FINITE QUANTUM GROUPS OVER ABELIAN GROUPS OF PRIME EXPONENT

NICOLÁS ANDRUSKIEWITSCH AND HANS-JÜRGEN SCHNEIDER

1. Introduction

Since the discovery of quantum groups (Drinfeld, Jimbo) and their finite dimensional variations (Lusztig, Manin), these objects were studied from different points of view and had many applications. The present paper is part of a series where we intend to show that important classes of Hopf algebras are quantum groups and therefore belong to Lie theory.

We will assume that the ground-field 𝕜 is algebraically closed field of characteristic 0. One of our main results is the explicit construction of a general family of pointed Hopf algebras from Dynkin diagrams (Theorem 5.17). All the Frobenius-Lusztig kernels and their parabolic subalgebras belong to this family, but in addition we get many new examples. We show that any finite dimensional pointed Hopf algebra with group of prime exponent (greater than 17) is indeed in this family; see our main Theorem below. An important step in the proof follows from another main result (Theorem 7.6), where we show that a large family of finite dimensional pointed Hopf algebras is generated by group-like and skew-primitive elements, giving additional support to a conjecture in [AS2].

If A is a Hopf algebra, then we denote by G(A) the group of group-like elements of A. Recall that A is pointed if 𝕜G(A) is the largest cosemisimple subcoalgebra of A, or equivalently if any irreducible A-comodule is one-dimensional.

Let Γ be a finite abelian group and Γ̂ the group of its characters. We denote the unit element in Γ̂ by ε.

To state our main result, we have to introduce some notation. A linking datum D of finite Cartan type for Γ is a collection D consisting of

a Cartan matrix of finite type (a_{ij})_{1≤i,j≤θ} of size θ × θ for some θ ≥ 1, [K],

elements g_{1}, …, g_{θ} ∈ Γ,

characters χ_{1}, …, χ_{θ} ∈ Γ̂ satisfying

(1.1)  χ_{i}(g_{i}) ≠ 1,  for all 1 ≤ i ≤ θ,
(1.2)  χ_{j}(g_{i})χ_{i}(g_{j}) = χ_{i}(g_{i})^{a_{ij}},  for all 1 ≤ i, j ≤ θ,

and a family (λ_{ij})_{1≤i<j≤θ, i≁j} of elements in 𝕜 such that λ_{ij} is arbitrary if g_{i}g_{j} ≠ 1 and χ_{i}χ_{j} = ε, but 0 otherwise.

Date: November 16. 2001 (final version).

1991 Mathematics Subject Classification. Primary: 17B37; Secondary: 16W30.

This paper is available at www.mate.uncor.edu/andrus/articulos.html. This work was partially supported by CONICET, Agencia Córdoba Ciencia-CONICOR, DAAD, the Graduiertenkolleg of the Math. Institut (Universität München) and Secyt (UNC).

The elements (λ_{ij})_{1≤i<j≤θ, i≁j} are called the linking elements of D.

Here, by i ∼ j, resp. i ≁ j, 1 ≤ i, j ≤ θ we mean that i and j belong to the same connected component, resp. to different connected components of the Dynkin diagram corresponding to (a_{ij}).

Now we fix a prime p > 2 and a natural number s. We consider finite abelian groups of the form Γ (s) := (ℤ/(p))^{s}.

Let D be a linking datum of finite Cartan type for Γ (s) with Cartan matrix (a_{ij})_{1≤i,j≤θ} and linking elements (λ_{ij})_{1≤i<j≤θ, i≁j}.

We define the algebra 𝔲(D) by generators a_{1}, …, a_{θ}, y_{1}, …, y_{s} and relations

(1.3)  y_{h}^{p} = 1,  y_{m}y_{h} = y_{h}y_{m},  for all 1 ≤ m, h ≤ s,
(1.4)  ga_{j} = χ_{j}(g)a_{j}y_{h},  for all g ∈ Γ, 1 ≤ j ≤ θ,
(1.5)  (ad a_{i})^{1−a_{ij}}a_{j} = 0,  for all 1 ≤ i ≠ j ≤ θ, i ∼ j,
(1.6)  a_{i}a_{j} − χ_{j}(g_{i})a_{j}a_{i} = λ_{ij}(1 − g_{i}g_{j}),  for all 1 ≤ i < j ≤ θ, i ≁ j;
(1.7)  a_{α}^{p} = 0,  for all α ∈ Φ^{+}.

To formulate these relations we used the following natural interpretation of elements g ∈ Γ as words in the generators y_{h}, 1 ≤ h ≤ s. Let Y_{h}, 1 ≤ h ≤ s be a ℤ/(p)-basis of Γ, and write g = Y_{1}^{t_{1}} ⋯ Y_{s}^{t_{s}}, where t_{1}, ⋯, t_{s} are natural numbers. Then in the relations above replace g by the formal expression y_{1}^{t_{1}} ⋯ y_{s}^{t_{s}}.

In (1.5), ad a_{i} is the adjoint action of a_{i}, that is for all x ∈ 𝔲(D),

(ad a_{i})x = a_{i}_{(1)}xS(a_{i}_{(2)}) = a_{i}x − g_{i}xg_{i}^{−1}a_{i}.

In this way the left hand side of (1.5) is meant as a well-defined expression in the generators.

In (1.7), Φ^{+} is the set of positive roots of the root system associated to the Cartan matrix (a_{ij}); the ”root vectors” a_{α} are defined in Subsection 4.1 below.

