-Schack bicomplex is provided in [45]). A special instance of Gerstenhaber-Schack cohomology is bialgebra cohomology, given by H_{b}^{∗}(A) = H_{GS}^{∗}(A, ℂ). As an example, we have by [46], H_{b}^{∗}(ℂΓ) ≃ H^{∗}(ℂΓ, ℂ) for any discrete group Γ. The bialgebra cohomology of O(G) for a connected reductive algebraic group G is also described in [46], Theorem 9.2, and some finite-dimensional examples are computed in [58]. Applications to deformations of pointed Hopf algebras are given in [41]. A key result, due to Taillefer [57, 56], shows that Gerstenhaber-Schack cohomology is in fact an Ext-functor:

H_{GS}^{∗}(A, V) ≃ Ext_{YD_{A}^{A}}^{∗}(ℂ, V)

We will use this description as a definition (we will recall and use the definition based on a bicomplex in the proof of the forthcoming Proposition 5.3).

Definition 5.1. The Gerstenhaber-Schack cohomological dimension of a Hopf algebra A is defined to be

cd_{GS}(A) = sup{n : H_{GS}^{n}(A, V) ≠ 0 for some V ∈ YD_{A}^{A}} ∈ ℕ ∪ {∞}

If A and B are Hopf algebras having equivalent tensor categories of comodules, then the given tensor equivalence F : M^{A} ≃^{⊗} M^{B} induces a tensor equivalence F̂ : YD_{A}^{A} ≃^{⊗} YD_{B}^{B} (see e.g. [11, 10], this is easy to see thanks to the description of the category of Yetter-Drinfeld modules as the weak center of the category of comodules, see [50]). Hence we get, for any Yetter-Drinfeld module V over A, an isomorphism

H_{GS}^{∗}(A, V) ≃ H_{GS}^{∗}(B, F̂(V))

and moreover cd_{GS}(A) = cd_{GS}(B). These properties are what we call the monoidal invariance of Gerstenhaber-Schack cohomology.

5.2 Complexes to compute Gerstenhaber-Schack cohomology.

We now discuss the description of complexes that compute Gerstenhaber-Schack cohomology in particular cases. Recall that a Hopf algebra A is said to be co-Frobenius if there exists a nonzero A-colinear map A → ℂ. By [39], A is co-Frobenius if and only if the category M^{A} of right comodules has enough projectives. Finite-dimensional Hopf algebras are co-Frobenius, as well as cosemisimple Hopf algebras. See [1, 2] for more examples.

Proposition 5.2. Let A be a co-Frobenius Hopf algebra and let

P_{.} = ⋯ P_{n+1} → P_{n} → ⋯ → P_{1} → P_{0} → 0

be a resolution of ℂ by projective objects of YD_{A}^{A}. We have, for any Yetter-Drinfeld module V over A, an isomorphism

H_{GS}^{∗}(A, V) ≃ H^{∗}(Hom_{YD}_{A}^{A}(P_{.}, V))

and we have

cd_{GS}(A) = inf{n : ℂ admits a projective resolution of length n in YD_{A}^{A}}

Proof. We know, since A is co-Frobenius, that YD_{A}^{A} has enough projective objects (Corollary 3.4 in [10]). Thus the description of H_{GS}^{∗}(A, −) as an Ext functor [57] yields that if P_{.} is a a resolution of ℂ by projective objects of YD_{A}^{A}, we have

H_{GS}^{∗}(A, V) ≃ H^{∗}(Hom_{YD}_{A}^{A}(P_{.}, V))

for any Yetter-Drinfeld module V. The proof of the last statement is similar to the one for group cohomology, see [14, Chapter VIII, Lemma 2.1].

Recall [10] that for any n ∈ ℕ, the Yetter-Drinfeld module A^{⊠n} is defined as follows:

A^{⊠0} = ℂ, A^{⊠1} = ℂ⊠A = A_{coad}, A^{⊠2} = A^{⊠1} ⊠A, …, A^{⊠(n+1)} = A^{⊠n} ⊠A, …

After the obvious vector space identification of A^{⊠n} with A^{⊗n}, the right Amodule structure of A^{⊠n} is given by right multiplication and its comodule structure is given by

ad_{r}^{(n)} : A^{⊠n} ⟶ A^{⊠n} ⊗ A
a_{1} ⊗ ⋯ ⊗ a_{n} ⟼ a_{1(2)} ⊗ ⋯ ⊗ a_{n(2)} ⊗ S(a_{1(1)} ⋯ a_{n(1)})a_{1(3)} ⋯ a_{n(3)}

Proposition 5.3. Let A be a Hopf algebra and let V be a Yetter-Drinfeld module over A. Assume that one of the following conditions holds.

1. A is cosemisimple.

2. V is relative injective.

Then the Gerstenhaber-Schack cohomology H_{GS}^{∗}(A, V) is the cohomology of the complex

0 → Hom^{A}(ℂ, V) ⟶^{∂}Hom^{A}(A^{⊠1}, V) →^{∂} ⋯

→^{∂} Hom^{A}(A^{⊠n}, V) ⟶^{∂} Hom^{A}(A^{⊠n+1}, V) ⟶^{∂} ⋯

where the differential ∂ : Hom^{A}(A^{⊠n}, V) ⟶ Hom^{A}(A^{⊠n+1}, V) is given by

∂(f)(a_{1} ⊗ ⋯ ⊗ a_{n+1}) =ε(a_{1})f(a_{2} ⊗ ⋯ ⊗ a_{n+1})+

∑_{i=1}^{n}(−1)^{i}f(a_{1} ⊗ ⋯ ⊗ a_{i}a_{i+1} ⊗ ⋯ ⊗ a_{n+1})

+ (−1)^{n+1}f(a_{1} ⊗ ⋯ ⊗ a_{n}) · a_{n+1}

Proof. By [10], Proposition 3.6, the standard resolution of ℂ_{ε} yields in a fact resolution of ℂ by free Yetter-Drinfeld modules in the category YD_{A}^{A}

