COHOMOLOGY OF FINITE DIMENSIONAL POINTED HOPF ALGEBRAS

M. MASTNAK, J. PEVTSOVA, P. SCHAUENBURG, AND S. WITHERSPOON

Abstract. We prove finite generation of the cohomology ring of any finite dimensional pointed Hopf algebra, having abelian group of grouplike elements, under some mild restrictions on the group order. The proof uses the recent classification by Andruskiewitsch and Schneider of such Hopf algebras. Examples include all of Lusztig’s small quantum groups, whose cohomology was first computed explicitly by Ginzburg and Kumar, as well as many new pointed Hopf algebras.

We also show that in general the cohomology ring of a Hopf algebra in a braided category is braided commutative. As a consequence we obtain some further information about the structure of the cohomology ring of a finite dimensional pointed Hopf algebra and its related Nichols algebra.

1. Introduction

Venkov [28] and Evens [16] independently proved that the cohomology ring of a finite group with coefficients in a field of positive characteristic is finitely generated. This fundamental result opened the door to using geometric methods in the study of cohomology and modular representations of finite groups. This geometric approach was pioneered by Quillen and expanded by Carlson [13], and Avrunin and Scott [6]. Friedlander and Suslin [18] vastly generalized the result of Venkov and Evens, proving that the cohomology ring of any finite group scheme (equivalently, finite dimensional cocommutative Hopf algebra) over a field of positive characteristic is finitely generated. In a different direction, Ginzburg and Kumar [19] computed the cohomology ring of each of Lusztig’s small quantum groups u_{q}(𝔤), defined over ℂ, under some restrictions on the parameters; it is the coordinate ring of the nilpotent cone of the Lie algebra 𝔤, and thus is finitely generated. The result is parallel to an earlier result of Friedlander and Parshall [17] who computed the cohomology ring of a restricted Lie algebra in positive characteristic. Recently, Bendel, Nakano, Parshall, and Pillen [8] calculated the cohomology of a small quantum group under significantly loosened restrictions on the parameters. The broad common feature of these works is that they investigate cohomology of certain nonsemisimple finite dimensional Hopf algebras. Hence, these results lead one to ask whether the cohomology ring of any finite dimensional Hopf algebra is finitely generated. A positive answer would simultaneously generalize the known results for cocommutative Hopf algebras and for small quantum groups. More generally, Etingof and Ostrik [15] conjectured finite generation of cohomology in the context of finite tensor categories.

Date: January 28, 2009.

In this paper, we begin the task of proving this conjecture for more general classes of noncocommutative Hopf algebras over a field of characteristic 0. We prove finite generation of the cohomology ring of any finite dimensional pointed Hopf algebra, with abelian group of grouplike elements, under some mild restrictions on the group order. Pointed Hopf algebras are precisely those whose coradicals are group algebras, and in turn these groups determine a large part of their structure. We use the recent classification of these Hopf algebras by Andruskiewitsch and Schneider [5]. Each has a presentation by generators and relations similar to those of the small quantum groups, yet they are much more general. Due to this similarity, some of the techniques of Ginzburg and Kumar [19] yield results in this general setting. However some differences do arise, notably that cohomology may no longer vanish in odd degrees, and that useful connections to Lie algebras have not been developed. Thus we must take a somewhat different approach in this paper, which also yields new proofs of some of the results in [19]. Since we have less information available in this general setting, we prove finite generation without computing the full structure of the cohomology ring. However we do explicitly identify a subalgebra over which the cohomology is finite, and establish some results about its structure.

These structure results follow in part from our general result in Section 3, that the cohomology ring of a Hopf algebra in a braided category is always braided graded commutative. This generalizes the well-known result that the cohomology ring of a Hopf algebra is graded commutative, one proof of which follows from the existence of two definitions of its multiplication. We generalize that proof, giving a braided version of the Eckmann-Hilton argument, from which follows the braided graded commutativity result. We apply this to a Nichols algebra in Section 5, thus obtaining some details about the structure of the finitely generated cohomology ring of the corresponding pointed Hopf algebra.

Of course one hopes for results in yet greater generality. However the structure of finite dimensional noncommutative, noncocommutative, nonsemisimple Hopf algebras, other than those treated in this paper, is largely unknown. There are a very small number of known (nontrivial) finite dimensional pointed Hopf algebras having nonabelian groups of grouplike elements (see for example [1, 2]). Even fewer examples are known of nonpointed, nonsemisimple Hopf algebras [12]. To prove finite generation of cohomology in greater generality, it may be necessary to find general techniques, such as the embedding of any finite group scheme into GL_{n} used by Friedlander and Suslin [18], rather than depending on structural knowledge of the Hopf algebras as we do here.

Our proof of finite generation is a two-step reduction to type A_{1} × ⋯ × A_{1}, in which case the corresponding Nichols algebra is a quantum complete intersection. For these algebras, we compute cohomology explicitly via a resolution constructed in Section 4. This resolution is adapted from [10, 21] where similar algebras were considered (but in positive characteristic). Each of our two reduction steps involves a spectral sequence associated to an algebra filtration. In the first step (Section 5) a Radford biproduct of the form B(V)#kΓ, for a group Γ and Nichols algebra B(V), has a filtration for which the associated graded algebra has type A_{1} × ⋯ × A_{1}. This filtration is generalized from De Concini and Kac [14]. We identify some permanent cycles and apply a lemma adapted from Friedlander and Suslin [18] to conclude finite generation. In the second reduction step (Section 6), any of Andruskiewitsch and Schneider’s pointed Hopf algebras u(D, λ, μ) has a filtration for which the associated graded algebra is a Radford biproduct, whose cohomology features in Section 5. Again we identify some permanent cycles and conclude finite generation. As a corollary we show that the Hochschild cohomology ring of u(D, λ, μ) is also finitely generated.

The first and last authors thank Ludwig-Maximilians-Universität München for its hospitality during the first stages of this project. The second author thanks MSRI for its hospitality and support during the final stage of this work. The first author was supported by an NSERC postdoctoral fellowship. The second author was partially supported by the NSF grants DMS-0629156 and DMS-0800940. The third author was supported by Deutsche Forschungsgemeinschaft through a Heisenberg Fellowship. The last author was partially supported by NSF grants DMS-0443476 and DMS-0800832, and NSA grant H98230-07-1-0038. The last author thanks D. J. Benson for very useful conversations.

2. Definitions and Preliminary Results

Let k be a field, usually assumed to be algebraically closed and of characteristic 0. All tensor products are over k unless otherwise indicated. Let Γ be a finite group.

Hopf algebras in Yetter-Drinfeld categories. A Yetter-Drinfeld module over kΓ is a Γ-graded vector space V = ⊕_{g∈Γ}V_{g} that is also a kΓ-module for which g · V_{h} = V_{ghg^{−1}} for all g, h ∈ Γ. The grading by the group Γ is equivalent to a kΓcomodule structure on V, that is a map δ : V → kΓ ⊗ V, defined by δ(v) = g ⊗ v for all v ∈ V_{g}. Let _{Γ}^{Γ}YD denote the category of all Yetter-Drinfeld modules over kΓ. The category has a tensor product: (V ⊗ W)_{g} = ⊕_{xy=g}V_{x} ⊗ W_{y} for all g ∈ Γ, and Γ acts diagonally on V ⊗ W, that is, g(v ⊗ w) = gv ⊗ gw for all g ∈ Γ, v ∈ V, and w ∈ W. There is a braiding c : V ⊗ W →^{∼} W ⊗ V for all V, W ∈ _{Γ}^{Γ}YD as follows: Let g ∈ Γ, v ∈ V_{g}, and w ∈ W. Then c(v ⊗ w) = gw ⊗ v. Thus _{Γ}^{Γ}YD is a braided monoidal category. (For details on the category _{Γ}^{Γ}YD, including the connection to Hopf algebras recalled below, see for example [5].)

Let C be any braided monoidal category. For simplicity, we will always assume the tensor product is strictly associative with strict unit object I. An algebra in C is an object R together with morphisms u : I → R and m : R ⊗ R → R in C, such that m is associative in the sense that m(m ⊗ 1_{R}) = m(1_{R} ⊗ m), and u is a unit in the sense that m(u ⊗ 1_{R}) = 1_{R} = m(1_{R} ⊗ u). The definition of a coalgebra in C is similar, with the arrows going in the opposite direction. Thus, an algebra (resp. coalgebra) in _{Γ}^{Γ}YD is simply an ordinary algebra (resp. coalgebra) with multiplication (resp. comultiplication) a graded and equivariant map. A braided Hopf algebra in C is an algebra as well as coalgebra in C such that its comultiplication and counit are algebra morphisms, and such that the identity morphism id: R → R has a convolution inverse s in C. When we say that the comultiplication Δ: R → R ⊗ R should be an algebra morphism, the braiding c of C arises in the definition of the algebra structure of R ⊗ R (so in particular, a braided Hopf algebra in _{Γ}^{Γ}YD is not an ordinary Hopf algebra). More generally, if A, B are two algebras in C, their tensor product A ⊗ B is defined to have multiplication m_{A⊗B} = (m_{A} ⊗ m_{B})(1_{A} ⊗ c ⊗ 1_{B}).

An example of a braided Hopf algebra in _{Γ}^{Γ}YD is the Nichols algebra B(V) associated to a Yetter-Drinfeld module V over kΓ; B(V) is the quotient of the tensor algebra T (V) by the largest homogeneous braided Hopf ideal generated by homogeneous elements of degree at least 2. For details, see [4, 5]. In this paper, we need only the structure of B(V) in some cases as are explicitly described below.

If R is a braided Hopf algebra in _{Γ}^{Γ}YD, then its Radford biproduct (or bosonization) R#kΓ is a Hopf algebra in the usual sense (that is, a Hopf algebra in _{k}^{k}YD). As an algebra, R#kΓ is just a skew group algebra, that is, R#kΓ is a free R-module with basis Γ on which multiplication is defined by (rg)(sh) = r(g·s)gh for all r, s ∈ R and g, h ∈ G. Comultiplication is given by Δ(rg) = r^{(1)}(r^{(2)})_{(−1)}g ⊗ (r^{(2)})_{(0)}g, for all r ∈ R and g ∈ Γ, where Δ(r) = ∑ r^{(1)} ⊗ r^{(2)} in R as a Hopf algebra in _{Γ}^{Γ}YD and δ(r) = ∑ r_{(−1)} ⊗ r_{(0)} denotes the kΓ-comodule structure.

The pointed Hopf algebras of Andruskiewitsch and Schneider. A pointed Hopf algebra H is one for which all simple comodules are one dimensional. This is equivalent to the condition H_{0} = kΓ where Γ = G(H) is the group of grouplike elements of H and H_{0} is the coradical of H (the initial term in the coradical filtration).

The Hopf algebras of Andruskiewitsch and Schneider in [5] are pointed, and are deformations of Radford biproducts. They depend on the following data: Let θ be a positive integer. Let (a_{ij})_{1≤i,j≤θ} be a Cartan matrix of finite type, that is the Dynkin diagram of (a_{ij}) is a disjoint union of copies of some of the diagrams A_{•}, B_{•}, C_{•}, D_{•}, E_{6}, E_{7}, E_{8}, F_{4}, G_{2}. In particular, a_{ii} = 2 for 1 ≤ i ≤ θ, a_{ij} is a nonpositive integer for i ≠ j, and a_{ij} = 0 implies a_{ji} = 0. Its Dynkin diagram is a graph with vertices labeled 1, …, θ: The vertices i and j are connected by a_{ij}a_{ji} edges, and if |a_{ij}| > |a_{ji}|, there is an arrow pointing from j to i.

Now assume Γ is abelian, and denote by Γ̂ its dual group of characters. For each i, 1 ≤ i ≤ θ, choose g_{i} ∈ Γ and χ_{i} ∈ Γ̂ such that χ_{i}(g_{i}) ≠ 1 and

(2.0.1)  χ_{j}(g_{i})χ_{i}(g_{j}) = χ_{i}(g_{i})^{a_{ij}}

(the Cartan condition) holds for 1 ≤ i, j ≤ θ. Letting q_{ij} = χ_{j}(g_{i}), this becomes q_{ij}q_{ji} = q_{ii}^{a_{ij}}. Call

(2.0.2)  D = (Γ, (g_{i})_{1≤i≤θ}, (χ_{i})_{1≤i≤θ}, (a_{ij})_{1≤i,j≤θ})

a datum of finite Cartan type associated to Γ and (a_{ij}). The Hopf algebras of interest will be generated as algebras by Γ and symbols x_{1}, …, x_{θ}.

Let V be the vector space with basis x_{1}, …, x_{θ}. Then V has a structure of a Yetter-Drinfeld module over kΓ: V_{g} = Span_{k}{x_{i} | g_{i} = g} and g(x_{i}) = χ_{i}(g)x_{i} for 1 ≤ i ≤ θ and g ∈ Γ. This induces the structure of an algebra in _{Γ}^{Γ}YD on the tensor algebra T (V). In particular Γ acts by automorphisms on T (V), and T (V) is a Γ-graded algebra in which x_{i_{1}} ⋯ x_{i_{s}} has degree g_{i_{1}} ⋯ g_{i_{s}}. The braiding c: T (V) ⊗ T (V) → T (V) ⊗ T (V) is induced by c(x_{i} ⊗ y) = g_{i}(y) ⊗ x_{i}. Moreover, T (V) can be made a braided Hopf algebra in _{Γ}^{Γ}YD if we define comultiplication as the unique algebra map Δ: T (V) → T (V) ⊗ T (V) satisfying Δ(v) = v ⊗ 1 + 1 ⊗ v for all v ∈ V. We define the braided commutators

ad_{c}(x_{i})(y) = [x_{i}, y]_{c} := x_{i}y − g_{i}(y)x_{i},

for all y ∈ T (V), and similarly in quotients of T (V) by homogeneous ideals.

Let Φ denote the root system corresponding to (a_{ij}), and let Π denote a fixed set of simple roots. If α_{i}, α_{j} ∈ Π, write i ∼ j if the corresponding vertices in the Dynkin diagram of Φ are in the same connected component. Choose scalars λ = (λ_{ij})_{1≤i<j≤θ, i≁j}, called linking parameters, such that

λ_{ij} = 0 if g_{i}g_{j} = 1 or χ_{i}χ_{j} ≠ ε,

where ε is the identity element in the dual group Γ̂ (equivalently ε is the counit on kΓ, ε(g) = 1 for all g ∈ Γ). The Hopf algebra U(D, λ) defined by Andruskiewitsch and Schneider [5] is the quotient of T (V)#kΓ by relations corresponding to the equations

(group action)  gx_{i}g^{−1} = χ_{i}(g)x_{i} (g ∈ Γ, 1 ≤ i ≤ θ),
(Serre relations)  (ad_{c}(x_{i}))^{1−a_{ij}}(x_{j}) = 0 (i ≠ j, i ∼ j),
(linking relations)  (ad_{c}(x_{i}))(x_{j}) = λ_{ij}(1 − g_{i}g_{j}) (i < j, i ≁ j).