Our main theorem is

Theorem 1.8. (a). Let D be a linking datum of finite Cartan type for Γ (s) with Cartan matrix (a_{ij}), and assume that p > 3 if (a_{ij}) has a connected component of type G_{2}. Then 𝔲(D) has a unique Hopf algebra structure determined by

(1.9)  Δy_{h} = y_{h} ⊗ y_{h},  Δa_{i} = a_{i} ⊗ 1 + g_{i} ⊗ a_{i},  for all 1 ≤ h ≤ s, 1 ≤ i ≤ θ.

The Hopf algebra 𝔲(D) is pointed, G(𝔲(D)) ≃ Γ (s) and dim 𝔲(D) = p^{s+|Φ^{+}|}.

(b). Let p > 17. Let A be a pointed finite-dimensional Hopf algebra such that G(A) ≃ Γ (s). Then there exist a linking datum D of finite Cartan type for Γ (s) such that A ≃ 𝔲(D).

Remarks 1.10. (i). In Section 5 we define the notion of a ”linking datum” for a general finite abelian group Γ. In the situation of the main theorem it is always possible to reduce to the case of linking data with all entries λ_{ij} equal to 0 or 1. Thus it follows from Theorem 1.8 that there are only finitely many isomorphism classes of finite dimensional Hopf algebras with fixed coradical 𝕜Γ (s). For more general finite abelian groups, this is no longer true [AS1], [BDG], [G].

(ii). The dimensions of the Hopf algebras in Theorem 1.8 are very special numbers. This phenomenon is shown in general for arbitrary finite groups Γ in Theorem 7.9.

(iii). Let (a_{ij})_{1≤i,j≤θ} be a finite Cartan matrix. The problem of actually finding all the collections g_{i} ∈ Γ (s), χ_{i} ∈ { \atop Γ (s)}, 1 ≤ i ≤ θ, such that (1.1) and (1.2) hold has been discussed in [AS2]. It can be stated as the problem of finding all the solutions of a system of algebraic equations over ℤ/(p) and it is in principle solvable. Note that in particular

θ ≤ 2s\frac{p − 1}{p − 2},

see [AS2, Prop. 8.3].

(iv). The question of finding all the possible linking elements attached to a fixed collection g_{i}, χ_{i}, 1 ≤ i ≤ θ, (a_{ij})_{1≤i,j≤θ}, is also of combinatorial nature, see Section 5, and also [Di]. Once these two problems are solved effectively, the isomorphism classes of the Hopf algebras 𝔲(D) can be determined using [AS2, Prop. 6.3], [AS3, Lemma 1.2].

(v). As a consequence of Theorem 1.8 one obtains the complete classification of all finite dimensional pointed Hopf algebras with group of group-likes Γ (1) = ℤ/(p), p ≠ 5, 7. It is the list given in [AS2, Theorem 1.3] plus the Frobenius-Lusztig kernels as described in [AS1]. Indeed, replacing in the proof of Theorem 1.8 [AS2, Cor. 1.2] by [AS2, Th. 1.3] we get the classification for all primes p ≠ 5 or 7, in view of Theorem 6.8 and [AS3, Lemma 4.2]. The only cases not covered are p = 5, type B_{2} and p = 7, type G_{2}. This result was independently obtained by Musson [Ms] using different methods starting from our previous article [AS2].

(vi). Up to now, the determination of all finite dimensional pointed Hopf algebras A with G(A) ≃ Γ, for a fixed group Γ, was known only for Γ = ℤ/(2) [N]. Other classification results of pointed Hopf algebras are known for some fixed dimension d: d = p^{2} is easy and follows from [N], [NZ]; d = p^{3} was done in [AS1], and by different methods in [CD], [SvO]; d = p^{4} in [AS3] (and does not seem to be possible via the other methods); d = 16 in [CDR], d = 32 in [Gñ1]; results on the case when Γ has exponent 2 can be found in [AD].

(vii). The classification of all coradically graded pointed Hopf algebras of dimension p^{5} was obtained in [Gñ2]. It is not difficult to deduce the classification of all pointed Hopf algebras of dimension p^{5} using Theorem 1.8 and results in [AS3].

(viii). The Hopf algebras 𝔲(D) can be defined for any Cartan datum of finite type D of an arbitrary finite abelian group. Part (a) of Theorem 1.8 is a special case of the general Theorem 5.17. For suitable choices of D, the Frobenius-Lusztig kernels and their parabolic subalgebras are of the form 𝔲(D). See example 5.12. Otherwise Theorem 5.17 provides many new examples of finite dimensional Hopf algebras arising from exotic linking data.

(ix). The results of this paper heavily depend on our paper [AS2] and on previous work on quantum groups [L1], [L2], [Ro1], [Ro2], [dCP], [Mu].

Conventions. Our reference for the theory of Hopf algebras is [Mo]. The notation for Hopf algebras is standard: Δ, S, ϵ, denote respectively the comultiplication, the antipode, the counit; we use Sweedler’s notation but dropping the summation symbol.

If C is a coalgebra then G(C) denotes the set of its group-like elements and C_{0} ⊂ C_{1} ⊂ … its coradical filtration. So that C_{0} is the coradical of C. As usual, P_{g,h}(C) denotes the space of (g, h)- skew primitives of C, g, h ∈ G(C). If C has a distinguished group-like 1, then we set P(C) = P_{1,1}(C), the space of primitive elements of C.