⋯ ⟶ A^{⊠n+1} ⟶ A^{⊠n} ⟶ ⋯ ⟶ A^{⊠2} ⟶ A^{⊠1} ⟶ 0

where each differential is given by

A^{⊠n+1} ⟶A^{⊠n}
a_{1} ⊗ ⋯ ⊗ a_{n+1} ⟼ε(a_{1})a_{2} ⊗ ⋯ ⊗ a_{n+1}+

∑_{i=1}^{n}(−1)^{i}a_{1} ⊗ ⋯ ⊗ a_{i}a_{i+1} ⊗ ⋯ ⊗ a_{n+1}

If A is cosemimple, then free Yetter-Drinfeld modules are projective, and we get, after standard identification using the fact that the free functor is left adjoint, the result by Proposition 5.2. To prove the result if the second condition holds, we recall the definition of Gerstenhaber-Schack cohomology using a bicomplex [53]. Let V, W be objects in YD_{A}^{A}, let P_{•} → V → 0 be a relative projective resolution of V (this means that the objects P_{q}, q ≥ 0, are relative projective and the the sequence P_{•} → V → 0 splits as a sequence of comodules), and let 0 → W → I^{•} be a relative injective resolution of W (this means that the objects I^{p}, p ≥ 0, are relative injective and the the sequence 0 → W → I^{•} splits as a sequence of modules). We then can form, in a standard way, the bicomplex C^{•,•}(V, W) = Hom_{YD}_{A}^{A}(P_{•}, I^{•}). The uniqueness, up to homotopy, of the previous resolutions ([53], chapter 10) shows that the cohomology of the bicomplex C^{•,•}(V, W) = Hom_{YD}_{A}^{A}(P_{•}, I^{•}) is independent of the choice of these resolutions, and is the Gertenhaber-Schack cohomology of the Yetter-Drinfeld modules V and W (see [56, 57] as well). When V = ℂ, we get the Gerstenhaber Schackcohomology H_{GS}^{∗}(A, W) as defined in Subsection 5.1, by [57]. Assuming that W is relative injective, we can use the relative injective resolution

0 → W → W → 0 → ⋯ → 0 ⋯

together with the standard resolution of the trivial object ℂ as above (which is indeed a relative projective resolution of ℂ), and we get a bicomplex with only one non-zero column, which is, again using the fact that the free functor is left adjoint, easily identified with the complex in the statement of the proposition.

Remark 5.4. When V = ℂ is the trivial Yetter-Drinfeld module, the complex in Theorem 5.3 is the same as the one defined in [26] in the study of additive deformations of Hopf algebras, which are of interest in quantum probability. This complex is also the complex that defines the so-called Davydov-Yetter cohomology of the tensor category M^{A} ([18, 60], see [24], Chapter 7).

Remark 5.5. Let V be a Yetter-Drinfeld module over A. The complex in Proposition 5.3 is a subcomplex of the complex that computes the Hochshild cohomology H^{∗}(A,_{ε}V), where the left A-module structure on _{ε}V is the one induced by the counit and the right module structure is the original one. We thus have a linear map

H_{GS}^{∗}(A, V) → H^{∗}(A,_{ε}V) ≃ Ext_{A}(ℂ_{ε}, V)

which is not injective in general. Indeed for q ∈ ℂ^{∗} generic (q = ±1 or not a root of unity), we have H_{GS}^{3}(O(SL_{q}(2)), ℂ) ≃ ℂ (see [10]), while H^{3}(O(SL_{q}(2)),_{ε}ℂ_{ε}) = 0 if q^{2} ≠ 1 (see e.g. [31]). In Subsection 5.4 we provide some conditions that ensure that the above map is injective.

5.3 Relation with Hochschild cohomology

We are ready to show that the Gerstenhaber-Schack cohomology of a Hopf algebra determines its Hochschild cohomology.

Theorem 5.6. Let A be a Hopf algebra and let M be an A-bimodule. Endow M ⊗ A with a Yetter-Drinfeld module structure defined by (a, b ∈ A, m ∈ M)

m ⊗ a ↦ m ⊗ a_{(1)} ⊗ a_{(2)}, (m ⊗ a) ← b = S(b_{(2)}).m.b_{(3)} ⊗ S(b_{(1)})ab_{(4)}

and denote by M′#A the resulting Yetter-Drinfeld module. Then we have an isomorphism

H^{∗}(A, M) ≃ H_{GS}^{∗}(A, M′#A)

In particular we have cd(A) ≤ cd_{GS}(A).

Proof. The Yetter-Drinfeld module M′#A is the co-free Yetter-Drinfeld associated to the right A-module M′ of Section 2. It is thus a relative injective Yetter-Drinfeld module (Proposition 4.3), and we can use the complex of Proposition 5.3 to compute its Gerstenhaber-Schack cohomology. Recall that since H^{∗}(A, M) ≃ Ext_{A}^{∗}(ℂ_{ε}, M′) (Section 2), the complex to compute H^{∗}(A, M) is

0 ⟶Hom(ℂ, M′) ⟶^{∂} Hom(A, M′) ⟶^{∂} ⋯

⋯ ⟶^{∂} Hom(A^{⊗n}, M′) ⟶^{∂} Hom(A^{⊗n+1}, M′) ⟶^{∂} ⋯

where the differential ∂ : Hom(A^{⊗n}, M′) ⟶ Hom(A^{⊗n+1}, M′) is given by

∂(f)(a_{1} ⊗ ⋯ ⊗ a_{n+1}) =ε(a_{1})f(a_{2} ⊗ ⋯ ⊗ a_{n+1})

+ ∑_{i=1}^{n}(−1)^{i}f(a_{1} ⊗ ⋯ ⊗ a_{i}a_{i+1} ⊗ ⋯ ⊗ a_{n+1})

+ (−1)^{n+1}S(a_{n+1(1)}) · f(a_{1} ⊗ ⋯ ⊗ a_{n}) · a_{n+1(2)}

For all n ≥ 0, we have linear isomorphisms

Hom^{A}(A^{⊠n}, M′#A) ⟶ Hom(A^{⊗n}, M′)
f ⟼ (id_{M} ⊗ ε)f

For f ∈ Hom(A^{⊠n}, M′#A) and a_{1}, …, a_{n} ∈ A, with f(a_{1}⊗⋯⊗a_{n}) = ∑_{i} m_{i}⊗ b_{i}, we have

id_{M}⊗ε(f(a_{1} ⊗ ⋯ ⊗ a_{n}) ← a_{n+1})

= id_{M} ⊗ ε (∑_{i} S(a_{n+1(2)}).m_{i}.a_{n+1(3)} ⊗ S(a_{n+1(1)})b_{i}a_{n+1(4)})

= ∑_{i} ε(b_{i})S(a_{n+1(1)}).m_{i}.a_{n+1(2)}

= S(a_{n+1(1)}). ((id_{M} ⊗ ε)(f(a_{1} ⊗ ⋯ ⊗ a_{n+1})) .a_{n+1(2)}

From this computation it follows easily that the previous isomorphisms commute with the differentials (as already said, the one for Gerstenhaber-Schack cohomology being given by the complex of Proposition 5.3), and hence the complexes that define both cohomologies are isomorphic.

We get the results announced in the introduction, providing a partial answer to Question 1.1.