The coalgebra structure of U(D, λ) is given by

Δ(g) = g ⊗ g, Δ(x_{i}) = x_{i} ⊗ 1 + g_{i} ⊗ x_{i},

ε(g) = 1, ε(x_{i}) = 0, s(g) = g^{−1}, s(x_{i}) = −g_{i}^{−1}x_{i}, for all g ∈ Γ, 1 ≤ i ≤ θ.

Example 2.1. Let 𝔤 be a simple Lie algebra of rank n, let (a_{ij}) be two block diagonal copies of the corresponding Cartan matrix, and let θ = 2n. Let q be a primitive ℓth root of unity, ℓ odd. Let Γ = (ℤ/ℓℤ)^{n} with generators g_{1}, …, g_{n} and define χ_{i} ∈ Γ̂ by χ_{i}(g_{j}) = q^{⟨α_{i}, α_{j}⟩}. Let g_{i+n} = g_{i} and χ_{i+n} = χ_{i}^{−1} for 1 ≤ i ≤ n. Let λ_{ij} = (q^{−1} − q)^{−1}δ_{j,i+n} for 1 ≤ i < j ≤ n. Then U(D, λ) is a quotient of the quantum group U_{q}(𝔤). An epimorphism U_{q}(𝔤) → U(D, λ) is given by K_{i} ↦ g_{i}, E_{i} ↦ x_{i} and F_{i} ↦ x_{i+n}g_{i}^{−1}.

The Hopf algebra U(D, λ) has finite dimensional quotients that we discuss next. As in [5] we make the assumptions:

(2.1.1)

The order of χ_{i}(g_{i}) is odd for all i,

and is prime to 3 for all i in a connected component of type G_{2}.

It then follows from the Cartan condition (2.0.1) that the order of χ_{i}(g_{i}) is the same as the order of χ_{j}(g_{j}) if i ∼ j. That is, this order is constant in each connected component J of the Dynkin diagram; denote this common order by N_{J}. It will also be convenient to denote it by N_{β_{j}} or N_{j} for each positive root β_{j} in J (this standard notation is defined below). Let α ∈ Φ^{+}, α = ∑_{i=1}^{θ} n_{i}α_{i}, and let ht(α) = ∑_{i=1}^{θ} n_{i},

(2.1.2)  g_{α} = ∏_{i=1}^{θ} g_{i}^{n_{i}}, and χ_{α} = ∏_{i=1}^{θ} χ_{i}^{n_{i}}.

There is a unique connected component J_{α} of the Dynkin diagram of Φ for which n_{i} ≠ 0 implies i ∈ J_{α}. We write J = J_{α} when it is clear which α is intended.

Let W be the Weyl group of the root system Φ. Let w_{0} = s_{i_{1}} ⋯ s_{i_{r}} be a reduced decomposition of the longest element w_{0} ∈ W as a product of simple reflections. Let

β_{1} = α_{i_{1}}, β_{2} = s_{i_{1}}(α_{i_{2}}), …, β_{r} = s_{i_{1}}s_{i_{2}} ⋯ s_{i_{r−1}}(α_{i_{r}}).

Then β_{1}, …, β_{r} are precisely the positive roots Φ^{+} [24]. Corresponding root vectors x_{β_{j}} ∈ U(D, λ) are defined in the same way as for the traditional quantum groups: In case D corresponds to the data for a quantum group U_{q}(𝔤) (see Example 2.1), let

x_{β_{j}} = T_{i_{1}}T_{i_{2}} ⋯ T_{i_{j−1}}(x_{i_{j}}),

where the T_{i_{j}} are Lusztig’s algebra automorphisms of U_{q}(𝔤) [24]. The x_{β_{j}} may be expressed as iterated braided commutators. If β_{j} is a simple root α_{l}, then x_{β_{j}} = x_{l}. In our more general setting, as in [5], define the x_{β_{j}} to be the analogous iterated braided commutators. More precisely, we describe how to obtain these elements in case the Dynkin diagram is connected; in the general case construct the root vectors separately for each connected component. Let I be the ideal of T (V) generated by elements corresponding to the Serre relations. Then [5, Lemmas 1.2 and 1.7] may be used to obtain a linear isomorphism between T (V)/I and the upper triangular part of some U_{q}(𝔤) with the same Dynkin diagram. This linear isomorphism preserves products and braided commutators up to nonzero scalar multiples, and thus yields x_{β_{j}} in our general setting as an iterated braided commutator. (See the proof of Lemma 2.4 below for more details on this linear isomorphism.)

Choose scalars (μ_{α})_{α∈Φ^{+}}, called root vector parameters, such that

(2.1.3)  μ_{α} = 0 if g_{α}^{N_{α}} = 1 or χ_{α}^{N_{α}} ≠ ε.

The finite dimensional Hopf algebra u(D, λ, μ) is the quotient of U(D, λ) by the ideal generated by all

(2.1.4)  (root vector relations)  x_{α}^{N_{α}} − u_{α}(μ) (α ∈ Φ^{+})

where u_{α}(μ) ∈ kΓ is defined inductively on Φ^{+} in [5, Defn. 2.14]. In this paper, we do not need the details of the construction of the elements u_{α}(μ) in the group algebra. We only need the fact that if μ_{α} = 0 for all α ∈ Φ^{+}, then u_{α}(μ) = 0 for all α ∈ Φ^{+} [5] (see for example Lemma 6.1 below). It is interesting to note that if α is a simple root, then u_{α}(μ) := μ_{α}(1 − g_{α}^{N_{α}}).

Example 2.2. Let D, λ be the data from Example 2.1. Then there is an isomorphism u_{q}(𝔤) ≃ u(D, λ, 0), induced by the epimorphism U_{q}(𝔤) → U(D, λ) given in that example.

The following theorem is [5, Classification Theorem 0.1], and requires k to be algebraically closed of characteristic 0.

Theorem 2.3 (Andruskiewitsch-Schneider). The Hopf algebras u(D, λ, μ) are finite dimensional and pointed. Conversely, if H is a finite dimensional pointed Hopf algebra having abelian group of grouplike elements with order not divisible by primes less than 11, then H ≃ u(D, λ, μ) for some D, λ, μ.

Two filtrations. Note that u(D, λ, μ) is a (coradically) filtered Hopf algebra, with deg(x_{i}) = 1 (1 ≤ i ≤ θ) and deg(g) = 0 (g ∈ Γ). The associated graded Hopf algebra is isomorphic to the graded Hopf algebra u(D, 0, 0). There is an isomorphism u(D, 0, 0) ≃ B(V)#kΓ, the Radford biproduct (or bosonization) of the Nichols algebra B(V) of the Yetter-Drinfeld module V over kΓ. For details, see [5]. Note that B(V) is isomorphic to the subalgebra of u(D, 0, 0) generated by x_{1}, …, x_{θ}.

In Section 6, we prove that the cohomology of u(D, λ, μ) is finitely generated by using a spectral sequence relating its cohomology to that of B(V)#kΓ. In Section 5, we prove that the cohomology of this (coradically) graded bialgebra B(V)#kΓ is finitely generated, by using a spectral sequence relating its cohomology to that of a much simpler algebra: We put a different filtration on it (see Lemma 2.4 below) for which the associated graded algebra has type A_{1} × ⋯ × A_{1}. (That is, the Dynkin diagram is a disjoint union of diagrams A_{1}.) In Section 4, we give the cohomology for type A_{1} × ⋯ × A_{1} explicitly as a special case of the cohomology of a quantum complete intersection.

By [5, Thm. 2.6], the Nichols algebra B(V) has PBW basis all

(2.3.1)  x^{a} = x_{β_{1}}^{a_{1}} ⋯ x_{β_{r}}^{a_{r}} (0 ≤ a_{i} < N_{β_{i}}),

and further

(2.3.2)  [x_{α}, x_{β}^{N_{β}}]_{c} = 0

for all α, β ∈ Φ^{+}. As in [14], put a total order on the PBW basis elements as follows: The degree of such an element is

d(x_{β_{1}}^{a_{1}} ⋯ x_{β_{r}}^{a_{r}}) = (∏ a_{i} ht(β_{i}), a_{r}, …, a_{1}) ∈ ℕ^{r+1}.

Order the elements (2.3.1) lexicographically by degree where

(0, …, 0, 1) < (0, …, 0, 1, 0) < ⋯ < (1, 0, …, 0).

Lemma 2.4. In the Nichols algebra B(V), for all i < j,

[x_{β_{j}}, x_{β_{i}}]_{c} = ∑_{a∈ℕ^{p}} ρ_{a}x^{a}

where the ρ_{a} are scalars for which ρ_{a} = 0 unless d(x^{a}) < d(x_{β_{i}}x_{β_{j}}).

Proof. First note that if i ≁ j, then [x_{β_{j}}, x_{β_{i}}]_{c} = 0 by the Serre relations, since a_{ij} = 0. Thus we may assume now that i ∼ j. Then the lemma is just the translation of [14, Lemma 1.7], via twisting by a cocycle and specializing q, into this more general setting. The twisting method is described in [5] and is used there to prove that (2.3.1) is a basis of B(V). In particular, [5, Lemma 2.3] states that there exist integers d_{i} ∈ {1, 2, 3}, 1 ≤ i ≤ θ, and q ∈ k such that for all 1 ≤ i, j ≤ θ, q_{ii} = q^{2d_{i}} and d_{i}a_{ij} = d_{j}a_{ji}. This allows us to define a matrix (q′_{ij}) by q′_{ij} = q^{d_{i}a_{ij}}, so that q_{ij}q_{ji} = q′_{ij}q′_{ji} and q_{ii} = q′_{ii}. Let V′ ∈ _{Γ}^{Γ}YD with basis x′_{1} …, x′_{θ} and structure given by the data {q′_{ij}}. Now {q′_{ij}} are classical quantum group parameters for the positive part of U_{q}(𝔤), where 𝔤 corresponds to the Dynkin diagram. Thus [14, Lemma 1.7] applies. Further, [5, Lemma 1.2] states that there is a cocycle σ : ℤΠ × ℤΠ → k^{×} and a k-linear isomorphism ϕ : T (V) → T (V′) with ϕ(x_{i}) = x′_{i} and such that if x ∈ T (V)_{g_{i}} and y ∈ T (V)_{g_{j}}, then

ϕ(xy) = σ(α_{i}, α_{j})ϕ(x)ϕ(y) and ϕ([x, y]_{c}) = σ(α_{i}, α_{j})[ϕ(x), ϕ(y)]_{c′}.

Thus ϕ preserves the PBW basis up to nonzero scalar multiples, and it preserves the total order on the PBW basis. Since it also preserves braided commutators up to nonzero scalar, the lemma now holds as a consequence of the same result [14, Lemma 1.7] for the parameters {q′_{ij}}. □

By Lemma 2.4, the above ordering induces a filtration F on B(V) for which the associated graded algebra Gr B(V) has relations [x_{β_{j}}, x_{β_{i}}]_{c} = 0 for all i < j, and x_{β_{i}}^{N_{i}} = 0. In particular, Gr B(V) is of type A_{1} × ⋯ × A_{1}. We may put a corresponding Hopf algebra structure on (Gr B(V))#kΓ as follows. If β_{i} = ∑_{j=1}^{θ} n_{j}α_{j}, let g_{β_{i}} = g_{1}^{n_{1}} ⋯ g_{θ}^{n_{θ}} as in (2.1.2). Now identify β_{1}, …, β_{r} with the simple roots of type A_{1} × ⋯ × A_{1}. For i < j, define q_{β_{i}β_{j}} = χ_{β_{j}}(g_{β_{i}}), q_{β_{j}β_{i}} = q_{β_{i}β_{j}}^{−1}, and q_{β_{i}β_{i}} = 1. Then the Cartan condition (2.0.1) holds for the scalars q_{β_{i}β_{j}} in this type A_{1} × ⋯ × A_{1}, and (Gr B(V))#kΓ is a Hopf algebra for which

Δ(x_{β_{i}}) = x_{β_{i}} ⊗ 1 + g_{β_{i}} ⊗ x_{β_{i}}.

For our spectral sequence constructions, we rewrite the total order on the PBW basis elements (2.3.1) explicitly as an indexing by positive integers. We may set

deg(x_{β_{1}}^{a_{1}} ⋯ x_{β_{r}}^{a_{r}}) = N_{β_{1}} ⋯ N_{β_{r}} ∏ a_{i} ht(β_{i}) + N_{β_{1}} ⋯ N_{β_{r−1}}a_{r} + ⋯ + N_{β_{1}}a_{2} + a_{1}.

A case-by-case argument shows that d(x_{β_{1}}^{a_{1}} ⋯ x_{β_{r}}^{a_{r}}) < d(x_{β_{1}}^{b_{1}} ⋯ x_{β_{r}}^{b_{r}}) if, and only if, deg(x_{β_{1}}^{a_{1}} ⋯ x_{β_{r}}^{a_{r}}) < deg(x_{β_{1}}^{b_{1}} ⋯ x_{β_{r}}^{b_{r}}).

Hochschild cohomology. In this paper, we are interested in the cohomology ring H^{∗}(u(D, λ, μ), k) := Ext_{u(D,λ,μ)}^{∗}(k, k), where k is the trivial module given by the counit ε. If A is any k-algebra with an augmentation ε : A → k, note that Ext_{A}^{∗}(k, k) is isomorphic to Hochschild cohomology with trivial coefficients, Ext_{A⊗A^{op}}^{∗}(A, k): This is due to an equivalence of the bar complexes for computing these Ext algebras. That is, these bar complexes are each equivalent to the reduced complex

(2.4.1) C^{•} :  0 → Hom_{k}(k, k) → Hom_{k}(A^{+}, k) → Hom_{k}((A^{+})^{2}, k) → ⋯,

where A^{+} = ker ε is the augmentation ideal of A. The differential is given by δ_{n+1}(f) = ∑_{i=0}^{n−1}(−1)^{i+1}f ∘(1^{i}⊗m⊗1^{n−i−1}) for all f : (A^{+})^{⊗n} → k. We will exploit this equivalence. This complex arises, for example, by applying Hom_{A}(−, k) to the free A-resolution of k:

⋯ → A ⊗ (A^{+})^{⊗2} ⟶^{∂_{2}} A ⊗ A^{+} ⟶^{∂_{1}} A ⟶^{ε} k → 0,

where

(2.4.2)  ∂_{i}(a_{0} ⊗ ⋯ ⊗ a_{i}) = ∑_{j=0}^{i−1}(−1)^{j}a_{0} ⊗ ⋯ ⊗ a_{j}a_{j+1} ⊗ ⋯ ⊗ a_{i}.