If A is an algebra and (x_{i})_{i∈I} is a family of elements of A then 𝕜⟨x_{i}⟩_{i∈I} or simply 𝕜⟨x_{i}⟩, resp. ⟨x_{i}⟩_{i∈I} or ⟨x_{i}⟩ denotes the subalgebra, resp. the two-sided ideal, generated by the x_{i}’s.

Let H be a Hopf algebra. A Yetter-Drinfeld module over H is a vector space V provided with structures of left H-module and left H-comodule such that δ(h.v) = h_{(1)}v_{(−1)}Sh_{(3)} ⊗ h_{(2)}.v_{(0)}. We denote by _{H}^{H}YD the (braided) category of Yetter-Drinfeld modules over H.

Assume that H = 𝕜Γ where Γ is a finite abelian group. We denote _{Γ}^{Γ}YD :=_{H}^{H} YD. Let g ∈ Γ, χ ∈ Γ̂ and V a module, resp. a comodule, resp. a Yetter-Drinfeld module, over Γ. Then we denote V^{χ} = {v ∈ V : h.v = χ(h)v, ∀h ∈ Γ}, resp. V_{g} = {v ∈ V : δ(v) = g ⊗ v}, resp. V_{g}^{χ} := V_{g} ∩ V^{χ}. If V is a locally finite Yetter-Drinfeld module, then V = ⊕_{g∈Γ,χ∈Γ̂}V_{g}^{χ}. Conversely, a vector space V provided with a direct sum decomposition V = ⊕_{g∈Γ,χ∈Γ̂}V_{g}^{χ} has an evident Yetter-Drinfeld module structure.

2. Outline of the paper and proof of the main result

Theorem 1.8 follows from Theorems 4.5, 5.17, 6.8, 6.10 and Corollary 7.7 in the present article, along the guidelines proposed in [AS1]. We give now the proof of Theorem 1.8 assuming those results which hold over arbitrary finite abelian groups. This section serves also as a guide to the different sections of the paper.

2.1. The proof. Let A be a finite dimensional pointed Hopf algebra with G(A) ≃ Γ (s). Let gr A := ⊕_{n≥0}gr A(n), where gr A(0) = A_{0}, gr A(n) = A_{n}/A_{n−1}, if n > 0 be the graded coalgebra associated to the coradical filtration of A. Then gr A is a graded Hopf algebra [Mo] and both the inclusion ι : A_{0} ↪ gr A and the graded projection π : gr A → A_{0} are Hopf algebra maps. Let R := gr A^{coπ} = {x ∈ gr A : (id ⊗π)Δ(x) = x ⊗ 1}; it is a graded braided Hopf algebra in _{Γ (s)}^{Γ (s)}YD with the grading inherited from gr A: R = ⊕_{n≥0}R(n), R(n) := R ∩ gr A(n). Notice that gr A can be reconstructed from R as a biproduct:

gr A ≃ R#𝕜Γ (s).

The braided Hopf algebra R is called the diagram of A. One has

R(0) = 𝕜1, (2.1) R(1) = P(R), (2.2) and we know from Corollary 7.7 below that

(2.3) R is generated as an algebra by R(1).

Let V := R(1); it is a Yetter-Drinfeld submodule of R. Since R satisfies (2.1), (2.2) and (2.3) we know that R ≃ 𝔅(V) is a Nichols algebra, see Subsection 3.2. Now there exists a basis x_{1}, …, x_{θ}

of V and g_{1}, …, g_{θ} ∈ Γ (s), χ_{1}, …, χ_{θ} ∈ { \atop Γ (s)} such that x_{i} ∈ V_{g_{i}}^{χ_{i}}, 1 ≤ i ≤ θ. Since A is finite

dimensional, χ_{i}(g_{i}) ≠ 1 for all i [AS1, Lemma 3.1] and there is a finite Cartan matrix (a_{ij})_{1≤i,j≤θ} such that (1.2) holds [AS2, Cor. 1.2].

To give an explicit description of 𝔅(V), we introduce root vectors in 𝔅(V) generalizing the root vectors defined in [L1]. We note that Lusztig’s root vectors can be described up to a non-zero scalar as an iterated braided commutator of simple root vectors. We then define the root vectors in the general case by exactly the same iterated braided commutator with respect to our more general braiding. As one of our main results, we obtain a presentation by generators and relations and a PBW basis for 𝔅(V) from the corresponding Theorem for Frobenius-Lusztig kernels, using Drinfeld’s twisting essentially in the same way as in [AS2]. See Theorem 4.5. We can then deduce part (a) of Theorem 1.8. For connected Dynkin diagrams it is a consequence of Theorem 4.5; in the non-connected case we apply the idea of twisting the algebra structure by a 2-cocycle which is given by a Hopf algebra map [DT]. See Theorem 5.17.

It follows from Theorem 4.5 that gr A can be presented as an algebra by generators y_{1}, …, y_{s} (homogeneous of degree 0) and x_{1}, …, x_{θ} (homogeneous of degree 1), and relations

(2.4)  y_{h}^{p} = 1,  y_{m}y_{h} = y_{h}y_{m},  for all 1 ≤ m, h ≤ s,
(2.5)  y_{h}x_{j} = χ_{j}(y_{h})x_{j}y_{h},  for all 1 ≤ h ≤ s, 1 ≤ j ≤ θ,
(2.6)  (ad x_{i})^{1−a_{ij}}x_{j} = 0,  for all 1 ≤ i ≠ j ≤ θ;
(2.7)  x_{α}^{p} = 0,  for all α ∈ Φ^{+};

and where the Hopf algebra structure is determined by

(2.8)  Δy_{h} = y_{h} ⊗ y_{h},  Δx_{i} = x_{i} ⊗ 1 + g_{i} ⊗ x_{i},  for all 1 ≤ h ≤ s, 1 ≤ i ≤ θ.