Corollary 5.7. Let A and B be Hopf algebras such that there exists an equivalence of linear tensor categories M^{A} ≃^{⊗} M^{B}. Then there exist two functors

F_{1} :_{A}M_{A} → YD_{B}^{B} and F_{2} :_{B}M_{B} → YD_{A}^{A}

such that for any A-bimodule M and any B-bimodule N, we have

H^{∗}(A, M) ≃ H_{GS}^{∗}(B, F_{1}(M)) and H^{∗}(B, N) ≃ H_{GS}^{∗}(A, F_{2}(N))

In particular we have max(cd(A), cd(B)) ≤ cd_{GS}(A) = cd_{GS}(B).

Proof. The construction in the previous theorem clearly yields a functor AM_{A} → YD_{A}^{A}, that we compose with the functor YD_{A}^{A} → YD_{B}^{B} from the discussion at the end of subsection 5.1, to get the announced functor F_{1}, and similarly the functor F_{2}. The last claim follows immediately.

Remark 5.8. Recall that Question 1.2, motivated by Theorem 5.6, asks if cd(A) = cd_{GS}(A) for any Hopf algebra A. Question 1.2 has indeed a positive answer in the finite-dimensional case: if A is semisimple, then it is cosemisimple by the Larson-Radford theorem [38], and hence YD_{A}^{A} is semisimple (since the Drinfeld double D(A) is then semisimple, see [48]), so we have cd(A) = 0 = cd_{GS}(A). If A is not semisimple, then cd(A) = ∞ = cd_{GS}(A). It thus follows that a positive answer to Question 1.2 would provide a natural infinite-dimensional generalization to the above mentioned Larson-Radford theorem. The characteristic zero assumption is indeed necessary: if A is a finitedimensional semisimple non cosemisimple Hopf algebra, the base field being then necessarily of characteristic > 0 [38], then cd(A) = 0 < cd_{GS}(A) = ∞. See the next subsection for some partial results in the cosemisimple case.

5.4 Cosemisimple Hopf algebras

We now provide some more precise partial answers to Questions 1.1 and 1.2 when the Hopf algebra is cosemisimple and of Kac type (recall that this means that S^{2} = id).

Proposition 5.9. Let A be a cosemisimple Hopf algebra of Kac type, and let V be a Yetter-Drinfeld module over A. Then the natural linear map

H_{GS}^{∗}(A, V) → H^{∗}(A,_{ε}V)

arising from Proposition 5.3 is injective.

Proof. Let h be the Haar integral on A. Recall that for any A-comodules V and W, we have a surjective averaging operator

M : Hom(V, W) ⟶ Hom^{A}(V, W)
f ⟼ M(f), M(f)(v) = h (f(v_{(0)})_{(1)}S(v_{(1)})) f(v_{(0)})_{(0)}

with f ∈ Hom^{A}(V, W) if and only if M(f) = f. Now let V be our given Yetter-Drinfeld module, and let f ∈ Hom(A^{⊗n}, V). We thus have M(f) ∈ Hom^{A}(A^{⊠n}, V), with

M(f)(a_{1} ⊗ ⋯ ⊗ a_{n}) =
h (f(a_{1(2)} ⊗ ⋯ ⊗ a_{n(2)})_{(1)}S(a_{1(3)} ⋯ a_{n(3)})S^{2}(a_{1(1)} ⋯ a_{n(1)}))
f(a_{1(2)} ⊗ ⋯ ⊗ a_{n(2)})_{(0)}

It is a tedious but straightforward verification to check that, under our assumption, we have ∂(M(f)) = M(∂(f)). To convince the reader, we present the verification at n = 2. Let f ∈ Hom(A^{⊗2}, V). We have

∂(M(f))(a ⊗ b ⊗ c) =

ε(a)h (f(b_{(2)} ⊗ c_{(2)})_{(1)}S(b_{(3)}c_{(3)})S^{2}(b_{(1)}c_{(1)})) f(b_{(2)} ⊗ c_{(2)})_{(0)}

− h (f(a_{(2)}b_{(2)} ⊗ c_{(2)})_{(1)}S(a_{(3)}b_{(3)}c_{(3)})S^{2}(a_{(1)}b_{(1)}c_{(1)})) f(a_{(2)}b_{(2)} ⊗ c_{(2)})_{(0)}

+ h (f(a_{(2)} ⊗ b_{(2)}c_{(2)})_{(1)}S(a_{(3)}b_{(3)}c_{(3)})S^{2}(a_{(1)}b_{(1)}c_{(1)})) f(a_{(2)} ⊗ b_{(2)}c_{(2)})_{(0)}

− h (f(a_{(2)} ⊗ b_{(2)})_{(1)}S(a_{(3)}b_{(3)})S^{2}(a_{(1)}b_{(1)})) f(a_{(2)} ⊗ b_{(2)})_{(0)} · c

On the other hand we have

M(∂(f))(a ⊗ b ⊗ c) =

h (∂(f)(a_{(2)} ⊗ b_{(2)} ⊗ c_{(2)})_{(1)}S(a_{(3)}b_{(3)}c_{(3)})S^{2}(a_{(1)}b_{(1)}c_{(1)})) ∂(f)(a_{(2)} ⊗ b_{(2)} ⊗ c_{(2)})_{(0)}

= h (ε(a_{(2)})f(b_{(2)} ⊗ c_{(2)})_{(1)}S(a_{(3)}b_{(3)}c_{(3)})S^{2}(a_{(1)}b_{(1)}c_{(1)})) f(b_{(2)} ⊗ c_{(2)})_{(0)}

− h (f(a_{(2)}b_{(2)} ⊗ c_{(2)})_{(1)}S(a_{(3)}b_{(3)}c_{(3)})S^{2}(a_{(1)}b_{(1)}c_{(1)})) f(a_{(2)}b_{(2)} ⊗ c_{(2)})_{(0)}

+ h (f(a_{(2)} ⊗ b_{(2)}c_{(2)})_{(1)}S(a_{(3)}b_{(3)}c_{(3)})S^{2}(a_{(1)}b_{(1)}c_{(1)})) f(a_{(2)} ⊗ b_{(2)}c_{(2)})_{(0)}

− h ((f(a_{(2)} ⊗ b_{(2)}) · c_{(2)})_{(1)}S(a_{(3)}b_{(3)}c_{(3)})S^{2}(a_{(1)}b_{(1)}c_{(1)}))

((f(a_{(2)} ⊗ b_{(2)}) · c_{(2)})_{(0)}

Using the Yetter-Drinfeld condition, the last expression equals

h (S(c_{(2)})f(a_{(2)} ⊗ b_{(2)})_{(1)}c_{(4)}S(a_{(3)}b_{(3)}c_{(5)})S^{2}(a_{(1)}b_{(1)}c_{(1)})) (f(a_{(2)}⊗b_{(2)})_{(0)}·c_{(3)}

The fact that S^{2} = id and that the Haar integral is a trace (since S^{2} = id) then shows that this last expression equals the last one in the computation of ∂(M(f))(a ⊗ b ⊗ c), and shows that indeed ∂(M(f)) = M(∂(f)). Now let f ∈ Hom^{A}(A^{⊠n}, V) be such that f = ∂(μ) for some μ ∈ Hom(A^{⊗n−1}, V). Then M(f) = M(∂(μ)) = ∂(M(μ))), with M(μ) ∈ Hom^{A}(A^{⊠n−1}, V), and hence f = 0 in H_{GS}^{n}(A, V): our claim is proved.