Then ∂_{n+1}^{∗} = δ_{n+1}. Equivalently, we may apply Hom_{A⊗A^{op}}(−, k) to the free A⊗A^{op}- resolution of A:

⋯ → A ⊗ (A^{+})^{⊗2} ⊗ A ⟶^{d_{2}} A ⊗ A^{+} ⊗ A ⟶^{d_{1}} A ⊗ A ⟶^{ε} A → 0,

d_{i}(a_{0} ⊗ ⋯ ⊗ a_{i+1}) = ∑_{j=0}^{i}(−1)^{j}a_{0} ⊗ ⋯ ⊗ a_{j}a_{j+1} ⊗ ⋯ ⊗ a_{i+1}.

A finite generation lemma. In Sections 5 and 6, we will need the following general lemma adapted from [18, Lemma 1.6]. Recall that an element a ∈ E_{r}^{p,q} is called a permanent cycle if d_{i}(a) = 0 for all i ≥ r.

Lemma 2.5. (a) Let \xymatrix{E_{1}^{p,q} \ar@{=>}[r] & E_{∞}^{p+q}} be a multiplicative spectral sequence of kalgebras concentrated in the half plane p + q ≥ 0, and let A^{∗,∗} be a bigraded commutative k-algebra concentrated in even (total) degrees. Assume that there exists a bigraded map of algebras \xymatrix{ϕ : A^{∗,∗} \ar[r] & E_{1}^{∗,∗}} such that

(i) ϕ makes E_{1}^{∗,∗} into a Noetherian A^{∗,∗}-module, and (ii) the image of A^{∗,∗} in E_{1}^{∗,∗} consists of permanent cycles.

Then E_{∞}^{∗} is a Noetherian module over Tot(A^{∗,∗}).

(b) Let \xymatrix{Ẽ_{1}^{p,q} \ar@{=>}[r] & Ẽ_{∞}^{p+q}} be a spectral sequence that is a bigraded module over the

spectral sequence E^{∗,∗}. Assume that Ẽ_{1}^{∗,∗} is a Noetherian module over A^{∗,∗} where A^{∗,∗} acts on Ẽ_{1}^{∗,∗} via the map ϕ. Then Ẽ_{∞}^{∗} is a finitely generated E_{∞}^{∗}-module.

Proof. Let Λ_{r}^{∗,∗} ⊂ E_{r}^{∗,∗} be the bigraded subalgebra of permanent cycles in E_{r}^{∗,∗}. Observe that d_{r}(E_{r}^{∗,∗}) is an A^{∗,∗}-invariant left ideal of Λ_{r}^{∗,∗}. Indeed, let a ∈ A^{p,q} and x ∈ E_{r}^{s,t}. We have d_{r}(\overline{ϕ(a)}x) = d_{r}(\overline{ϕ(a)})x + (−1)^{p+q}\overline{ϕ(a)}d_{r}(x) = \overline{ϕ(a)}d_{r}(x) since \overline{ϕ(a)} ∈ A^{∗,∗} is assumed to be a permanent cycle of even total degree. A similar computation shows that Λ_{1}^{∗,∗} is an A^{∗,∗}-submodule of E_{1}^{∗,∗}. By induction, Λ_{r+1}^{∗,∗} = Λ_{r}^{∗,∗}/d_{r}(E_{r}^{∗,∗}) is an A^{∗,∗}-module for any r ≥ 1. We get a sequence of surjective maps of A^{∗,∗}-modules:

(2.5.1)  \xymatrix{Λ_{1}^{∗,∗} \ar@{->>}[r] & … \ar@{->>}[r] & Λ_{r}^{∗,∗} \ar@{->>}[r] & Λ_{r+1}^{∗,∗} \ar@{->>}[r] & …}

Since Λ_{1}^{∗,∗} is an A^{∗,∗}-submodule of E_{1}^{∗,∗}, it is Noetherian as an A^{∗,∗}-module. Therefore, the kernels of the maps \xymatrix{Λ_{1}^{∗,∗} \ar@{->>}[r] & Λ_{r}^{∗,∗}} are Noetherian for all r ≥ 1. These kernels form an increasing chain of submodules of Λ_{1}^{∗,∗}, hence, by the Noetherian property, they stabilize after finitely many steps. Consequently, the sequence (2.5.1) stabilizes after finitely many steps. We conclude that Λ_{r}^{∗,∗} = E_{∞}^{∗,∗} for some r. Therefore E_{∞}^{∗,∗} is a Noetherian A^{∗,∗}-module. By [16, Proposition 2.1], E_{∞}^{∗} is a Noetherian module over Tot(A^{∗,∗}), finishing the proof of (a).

Let Λ̃_{r}^{∗,∗} ⊂ Ẽ_{r}^{∗,∗} be the subspace of permanent cycles. Arguing as above we can show that Λ̃_{r}^{∗,∗} is an A^{∗,∗}-submodule of Ẽ_{r}^{∗,∗}, and, moreover, that there exists r such that Λ̃_{r}^{∗,∗} = Ẽ_{∞}^{∗,∗}. Hence, Ẽ_{∞}^{∗,∗} is Noetherian over A^{∗,∗}. Applying [16, Proposition 2.1] once again, we conclude that Ẽ_{∞}^{∗} is Noetherian and hence finitely generated over Tot(A^{∗,∗}). Therefore, it is finitely generated over E_{∞}^{∗}. □

3. Braided graded commutativity of cohomology

We will show that Hochschild cohomology of a braided Hopf algebra in _{Γ}^{Γ}YD is a braided graded commutative algebra (more precisely, its opposite algebra is). This will be done in a more general categorical setting, so it will make sense to relax the overall requirement that k is a field.

To prepare for the main result of this section, we need some information on the Alexander-Whitney map. We follow Weibel [29, Section 8.5]. The case of a tensor product of simplicial modules over a commutative ring (rather than simplicial objects in a tensor category, or bisimplicial objects) can also be found in Mac Lane [25, Ch. VII, Section 8].

Consider a bisimplicial object A = (A_{kℓ}, ∂^{h}, ∂^{v}) in an abelian category. Associated to it we have the diagonal simplicial object diag A with components A_{nn}; to this in turn we associate the usual chain complex, whose differentials we denote by d_{n}^{diag}. Also associated to A is the double complex CA, whose horizontal and vertical differentials we denote by d_{k,ℓ}^{h}: A_{kℓ} → A_{k−1,ℓ} and d_{k,ℓ}^{v}: A_{kℓ} → A_{k,ℓ−1}. There is a natural chain morphism, unique up to natural chain homotopy,

f: diag A → Tot CA

for which f_{0} is the identity. We note that f being a chain morphism means that the components f_{kℓ}: A_{nn} → A_{kℓ} for n = k + ℓ satisfy

f_{kℓ} d_{n+1}^{diag} = d_{k+1,ℓ}^{h}f_{k+1,ℓ} + d_{k,ℓ+1}^{v}f_{k,ℓ+1}.

One possible choice for f is the Alexander-Whitney map, whose components are

f_{kℓ} = ∂_{k+1}^{h} … ∂_{n}^{h} \underbrace{∂_{0}^{v} … ∂_{0}^{v}}_{k}: A_{nn} → A_{kℓ}

for n = k + ℓ. We claim that

f′_{kℓ} = (−1)^{kℓ}∂_{0}^{h} … ∂_{0}^{h}∂_{ℓ+1}^{v} … ∂_{n}^{v}

is another choice, and therefore chain homotopic to f. For this it suffices to check that f′ (which is clearly natural) is a chain morphism. Instead of doing this directly (by more or less the same calculations usually done for the Alexander-Whitney map), we consider the front-to-back dual version of A. We denote this by Ã. The horizontal and vertical face operators are ˜∂_{k}^{h} = ˜∂_{n−k}^{h} and ˜∂_{k}^{v} = ˜∂_{n−k}^{v}. The horizontal and vertical differentials in CÃ are d̃_{k,ℓ}^{h} = (−1)^{k}d_{k,ℓ}^{h} and d̃_{k,ℓ}^{v} = (−1)^{ℓ}d_{k,ℓ}^{v}, the differentials in diag Ã are d̃_{n}^{diag} = (−1)^{n}d_{n}^{diag}. The Alexander-Whitney map f̃ of Ã is related to f′ by f′_{kℓ} = (−1)^{kℓ}f̃_{kℓ}. Therefore

f′_{kℓ} d_{n+1}^{diag} = (−1)^{kℓ}f̃_{kℓ}(−1)^{n+1}d̃_{n+1}^{diag}
= (−1)^{kℓ+k+ℓ+1}(d̃_{k+1,ℓ}^{h}f̃_{k+1,ℓ} + d̃_{k,ℓ+1}^{v}f̃_{k,ℓ+1})
= (−1)^{k+1}d̃_{k+1,ℓ}^{h}(−1)^{(k+1)ℓ}f̃_{k+1,ℓ} + (−1)^{ℓ+1}d̃_{k,ℓ+1}^{v}(−1)^{k(ℓ+1)}f̃_{k,ℓ+1}
= d_{k+1,ℓ}^{h}f′_{k+1,ℓ} + d_{k,ℓ+1}^{v}f′_{k,ℓ+1}.

We will use the above to treat the cohomology of braided Hopf algebras with trivial coefficients.

Let C be a braided monoidal category with braiding c; we denote the unit object by I. We will assume that the tensor product in C is strictly associative and unital, so we can perform some calculations in the graphical calculus (see for example [30] for more extensive examples of graphical calculations or [22] for more rigorous exposition). Our standard notations for braiding, multiplication, and unit will be

c = c_{XY} = X Y Y X ,  m = m_{A} = [figure: A; A; A],  η = η_{A} = A^{}.

Thus, if A, B are algebras in C, their tensor product A ⊗ B is defined to have multiplication

m_{A⊗B} = [figure: A; B; A; B; A; B].

We will need the following universal property of the tensor product algebra: Given an algebra R in C and algebra morphisms f_{X}: X → R for X ∈ {A, B} satisfying m_{R}(f_{B} ⊗f_{A}) = m_{R}(f_{A} ⊗f_{B})c, there exists a unique algebra morphism f: A⊗B → R with f_{A} = f(1_{A} ⊗ η_{B}) and f_{B} = f(η_{A} ⊗ 1_{B}), namely f = m_{R}(f_{A} ⊗ f_{B}).

We will denote by A \overline{⊗} B the tensor product algebra of A, B ∈ C taken with respect to the inverse of the braiding (i.e. in the braided category (B, c^{−1}).). It possesses an obvious analogous universal property. The equation

[figure: B; A; B; A; A; B] = [figure: B; A; B; A; A; B]

shows that c: B \overline{⊗} A → A ⊗ B is an isomorphism of algebras in C.

The following lemma is a version of the Eckmann-Hilton argument for algebras in braided monoidal categories.

Lemma 3.1. Let (A, m) be an algebra in the braided monoidal category C, equipped with a second multiplication ∇: A ⊗ A → A that shares the same unit η: I → A. Suppose that ∇: A ⊗ A → A is multiplicative with respect to m. Then the two multiplications coincide, and they are commutative.

Proof. We use the graphical symbols

m = [figure: A; A; A]  ∇ = [figure: A; A; A]

to distinguish the two multiplications.

The condition that ∇ is multiplicative then reads

[figure: A; A; A; A; A] = [figure: A; A; A; A; A]

which we use twice in the calculation

[figure: A; A; A] = [figure: A; A; A] = [figure: A; A; A] = [figure: A; A; A] = [figure: A; A; A] = [figure: A; A; A] = [figure: A; A; A]

□

Remark 3.2. It is not used in the proof that (A, m) is an (associative) algebra. It would suffice to require that we are given two multiplications that share the same unit, and that one is multiplicative with respect to the other according to the same formula that results from the definition of a tensor product of algebras.

The above generalities on tensor products of algebras in a braided category will be applied below to an ℕ-graded algebra that occurs as the cohomology of an algebra. To fix the terminology, we will denote the monoidal category of ℕ-graded objects in C by C^{ℕ}, with braiding c_{gr}, and we will call a commutative algebra in C^{ℕ} a braided graded commutative algebra.

Definition 3.3. Let C be a monoidal category, and A an augmented algebra in C. The simplicial object S_{•}A is defined by S_{n}A = A^{⊗n} with the faces ∂_{k}^{n}: S_{n}A → S_{n−1}A defined by ∂_{0}^{n} = ε ⊗ (1_{A})^{⊗(n−1)}, ∂_{i}^{n} = (1_{A})^{⊗(i−1)} ⊗ m ⊗ (1_{A})^{⊗(n−i−1)} for 0 < i < n and ∂_{n}^{n} = (1_{A})^{⊗(n−1)} ⊗ε and the degeneracies σ_{k}^{n}: S_{n}A → S_{n+1}A defined by inserting units in appropriate places.

If C is abelian, then S_{•}A has an associated chain complex, which we will also denote by S_{•}A.

Recall that a monoidal category C is (right) closed if for each object V, the endofunctor − ⊗ V is left adjoint. Its right adjoint is denoted \underline{hom}(V, —) and called an inner hom-functor. If C is right closed, then \underline{hom}(V, Y) is a bifunctor, covariant in Y and contravariant in V.

More generally, one can denote by \underline{hom}(V, Y) an object, if it exists, such that C(X ⊗ V, Y) ≃ C(X, \underline{hom}(V, Y)) as functors of X ∈ C. By Yoneda’s Lemma, such hom-objects are unique, and bifunctorial in those objects V and Y for which they exist.

Now consider a monoidal functor F: C → C′. If V, Y ∈ C are such that the hom objects \underline{hom}(V, Y) in C and \underline{hom}(FV, FY) in C′ exist, then there is a canonical morphism F\underline{hom}(V, Y) → \underline{hom}(FV, FY). If this is an isomorphism, we say that F preserves the hom object \underline{hom}(V, Y); if both categories are right closed, and all hom objects are preserved, we say that F preserves inner hom-functors.

Definition 3.4. Let C be an abelian monoidal category, and A an augmented algebra in C such that all hom objects \underline{hom}(A^{⊗n}, I) exist (e.g. if C is closed).