By [AS1, Lemma 5.4], we can choose a_{i} ∈ P_{g_{i},1}(A)^{χ_{i}} such that the class of a_{i} in gr A(1) coincides with x_{i}. We also keep the notation y_{j} for the generators of G(A). It is clear that relations (1.3) and (1.4) hold. Now relations (1.5) and (1.6), resp. (1.7), hold because of Theorem 6.8, resp. Lemma 6.9.

The Theorem now follows from Theorem 6.10. □

2.2. The general case. There are several obstructions to extend Theorem 1.8 to general finite abelian groups. First, it is open whether the diagram of a finite dimensional pointed Hopf algebra is generated in degree one, i.e. when it is a Nichols algebra; second, there are finite dimensional Nichols algebras which are not of Cartan type [N].

For liftings of gr A when R is a Nichols algebra of Cartan type, the quantum Serre relations of connected vertices in general still hold as we show in Theorem 6.8 below; however the powers of the root vectors are not necessarily 0. We should have a_{α}^{N_{α}} = u_{α} ∈ 𝕜G(A); the determination of u_{α} when α is a non-simple root was done in [AS3] for type A_{2}, in [BDR] for type B_{2}, and in [AS4] for type A_{n} for any n (up to some exceptional cases concerning the orders of the roots of unity).

3. Braided Hopf algebras

3.1. Biproducts. Let R be a braided Hopf algebra in _{H}^{H}YD; this means that R is an algebra and a coalgebra in _{H}^{H}YD and that the comultiplication Δ_{R} : R → R ⊗ R is an algebra map when in R ⊗ R the multiplication twisted by the braiding c is considered; in addition R admits an antipode. To avoid confusions we use the following variant of Sweedler’s notation for the comultiplication of R: Δ_{R}(r) = r^{(1)} ⊗ r^{(2)}. Let A = R#H be the biproduct or bosonization of R [Mj], [Ra]. Recall that the multiplication and comultiplication of A are given by

(r#h)(s#f) = r(h_{(1)}.s)#h_{(2)}f, Δ(r#h) = r^{(1)}#(r^{(2)})_{(−1)}h_{(1)} ⊗ (r^{(2)})_{(0)}#h_{(2)}.

The maps π : A → H and ι : H → A, π(r#h) = ϵ(r)h, ι(h) = 1#h, are Hopf algebra homomorphisms; we have R = {a ∈ A : (id ⊗π)Δ(a) = a ⊗ 1}. Conversely, let A, H be Hopf algebras provided with Hopf algebra homomorphisms π : A → H and ι : H → A. Then R = {a ∈ A : (id ⊗π)Δ(a) = a ⊗ 1} is a braided Hopf algebra in _{H}^{H}YD. The action . of H on R is the restriction of the adjoint action (composed with ι) and the coaction is (π ⊗ id )Δ; R is a subalgebra of A and the comultiplication is Δ_{R}(r) = r_{(1)}ιπS(r_{(2)}) ⊗ r_{(3)}. These constructions are inverse to each other. We shall mostly omit ι in what follows.

Let ϑ : A → R be the map given by ϑ(a) = a_{(1)}πS(a_{(2)}). Then

(3.1)  ϑ(ab) = a_{(1)}ϑ(b)πS(a_{(2)}),

for all a, b ∈ A and ϑ(h) = ε(h) for all h ∈ H; therefore, for all a ∈ A, h ∈ H, we have ϑ(ah) = ϑ(a)ε(h) and

(3.2)  ϑ(ha) = h.ϑ(a) = ϑ(h_{(1)}aπS(h_{(2)})).

Notice also that ϑ induces a coalgebra isomorphism A/AH^{+} ≃ R. In fact, the isomorphism A → R#H can be expressed explicitly as

a ↦ ϑ(a_{(1)})#π(a_{(2)}), a ∈ A.

If A is a Hopf algebra, the well-known adjoint representation ad of A on itself is given by ad x(y) = x_{(1)}yS(x_{(2)}). If R is a braided Hopf algebra in _{H}^{H}YD then there is also a braided adjoint representation ad_{c} of R on itself given by

ad_{c}x(y) = μ(μ ⊗ S)(id ⊗c)(Δ ⊗ id )(x ⊗ y),

where μ is the multiplication and c ∈ End (R ⊗ R) is the braiding. Note that if x ∈ P(R) then the braided adjoint representation of x is just

(3.3)  ad_{c}x(y) = μ(id −c)(x ⊗ y) =: [x, y]_{c}.

The element [x, y]_{c} defined by the second equality for any x and y, regardless of whether x is primitive, will be called a braided commutator.

When A = R#H, then for all b, d ∈ R,

(3.4)  ad_{(b#1)}(d#1) = (ad_{c}b(d))#1.