We thus get the following partial answers to Questions 1.2 and 1.1.

Corollary 5.10. Let A be cosemisimple Hopf algebra of Kac type. Then cd(A) = cd_{GS}(A).

Proof. We have cd(A) ≤ cd_{GS}(A) by Theorem 5.6, and the previous proposition ensures that cd_{GS}(A) ≤ cd(A).

Corollary 5.11. Let A and B be cosemisimple Hopf algebras such that there exists an equivalence of linear tensor categories M^{A} ≃^{⊗} M^{B}. If A is of Kac type, then we have cd(A) ≥ cd(B), and if A and B both are of Kac type, then cd(A) = cd(B).

Proof. We have, combining Theorem 5.6 and the previous corollary,

cd(A) = cd_{GS}(A) = cd_{GS}(B) ≥ cd(B)

with cd(B) = cd_{GS}(B) if B is of Kac type as well.

See the next section for examples that are not of Kac type.

6 Application to quantum symmetry algebras

In this section we provide applications of the previous considerations to quantum symmetry algebras.

6.1 The universal Hopf algebra of a non-degenerate bilinear form and its adjoint subalgebra

Let E ∈ GL_{n}(ℂ). Recall that the algebra B(E) [21] is presented by generators (u_{ij})_{1≤i,j≤n} and relations

E^{−1}u^{t}Eu = I_{n} = uE^{−1}u^{t}E,

where u is the matrix (u_{ij})_{1≤i,j≤n}. It has a Hopf algebra structure defined by

Δ(u_{ij}) = ∑_{k=1}^{n} u_{ik} ⊗ u_{kj}, ε(u_{ij}) = δ_{ij}, S(u) = E^{−1}u^{t}E

The Hopf algebra B(E) represents the quantum symmetry group of the bilinear form associated to the matrix E. It can also be constructed as a quotient of the FRT bialgebra associated to Yang-Baxter operators constructed by Gurevich [30]. For the matrix

E_{q} = (\begin{matrix}0 & 1 \\ −q^{−1} & 0 \end{matrix})

we have B(E_{q}) = O(SL_{q}(2)), and thus the Hopf algebras B(E) are natural generalizations of O(SL_{q}(2)). It is shown in [9] that for q ∈ ℂ^{∗} satisfying tr(E^{−1}E^{t}) = −q − q^{−1}, the tensor categories of comodules over B(E) and O(SL_{q}(2)) are equivalent. Thus B(E) is cosemisimple if and only if the corresponding q is not a root of unity or q = ±1. It was proved in [10] that if n ≥ 2, then cd(B(E)) = 3 (Theorem 6.1 and Proposition 6.4 in [10], see e.g. [31] for the case E = E_{q} and [17] for the case E = I_{n}), and the bialgebra cohomology of B(E) was computed there in the cosemisimple case. As a preliminary step towards the study of quantum symmetry algebras of semisimple algebras, we now study the adjoint subalgebra B_{+}(E) of B(E). The algebra B_{+}(E) is, by definition, the subalgebra of B(E) generated by the elements u_{ij}u_{kl}, 1 ≤ i, j, k, l ≤ n. It is easily seen to be a Hopf subalgebra. Also it is easily seen that B_{+}(E) = B(E)^{coℂℤ_{2}}, where p is the cocentral Hopf algebra map B(E) → ℂℤ_{2}, u_{ij} ↦ δ_{ij}g, where g stands for the generator of ℤ_{2}, the cyclic group of order 2. The Hopf algebra B_{+}(E) is cosemisimple if and only if B(E) is.

Lemma 6.1. Assume that tr(E^{−1}E^{t}) ≠ 0. Then there exists a linear map σ : ℂℤ_{2} → B(E) satisfying the conditions of Proposition 4.8.

Proof. Consider the matrix F = E(E^{t})^{−1} = (α_{ij}). We have tr(F) = tr(E^{−1}E^{t}) = t ≠ 0. Consider the element x = t^{−1} ∑_{ij} α_{ij}u_{ij} ∈ B(E) and let σ : ℂℤ_{2} → B(E) be the unique linear map such that σ(1) = 1 and σ(g) = x. It is straightforward to check that σ indeed satisfies the conditions of Proposition 4.8.

Theorem 6.2. Let E ∈ GL_{n}(ℂ) with n ≥ 2. Then we have cd(B_{+}(E)) = 3 ≤ cd_{GS}(B_{+}(E)), and if moreover B_{+}(E) is cosemisimple, then cd_{GS}(B_{+}(E)) = 3.

Proof. We have, by Proposition 2.4, a strict exact sequence of Hopf algebras

ℂ → B_{+}(E) → B(E) → ℂℤ_{2} → ℂ

so it follows from Proposition 3.2 that cd(B_{+}(E)) = cd(B(E)) = 3. By Theorem 5.6 we have cd_{GS}(B_{+}(E)) ≥ 3. Consider now the exact sequence of free Yetter-Drinfeld modules over B(E) from [10]:

0 → ℂ⊠B(E) →^{ϕ_{1}} (V_{E}^{∗}⊗V_{E})⊠B(E) →^{ϕ_{2}} (V_{E}^{∗}⊗V_{E})⊠B(E) ⟶^{ϕ_{3}} ℂ⊠B(E) ⟶^{ε} ℂ → 0

All the B(E)-comodules involved in the left terms are in fact comodules over B_{+}(E), so we have, by Proposition 4.5, an exact sequence of Yetter-Drinfeld modules over B_{+}(E). Assume now that B_{+}(E) is cosemisimple. The previous lemma ensures that we are in the situation of Proposition 4.8, so all the terms in the sequence (except the last one of course) are projective Yetter-Drinfeld modules over B_{+}(E). We conclude from Proposition 5.2 that cd_{GS}(B_{+}(E)) ≤ 3, and hence that cd_{GS}(B_{+}(E)) = 3.

To compute the bialgebra cohomology of B_{+}(E) in the cosemisimple case, we

need some preliminaries. We specialize at E_{q} = (\begin{matrix}0 & 1 \\ −q^{−1} & 0 \end{matrix}) and we put A =

B(E_{q}) = O(SL_{q}(2)) (with its standard generators a, b, c, d) and B = B_{+}(E_{q}). In the next lemma we only assume that q + q^{−1} ≠ 0. Recall from Subsection 4.4 that if W is a B-comodule, then W ⊠ A is a Yetter-Drinfeld module over B.