The Hochschild cohomology of A in C (with trivial coefficients) is the cohomology of the cochain complex \underline{hom}(S_{•}A, I). Thus the Hochschild cohomology consists of objects in the category C. We denote it by H^{∗}(A).

Remark 3.5. We will only need the results in this section for the case C = _{Γ}^{Γ}YD, but we will indicate more generally how they apply to ordinary algebras and (braided) Hopf algebras. For this, it is not necessary always to stick to the paper’s general assumption that we work over a base field, therefore we assume now that k is an arbitrary commutative base ring.

(1) In the case of an ordinary augmented k-algebra A, Definition 3.4 recovers the ordinary definition of Hochschild cohomology with trivial coefficients, that is H^{∗}(A) = H^{∗}(A, k) (see §2), which motivates our notation; general coefficients are not considered here.

(2) Assume that F: C → C′ is an exact monoidal functor that preserves the hom objects \underline{hom}(S_{•}A, I). Then F preserves Hochschild cohomology of A in the sense that H^{∗}(F(A)) ≃ F(H^{∗}(A)).

(3) The category of (say, left) G-modules for a k-Hopf algebra G has inner homobjects preserved by the underlying functor to the category of k-modules. Also, if G is a finitely generated projective Hopf algebra over k, the categories of (say, left) G-comodules, and of Yetter-Drinfeld modules over G have inner hom-objects that are preserved by the underlying functors to the category of k-modules. More concretely, the hom-object \underline{hom}(V, Y) is Hom_{k}(V, Y) with G-module and comodule structures induced by the antipode and its inverse.

(4) Recall that a dual object (V^{∗}, e, d) of an object V is an object V^{∗} with morphisms e: V^{∗}⊗V → I and d: I → V ⊗V^{∗} that satisfy (1_{V} ⊗e)(d⊗1_{V}) = 1_{V} and (e⊗1_{V^{∗}})(1_{V^{∗}} ⊗d) = 1_{V^{∗}}. If an object V in C has a dual, then all the inner hom-objects \underline{hom}(V, Y) exist and are given by \underline{hom}(V, Y) = Y ⊗ V^{∗}. Also, monoidal functors always preserve dual objects (by and large because these are given by morphisms and relations) and thus, if V has a dual, then the inner hom-objects \underline{hom}(V, Y) are preserved by every monoidal functor.

(5) A module V over a commutative ring k has a dual in the category of kmodules if and only if it is finitely generated projective; then V^{∗} is the dual module, e is the evaluation, and d maps 1_{k} to the canonical element, in the field case obtained by tensoring basis elements with the elements of the dual basis. If G is a k-Hopf algebra with bijective antipode, then in the categories of G-modules, G-comodules, and Yetter-Drinfeld modules over G an object V has a dual if and only if V is finitely generated projective over k.

(6) Let G be a flat Hopf algebra over k with bijective antipode, so that the category C of Yetter-Drinfeld modules over G is abelian braided monoidal with an exact underlying functor to the category C′ of k-modules. Then by the above remarks the underlying functor will preserve the Hochschild cohomology of A in C whenever G is finitely generated projective over k, or A is finitely generated projective. Thus, in either of these cases the Hochschild cohomology of A defined as above within the Yetter-Drinfeld category is the same as the ordinary Hochschild cohomology with trivial coefficients, endowed with a Yetter-Drinfeld module structure induced by that of A.

Remark 3.6. In a closed monoidal category C, there is a natural morphism ξ: \underline{hom}(B, I)⊗\underline{hom}(A, I) → \underline{hom}(A⊗B, I) which is suitably coherent with respect to higher tensor products. Under the adjunction

C(\underline{hom}(B, I) ⊗ \underline{hom}(A, I), \underline{hom}(A ⊗ B, I)) ≃ C(\underline{hom}(B, I) ⊗ \underline{hom}(A, I) ⊗ A ⊗ B, I)

ξ corresponds to

\underline{hom}(B, I) ⊗ \underline{hom}(A, I) ⊗ A ⊗ B ⟶^{1⊗e⊗1} \underline{hom}(B, I) ⊗ B ⟶^{e} I,

where e: \underline{hom}(X, I) ⊗ X → I is the natural “evaluation” morphism. Of course, to define the morphism in question, it is enough to assume that all the inner hom objects that occur in it exist.

If all inner hom functors \underline{hom}(V, I) exist, the isomorphisms ξ make \underline{hom}(—, I) into a contravariant weak monoidal functor.

If A and B have dual objects in C, then one can show that ξ above is an isomorphism; this can be attributed to the fact that (B^{∗} ⊗A^{∗}, ẽ, d̃) is a dual object for A ⊗ B with

ẽ = (B^{∗} ⊗ A^{∗} ⊗ A ⊗ B ⟶^{1⊗e⊗1} B^{∗} ⊗ B ⟶^{e} I)

d̃= (I ⟶^{d} A ⊗ A^{∗} ⟶^{1⊗d⊗1} A ⊗ B ⊗ B^{∗} ⊗ A^{∗})

Definition 3.7. The cup product on H^{∗}(A) is the collection of morphisms H^{n}(A)⊗ H^{m}(A) → H^{m+n}(A) induced by the morphisms

\underline{hom}(S_{n}A, I) ⊗ \underline{hom}(S_{m}A, I) ⟶^{ξ} \underline{hom}(S_{m}A ⊗ S_{n}A, I) = \underline{hom}(S_{m+n}(A), I).

Remark 3.8. Thus, in the category of modules over a commutative base ring k, our definition of cup product recovers the opposite of the usual cup product in Hochschild cohomology.

From now on we assume that C is braided.

Lemma 3.9. Let A, B be two augmented algebras in C. Denote by S_{•}A × S_{•}B the bisimplicial object obtained by tensoring the two simplicial objects S_{•}A and S_{•}B. An isomorphism S_{•}(A ⊗ B) → diag(S_{•}A × S_{•}B) of simplicial objects is given by the morphisms

g_{n}: S_{n}(A ⊗ B) = (A ⊗ B)^{⊗n} → A^{⊗n} ⊗ B^{⊗n} = diag_{n}(S_{•}A ⊗ S_{•}B)

that are composed from instances of the braiding c_{B,A}: B ⊗ A → A ⊗ B.

Proof. For example we have

g_{3} = A B A B A B A A A B B B and g_{4} = A B A B A B A B A A A A B B B B

and therefore

(∂_{2} ⊗ ∂_{2})g_{4} = [figure: A; B; A; B; A; B; A; B; A; A; A; B; B; B] = [figure: A; B; A; B; A; B; A; B; A; A; A; B; B; B] = g_{3}∂_{2}

The general claim (∂_{k} ⊗ ∂_{k})g_{n} = g_{n−1}∂_{k} is a merely larger version of the above example for 0 < k < n, and simpler (with augmentations instead of multiplications) for k ∈ {0, n}. □

According to the discussion at the beginning of the section we have two chain homotopic morphisms

diag(S_{•}A × S_{•}B) ≃ S_{•}A ⊗ S_{•}B.

Here, the right hand side denotes the tensor product of chain complexes. Explicitly, the two morphisms are given by

f_{kℓ} = (1_{A})^{⊗k} ⊗ ε^{⊗ℓ} ⊗ ε^{⊗k} ⊗ (1_{B})^{⊗ℓ}: S_{n}A ⊗ S_{n}B → S_{k}A ⊗ S_{ℓ}B
f′_{kℓ} = (−1)^{kℓ}ε^{⊗ℓ} ⊗ (1_{A})^{⊗k} ⊗ (1_{B})^{⊗ℓ} ⊗ ε^{⊗k}: S_{n}A ⊗ S_{n}B → S_{k}A ⊗ S_{ℓ}B

for k + ℓ = n.

Upon application of \underline{hom}(—, I), each of these morphisms, composed with the map g^{−1} from the preceding lemma, yields a morphism

(3.9.1) \underline{hom}(S_{•}B, I)⊗\underline{hom}(S_{•}A, I) → \underline{hom}(S_{•}A⊗S_{•}B, I) → \underline{hom}(S_{•}(A⊗B), I).

Both these maps, being homotopic, yield the same morphism

T′: H^{•}(B) ⊗ H^{•}(A) → H^{•}(A ⊗ B),

which we compose with the inverse of the braiding in C to obtain a morphism

T = T′c_{gr}^{−1} : H^{•}(A) ⊗ H^{•}(B) → H^{•}(A ⊗ B).

Lemma 3.10. The morphism

T: H^{•}(A) ⊗ H^{•}(B) → H^{•}(A ⊗ B)

is a morphism of graded algebras in C.

Proof. We need to show that

T′: H^{•}(B) \overline{⊗} H^{•}(A) → H^{•}(A ⊗ B)

is a morphism of algebras. To do this, we consider the algebra morphisms

H^{•}(A) → H^{•}(A ⊗ B) ← H^{•}(B).

We need to show that the composite

H^{•}(A) ⊗ H^{•}(B) → H^{•}(A ⊗ B) ⊗ H^{•}(A ⊗ B) → H^{•}(A ⊗ B),

where the last morphism is multiplication, is the map in the statement of the lemma, while the composite

H^{•}(B) ⊗ H^{•}(A) → H^{•}(A ⊗ B) ⊗ H^{•}(A ⊗ B) → H^{•}(A ⊗ B)

is the same, composed with the inverse braiding. But for this we only need to observe that

S_{n}(A ⊗ B) = S_{k}(A ⊗ B) ⊗ S_{ℓ}(A ⊗ B) ⟶^{S_{k}(1_{A}⊗ε)⊗S_{ℓ}(ε⊗1_{B})} S_{k}(A) ⊗ S_{ℓ}(B)

equals

S_{n}(A ⊗ B) ⟶^{g} S_{n}(A) ⊗ S_{n}(B) ⟶^{f_{kℓ}} S_{k}(A) ⊗ S_{ℓ}(B),

whereas

S_{n}(A ⊗ B) = S_{k}(A ⊗ B) ⊗ S_{ℓ}(A ⊗ B) ⟶^{S_{k}(ε⊗1_{B})⊗S_{ℓ}(1_{A}⊗ε)} S_{k}(B) ⊗ S_{ℓ}(A)

equals

S_{n}(A ⊗ B) ⟶^{g} S_{n}(A) ⊗ S_{n}(B) ⟶^{f′_{ℓk}} S_{ℓ}(A) ⊗ S_{k}(B) ⟶^{c_{gr}^{−1}} S_{k}(B) ⊗ S_{ℓ}(A).

□

Remark 3.11. In the case which is of main interest to us, k is a field, C is the category of Yetter-Drinfeld modules over a Hopf algebra G, and A, B are finite dimensional. We briefly sketch why these conditions imply that T is an isomorphism (see also [20, Prop.4.5]).

First, both in the construction of T′ (when homology is applied to the sequence 3.9.1) and in that of the cup product in Definition 3.7 we have tacitly used a version of a Künneth-type morphism

H(C^{•}) ⊗ H(D^{•}) → H(C^{•} ⊗ D^{•})

for chain complexes C^{•}, D^{•} in a closed monoidal category C. Under suitable flatness hypotheses on the objects and homologies that occur, this map will be an isomorphism. For example, this is the case if the tensor product in C is assumed to be exact in each argument which holds in our situation.

Second, the morphism ξ that occurs in the construction of T′ will be an isomorphism by Remark 3.6 since the algebras A and B clearly have duals in C.

Therefore, in our case T is an isomorphism.

Theorem 3.12. Let R be a bialgebra in the abelian braided monoidal category C. Then Hochschild cohomology H^{∗}(R) is a braided graded commutative algebra in C, assuming it is defined (i.e. the necessary hom-objects exist).

Proof. Comultiplication Δ on the bialgebra R yields a second multiplication

∇ := (H^{∗}(R) ⊗ H^{∗}(R) → H^{∗}(R ⊗ R) ⟶^{H^{∗}(Δ)} H^{∗}(R))

on H^{∗}(R). By construction, ∇ is an algebra map with respect to the algebra structure of H^{∗}(R) given by the cup product. Also, the two multiplications share the same unit, induced by the counit of R. Therefore, the two multiplications coincide and are commutative by Lemma 3.1. □

Corollary 3.13. Let G be a Hopf algebra with bijective antipode, and R a bialgebra in the category C of Yetter-Drinfeld modules over G. Assume that either G or R is finite dimensional. Then the Hochschild cohomology of R in C is a braided graded commutative algebra. Consequently, the opposite of ordinary Hochschild cohomology is a braided commutative Yetter-Drinfeld module algebra.

Proof. In fact the assumptions on G or R are designed to ensure that the hypotheses on inner hom objects in the results above are satisfied: If G is finitely generated projective, the category of Yetter-Drinfeld modules as a whole is closed, with inner hom-functors preserved by the underlying functors. If G is not finitely generated projective, but R is, then still R and all its tensor powers have dual objects in the category of Yetter-Drinfeld modules over G, that are preserved by the underlying functors. □

4. Cohomology of quantum complete intersections

Let θ be a positive integer, and for each i, 1 ≤ i ≤ θ, let N_{i} be an integer greater than 1. Let q_{ij} ∈ k^{×} for 1 ≤ i < j ≤ θ. Let S be the k-algebra generated by x_{1}, …, x_{θ} subject to the relations

(4.0.1)  x_{i}x_{j} = q_{ij}x_{j}x_{i} for all i < j and x_{i}^{N_{i}} = 0 for all i.

It is convenient to set q_{ji} = q_{ij}^{−1} for i < j. For B(V), the Nichols algebra constructed in Section 2, Lemma 2.4 implies that Gr B(V) has this form.

We will compute H^{∗}(S, k) = Ext_{S}^{∗}(k, k) for use in later sections. The structure of this ring may be determined by using the braided Künneth formula of Grunenfelder and Mastnak [20, Prop. 4.5 and Cor. 4.7] or the twisted tensor product formula of Bergh and Oppermann [11, Thms. 3.7 and 5.3]. We give details, using an explicit free S-resolution of k, in order to record needed information at the chain level. Our resolution was originally adapted from [10] (see also [21]): It is a braided tensor product of the periodic resolutions (4.0.2)

⋯⟶^{x_{i}^{N_{i}−1}·}k[x_{i}]/(x_{i}^{N_{i}})⟶^{x_{i}·}k[x_{i}]/(x_{i}^{N_{i}})⟶^{x_{i}^{N_{i}−1}·}k[x_{i}]/(x_{i}^{N_{i}})⟶^{x_{i}·}k[x_{i}]/(x_{i}^{N_{i}})→^{ε}k →0,

one for each i, 1 ≤ i ≤ θ.