3.2. Nichols algebras. Let H be a Hopf algebra and let R = ⊕_{n∈ℕ}R(n) be a graded braided Hopf algebra in _{H}^{H}YD. We say that R is a Nichols algebra if 2.1, 2.2 and 2.3 hold, cf. [N], [AS2], [AG]. A Nichols algebra R is uniquely determined by the Yetter-Drinfeld module P(R); given a Yetter-Drinfeld module V, there exists a unique (up to isomorphism) Nichols algebra R with P(R) ≃ V. It will be denoted 𝔅(V). In fact, the kernel of the canonical map ϖ : T (V) → 𝔅(V) can be described in several different ways. For instance, Ker ϖ = ⊕_{n≥0}Ker S_{n} where S_{n} is the ”quantum antisymmetrizer” defined from the braiding c; so that 𝔅(V) is a ”quantum shuffle algebra” and as algebra and coalgebra only depends on the braiding c : V ⊗ V → V ⊗ V. See [N], [Wo], [L3], [Ro1], [Ro2], [Sch].

Let H = 𝕜Γ where Γ is a finite abelian group. Let V be a finite dimensional Yetter-Drinfeld module over Γ. Then there exist a basis x_{1}, …, x_{θ} of V and elements g_{1}, …, g_{θ} ∈ Γ, χ_{1}, …, χ_{θ} ∈ Γ̂ such that

(3.5)  x_{j} ∈ V_{g_{j}}^{χ_{j}},  for all 1 ≤ j ≤ θ.

In what follows we shall only consider Yetter-Drinfeld modules V such that χ_{i}(g_{i}) ≠ 1, 1 ≤ i ≤ θ. The braiding c is given with respect to the basis x_{i} ⊗ x_{j} by c(x_{i} ⊗ x_{j}) = b_{ij} x_{j} ⊗ x_{i}, where

(b_{ij})_{1≤i,j≤θ} = (χ_{j}(g_{i}))_{1≤i,j≤θ}.

Remark 3.6. Let V, resp. Ṽ, be a finite dimensional Yetter-Drinfeld module over Γ, resp. Γ̃, with a basis x_{1}, …, x_{θ} such that x_{i} ∈ V_{g_{i}}^{χ_{i}}, resp. with a basis x̃_{1}, …, x̃_{θ} such that x̃_{i} ∈ V χ̃_{i} ; g̃_{i}. Assume that χ_{i}(g_{j}) = χ̃_{i}(g̃_{j}) for all 1 ≤ i, j ≤ θ. Then there exists a unique algebra and coalgebra isomorphism 𝔅(V) → 𝔅(Ṽ) such that x_{i} ↦ x̃_{i} for all 1 ≤ i ≤ θ.

Definition 3.7. We shall say that a braiding given by a matrix b = (b_{ij})_{1≤i,j≤θ} whose entries are roots of unity is of Cartan type if for all i, j, b_{ii} ≠ 1 and there exists a_{ij} ∈ ℤ such that

b_{ij}b_{ji} = b_{ii}^{a_{ij}}.

The integers a_{ij} are uniquely determined by the following rules:

• If i = j we take a_{ii} = 2;

• if i ≠ j, we select the unique a_{ij} such that − ord b_{ii} < a_{ij} ≤ 0. Then (a_{ij}) is a generalized Cartan matrix [K]. We shall say a Yetter-Drinfeld module V is of

Cartan type, resp. finite Cartan type, if its corresponding braiding is of Cartan type, resp. the same plus the matrix (a_{ij}) is of finite type.

3.3. The twisting functor. Let H be a Hopf algebra and let F be an invertible element in H ⊗ H such that

(3.8)  F_{12}(Δ ⊗ id )F = F_{23}(id ⊗Δ)F,  (ε ⊗ id )(F) = 1 = (id ⊗ε)(F).

Then H_{F}, the same algebra H but with the comultiplication Δ_{F} := F ΔF^{−1}, is again a Hopf algebra [Dr]. We shall write F = F^{1} ⊗ F^{2}, F^{−1} = G^{1} ⊗ G^{2}; the new comultiplication will be denoted by Δ_{F} (h) = h_{(1,F)} ⊗ h_{(2,F)}.

Let now R be a braided Hopf algebra in _{H}^{H}YD, let A = R#H be its bosonization and consider the Hopf algebra A_{F}. It follows from the definitions that π : A_{F} → H_{F} and ι : H_{F} → A_{F} are also Hopf algebra homomorphisms. Hence

R_{F} := {a ∈ A_{F} : (id ⊗π)Δ_{F} (a) = a ⊗ 1}

is a braided Hopf algebra in the category _{H_{F}}^{H_{F}} YD. We consider the corresponding map ϑ_{F} and define ψ : R → R_{F} by

(3.9)  ψ(r) = ϑ_{F} (r),  r ∈ R.

The map ψ was defined in [AS2] in the case H = 𝕜Γ is the group algebra of a finite abelian group. The following Lemma generalizes [AS2, Lemma 2.3]; part (iii), new even for H = 𝕜Γ, will be needed in the sequel.

Lemma 3.10. (i). ψ is an isomorphism of H-modules. (Recall that H = H_{F} as algebras).

(ii). If r, s ∈ R then

(3.11)  ψ(rs) = F^{1}.ψ(r) F^{2}.ψ(s).

(iii). If r ∈ R then

(3.12)  Δ_{R_{F}} ψ(r) = F^{1}.ψ(r^{(1)}) ⊗ F^{2}.ψ(r^{(2)}).

(iv). If R is a graded braided Hopf algebra, then R_{F} also is and ψ is a graded map. If R is a coradically graded braided Hopf algebra (resp. a Nichols algebra), then R_{F} also is.