Lemma 6.3. We have, for any B-comodule W, a vector space isomorphism

Hom_{YD}_{B}^{B}(W ⊠ A, ℂ) ⟶ Hom^{B}(W, ℂ) ⊕ Hom^{B}(W, ℂ)
ψ ⟼ (ψ(− ⊗ 1), ψ(− ⊗ χ))

where χ = q^{−1}a + qd.

Proof. Let ψ ∈ Hom_{YD}_{B}^{B}(W ⊠ A, ℂ). That both ψ(− ⊗ 1) and ψ(− ⊗ χ) are B-comodule maps follow from the fact that 1 and χ are coinvariant for the co-adjoint action of A. We have, for any w ∈ W, using the B-linearity

ψ(w ⊗ b) = ψ(w ⊗ b(ad − q^{−1}bc)) = ψ(w ⊗ bad) = qψ(w ⊗ abd) = 0

and similarly ψ(w ⊗ c) = 0. We also have

ψ(w ⊗ d) = ψ(w ⊗ d(ad − q^{−1}bc))) = ψ(w ⊗ dad) = ψ(w ⊗ ad^{2}) = ψ(w ⊗ a)

These identities, together with the fact that A = B ⊕ B′, where B′ = XB and X = {a, b, c, d}, show that the map in the statement of the lemma is injective. For (ψ_{1}, ψ_{2}) ∈ Hom^{B}(W, ℂ)⊕Hom^{B}(W, ℂ), we define a linear map ψ : W ⊗A → ℂ by

ψ(w ⊗ (y + y′)) = ψ_{1}(w)ε(y) + (q + q^{−1})^{−1}ψ_{2}(w)ε(y′), y ∈ B, y ∈ B′

It is clear that ψ is A-linear and a direct verification to check that ψ is a map of B-comodules, for the co-action of W ⊠A. Hence we have ψ ∈ Hom_{YD}_{B}^{B}(W ⊠ A, ℂ), and clearly ψ(− ⊗ 1) = ψ_{1} and ψ(− ⊗ χ) = ψ_{2}. Therefore our map is surjective, and we are done.

Theorem 6.4. Let E ∈ GL_{n}(ℂ) with n ≥ 2. If B_{+}(E) is cosemisimple, then

H_{b}^{n}(B_{+}(E)) ≃ \begin{cases}0 & if n ≠ 0, 3 \\ ℂ & if n = 0, 3 \end{cases}

Proof. The monoidal invariance of bialgebra cohomology enables us to assume that E = E_{q} as in the previous discussion, of which we keep the notations. We denote by V the fundamental A-comodule of dimension 2, of which we fix a basis e_{1}, e_{2}. We have an exact sequence of Yetter-Drinfeld modules over A (and over B)

0 → ℂ ⊠ A ⟶^{ϕ_{1}} (V^{∗} ⊗ V) ⊠ A ⟶^{ϕ_{2}} (V^{∗} ⊗ V) ⊠ A ⟶^{ϕ_{3}} ℂ ⊠ A ⟶^{ε} ℂ → 0

with for any x ∈ A (see the proof of Lemma 5.6 in [10])

ϕ_{1}(x) = e_{1}^{∗} ⊗ e_{1}⊗((−q^{−1} + qd)x) + e_{1}^{∗} ⊗ e_{2} ⊗ (−cx)
+ e_{2}^{∗} ⊗ e_{1} ⊗ (−bx) + e_{2}^{∗} ⊗ e_{2} ⊗ ((−q + q^{−1}a)x)
ϕ_{2}(e_{1}^{∗} ⊗ e_{1} ⊗ x) = e_{1}^{∗} ⊗ e_{1} ⊗ x + e_{2}^{∗} ⊗ e_{1} ⊗ (−qbx) + e_{2}^{∗} ⊗ e_{2} ⊗ ax
ϕ_{2}(e_{1}^{∗} ⊗ e_{2} ⊗ x) = e_{1}^{∗} ⊗ e_{1} ⊗ bx + e_{1}^{∗} ⊗ e_{2} ⊗ (1 − q^{−1}a)x
ϕ_{2}(e_{2}^{∗} ⊗ e_{1} ⊗ x) = e_{2}^{∗} ⊗ e_{1} ⊗ (1 − qd)x + e_{2}^{∗} ⊗ e_{2} ⊗ cx
ϕ_{2}(e_{2}^{∗} ⊗ e_{2} ⊗ x) = e_{1}^{∗} ⊗ e_{1} ⊗ dx + e_{1}^{∗} ⊗ e_{2} ⊗ (−q^{−1}cx) + e_{2}^{∗} ⊗ e_{2} ⊗ x
ϕ_{3}(e_{1}^{∗} ⊗ e_{1} ⊗ x) = (a − 1)x, ϕ_{3}(e_{1}^{∗} ⊗ e_{2} ⊗ x) = bx,
ϕ_{3}(e_{2}^{∗} ⊗ e_{1} ⊗ x) = cx, ϕ_{3}(e_{2}^{∗} ⊗ e_{2} ⊗ x) = (d − 1)x

and by Lemma 6.3, Proposition 4.8 and Proposition 5.2, the bialgebra cohomology of B is the cohomology of the complex

0 →Hom_{YD}_{B}^{B}(ℂ ⊠ A, ℂ) →^{ϕ_{3}^{t}} Hom_{YD}_{B}^{B}(V^{∗} ⊗ V) ⊠ A, ℂ)

→^{ϕ_{2}^{t}} Hom_{YD}_{B}^{B}(V^{∗} ⊗ V) ⊠ A, ℂ) →^{ϕ_{1}^{t}} Hom_{YD}_{B}^{B}(ℂ ⊠ A, ℂ) → 0

We have, by the previous lemma, Hom_{YD}_{B}^{B}(ℂ ⊠ A, ℂ) ≃ ℂ^{2}, and

Hom_{YD}_{B}^{B}(V^{∗} ⊗ V) ⊠ A, ℂ) ≃ Hom^{B}(V^{∗} ⊗ V, ℂ) ⊕ Hom^{B}(V^{∗} ⊗ V, ℂ) ≃ ℂ^{2}

Therefore the previous complex is isomorphic to a complex of the form

0 ⟶ ℂ^{2} ⟶ ℂ^{2} ⟶ ℂ^{2} ⟶ ℂ^{2} ⟶ 0

The reader will easily write down explicitly this complex and compute its cohomology, yielding the announced result for the bialgebra cohomology of B.