Specifically, let K_{•} be the following complex of free S-modules. For each θ-tuple (a_{1}, …, a_{θ}) of nonnegative integers, let Φ(a_{1}, …, a_{θ}) be a free generator in degree a_{1} + ⋯ + a_{θ}. Thus K_{n} = ⊕_{a_{1}+···+a_{θ}=n}SΦ(a_{1}, …, a_{θ}). For each i, 1 ≤ i ≤ θ, let σ_{i}, τ_{i} : ℕ → ℕ be the functions defined by

σ_{i}(a) = \{ \begin{matrix}1, & if a is odd \\ N_{i} − 1, & if a is even, \end{matrix}

and τ_{i}(a) = ∑_{j=1}^{a} σ_{i}(j) for a ≥ 1, τ(0) = 0. Let

d_{i}(Φ(a_{1}, …, a_{θ})) = (∏_{ℓ<i}(−1)^{a_{ℓ}}q_{ℓi}^{σ_{i}(a_{i})τ_{ℓ}(a_{ℓ})}) x_{i}^{σ_{i}(a_{i})}Φ(a_{1}, …, a_{i} − 1, …, a_{θ})

if a_{i} > 0, and d_{i}(Φ(a_{1}, …, a_{θ})) = 0 if a_{i} = 0. Extend each d_{i} to an S-module homomorphism. Note that d_{i}^{2} = 0 for each i since x_{i}^{N_{i}} = 0 and σ_{i}(a_{i})+σ_{i}(a_{i} −1) = N_{i}. If i < j, we have d_{i}d_{j}(Φ(a_{1}, …, a_{θ}))

= d_{i} ((∏_{ℓ<j}(−1)^{a_{ℓ}}q_{ℓj}^{σ_{j}(a_{j})τ_{ℓ}(a_{ℓ})}) x_{j}^{σ_{j}(a_{j})}Φ(a_{1}, …, a_{j} − 1, …, a_{θ}))

= (∏_{ℓ<j}(−1)^{a_{ℓ}}q_{ℓj}^{σ_{j}(a_{j})τ_{ℓ}(a_{ℓ})}) (∏_{m<i}(−1)^{a_{m}}q_{mi}^{σ_{i}(a_{i})τ_{m}(a_{m})}) ·
x_{j}^{σ_{j}(a_{j})}x_{i}^{σ_{i}(a_{i})}Φ(a_{1}, …, a_{i} − 1, …, a_{j} − 1, …, a_{θ}).

Now d_{j}d_{i}(Φ(a_{1}, …, a_{θ})) is also an S-multiple of Φ(a_{1}, …, a_{i}−1, …, a_{j} −1, …, a_{θ}); comparing the two, we see that the term in which ℓ = i has a scalar factor that changes from (−1)^{a_{i}}q_{ij}^{σ_{j}(a_{j})τ_{i}(a_{i})} to (−1)^{a_{i}−1}q_{ij}^{σ_{j}(a_{j})τ_{i}(a_{i}−1)}, and x_{j}^{σ_{j}(a_{j})}x_{i}^{σ_{i}(a_{i})} is replaced by x_{i}^{σ_{i}(a_{i})}x_{j}^{σ_{j}(a_{j})} = q_{ij}^{σ_{j}(a_{j})σ_{i}(a_{i})}x_{j}^{σ_{j}(a_{j})}x_{i}^{σ_{i}(a_{i})}. Since τ_{i}(a_{i}) = τ_{i}(a_{i} − 1) + σ_{i}(a_{i}), this shows that

d_{i}d_{j} + d_{j}d_{i} = 0.

Letting

(4.0.3)  d = d_{1} + ⋯ + d_{θ},

we now have d^{2} = 0, so K_{•} is indeed a complex.

Next we show that K_{•} is a resolution of k by giving a contracting homotopy: Let α ∈ S, and fix ℓ, 1 ≤ ℓ ≤ θ. Write α = ∑_{j=0}^{N_{i}−1} α_{j}x_{ℓ}^{j} where α_{j} is in the subalgebra of S generated by the x_{i} with i ≠ ℓ. Define s_{ℓ}(αΦ(a_{1}, …, a_{θ})) to be the sum ∑_{j=0}^{N_{i}−1} s_{ℓ}(α_{j}x_{ℓ}^{j}Φ(a_{1}, …, a_{θ})), where

s_{ℓ}(α_{j}x_{ℓ}^{j}Φ(a_{1}, …, a_{θ}))

= \{ \begin{matrix}δ_{j>0}(∏_{m<ℓ}(−1)^{a_{m}}q_{mℓ}^{−σ_{ℓ}(a_{ℓ}+1)τ_{m}(a_{m})})α_{j}x_{ℓ}^{j−1}Φ(a_{1}, …, a_{ℓ} + 1, …, a_{θ}), \\ if a_{ℓ} is even \\ δ_{j,N_{ℓ}−1}(∏_{m<ℓ}(−1)^{a_{m}}q_{mℓ}^{−σ_{ℓ}(a_{ℓ}+1)τ_{m}(a_{m})})α_{j}Φ(a_{1}, …, a_{ℓ} + 1, …, a_{θ}), \\ if a_{ℓ} is odd \end{matrix}

where δ_{j>0} = 1 if j > 0 and 0 if j = 0. Calculations show that for all i, 1 ≤ i ≤ θ,

(s_{i}d_{i} + d_{i}s_{i})(α_{j}x_{i}^{j}Φ(a_{1}, …, a_{θ})) = \{ \begin{matrix}α_{j}x_{i}^{j}Φ(a_{1}, …, a_{θ}), & if j > 0 or a_{i} > 0 \\ 0, & if j = 0 and a_{i} = 0 \end{matrix}

and s_{ℓ}d_{i} + d_{i}s_{ℓ} = 0 for all i, ℓ with i ≠ ℓ. For each x_{1}^{j_{1}} ⋯ x_{θ}^{j_{θ}}Φ(a_{1}, …, a_{θ}), let c = c_{j_{1},...,j_{θ},a_{1},...,a_{θ}} be the cardinality of the set of all i (1 ≤ i ≤ θ) such that j_{i}a_{i} = 0. Define

s(x_{1}^{j_{1}} ⋯ x_{θ}^{j_{θ}}Φ(a_{1}, …, a_{θ})) = \frac{1}{θ − c_{j_{1},...,j_{θ},a_{1},...,a_{θ}}} (s_{1}+⋯+s_{θ})(x_{1}^{j_{1}} ⋯ x_{θ}^{j_{θ}}Φ(a_{1}, …, a_{θ})).

Then sd + ds = id on each K_{n}, n > 0. That is, K_{•} is exact in positive degrees. To show that K_{•} is a resolution of k, put k in degree −1, and let the corresponding differential be the counit map ε : S → k. It can be shown directly that K_{•} then becomes exact at K_{0} = S: The kernel of ε is spanned over the field k by the elements x_{1}^{j_{1}} ⋯ x_{θ}^{j_{θ}}Φ(0, …, 0), 0 ≤ j_{i} ≤ N_{i}, with at least one j_{i} ≠ 0. Let x_{1}^{j_{1}} ⋯ x_{θ}^{j_{θ}}Φ(0, …, 0) be such an element, and let i be the smallest positive integer such that j_{i} ≠ 0. Then d(x_{i}^{j_{i}−1} ⋯ x_{θ}^{j_{θ}}Φ(0, …, 1, …, 0)) is a nonzero scalar multiple of x_{i}^{j_{i}} ⋯ x_{θ}^{j_{θ}}Φ(0, …, 0). Thus ker(ε) = im(d), and K_{•} is a free resolution of k as an S-module.

Next we will use K_{•} to compute Ext_{S}^{∗}(k, k). Applying Hom_{S}(−, k) to K_{•}, the induced differential d^{∗} is the zero map since x_{i}^{σ_{i}(a_{i})} is always in the augmentation ideal. Thus the cohomology is the complex Hom_{S}(K_{•}, k), and in degree n this is a vector space of dimension \binom{n+θ−1}{θ−1}. Now let ξ_{i} ∈ Hom_{S}(K_{2}, k) be the function dual to Φ(0, …, 0, 2, 0, …, 0) (the 2 in the ith place) and η_{i} ∈ Hom_{S}(K_{1}, k) be the function dual to Φ(0, …, 0, 1, 0, …, 0) (the 1 in the ith place). By abuse of notation, identify these functions with the corresponding elements in H^{2}(S, k) and H^{1}(S, k), respectively. We will show that the ξ_{i}, η_{i} generate H^{∗}(S, k), and determine the relations among them. In order to do this we will abuse notation further and denote by ξ_{i} and η_{i} the corresponding chain maps ξ_{i} : K_{n} → K_{n−2} and η_{i} : K_{n} → K_{n−1} defined by

ξ_{i}(Φ(a_{1}, …, a_{θ})) = ∏_{ℓ>i} q_{iℓ}^{N_{i}τ_{ℓ}(a_{ℓ})}Φ(a_{1}, …, a_{i} − 2, …, a_{θ})
η_{i}(Φ(a_{1}, …, a_{θ})) = ∏_{ℓ<i} q_{ℓi}^{(σ_{i}(a_{i})−1)τ_{ℓ}(a_{ℓ})} ∏_{ℓ>i}(−1)^{a_{ℓ}}q_{iℓ}^{τ_{ℓ}(a_{ℓ})}x_{i}^{σ_{i}(a_{i})−1}Φ(a_{1}, …, a_{i} − 1, …, a_{θ}).

Calculations show that these are indeed chain maps. The ring structure of the subalgebra of H^{∗}(S, k) generated by ξ_{i}, η_{i} is given by composition of these chain maps. Direct calculation shows that the relations given in Theorem 4.1 below hold. (Note that if N_{i} ≠ 2 the last relation implies η_{i}^{2} = 0.) Alternatively, in case S = B(V), a Nichols algebra defined in Section 2, we may apply Corollary 3.13 to obtain the relations in Theorem 4.1 below, performing calculations that are a special case of the ones in the proof of Theorem 5.4. Thus any element in the algebra generated by the ξ_{i} and η_{i} may be written as a linear combination of elements of the form ξ_{1}^{b_{1}} ⋯ ξ_{θ}^{b_{θ}}η_{1}^{c_{1}} ⋯ η_{θ}^{c_{θ}} with b_{i} ≥ 0 and c_{i} ∈ {0, 1}. Such an element takes Φ(2b_{1} + c_{1}, …, 2b_{θ} + c_{θ}) to a nonzero scalar multiple of Φ(0, …, 0) and all other S-basis elements of K_{∑(2b_{i}+c_{i})} to 0. Since the dimension of H^{n}(S, k) is \binom{n+θ−1}{θ−1}, this shows that the ξ_{1}^{b_{1}} ⋯ ξ_{θ}^{b_{θ}}η_{1}^{c_{1}} ⋯ η_{θ}^{c_{θ}} form a k-basis for H^{∗}(S, k). We have proven:

Theorem 4.1. Let S be the k-algebra generated by x_{1}, …, x_{θ}, subject to relations (4.0.1). Then H^{∗}(S, k) is generated by ξ_{i}, η_{i} (i = 1, …, θ) where deg ξ_{i} = 2 and deg η_{i} = 1, subject to the relations

(4.1.1)  ξ_{i}ξ_{j} = q_{ji}^{N_{i}N_{j}}ξ_{j}ξ_{i}, η_{i}ξ_{j} = q_{ji}^{N_{j}}ξ_{j}η_{i}, and η_{i}η_{j} = −q_{ji}η_{j}η_{i}.

Note that although the relations (4.1.1) can be obtained as a consequence of Corollary 3.13, in order to obtain the full statement of the theorem, we needed more information.

Remark 4.2. We obtain [19, Prop. 2.3.1] as a corollary: In this case we replace the generators x_{i} of S by the generators E_{α} (α ∈ Δ^{+}) of Gr u_{q}^{+}, whose relations are

E_{α}E_{β} = q^{⟨α,β⟩}E_{β}E_{α} (α ≻ β), E_{α}^{ℓ} = 0 (α ∈ Δ^{+}).

Theorem 4.1 then implies that H^{∗}(Gr u_{q}^{+}) is generated by ξ_{α}, η_{α} (α ∈ Δ^{+}) with relations

ξ_{α}ξ_{β} = ξ_{β}ξ_{α}, η_{α}ξ_{β} = ξ_{β}η_{α}, η_{α}η_{β} = −q^{−⟨α,β⟩}η_{β}η_{α} (α ≻ β), η_{α}^{2} = 0,

which is precisely [19, Prop. 2.3.1].

Now assume a finite group Γ acts on S by automorphisms for which x_{1}, …, x_{θ} are eigenvectors. For each i, 1 ≤ i ≤ θ, let χ_{i} be the character on Γ for which gx_{i} = χ_{i}(g)x_{i}. As the characteristic of k is 0, we have

Ext_{S#kΓ}^{∗}(k, k) ≃ Ext_{S}^{∗}(k, k)^{Γ},

where the action of Γ at the chain level on (4.0.2) is as usual in degree 0, but shifted in higher degrees so as to make the differentials commute with the action of Γ. Specifically, note that the following action of Γ on K_{•} commutes with the differentials:

g · Φ(a_{1}, …, a_{θ}) = ∏_{ℓ=1}^{θ} χ_{ℓ}(g)^{τ_{ℓ}(a_{ℓ})}Φ(a_{1}, …, a_{θ})

for all g ∈ Γ, and a_{1}, …, a_{θ} ≥ 0. Then the induced action of Γ on generators ξ_{i}, η_{i} of the cohomology ring H^{∗}(S, k) is given explicitly by

(4.2.1)  g · ξ_{i} = χ_{i}(g)^{−N_{i}}ξ_{i} and g · η_{i} = χ_{i}(g)^{−1}η_{i}.

5. Coradically graded finite dimensional pointed Hopf algebras

Let D be arbitrary data as in (2.0.2), V the corresponding Yetter-Drinfeld module, and R = B(V) its Nichols algebra, and described in Section 2. By Lemma 2.4, there is a filtration on R for which S = Gr R is of type A_{1} × ⋯ × A_{1}, given by generators and relations of type (4.0.1). Thus H^{∗}(S, k) is given by Theorem 4.1. As the filtration is finite, there is a convergent spectral sequence associated to the filtration (see [29, 5.4.1]):

(5.0.2)  \xymatrix{E_{1}^{p,q} = H^{p+q}(Gr_{(p)} R, k) \ar@{=>}[r] & H^{p+q}(R, k).}

It follows that the E_{1}-page of the spectral sequence is given by Theorem 4.1 with grading corresponding to the filtration on R. We will see that by (5.0.6) and Lemma 5.1 below, the generators ξ_{i} are in degrees (p_{i}, 2 − p_{i}), where

(5.0.3)  p_{i} = N_{β_{1}} ⋯ N_{β_{i}}(N_{β_{i}} ⋯ N_{β_{r}} ht(β_{i}) + 1).