Proof. (i) follows from (3.2): ψ(h.r) = ϑ_{F} (h.r) = ϑ_{F} (hr) = h.ϑ_{F} (r) = h.ψ(r). Now we prove (ii):

ψ(rs) = ϑ_{F} (rs) = r_{(1,F)}ϑ_{F} (s)π(S_{F} (r_{(2,F)}))
= r_{(1,F)}π(S_{F} (r_{(2,F)}))π(r_{(3,F)})ϑ_{F} (s)π(S_{F} (r_{(4,F)}))
= ψ(r_{(1,F)})π(r_{(2,F)}).ψ(s) = ψ(F^{1}r_{(1)}G^{1})π(F^{2}r_{(2)}G^{2}).ψ(s)
= ψ(F^{1}r_{(1)})ε(G^{1})π(F^{2}r_{(2)}G^{2}).ψ(s) = F^{1}.ψ(r_{(1)})π(F^{2})π(r_{(2)}).ψ(s)
= F^{1}.ψ(r)π(F^{2}).ψ(s),

as claimed. Here we have used (3.1), the definitions and (3.8). For the proof of (iii), we first observe that, if r ∈ R, then

(3.13)  ψ(r^{(1)}) ⊗ ψ(r^{(2)}) = ϑ_{F} (r_{(1)}πS(r_{(2)})) ⊗ ϑ_{F} (r_{(3)}) = ψ(r_{(1)}) ⊗ ψ(r_{(2)}).

Using that ϑ_{F} is a coalgebra map, (3.2) and (3.13), we conclude that

Δ_{R_{F}} ψ(r) = Δ_{R_{F}} ϑ_{F} (r) = ϑ_{F} (r_{(1,F)}) ⊗ ϑ_{F} (r_{(2,F)})
= ϑ_{F} (F^{1}r_{(1)}G^{1}) ⊗ ϑ_{F} (F^{2}r_{(2)}G^{2}) = ϑ_{F} (F^{1}r_{(1)}) ⊗ ϑ_{F} (F^{2}r_{(2)})
= F^{1}.ϑ_{F} (r_{(1)}) ⊗ F^{2}.ϑ_{F} (r_{(2)}) = F^{1}.ψ(r^{(1)}) ⊗ F^{2}.ψ(r^{(2)});

The proof of (iv) has no difference with the proof of the analogous statement in [AS2, Lemma 2.3]. □

We now consider the special case when H = 𝕜Γ, Γ a finite abelian group. Let ω : Γ̂ × Γ̂ → 𝕜^{×} be a 2-cocycle, i.e. ω(τ, 1) = ω(1, τ) = 1 and ω(τ, ζ)ω(τζ, η) = ω(τ, ζη)ω(ζ, η). The cocycle ω allows to define a map Ψ : Γ̂ × Γ → Γ by

(3.14)  ⟨τ, Ψ(χ, g)⟩ = ω(τ, χ)ω(χ, τ)^{−1}⟨τ, g⟩,  τ ∈ Γ̂.

We identify H with the Hopf algebra 𝕜Γ̂ of functions on the group Γ̂; we denote by δ_{τ} ∈ H the function given by δ_{τ}(ζ) = δ_{τ,ζ}, τ, ζ ∈ Γ̂. Then δ_{τ} = \frac{1}{|Γ|} ∑_{g∈Γ}⟨τ, g^{−1}⟩g. Let F ∈ H ⊗ H be given by

F = ∑_{τ,ζ∈Γ̂} ω(τ, ζ)δ_{τ} ⊗ δ_{ζ}.

Then F satisfies (3.8); note that H = H_{F}. Let now R be a braided Hopf algebra in _{Γ}^{Γ}YD; we can consider the Hopf algebras A = R#𝕜Γ and A_{F}, the braided Hopf algebra R_{F} ∈ _{Γ}^{Γ}YD and the map ψ : R → R_{F}. We have

ψ(r) = ∑_{τ∈Γ̂} ω(χ, τ)^{−1}r#δ_{τ}, r ∈ R^{χ}; ψ(R_{g}^{χ}) = R_{Ψ(χ,g)}^{ψ}.

See [AS2, Lemma 2.3]. Note that (3.11) is now ψ(rs) = ω(χ, τ)ψ(r)ψ(s), r ∈ R^{χ}, s ∈ R^{τ}. Lemma 3.15. If r ∈ P (R)_{g}^{χ} and s ∈ R^{τ} then

(3.16)  ψ ([r, s]_{c}) = ω(χ, τ)[ψ(r), ψ(s)]_{c}.

Proof. We have

ψ ([r, s]_{c}) = ψ (rs − τ(g)sr)
= ω(χ, τ)ψ(r)ψ(s) − ω(τ, χ)τ(g)ψ(s)ψ(r)
= ω(χ, τ) (ψ(r)ψ(s) − ⟨τ, Ψ(χ, g)⟩ψ(s)ψ(r))
= ω(χ, τ)[ψ(r)), ψ(s)]_{c},

where we used (3.14). □

Remark 3.17. It is possible to show that (ψ ⊗ ψ)c(r ⊗ s) = F.c_{F} (ψ(r) ⊗ ψ(s)), for all r ∈ R_{g}^{χ}, s ∈ R^{τ}.