6.2 Bialgebra cohomology and cohomological dimensions of A_{aut}(R, φ)

Let (R, φ) be a finite-dimensional measured algebra: this means that R is a finite-dimensional algebra and φ : R → ℂ is a linear map (a measure on R) such that the associated bilinear map R×R → ℂ, (x, y) ↦ φ(xy) is non-degenerate. Thus a finite-dimensional measured algebra is a Frobenius algebra together with a fixed measure. A coaction of a Hopf algebra A on a finite-dimensional measured algebra (R, φ) is an A-comodule structure on R making it into an A-comodule algebra and such that φ : R → ℂ is A-colinear. It is well-known that there exists a universal Hopf algebra coacting on (R, φ) (see [61] in the compact case with R semisimple and [8] in general), that we denote A_{aut}(R, φ) and call the quantum symmetry algebra of (R, φ). The following particular cases are of special interest.

1. For R = ℂ^{n} and φ = φ_{n} the canonical integration map (with φ_{n}(e_{i}) = 1 for e_{1}, …, e_{n} the canonical basis of ℂ^{n}), we have A_{aut}(ℂ^{n}, φ_{n}) =: A_{s}(n), the coordinate algebra on the quantum permutation group [61], presented by generators x_{ij}, 1 ≤ i, j ≤ n, submitted to the relations

∑_{l=1}^{n} x_{li} = 1 = ∑_{l=1}^{n} x_{il}, x_{ik}x_{ij} = δ_{kj}x_{ij}, x_{ki}x_{ji} = δ_{kj}x_{ji}, 1 ≤ i, j, k ≤ n

Its Hopf algebra structure is defined by

Δ(x_{ij}) = ∑_{k=1}^{n} x_{ik} ⊗ x_{kj}, ε(x_{ij}) = δ_{ij}, S(x_{ij}) = x_{ji}

The Hopf algebra A_{s}(n) is infinite-dimensional if n ≥ 4 [61].

2. For R = M_{2}(ℂ) and q ∈ ℂ^{∗}, let tr_{q} : M_{2}(ℂ) → ℂ be the q-trace, i.e. tr_{q}(g) = qg_{11} + q^{−1}g_{22} for g = (g_{ij}) ∈ M_{2}(ℂ). Then we have A_{aut}(M_{2}(ℂ), tr_{q}) ≃ O(PSL_{q}(2)), the latter algebra being B_{+}(E_{q}) in the notation of the previous subsection (it is often denoted O(SO_{q^{1/2}}(3)), see e.g. [35]). The above isomorphism A_{aut}(M_{2}(ℂ), tr_{q}) → O(PSL_{q}(2)) is constructed using the universal property of A_{aut}(M_{2}(ℂ), tr_{q}), and the verification that it is indeed injective is a long and tedious computation, as in [20].

Let (R, φ) be a finite-dimensional measured algebra. Since φ ∘ m is nondegenerate, where m is the multiplication of R, there exists a linear map δ : ℂ → R ⊗ R such that (R, φ ∘ m, δ) is a left dual for R, i.e.

((φ ∘ m) ⊗ id_{R}) ∘ (id_{R} ⊗ δ) = id_{R} = (id_{R} ⊗ (φ ∘ m)) ∘ (δ ⊗ id_{R})

Following [43], we put

φ̃ = φ ∘ m ∘ (m ⊗ id_{R}) ∘ (id_{R} ⊗ δ) : R → ℂ

and we say that (R, φ) (or φ) is normalizable if φ(1) ≠ 0 and if there exists λ ∈ ℂ^{∗} such that φ̃ = λφ. Using the definition of Frobenius algebra in terms of coalgebras, the coproduct is Δ = (m ⊗ id_{R}) ∘ (id_{R} ⊗ δ) = (id_{R} ⊗ m) ∘ (δ ⊗ id_{R}), and we have φ̃ = φ ∘ m ∘ Δ. The condition that φ is normalizable is equivalent to require, in the language of [33, Definition 3.1], that R/ℂ is a strongly separable extension with Frobenius system (φ, x_{i}, y_{i}), where δ(1) = ∑_{i} x_{i} ⊗ y_{i}. It thus follows that if φ is normalizable, then R is necesarily a separable (semisimple) algebra. Conversely, if R is semisimple, writing R as a direct product of matrix algebras, one easily sees the conditions that ensure that φ is normalizable, see [43]. It is shown in [43] (Corollary 4.9), generalizing earlier results from [5, 6, 19], that if (R, φ) is a finite-dimensional semisimple measured algebra with dim(R) ≥ 4 and φ normalizable, then there exists q ∈ ℂ^{∗} with q + q^{−1} ≠ 0 such that

M^{A_{aut}(R,φ)} ≃^{⊗} M^{O(PSL_{q}(2))}

The parameter q is determined as follows. First consider λ ∈ ℂ^{∗} such that φ̃ = λφ and choose μ ∈ ℂ^{∗} such that μ^{2} = λφ(1). Then q is any solution of the equation q + q^{−1} = μ (recall that O(PSL_{q}(2)) = O(PSL_{−q}(2)), so the choice of μ does not play any role). As an example, for (ℂ^{n}, φ_{n}) as above (and n ≥ 4), φ_{n} is normalizable with the corresponding λ equal to 1, and q is any solution of the equation q +q^{−1} = \sqrt{n}.

Theorem 6.5. Let (R, φ) be a finite-dimensional semisimple measured algebra with dim(R) ≥ 4 and φ normalizable. Assume that A_{aut}(R, φ) is cosemisimple. Then we have

H_{b}^{n}(A_{aut}(R, φ)) ≃ \begin{cases}0 & if n ≠ 0, 3 \\ ℂ & if n = 0, 3 \end{cases}

and cd(A_{aut}(R, φ)) ≤ cd_{GS}(A_{aut}(R, φ)) = 3, with equality if φ is a trace. In particular we have cd(A_{s}(n)) = 3 = cd_{GS}(A_{s}(n)) for any n ≥ 4.

Proof. The proof follows immediately from the combination of the above monoidal equivalence, the monoidal invariance of Gerstenhaber-Schack cohomology, Theorem 6.2, Theorem 6.4, Theorem 5.6, and Corollary 5.10 (A_{aut}(R, φ) being of Kac type when φ is a trace).

Note that the length 3 resolution of the trivial Yetter-Drinfeld module over O(PSL_{q}(2)) by relative projective Yetter-Drinfeld modules considered in the previous subsection (see the proof of Theorem 6.4) transports to a length 3 resolution of the trivial Yetter-Drinfeld module over A_{aut}(R, φ) by relative projective Yetter-Drinfeld modules (see Theorem 4.1 in [10]), and in particular this yields a length 3 projective resolution of the trivial module over A_{aut}(R, φ). We have not been able to write down this resolution explicitly enough to compute Hochschild cohomology groups and show that one always has cd(A_{aut}(R, φ)) = 3. We believe that this is true however.

Remark 6.6. It follows that the L^{2}-Betti numbers ([36]) β_{k}^{(2)}(A_{s}(n)) vanish for k ≥ 4, and we have as well β_{0}^{(2)}(A_{s}(n)) = 0 by [37].