Since the PBW basis elements (2.3.1) are eigenvectors for Γ, the action of Γ on R preserves the filtration, and we further get a spectral sequence converging to the cohomology of u(D, 0, 0) ≃ R#kΓ:

(5.0.4)  \xymatrix{H^{p+q}(Gr_{(p)} R, k)^{Γ} \ar@{=>}[r] & H^{p+q}(R, k)^{Γ} ≃ H^{p+q}(R#kΓ, k).}

Moreover, if M is a finitely generated R#kΓ-module, there is a spectral sequence converging to the cohomology of R with coefficients in M:

(5.0.5)  \xymatrix{H^{p+q}(Gr_{(p)} R, M) \ar@{=>}[r] & H^{p+q}(R, M),}

also compatible with the action of Γ.

We wish to apply Lemma 2.5 to the spectral sequence (5.0.4) for the filtered algebra R. In order to do so, we must find some permanent cycles.

The Hochschild cohomology of R in degree 2, with trivial coefficients, was studied in [26]: There is a linearly independent set of 2-cocycles ξ_{α} on R, indexed by the positive roots α ([26, Theorem 6.1.3]). We will use the notation ξ_{α} in place of the notation f_{α} used there. As shown in [26], these 2-cocycles may be expressed as functions at the chain level in the following way. Let x^{a}, x^{b} denote arbitrary PBW basis elements (2.3.1) of u(D, 0, 0). Let x̃^{a}, x̃^{b} denote corresponding elements in the infinite dimensional algebra U(D, 0) arising from the section of the quotient map U(D, 0) → u(D, 0, 0) for which PBW basis elements are sent to PBW basis elements. Then

(5.0.6)  ξ_{α}(x^{a} ⊗ x^{b}) = c_{α}

where c_{α} is the coefficient of x̃_{α}^{N_{α}} in the product x̃^{a} · x̃^{b}, and x^{a}, x^{b} range over all pairs of PBW basis elements. A direct proof that these are 2-cocycles is in [26, §6.1]; alternatively the proof in Lemma 6.2 below, that analogous functions f_{α} in higher degrees are cocycles for u(D, λ, 0), applies with minor modifications to the ξ_{α} in this context.

We wish to relate these functions ξ_{α} to elements on the E_{1}-page of the spectral sequence (5.0.2), found in the previous section. Recall that β_{1}, …, β_{r} enumerate the positive roots, and the integer p_{i} is defined in (5.0.3). By (5.0.6) and Lemma 2.4, we have ξ_{β_{i}} ↓_{F_{p_{i}−1}(R⊗R)}= 0 but ξ_{β_{i}} ↓_{F_{p_{i}}(R⊗R)}≠ 0. Note that the filtration on R induces a filtration on R^{+}, and thus a (decreasing) filtration on the complex C^{•} defined in (2.4.1), given by F^{p}C^{n} = {f : (R^{+})^{⊗n} → k | f ↓_{F_{p−1}((R^{+})^{⊗n})}= 0}. We conclude that ξ_{β_{i}} ∈ F^{p_{i}}C^{2} but ξ_{β_{i}} ∉ F^{p_{i}+1}C^{2}. Denoting the corresponding cocycle by the same letter, we further conclude that ξ_{β_{i}} ∈ im{H^{2}(F^{p_{i}}C^{•}) → H^{2}(C^{•})} = F^{p_{i}} H^{2}(R, k), but ξ_{β_{i}} ∉ im{H^{2}(F^{p_{i+1}}C^{•}) → H^{2}(C^{•})} = F^{p_{i}+1} H^{2}(R, k). Hence, ξ_{β_{i}} can be identified with the corresponding nontrivial homogeneous element in the associated graded complex:

ξ̃_{β_{i}} ∈ F^{p_{i}} H^{2}(R, k)/F^{p_{i}+1} H^{2}(R, k) ≃ E_{∞}^{p_{i},2−p_{i}}.

Since ξ_{β_{i}} ∈ F^{p_{i}}C^{2} but ξ_{β_{i}} ∉ F^{p_{i}+1}C^{2}, it induces an element ξ̄_{β_{i}} ∈ E_{0}^{p_{i},2−p_{i}} = F^{p_{i}}C^{2}/F^{p_{i}+1}C^{2}. Since ξ̄_{β_{i}} is induced by an actual cocycle in C^{•}, it will be in the kernels of all the differentials of the spectral sequence. Hence, the residue of ξ̄_{β_{i}} will be in the E_{∞}–term where it will be identified with the non-zero element ξ̃_{β_{i}} since these classes are induced by the same cocycle in C^{•}. We conclude that ξ̄_{β_{i}} ∈ E_{0}^{p_{i},2−p_{i}}, and, correspondingly, its image in E_{1}^{p_{i},2−p_{i}} ≃ H^{2}(Gr R, k) which we denote by the same symbol, is a permanent cycle.

Note that the results of [26] apply equally well to S = Gr R to yield similar cocycles ξ̂_{α} via the formula (5.0.6), for each positive root α in type A_{1} × ⋯ × A_{1}. Now S = Gr R has generators corresponding to the root vectors of R, and we similarly identify elements in cohomology. Thus each ξ̂_{α}, α a positive root in type A_{1} × ⋯ × A_{1}, may be relabeled ξ̂_{β_{i}} for some i (1 ≤ i ≤ r), the indexing corresponding to β_{1}, …, β_{r} ∈ Φ^{+}. Comparing the values of \overline{ξ}_{β_{i}} and ξ̂_{β_{i}} on basis elements x^{a} ⊗ x^{b} of Gr R ⊗ Gr R we conclude that they are the same function. Hence ξ̂_{β_{i}} ∈ E_{1}^{p_{i},2−p_{i}} are permanent cycles.

We wish to identify these elements ξ̂_{β_{i}} ∈ H^{2}(Gr R, k) with the cohomology classes ξ_{i} in H^{∗}(S, k) constructed in Section 4, as we may thus exploit the algebra structure of H^{∗}(S, k) given in Theorem 4.1. We use the same symbols x_{β_{i}} to denote the corresponding root vectors in R and in S = Gr R, as this should cause no confusion.

Lemma 5.1. For each i (1 ≤ i ≤ r), the cohomology classes ξ_{i} and ξ̂_{β_{i}} coincide as elements of H^{2}(Gr R, k).

Proof. Let K_{•} be the chain complex defined in Section 4, a projective resolution of the trivial Gr R-module k. Elements ξ_{i} ∈ H^{2}(Gr R, k) and η_{i} ∈ H^{1}(Gr R, k) were defined via the complex K_{•}. We wish to identify ξ_{i} with elements of the chain complex C^{•} defined in (2.4.1), where A = R. To this end we define maps F_{1}, F_{2} making the following diagram commute, where S = Gr R:

⋯ →  K_{2}  ⟶^{d} K_{1}  ⟶^{d} K_{0} ⟶^{ε} k → 0
↓ F_{2}  ↓ F_{1}  ∥  ∥
⋯ → S ⊗ (S^{+})^{⊗2} ⟶^{∂_{2}} S ⊗ S^{+} ⟶^{∂_{1}} S ⟶^{ε} k → 0

In this diagram, the maps d are given in (4.0.3), and ∂_{i} in (2.4.2). Let Φ(⋯ 1_{i} ⋯) denote the basis element of K_{1} having a 1 in the ith position, and 0 in all other positions, Φ(⋯ 1_{i} ⋯ 1_{j} ⋯) (respectively Φ(⋯ 2_{i} ⋯)) the basis element of K_{2} having a 1 in the ith and jth positions (i ≠ j), and 0 in all other positions (respectively a 2 in the ith position and 0 in all other positions). Let

F_{1}(Φ(⋯ 1_{i} ⋯)) = 1 ⊗ x_{β_{i}},
F_{2}(Φ(⋯ 2_{i} ⋯)) = ∑_{a_{i}=0}^{N_{i}−2} x_{β_{i}}^{a_{i}} ⊗ x_{β_{i}} ⊗ x_{β_{i}}^{N_{i}−a_{i}−1},
F_{2}(Φ(⋯ 1_{i} ⋯ 1_{j} ⋯)) = 1 ⊗ x_{β_{i}} ⊗ x_{β_{j}} − q_{ij} ⊗ x_{β_{j}} ⊗ x_{β_{i}}.

Direct computations show that the two nontrivial squares in the diagram above commute. So F_{1}, F_{2} extend to maps F_{i} : K_{i} → S ⊗ (S^{+})^{⊗i}, i ≥ 1, providing a chain map F_{•} : K_{•} → S ⊗ (S^{+})^{⊗•}, thus inducing isomorphisms on cohomology.

We now verify that the maps F_{1}, F_{2} make the desired identifications. By ξ_{β_{i}} we mean the function on the reduced bar complex, ξ_{β_{i}}(1 ⊗ x^{a} ⊗ x^{b}) := ξ_{β_{i}}(x^{a} ⊗ x^{b}), defined in (5.0.6). Then

F_{2}^{∗}(ξ_{β_{i}})(Φ(⋯ 2_{i} ⋯)) = ξ_{β_{i}}(F_{2}(Φ(⋯ 2_{i} ⋯)))
= ξ_{β_{i}}( ∑_{a_{i}=0}^{N_{i}−2} x_{β_{i}}^{a_{i}} ⊗ x_{β_{i}} ⊗ x_{β_{i}}^{N_{i}−a_{i}−1})

= ∑_{a_{i}=0}^{N_{i}−2} ε(x_{β_{i}}^{a_{i}})ξ_{β_{i}}(1 ⊗ x_{β_{i}} ⊗ x_{β_{i}}^{N_{i}−a_{i}−1})
= ξ_{β_{i}}(x_{β_{i}} ⊗ x_{β_{i}}^{N_{i}−1}) = 1.

Further, it may be checked similarly that F_{2}^{∗}(ξ_{β_{i}})(Φ(⋯ 1_{i} ⋯ 1_{j} ⋯)) = 0 for all i, j and F_{2}^{∗}(ξ_{β_{i}})(Φ(⋯ 2_{j} ⋯)) = 0 for all j ≠ i. Therefore F_{2}^{∗}(ξ_{β_{i}}) is the dual function to Φ(⋯ 2_{i} ⋯), which is precisely ξ_{i}. □

Similarly, we identify the elements η_{i} defined in Section 4 with functions at the chain level in cohomology: Define

(5.1.1)  η_{α}(x^{a}) = \{ \begin{matrix}1, & if x^{a} = x_{α} \\ 0, & otherwise \end{matrix} .

The functions η_{α} represent a basis of H^{1}(S, k) ≃ Hom_{k}(S^{+}/(S^{+})^{2}, k). Similarly functions corresponding to the simple roots β_{i} only represent a basis of H^{1}(R, k) ≃ Hom_{k}(R^{+}/(R^{+})^{2}, k). A computation shows that F_{1}^{∗}(η_{β_{i}})(Φ(⋯ 1_{j} ⋯)) = δ_{ij}, so that F_{1}^{∗}(η_{β_{i}}) is the dual function to Φ(⋯ 1_{i} ⋯). Therefore η_{i} and η_{β_{i}} coincide as elements of H^{1}(S, k).

For each α ∈ Φ^{+}, let M_{α} be any positive integer for which χ_{α}^{M_{α}} = ε. (For example, let M_{α} be the order of χ_{α}.) Note that ξ_{α}^{M_{α}} is Γ-invariant: By (4.2.1), g · ξ_{α}^{M_{α}} = χ_{α}^{−M_{α}N_{α}}(g)ξ_{α}^{M_{α}} = ξ_{α}^{M_{α}}.

Recall the notation u(D, 0, 0) ≃ R#kΓ, where R = B(V) is the Nichols algebra.

Lemma 5.2. The cohomology algebra H^{∗}(u(D, 0, 0), k) is finitely generated over the subalgebra generated by all ξ_{α}^{M_{α}} (α ∈ Φ^{+}).

Proof. Let \xymatrix{E_{1}^{∗,∗} \ar@{=>}[r] & H^{∗}(R, k)} be the spectral sequence (5.0.2), and let B^{∗,∗} be the bigraded subalgebra of E_{1}^{∗,∗} generated by the elements ξ_{i}. By Lemma 5.1 and the discussion prior to it, B^{∗,∗} consists of permanent cycles. Since ξ_{i} is ξ̄_{β_{i}} by Lemma 5.1, it is in bidegree (p_{i}, 2 − p_{i}). Let A^{∗,∗} be the subalgebra of B^{∗,∗} generated by \overline{ξ}_{α}^{M_{α}} (α ∈ Φ^{+}). Then A^{∗,∗} also consists of permanent cycles. Observe that A^{∗,∗} is a subalgebra of H^{∗}((Gr R)#kΓ, k), which is graded commutative since (Gr R)#kΓ is a Hopf algebra. Hence, A^{∗,∗} is commutative as it is concentrated in even (total) degrees. Finally, A^{∗,∗} is Noetherian since it is a polynomial algebra in the ξ_{α}^{M_{α}}. We conclude that the bigraded commutative algebra A^{∗,∗} satisfies the hypotheses of Lemma 2.5. By Theorem 4.1, the algebra E_{1}^{∗,∗} ≃ H^{∗}(Gr R, k) is generated by ξ_{i} and η_{i} where the generators η_{i} are nilpotent. Hence, E_{1}^{∗,∗} is a finitely generated module over B^{∗,∗}. The latter is clearly a finitely generated module over A^{∗,∗}. Hence E_{1}^{∗,∗} is a finitely generated module over A^{∗,∗}. Lemma 2.5 implies that H^{∗}(R, k) is a Noetherian Tot(A^{∗,∗})-module; moreover, the action of Γ on H^{∗}(R, k) is compatible with the action on A^{∗,∗} since the spectral sequence is compatible with the action of Γ. Therefore, H^{∗}(R#kΓ, k) ≃ H^{∗}(R, k)^{Γ} is a Noetherian Tot(A^{∗,∗})-module. Since Tot(A^{∗,∗}) is finitely generated, we conclude that H^{∗}(R#kΓ, k) is finitely generated. □

We immediately have the following theorem. The second statement of the theorem follows by a simple application of the second statement of Lemma 2.5.