From the previous considerations and Lemma 3.10 we immediately get Proposition 3.18. Let R be an algebra in _{Γ}^{Γ}YD, (x_{i})_{i∈I} a family of elements of R, x_{i} ∈ R_{g_{i}}^{χ_{i}} for some g_{i} ∈ Γ, χ_{i} ∈ Γ̂. Then:

(i). ψ(𝕜⟨x_{i}⟩) = 𝕜⟨ψ(x_{i})⟩, ψ(⟨x_{i}⟩) = ⟨ψ(x_{i})⟩.

(ii). If R has a presentation by generators x_{i} and relations t_{j}, where also the t_{j}’s are homogeneous then R_{F} has a presentation by generators ψ(x_{i}) and relations ψ(t_{j}).

(iii). If x_{i} is central and ω(χ_{i}, τ) = ω(τ, χ_{i}) for all τ such that R^{τ} ≠ 0, then ψ(x_{i}) is central. □

4. Root vectors and Quantum Serre relations

4.1. Root vectors. In this Section, we assume the following situation:

We fix a finite abelian group Γ, a finite Cartan matrix (a_{ij})_{1≤i,j≤θ} and g_{1}, …, g_{θ} ∈ Γ, χ_{1}, …, χ_{θ} ∈ Γ̂ such that (1.1) and (1.2) hold. Let d_{1}, …, d_{θ} ∈ {1, 2, 3} such that d_{i}a_{ij} = d_{j}a_{ji} for all i, j. We set q_{i} = χ_{i}(g_{i}), N_{i} the order of q_{i}. We assume, for all i and j, that the order of χ_{i}(g_{j}) is odd, and that N_{i} is not divisible by 3 if i belongs to a connected component of type G_{2}.

Let X be the set of connected components of the Dynkin diagram corresponding to (a_{ij}). We assume that for each I ∈ X, there exist c_{I}, d_{I} such that I = {j : c_{I} ≤ j ≤ d_{I}}; that is, after reordering the Cartan matrix is a matrix of blocks corresponding to the connected components. Let I ∈ X and i ∼ j in I; then N_{i} = N_{j}, hence N_{I} := N_{i} is well defined. Let Φ_{I}, resp. Φ_{I}^{+}, be the root system, resp. the subset of positive roots, corresponding to the Cartan matrix (a_{ij})_{i,j∈I}; then Φ = ⋃_{I∈X} Φ_{I}, resp. Φ^{+} = ⋃_{I∈X} Φ_{I}^{+} is the root system, resp. the subset of positive roots, corresponding to the Cartan matrix (a_{ij})_{1≤i,j≤θ}. Let α_{1}, …, α_{θ} be the set of simple roots.

Let W_{I} be the Weyl group corresponding to the Cartan matrix (a_{ij})_{i,j∈I}; we identify it with a subgroup of the Weyl group W corresponding to the Cartan matrix (a_{ij}). We fix a reduced decomposition of the longest element ω_{0,I} of W_{I} in terms of simple reflections. Then we obtain a reduced decomposition of the longest element ω_{0} = s_{i_{1}} … s_{i_{P}} of W from the expression of ω_{0} as product of the ω_{0,I}’s in some fixed order of the components, say the order arising from the order of the vertices. Therefore β_{j} := s_{i_{1}} … s_{i_{j−1}}(α_{i_{j}}) is a numeration of Φ^{+}.

We fix a finite dimensional Yetter-Drinfeld module V over Γ with a basis x_{1}, …, x_{θ} with x_{i} ∈ V_{g_{i}}^{χ_{i}}, 1 ≤ i ≤ θ.

Major examples of modules of Cartan type are the Frobenius-Lusztig kernels. Let N > 1 be an odd natural number and let q ∈ 𝕜 be a primitive N-th root of 1, not divisible by 3 in case (a_{ij}) has a component of type G_{2}. Let 𝔾 = ℤ/(N)^{θ} = ⟨e_{1}⟩ ⊕ ⋯ ⊕ ⟨e_{θ}⟩; let η_{j} ∈ 𝔾̂ be the unique character such that ⟨η(j), e(i)⟩ = q^{d_{i}a_{ij}}. Let 𝕍 be a Yetter-Drinfeld module over 𝔾 with a basis X_{1}, …, X_{θ} such that

X_{i} ∈ 𝕍_{η_{i}}^{e_{i}}, for all 1 ≤ i ≤ θ.

We denote by 𝔠 the braiding of 𝕍. Lusztig defined root vectors X_{α} ∈ 𝔅(𝕍), α ∈ Φ^{+} [L2]. One can see from [L3] that, up to a non-zero scalar, each root vector can be written as an iterated braided commutator in some sequence X_{ℓ_{1}}, …, X_{ℓ_{a}} of simple root vectors such as [[X_{ℓ_{1}}, [X_{ℓ_{2}}, X_{ℓ_{3}}]_{𝔠}]_{𝔠}, [X_{ℓ_{4}}, X_{ℓ_{5}}]_{𝔠}]_{𝔠}. This can also be seen in the situation in [Ri].

We now fix for each α ∈ Φ^{+} such a representation of X_{α} as an iterated braided commutator. In the general case of our V, we define root vectors x_{α} in the tensor algebra T (V), α ∈ Φ^{+}, as the same formal iteration of braided commutators in the elements x_{1}, …, x_{θ} instead of X_{1}, …, X_{θ} but with respect to the braiding c given by the general matrix (χ_{j}(g_{i})). Note that each x_{α} is homogeneous and has the same degree as X_{α}, where we mean the degree in the sense of [L3]. Also,

(4.1)  x_{α} ∈ T (V)_{g_{α}}^{χ_{α}},

where g_{α} = g_{1}^{b_{1}} … g_{θ}^{b_{θ}}, χ_{α} = χ_{1}^{b_{1}} … χ_{θ}^{b_{θ}}, where α = b_{1}α_{1} + ⋯ + b_{θ}α_{θ}.