7 Hopf algebras with a projection

It is natural to ask whether similar results to those of Section 2 hold for Gerstenhaber-Schack cohomological dimension. A positive answer to Question 1.2 would of course provide an affirmative answer. So far, our only positive result in this direction is the following one, in the setting of Hopf algebras with a projection [47, 40].

Proposition 7.1. Let B ⊂ A be a Hopf subalgebra. Assume that there exists a Hopf algebra map π : A → B such that π_{|B} = id_{B} and that A is cosemisimple. Then we have cd_{GS}(B) ≤ cd_{GS}(A).

Proof. The inclusion B ⊂ A together with the Hopf algebra map π : A → B induce a vector space preserving linear exact tensor functor

F : YD_{A}^{A} ⟶ YD_{B}^{B}

where if V is Yetter-Drinfeld module over A, then F (V) = V as a vector space, the B-module structure is the restriction of that of A, and the B-comodule structure is given by (id_{V} ⊗π)α, where α is the original co-action of A. We claim that it is enough to show that F sends (relative) projective Yetter-Drinfeld modules over A to (relative) projective Yetter-Drinfeld modules over B. Indeed, if we have a length n resolution of the trivial Yetter-Drinfeld module over A by (relative) projectives, the functor F will transform it into a a length n resolution of the trivial Yetter-Drinfeld module over B by (relative) projectives, and hence by Proposition 5.2, we have cd_{GS}(B) ≤ cd_{GS}(A).

As usual, put R = ^{coB}A = {a ∈ A | π(a_{(1)})⊗a_{(2)} = 1⊗a}. This is a subalgebra of A and we have (id ⊗ π)Δ(R) ⊂ R ⊗ B, which endows R with a right Bcomodule structure. For any a ∈ A, we have a_{(2)}πS^{−1}(a_{(1)}) ∈ R (since A is cosemisimple, its antipode is bijective), and thus we have a linear isomorphism [47, 40]

A ⟶ R ⊗ B
a ⟼ a_{(3)}πS^{−1}(a_{(2)}) ⊗ π(a_{(1)})

whose inverse is the restriction of the multiplication of A. Let V be a right A-comodule: it also has a right B-comodule structure obtained using the projection π : A → B, that we denote V_{π}. Consider now the map

F (V ⊠ A) ⟶ (V_{π} ⊗ R) ⊠ B
v ⊗ a ⟼ v ⊗ a_{(3)}πS^{−1}(a_{(2)}) ⊗ π(a_{(1)})

This is an isomorphism by the previous considerations, and it is a direct verification to check that it is a morphism of Yetter-Drinfeld modules over B. Hence the functor F sends free Yetter-Drinfeld modules over A to free Yetter-Drinfeld modules over B, and since it is additive, it sends, by Proposition 4.2, projective Yetter-Drinfeld modules over A to projective Yetter-Drinfeld modules over B. This concludes the proof.

As an illustration, consider the hyperoctahedral Hopf algebra A_{h}(n) [7]. This is the algebra presented by generators a_{ij}, 1 ≤ i, j ≤ n, submitted to the relations

∑_{l=1}^{n} a_{li}^{2} = 1 = ∑_{l=1}^{n} a_{il}^{2}, a_{ik}a_{ij} = 0 = a_{ji}a_{ki} if j ≠ k, 1 ≤ i, j, k ≤ n

Its Hopf algebra structure is given by the same formulas as those for A_{s}(n). There exist Hopf algebra maps i : A_{s}(n) → A_{h}(n), x_{ij} ↦ a_{ij}^{2}, π : A_{h}(n) → A_{s}(n), a_{ij} ↦ x_{ij}, such that πi = id. Hence we deduce from the previous proposition that cd_{GS}(A_{h}(n)) ≥ cd_{GS}(A_{s}(n)), and hence by Theorem 6.5, if n ≥ 4, we have cd_{GS}(A_{h}(n)) ≥ cd_{GS}(A_{s}(n)) = 3 (since A_{h}(n) is cosemisimple of Kac type, this could be deduced as well from the combination of Proposition 3.1 and Corollary 5.10).

References

[1] N. Andruskiewitsch, J. Cuadra, On the structure of (co-Frobenius) Hopf algebras, J. Noncommut. Geom. 7 (2013), no. 1, 83-104.

[2] N. Andruskiewitsch, S. Dascalescu, Co-Frobenius Hopf algebras and the coradical filtration, Math. Z. 243 (2003), no. 1, 145-154.

[3] N. Andruskiewitsch, J. Devoto, Extensions of Hopf algebras, St. Petersburg Math. J. 7 (1996), no. 1, 17-52.

[4] S. Arkhipov, D. Gaitsgory, Another realization of the category of modules over the small quantum group, Adv. Math. 173 (2003), no. 1, 114-143.

[5] T. Banica, Symmetries of a generic coaction, Math. Ann. 314 (1999), no. 4, 763-780.

[6] T. Banica, Quantum groups and Fuss-Catalan algebras, Comm. Math. Phys. 226 (2002), no. 1, 221-232.

[7] T. Banica, J. Bichon, B. Collins, The hyperoctahedral quantum group, J. Ramanujan Math. Soc. 22 (2007), no. 4, 345-384.

[8] J. Bichon, Galois reconstruction of finite quantum groups, J. Algebra 230 (2000), no. 2, 683-693.

[9] J. Bichon, The representation category of the quantum group of a nondegenerate bilinear form, Comm. Algebra 31 (2003), no. 10, 4831-4851.

[10] J. Bichon, Hochschild homology of Hopf algebras and free Yetter-Drinfeld resolutions of the counit, Compositio Math. 149 (2013), no. 4, 658-678.

[11] J. Bichon, Hopf-Galois objects and cogroupoids, Rev. Un. Mat. Argentina 55 (2014), no. 2, 11-69.

[12] K.A. Brown, K.R Goodearl, Lectures on algebraic quantum groups, Advanced Courses in Mathematics CRM Barcelona, Birkhäuser Verlag, 2002.

[13] K.A. Brown, J.J. Zhang, Dualising complexes and twisted Hochschild (co)homology for Noetherian Hopf algebras, J. Algebra 320 (2008), no. 5, 1814-1850.

[14] K.S. Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer-Verlag, New York-Berlin, 1982.

[15] S. Caenepeel, G. Militaru, S. Zhu, Crossed modules and Doi-Hopf modules, Israel J. Math. 100 (1997), 221-247.

[16] A. Chirvasitu, Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras, Algebra Number Theory 8 (2014), no. 5, 1179-1199.

[17] B. Collins, J. Härtel, A. Thom, Homology of free quantum groups. C. R. Math. Acad. Sci. Paris 347 (2009), no. 5-6, 271-276.

[18] A. Davydov, Twisting of monoidal structures, arXiv:q-alg/9703001.