Theorem 5.3. The algebra H^{∗}(u(D, 0, 0), k) is finitely generated. If M is a finitely generated u(D, 0, 0)-module, then H^{∗}(u(D, 0, 0), M) is a finitely generated module over H^{∗}(u(D, 0, 0), k).

Thanks to Corollary 3.13, we have some information about the algebra structure of the cohomology ring of the Nichols algebra R = B(V): Recall that q_{βα} = χ_{α}(g_{β}) (see (2.1.2)). Compare the following result with the graded case, Theorem 4.1.

Theorem 5.4. The following relations hold in H^{∗}(R, k) for all ξ_{α}, ξ_{β} (α, β ∈ Φ^{+}) and η_{α}, η_{β} (α, β ∈ Π):

ξ_{α}ξ_{β} = q_{βα}^{N_{α}N_{β}}ξ_{β}ξ_{α}, η_{α}ξ_{β} = q_{βα}^{N_{β}}ξ_{β}η_{α}, and η_{α}η_{β} = −q_{βα}η_{β}η_{α}.

Proof. As we shall see, this is a consequence of Corollary 3.13, since R is a braided Hopf algebra in _{Γ}^{Γ}YD.

Note first that as a function on the Yetter-Drinfeld module R ∈ _{Γ}^{Γ}YD the cocycle ξ_{α} is Γ-homogeneous of degree g_{α}^{−N_{α}} and spans a one-dimensional Γ-module with character χ_{α}^{−N_{α}}. To see this in full detail, let us rephrase the definition (5.0.6) of ξ_{α} as follows. One can write U(D, 0) as a Radford biproduct U(D, 0) = R̂#kΓ with a braided Hopf algebra R̂ ∈ _{Γ}^{Γ}YD such that R is a quotient of R̂. Let s: R → R̂ be the section of the surjection R̂ → R that maps PBW basis elements to PBW basis elements. Note that s is a map in _{Γ}^{Γ}YD. Finally, let p_{α}: R̂ → k be the function projecting r ∈ R̂ to the coefficient in r of the PBW basis element x̃_{α}^{N_{α}}. Then ξ_{α} is by definition the pullback of p_{α} under R ⊗ R ⟶^{s⊗s} R̂ ⊗ R̂ ⟶^{m} R̂. As a consequence, since p_{α} is clearly Γ-homogeneous of degree g_{α}^{−N_{α}} and satisfies g · p_{α} = χ_{α}^{−N_{α}}(g)p_{α} (cf. (4.2.1)), the statements on ξ_{α} follow.

On the other hand, it is easy to see from definition (5.1.1) that η_{α} has degree g_{α}^{−1} and spans a one-dimensional Γ-module with character χ_{α}^{−1}.

Let us denote the opposite of multiplication in H^{∗}(R, k) by ζ∘θ := θζ. According to Corollary 3.13, the opposite multiplication is braided graded commutative. In particular

ξ_{α}ξ_{β} = ξ_{β} ∘ ξ_{α} = (g_{β}^{−N_{β}} · ξ_{α}) ∘ ξ_{β}
= χ_{α}^{−N_{α}} (g_{β}^{−N_{β}}) ξ_{α} ∘ ξ_{β}
= χ_{α} (g_{β})^{N_{β}N_{α}} ξ_{α} ∘ ξ_{β}
= q_{βα}^{N_{β}N_{α}}ξ_{α} ∘ ξ_{β} = q_{βα}^{N_{β}N_{α}}ξ_{β}ξ_{α}

and

η_{α}ξ_{β} = ξ_{β}η_{α} = (g_{β}^{−N_{β}} · η_{α}) ∘ ξ_{β} = χ_{α}^{−1} (g_{β}^{−N_{β}}) η_{α} ∘ ξ_{β} = q_{βα}^{N_{β}}ξ_{β}η_{α}

as well as finally

η_{α}η_{β} = η_{β} ∘ η_{α} = − (g_{β}^{−1} · η_{α}) ∘ η_{β} = −χ_{α}^{−1} (g_{β}^{−1}) η_{α} ∘ η_{β} = −q_{βα}η_{β}η_{α}.
□

We give a corollary in a special case. It generalizes [19, Theorem 2.5(i)]. Recall that Π denotes a set of simple roots in the root system Φ.

Corollary 5.5. Assume there are no Γ-invariants in H^{∗}(Gr R, k) of the form ξ_{β_{1}}^{b_{1}} ⋯ ξ_{β_{r}}^{b_{r}}η_{β_{1}}^{c_{1}} ⋯ η_{β_{r}}^{c_{r}} for which c_{1} + ⋯ + c_{r} is odd. Then

H^{∗}(u(D, 0, 0), k) ≃ k⟨ξ_{α}, η_{β} | α ∈ Φ^{+}, β ∈ Π⟩^{Γ},

with the relations of Theorem 5.4, deg(η_{β}) = 1, deg(ξ_{α}) = 2. If there are no Γ-invariants with c_{1} + ⋯ + c_{r} ≠ 0, then H^{∗}(R#kΓ, k) ≃ k[ξ_{α} | α ∈ Φ^{+}]^{Γ}.

Proof. The hypothesis of the first statement implies that H^{i}(Gr R, k)^{Γ} = 0 for all odd integers i. Thus on the E_{1}-page, every other diagonal is 0. This implies E_{1} = E_{∞}. It follows that, as a vector space, H^{∗}(u(D, 0, 0), k) is exactly as stated. The algebra structure is a consequence of Theorem 5.4 and the fact that the cohomology of a Hopf algebra is graded commutative. The hypothesis of the last statement implies further that the Γ-invariant subalgebra of H^{∗}(Gr R, k) is spanned by elements of the form ξ_{β_{1}}^{b_{1}} ⋯ ξ_{β_{r}}^{b_{r}}. By graded commutativity of the cohomology ring of a Hopf algebra and the relations of Theorem 5.4, H^{∗}(R#kΓ, k) may be identified with the Γ-invariant subalgebra of a polynomial ring in variables ξ_{α}, with corresponding Γ-action. □

Remark 5.6. Assume the hypotheses of Corollary 5.5, and that q_{αα} ≠ −1 for all α ∈ Π. Then η_{α}^{2} = 0 for all α ∈ Π, and it follows that the maximal ideal spectrum of H^{∗}(u(D, 0, 0), k) is Spec k[ξ_{α} | α ∈ Φ^{+}]^{Γ} ≃ Spec k[ξ_{α} | α ∈ Φ^{+}]/Γ.

We give an example to show that cohomology may in fact be nonzero in odd degrees, in contrast to that of the small quantum groups of [19]. This complicates any determination of the explicit structure of cohomology in general. The simplest example occurs in type A_{1} × A_{1} × A_{1}, where there can exist a nonzero cycle in degree 3.

Example 5.7. Let Γ = ℤ/ℓℤ with generator g. Let q be a primitive ℓth root of unity. Let g_{1} = g_{2} = g_{3} = g, and choose χ_{1}, χ_{2}, χ_{3} so that the matrix (q_{ij}) = (χ_{j}(g_{i})) is

( \begin{matrix}q & q^{−1} & 1 \\ q & q^{−2} & q \\ 1 & q^{−1} & q \end{matrix} ) .

Let u(D, 0, 0) = R#kΓ be the pointed Hopf algebra of type A_{1} × A_{1} × A_{1} defined by this data. Let η_{1}, η_{2}, η_{3} represent elements of H^{1}(R, k) as defined by (5.1.1). The action of Γ is described in (4.2.1): g_{i} · η_{j} = q_{ij}^{−1}η_{j}. Since the product of all entries in any given row of the matrix (q_{ij}) is 1, we conclude that η_{1}η_{2}η_{3} is invariant under Γ. Hence, it is a nontrivial cocycle in H^{3}(u(D, 0, 0), k).

We also give an example in type A_{2} × A_{1} to illustrate, in particular, that the methods employed in [19] do not transfer to our more general setting. That is, for an arbitrary (coradically graded) pointed Hopf algebra, the first page of the spectral sequence (5.0.4) can have nontrivial elements in odd degrees. In the special case of a small quantum group (with some restrictions on the order of the root of unity), it is shown in [19] that this does not happen.

Example 5.8. Let Γ = ℤ/ℓℤ × ℤ/ℓℤ, with generators g_{1}, g_{2}. Let q be a primitive ℓth root of unity (ℓ odd), and let D be of type A_{2} × A_{1} so that the Cartan matrix is

( \begin{matrix}2 & −1 & 0 \\ −1 & 2 & 0 \\ 0 & 0 & 2 \end{matrix} ) .

Let χ_{1}, χ_{2} be as for u_{q}(sl_{3})^{+}, that is

\begin{aligned}χ_{1}(g_{1}) & = & q^{2}, & χ_{1}(g_{2}) & = & q^{−1}, \\ χ_{2}(g_{1}) & = & q^{−1}, & χ_{2}(g_{2}) & = & q^{2}. \end{aligned}

Now let g_{3} := g_{1}g_{2} and χ_{3} := χ_{1}^{−1}χ_{2}^{−1}. Then χ_{3}(g_{3}) = q^{−2} ≠ 1 and the Cartan condition holds, for example χ_{3}(g_{1})χ_{1}(g_{3}) = q^{−1}q = 1 = χ_{1}(g_{1})^{a_{13}} since a_{13} = 0. Let R = B(V), the Nichols algebra defined from this data.

The root vector corresponding to the nonsimple positive root is x_{12} = [x_{1}, x_{2}]_{c} = x_{1}x_{2} − q^{−1}x_{2}x_{1}. The relations among the root vectors other than x_{3} are now

x_{2}x_{1} = qx_{1}x_{2} − qx_{12}, x_{12}x_{1} = q^{−1}x_{1}x_{12}, x_{2}x_{12} = q^{−1}x_{12}x_{2}.

The associated graded algebra, which is of type A_{1} × A_{1} × A_{1} × A_{1}, thus has relations (excluding those involving x_{3} that do not change):

x_{2}x_{1} = qx_{1}x_{2}, x_{12}x_{1} = q^{−1}x_{1}x_{12}, x_{2}x_{12} = q^{−1}x_{12}x_{2}.

Thus the cohomology of the associated graded algebra Gr R#kΓ is the subalgebra of Γ-invariants of an algebra with generators

ξ_{1}, ξ_{12}, ξ_{2}, ξ_{3}, η_{1}, η_{12}, η_{2}, η_{3}

(see Theorem 4.1). The element η_{1}η_{2}η_{3} is Γ-invariant since g ·η_{i} = χ_{i}(g)^{−1}η_{i} for all i, and χ_{3} = χ_{1}^{−1}χ_{2}^{−1}. Therefore in the spectral sequence relating the cohomology of Gr R#kΓ to that of R#kΓ, there are some odd degree elements on the E_{1} page.

6. Finite dimensional pointed Hopf algebras

Let u(D, λ, μ) be one of the finite dimensional pointed Hopf algebras from the Andruskiewitsch-Schneider classification [5], as described in Section 2. With respect to the coradical filtration, its associated graded algebra is Gr u(D, λ, μ) ≃ u(D, 0, 0) ≃ R#kΓ where R = B(V) is a Nichols algebra, also described in Section 2. By Theorem 5.3, H^{∗}(u(D, 0, 0), k) is finitely generated. Below we apply the spectral sequence for a filtered algebra, employing this new choice of filtration:

(6.0.1)  \xymatrix{E_{1}^{p,q} = H^{p+q}(u(D, 0, 0), k) \ar@{=>}[r] & H^{p+q}(u(D, λ, μ), k).}

As a consequence of the following lemma, we may assume without loss of generality that all root vector relations (2.1.4) are trivial.

Lemma 6.1. (i) For all D, λ, μ, there is an isomorphism of graded algebras,

H^{∗}(u(D, λ, μ), k) ≃ H^{∗}(u(D, λ, 0), k).

(ii) Let M be a finitely generated u(D, λ, μ)-module. There exists a finitely generated u(D, λ, 0)-module M̃ and an isomorphism of H^{∗}(u(D, λ, μ), k)-modules

H^{∗}(u(D, λ, μ), M) ≃ H^{∗}(u(D, λ, 0), M̃),

where the action of H^{∗}(u(D, λ, μ), k) on H^{∗}(u(D, λ, 0), M̃) is via the isomorphism of graded algebras in (i).

Proof. Define the subset I ⊂ Γ as follows: g ∈ I if and only if there exists a root vector relation x_{α}^{N_{α}} − u_{α}(μ) (see (2.1.4)), such that the element u_{α}(μ) of kΓ has a nonzero coefficient of g when written as a linear combination of group elements. If x_{α}^{N_{α}} − u_{α}(μ) is a nontrivial root vector relation, then necessarily x_{α}^{N_{α}} commutes with elements of Γ, implying that χ_{α}^{N_{α}} = ε. From this and (2.3.2), we see that x_{α}^{N_{α}} is central in u(D, λ, μ). Since x_{α}^{N_{α}} = u_{α}(μ), this element of the group ring kΓ must also be central. Since Γ acts diagonally, each group element involved in u_{α}(μ) is necessarily central in u(D, λ, μ) as well.

Let Z = k⟨I⟩, a subalgebra of u(D, λ, μ), and let \overline{u} = u(D, λ, μ)/(g − 1 | g ∈ I). We have a sequence of algebras (see [19, §5.2]):

Z → u(D, λ, μ) → u(D, λ, μ)//Z ≃ \overline{u}.

Hence, there is a multiplicative spectral sequence

H^{p}(\overline{u}, H^{q}(Z, k)) ⟹ H^{p+q}(u(D, λ, μ), k).

Since the characteristic of k does not divide the order of the group, we have H^{q>0}(Z, k) = 0. Thus the spectral sequence collapses, and we get an isomorphism of graded algebras

(6.1.1)  H^{∗}(\overline{u}, k) ≃ H^{∗}(u(D, λ, μ), k).

Similarly, if M is any u(D, λ, μ)-module, then we have a spectral sequence of H^{∗}(\overline{u}, k)-modules

H^{p}(\overline{u}, H^{q}(Z, M)) ⟹ H^{p+q}(u(D, λ, μ), M).

The spectral sequence collapses, and we get an isomorphism

(6.1.2)  H^{∗}(\overline{u}, M^{Z}) ≃ H^{∗}(u(D, λ, μ), M)

which respects the action of H^{∗}(u(D, λ, μ), where H^{∗}(u(D, λ, μ) acts on the left side via the isomorphism (6.1.1).

Note that Z = k⟨I⟩ is also a central subalgebra of u(D, λ, 0). Arguing exactly as above, we get an isomorphism of graded algebras

H^{∗}(\overline{u}, k) ≃ H^{∗}(u(D, λ, 0), k),

which implies (i).

Let M be a u(D, λ, μ)-module. Let M̃ be a u(D, λ, 0)-module which we get by inflating the \overline{u}-module M^{Z} via the projection \xymatrix{u(D, λ, 0) \ar@{->>}[r] & \overline{u}} . Since (M̃)^{Z} ≃ M^{Z} by construction, we get an isomorphism of u(D, λ, 0)-modules

H^{∗}(\overline{u}, M^{Z}) ≃ H^{∗}(u(D, λ, 0), M̃)

using another spectral sequence argument. Combining with the isomorphism (6.1.2), we get (ii).

□

By Lemma 6.1, it suffices to work with the cohomology of u(D, λ, 0), in which all the root vectors are nilpotent. In this case we define some permanent cycles: As before, for each α ∈ Φ^{+}, let M_{α} be any positive integer for which χ_{α}^{M_{α}} = ε (for example, take M_{α} to be the order of χ_{α}). If α = α_{i} is simple, then χ_{α}(g_{α}) has order N_{α}, and so N_{α} divides M_{α}.

We previously identified an element ξ_{α} of H^{2}(R, k), where R = B(V). Now B(V) is no longer a subalgebra of u(D, λ, 0) in general, due to the potential existence of nontrivial linking relations, but we will show that still there is an element analogous to ξ_{α}^{M_{α}} in H^{2M_{α}}(u(D, λ, 0), k). For simplicity, let U = U(D, λ) and u = u(D, λ, 0) ≃ U(D, λ)/(x_{α}^{N_{α}} | α ∈ Φ^{+}) (see Section 2). Let U^{+}, u^{+} denote the augmentation ideals of U, u.

We will use a similar construction as in Section 4 of [26], defining functions as elements of the bar complex (2.4.1). For each α ∈ Φ^{+}, define a k-linear function f̃_{α} : (U^{+})^{2M_{α}} → k by first letting r_{1}, …, r_{2M_{α}} be PBW basis elements (2.3.1) and requiring

f̃_{α}(r_{1} ⊗ ⋯ ⊗ r_{2M_{α}}) = γ_{12}γ_{34} ⋯ γ_{2M_{α}−1,2M_{α}}

where γ_{ij} is the coefficient of x_{α}^{N_{α}} in the product r_{i}r_{j} as a linear combination of PBW basis elements. Now define f̃_{α} to be 0 whenever a tensor factor is in kΓ ∩ ker ε, and

f̃_{α}(r_{1}g_{1} ⊗ ⋯ ⊗ r_{2M_{α}}g_{2M_{α}}) = f̃_{α}(r_{1} ⊗^{g_{1}}r_{2} ⊗ ⋯ ⊗^{g_{1}···g_{2Mα−1}}r_{2M_{α}})

for all g_{1}, …, g_{2M_{α}} ∈ Γ. It follows from the definition of f̃_{α} and the fact that χ_{α}^{M_{α}} = ε that f̃_{α} is Γ-invariant. We will show that f̃_{α} factors through the quotient u^{+} of U^{+} to give a map f_{α} : (u^{+})^{⊗2M_{α}} → k. Precisely, it suffices to show that f̃_{α}(r_{1} ⊗⋯⊗r_{2M_{α}}) = 0 whenever one of r_{1}, …, r_{2M_{α}} is in the kernel of the quotient map π : U(D, λ) → u(D, λ, 0). Suppose r_{i} ∈ ker π, that is r_{i} = x_{β_{1}}^{a_{1}} ⋯ x_{β_{r}}^{a_{r}} and

for some j, a_{j} ≥ N_{β_{j}}. Since x_{β_{j}}^{N_{β_{j}}} is braided-central, r_{i} is a scalar multiple of

x_{β_{j}}^{N_{β_{j}}} x_{β_{1}}^{b_{1}} ⋯ x_{β_{r}}^{b_{r}} for some b_{1}, …, b_{r}. Now f̃_{α}(r_{1} ⊗ ⋯ ⊗ r_{2M_{α}}) is the product of the coefficients of x_{α}^{N_{α}} in r_{1}r_{2}, …, r_{2M_{α}−1}r_{2M_{α}}. However, the coefficient of x_{α}^{N_{α}} in each of r_{i−1}r_{i} and r_{i}r_{i+1} is 0: If α = β_{i}, then since r_{i−1}, r_{i+1} ∈ U^{+}, this product cannot have a nonzero coefficient for x_{α}^{N_{α}}. If α ≠ β_{i}, the same is true since x_{β_{j}}^{N_{β_{j}}} is a factor of r_{i−1}r_{i} and of r_{i}r_{i+1}. Therefore f̃_{α} factors to give a linear map f_{α} : (u^{+})^{2M_{α}} → k. In calculations, we define f_{α} via f̃_{α} and a choice of section of the quotient map π : U → u.

Lemma 6.2. For each α ∈ Φ^{+}, f_{α} is a cocycle. The f_{α} (α ∈ Φ^{+}) represent a linearly independent subset of H^{∗}(u(D, λ, 0), k).

Proof. We first verify that f̃_{α} is a cocycle on U: Let r_{0}, …, r_{2M_{α}} ∈ U^{+}, of positive degree. Then

d(f̃_{α})(r_{0} ⊗ ⋯ ⊗ r_{2M_{α}}) = ∑_{i=0}^{2M_{α}−1}(−1)^{i+1}f̃_{α}(r_{0} ⊗ ⋯ ⊗ r_{i}r_{i+1} ⊗ ⋯ ⊗ r_{2M_{α}}).

By definition of f̃_{α}, note that the first two terms cancel:

f̃_{α}(r_{0}r_{1} ⊗ r_{2} ⊗ ⋯ ⊗ r_{2M_{α}}) = f̃_{α}(r_{0} ⊗ r_{1}r_{2} ⊗ ⋯ ⊗ r_{2M_{α}}),

and similarly for all other terms, so d(f̃_{α})(r_{0}⊗⋯⊗r_{2M_{α}}) = 0. A similar calculation shows that d(f̃_{α})(r_{0}g_{0} ⊗⋯⊗r_{2M_{α}−1}g_{2M_{α}−1}) = 0 for all g_{0}, …, g_{2M_{α}−1} ∈ Γ. If there is an element of kΓ∩ker ε in one of the factors, we obtain 0 as well by the definition of f̃_{α}, a similar calculation.

Now we verify that f_{α} is a cocycle on the quotient u of U: Let r_{0}, …, r_{2M_{α}} ∈ u^{+}. Again we have

d(f_{α})(r_{0} ⊗ ⋯ ⊗ r_{2M_{α}}) = ∑_{i=0}^{2M_{α}−1}(−1)^{i+1}f_{α}(r_{0} ⊗ ⋯ ⊗ r_{i}r_{i+1} ⊗ ⋯ ⊗ r_{2M_{α}}).

We will show that f_{α}(r_{0}r_{1} ⊗ r_{2} ⊗ ⋯ ⊗ r_{2M_{α}}) = f_{α}(r_{0} ⊗ r_{1}r_{2} ⊗ ⋯ ⊗ r_{2M_{α}}), and similarly for the other terms. Let r̃_{i} denote the element of U corresponding to r_{i} under a chosen section of the quotient map π : U → u. Note that r̃_{0} · r̃_{1} = r̃_{0}r_{1} + y and r̃_{1} · r̃_{2} = r̃_{1}r_{2} + z for some y, z ∈ ker π. So

f_{α}(r_{0}r_{1} ⊗ r_{2} ⊗ ⋯ ⊗ r_{2M_{α}}) = f̃_{α}(r̃_{0}r_{1} ⊗ r̃_{2} ⊗ ⋯ ⊗ r̃_{2M_{α}})
= f̃_{α}((r̃_{0} · r̃_{1} − y) ⊗ r̃_{2} ⊗ ⋯ ⊗ r̃_{2M_{α}})
= f̃_{α}(r̃_{0} · r̃_{1} ⊗ r̃_{2} ⊗ ⋯ ⊗ r̃_{2M_{α}})
= f̃_{α}(r̃_{0} ⊗ r̃_{1} · r̃_{2} ⊗ ⋯ ⊗ r̃_{2M_{α}})
= f̃_{α}(r̃_{0} ⊗ (r̃_{1} · r̃_{2} + z) ⊗ ⋯ ⊗ r̃_{2M_{α}})
= f̃_{α}(r̃_{0} ⊗ r̃_{1}r_{2} ⊗ ⋯ ⊗ r̃_{2M_{α}})
= f_{α}(r_{0} ⊗ r_{1}r_{2} ⊗ ⋯ ⊗ r_{2M_{α}}).

Other computations for this case are similar to those for U. Thus f_{α} is a cocycle on u.

We prove that in a given degree, the f_{α} in that degree represent a linearly independent set in cohomology: Suppose ∑_{α} c_{α}f_{α} = ∂h for some scalars c_{α} and linear map h. Then for each α,

c_{α} = (∑ c_{α}f_{α})(x_{α} ⊗ x_{α}^{N_{α}−1} ⊗ ⋯ ⊗ x_{α} ⊗ x_{α}^{N_{α}−1})
= ∂h(x_{α} ⊗ x_{α}^{N_{α}−1} ⊗ ⋯ ⊗ x_{α} ⊗ x_{α}^{N_{α}−1})
= −h(x_{α}^{N_{α}} ⊗ x_{α} ⊗ ⋯ ⊗ x_{α}^{N_{α}−1}) + ⋯ − h(x_{α} ⊗ ⋯ ⊗ x_{α}^{N_{α}−1} ⊗ x_{α}^{N_{α}}) = 0

since x_{α}^{N_{α}} = 0 in u = u(D, λ, 0). □

These functions f_{α} correspond to their counterparts ξ_{α}^{M_{α}} defined on u(D, 0, 0), in the E_{1}-page of the spectral sequence (6.0.1), by observing what they do as functions at the level of chain complexes (2.4.1).

Theorem 6.3. The algebra H^{∗}(u(D, λ, μ), k) is finitely generated. If M is a finitely generated u(D, λ, μ)-module, then H^{∗}(u(D, λ, μ), M) is a finitely generated module over H^{∗}(u(D, λ, μ), k).

Proof. By Lemma 6.1, it suffices to prove the statements in the case μ = 0. We have E_{1}^{∗,∗} ≃ H^{∗}(u(D, 0, 0), k), where u(D, 0, 0) ≃ R#kΓ, R = B(V). By Lemma 5.2, E_{1}^{∗,∗} is finitely generated over its subalgebra that is generated by all ξ_{α}^{M_{α}}. By Lemma 6.2 and the above remarks, each ξ_{α}^{M_{α}} is a permanent cycle, corresponding to the cocycle f_{α} on u(D, λ, μ). The rest of the proof is an application of Lemma 2.5, where A^{∗,∗} is the subalgebra of E_{1}^{∗,∗} generated by the ξ_{α}^{M_{α}} (α ∈ Φ^{+}), similar to the proof of Lemma 5.2 and Theorem 5.3. □

Corollary 6.4. The Hochschild cohomology ring H^{∗}(u(D, λ, μ), u(D, λ, μ)) is finitely generated.

Proof. Apply Theorem 6.3 to the finitely generated u(D, λ, μ)-module u(D, λ, μ), under the adjoint action. By [19, Prop. 5.6], this is isomorphic to the Hochschild cohomology ring of u(D, λ, μ). □

In the special case of a small quantum group, we obtain the following finite generation result (cf. [8, Thm. 1.3.4]).

Corollary 6.5. Let u_{q}(𝔤) be a quantized restricted enveloping algebra such that the order ℓ of q is odd and prime to 3 if 𝔤 is of type G_{2}. Then H^{∗}(u_{q}(𝔤), k) is a finitely generated algebra. Moreover, for any finitely generated u_{q}(𝔤)-module M, H^{∗}(u_{q}(𝔤), M) is a finitely generated module over H^{∗}(u_{q}(𝔤), k).

Remark 6.6. The restrictions on ℓ correspond to the assumptions (2.1.1). However our techniques and results are more general: The restrictions are used in the classification of Andruskiewitsch and Schneider, but not in our arguments. We need only the filtration lemma of De Concini and Kac [14, Lemma 1.7] as generalized to our setting (Lemma 2.4) to guarantee existence of the needed spectral sequences. Our results should hold for all small quantum groups u_{q}(𝔤) having such a filtration, including those at even roots of unity q for which q^{2d} ≠ 1 (see [7] for the general theory at even roots of unity).

We illustrate the connection between our results and those in [19] with a small example. Our structure results are not as strong, however our finite generation result is much more general.

Example 6.7. As an algebra, u_{q}(sl_{2}) is generated by E, F, K, with relations K^{ℓ} = 1, E^{ℓ} = 0, F^{ℓ} = 0, KEK^{−1} = q^{2}E, KF K^{−1} = q^{−1}F, and

(6.7.1)  EF − F E = \frac{K − K^{−1}}{q − q^{−1}} ,

where q is a primitive ℓth root of 1, ℓ > 2. Consider the coradical filtration on u_{q}(sl_{2}), in which deg(K) = 0, deg(E) = deg(F) = 1. Note that Gr u_{q}(sl_{2}) is generated by E, F, K, with all relations being the same except that (6.7.1) is replaced by EF − F E = 0. This is an algebra of the type featured in Section 4: H^{∗}(Gr u_{q}(sl_{2}), k) ≃ k[ξ_{1}, ξ_{2}, η_{1}η_{2}]/((η_{1}η_{2})^{2}), since these are the invariants, under the action of Γ = ⟨K⟩, of the cohomology of the subalgebra of Gr u_{q}(sl_{2}) generated by E, F. By [19],

H^{∗}(u_{q}(sl_{2}), k) ≃ k[α, β, γ]/(αβ + γ^{2}),

the coordinate algebra of the nilpotent cone of sl_{2}. Identify α ∼ ξ_{1}, β ∼ ξ_{2}, γ ∼ η_{1}η_{2}: Then as maps, deg(α) = ℓ, deg(β) = ℓ, deg(γ) = 2, so αβ + γ^{2} = 0 will imply γ^{2} = 0 in the associated graded algebra, as expected.

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Department of Mathematics and Computer Science, Saint Mary’s University, Halifax, NS B3H 3C3, Canada

E-mail address: mmastnak@cs.smu.ca

Department of Mathematics, University of Washington, Seattle, WA 98195, USA

E-mail address: julia@math.washington.edu

Mathematisches Institut der Universität München, Theresienstr. 39, 80333 München, Germany

E-mail address: schauenburg@math.lmu.de

Department of Mathematics, Texas A&M University, College Station, TX 77843, USA

E-mail address: sjw@math.tamu.edu