[19] A. De Rijdt, N. Vander Vennet, Actions of monoidally equivalent compact quantum groups and applications to probabilistic boundaries, Ann. Inst. Fourier 60 (2010), no. 1, 169-216.

[20] M.S. Dijkhuizen, The double covering of the quantum group SO_{q}(3), Rend. Circ. Math. Palermo Serie II 37 (1994), 47-57.

[21] M. Dubois-Violette, G. Launer, The quantum group of a non-degenerate bilinear form, Phys. Lett. B 245, no.2 (1990), 175-177.

[22] M.J. Dunwoody, Accessibility and groups of cohomological dimension one, Proc. London Math. Soc. 38 (1979), no. 2, 193-215.

[23] P. Etingof, S. Gelaki, On finite-dimensional semisimple and cosemisimple Hopf algebras in positive characteristic. Internat. Math. Res. Notices 1998, no. 16, 851-864.

[24] P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik, Tensor categories. Mathematical Surveys and Monographs 205, American Mathematical Society, Providence, RI, 2015.

[25] P. Feng, B. Tsygan, Hochschild and cyclic homology of quantum groups. Comm. Math. Phys. 140 (1991), no. 3, 481-521.

[26] M. Gerhold, Additive deformations of Hopf algebras, J. Algebra 339 (2011), 114-134.

[27] M. Gerstenhaber, S. Schack, Bialgebra cohomology, deformations and quantum groups, Proc. Nat. Acad. Sci. USA 87 (1990), no. 1, 78-81.

[28] M. Gerstenhaber, S. Schack, Algebras, bialgebras, quantum groups, and algebraic deformations, Contemp. Math. 134 (1992), 51-92.

[29] V. Ginzburg, S. Kumar, Cohomology of quantum groups at roots of unity, Duke Math. J. 69 (1993), no. 1, 179-198.

[30] D.I. Gurevich, Algebraic aspects of the quantum Yang-Baxter equation, Leningrad Math. J. 2 (1991), no. 4, 801-828.

[31] T. Hadfield, U. Krähmer, Twisted homology of quantum SL(2), K-Theory 34 (2005), no. 4, 327-360.

[32] T. Hadfield, U. Krähmer, On the Hochschild homology of quantum SL(N), C. R. Math. Acad. Sci. Paris 343 (2006), no. 1, 9-13.

[33] L. Kadison, New examples of Frobenius extensions. University Lecture Series 14, American Mathematical Society, Providence, RI, 1999.

[34] C. Kassel, Quantum groups, GTM 155, Springer, 1995.

[35] A. Klimyk, K. Schmüdgen, Quantum groups and their representations, Texts and Monographs in Physics, Springer, 1997.

[36] D. Kyed, L^{2}-homology for compact quantum groups, Math. Scand. 103 (2008), no. 1, 111-129.

[37] D. Kyed, On the zeroth L^{2}-homology of a quantum group, Münster J. Math. 4 (2011), 119-128.

[38] R.G. Larson, D.E. Radford, Finite-dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple, J. Algebra 117 (1988), no. 2, 267-289.

[39] B.I. Lin, Semiperfect coalgebras, J. Algebra 49 (1977), no. 2, 357-373.

[40] S. Majid, Cross products by braided groups and bosonization, J. Algebra 163 (1994), no. 1, 165-190.

[41] M. Mastnak, S. Witherspoon, Bialgebra cohomology, pointed Hopf algebras, and deformations, J. Pure Appl. Algebra 213 (2009), no. 7, 1399-1417.

[42] S. Montgomery, Hopf algebras and their actions on rings, Amer. Math. Soc. 1993.

[43] C. Mrozinski, Quantum automorphism groups and SO(3)-deformations, J. Pure Appl. Algebra 219 (2015), no. 1, 1-32.

[44] E. Müller, H.J. Schneider, Quantum homogeneous spaces with faithfully flat module structures, Israel J. Math. 111 (1999), 157-190.

[45] F. Panaite, D. Stefan, Deformation cohomology for Yetter-Drinfeld modules and Hopf (bi)modules. Comm. Algebra 30 (2002), no. 1, 331-345.

[46] B. Parshall, J. Wang, On bialgebra cohomology, Bull. Soc. Math. Belg. Sér. A 42 (1990), no. 3, 607-642.

[47] D.E. Radford, The structure of Hopf algebras with a projection, J. Algebra 92 (1985), no. 2, 322-347.

[48] D.E. Radford, Minimal quasitriangular Hopf algebras, J. Algebra 157 (1993), no. 2, 285-315.

[49] P. Schauenburg, Hopf modules and Yetter-Drinfeld modules, J. Algebra 169 (1994), no. 3, 874-890.

[50] P. Schauenburg, Hopf-Galois and bi-Galois extensions, Fields Inst. Comm. 43 (2004), 469-515.

[51] H.J. Schneider, Normal basis and transitivity of crossed products for Hopf algebras, J. Algebra 152 (1992), no. 2, 289-312.

[52] H.-J. Schneider, Some remarks on exact sequences of quantum groups, Comm. Algebra 21 (1993), no. 9, 3337-3357.

[53] S. Shnider, S. Sternberg, Quantum groups. From coalgebras to Drinfeld algebras. A guided tour. Graduate Texts in Mathematical Physics, II. International Press, Cambridge, MA, 1993.

[54] D. Stefan, Hochschild cohomology on Hopf Galois extensions, J. Pure Appl. Algebra 103 (1995), no. 2, 221-233.

[55] D. Stefan, The set of types of n-dimensional semisimple and cosemisimple Hopf algebras is finite, J. Algebra 193 (1997), no. 2, 571-580.

[56] R. Taillefer, Cohomology theories of Hopf bimodules and cup-product, Algebr. Represent. Theory 7 (2004), no. 5, 471-490.

[57] R. Taillefer, Injective Hopf bimodules, cohomologies of infinite dimensional Hopf algebras and graded-commutativity of the Yoneda product, J. Algebra 276 (2004), no. 1, 259-279.

[58] R. Taillefer, Bialgebra cohomology of the duals of a class of generalized Taft algebras, Comm. Algebra (2007), no. 4, 1415-1420.

[59] M. Takeuchi, Relative Hopf modules-equivalences and freeness criteria, J. Algebra 60 (1979), no. 2, 452-471.

[60] D.N. Yetter, Abelian categories of modules over a (lax) monoidal functor, Adv. Math. 174 (2003), no. 2, 266-309.

[61] S. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), no. 1, 195-211.

[62] C. Weibel, An Introduction to Homological Algebra, Cambridge University Press, 1994.

Julien Bichon

Laboratoire de Mathématiques Université Blaise Pascal Campus des Cézeaux 3 place Vasarely 63178 Aubière Cedex France Julien.Bichon@math.univbpclermont.fr