Theorem 4.2. The Nichols algebra 𝔅(𝕍) is presented by generators X_{i}, 1 ≤ i ≤ θ, and relations

(4.3)  ad_{c}(X_{i})^{1−a_{ij}}(X_{j}) = 0,  for all i ≠ j,
(4.4)  X_{α}^{N} = 0,  for all α ∈ Φ^{+}.

Moreover, the following elements constitute a basis of 𝔅(𝕍):

X_{β_{1}}^{h_{1}}X_{β_{2}}^{h_{2}} … X_{β_{P}}^{h_{P}} , for all 0 ≤ h_{j} ≤ N − 1, 1 ≤ j ≤ P.

Proof. It follows from results of Lusztig [L1], [L2], Rosso [Ro1], [Ro2] and Müller [Mu] that 𝔅(𝕍) is the positive part of the so-called Frobenius-Lusztig kernel corresponding to the Cartan matrix (a_{ij}). See [AS2, Th. 3.1] for details. The presentation by generators and relations follows from the considerations in the last paragraph of p. 15 and the first paragraph of p. 16 in [AJS] referring to [dCP, §19, Corollary in p. 120]. The statement about the basis is [L1, L2]. □

4.2. Nichols algebras of Cartan type. We can now prove the first main result of the present paper, describing 𝔅(V) by generators and relations when V is of finite Cartan type, improving [AS2, Th. 1.1 (i)]. As in loc. cit., we use repeatedly Remark 3.6. Theorem 4.5. The Nichols algebra 𝔅(V) is presented by generators x_{i}, 1 ≤ i ≤ θ, and relations

(4.6)  ad_{c}(x_{i})^{1−a_{ij}}(x_{j}) = 0,  for all i ≠ j,
(4.7)  x_{α}^{N_{I}} = 0,  for all α ∈ Φ_{I}^{+}, I ∈ X .

Moreover, the following elements constitute a basis of 𝔅(V):

x_{β_{1}}^{h_{1}}x_{β_{2}}^{h_{2}} … x_{β_{P}}^{h_{P}} , for all 0 ≤ h_{j} ≤ N_{I} − 1, if β_{j} ∈ I, 1 ≤ j ≤ P.

Proof. (a) Let us first assume that the braiding is symmetric, that is χ_{i}(g_{j}) = χ_{j}(g_{i}) for all i, j. By [AS2, Lemma 4.2] we can assume moreover that the Cartan matrix (a_{ij}) is connected. From our assumptions on the orders of the χ_{i}(g_{j}) we then conclude that the braiding has the form χ_{j}(g_{i}) = q^{d_{i}a_{ij}} for all i, j where q is a root of unity of order N = χ_{i}(g_{i}). See [AS2, Lemma 4.3]. Hence the Theorem follows directly from Theorem 4.2 and Remark 3.6.

(b) In the case of an arbitrary braiding we know from Lemma 4.1 of [AS2] that there exists a finite abelian group 𝔾 satisfying:

• The braiding c of V can be realized from a Yetter-Drinfeld module structure over 𝔾 that we continue denoting by V, cf. Remark 3.6.

• There exists a cocycle ω : 𝔾̂ × 𝔾̂ → 𝕜^{×} with corresponding F ∈ 𝕜𝔾 ⊗ 𝕜𝔾 such that the braiding of V_{F} is symmetric. Let ψ : 𝔅(V) → 𝔅(V_{F}) be the isomorphism having the same meaning as in (3.9).

• The braiding of V_{F} is given in the basis ψ(x_{i}) ⊗ ψ(x_{j}) by a matrix (b_{ij}^{F}) such that b_{ii}^{F} = χ_{i}(g_{i}) and the order of (b_{ij}^{F}) is again odd for all i and j.

12 NICOLÁS ANDRUSKIEWITSCH AND HANS-JÜRGEN SCHNEIDER

If ϖ : T (V) → 𝔅(V), ϖ_{F} : T (V_{F}) → 𝔅(V_{F}) denote the canonical maps, then we have a commutative diagram

T (V) ⟶^{ϖ} 𝔅(V)
ψ↓  ↓^{ψ}
T (V_{F}) ⟶^{ϖ_{F}} 𝔅(V_{F}).

Clearly, ψ(Ker ϖ) = Ker ϖ_{F}; if (r_{j})_{j∈J} is a set of generators of the ideal Ker ϖ with r_{j} ∈ T (V)_{h_{j}}^{η_{j}} then by Proposition 3.18 (ψ(r_{j}))_{j∈J} is a set of generators of the ideal Ker ϖ_{F}. By the symmetric case (a), we know the generators of Ker φ_{F}. Let us denote X_{i} := ψ(x_{i}). Then by Lemma 3.15 and (3.11), we have ψ (ad_{c}(x_{i})^{1−a_{ij}}(x_{j})) = u_{ij}ad_{c}(X_{i})^{1−a_{ij}}(X_{j}) and ψ (x_{α}^{N_{I}}) = u_{α}X_{α}^{N_{I}}, α ∈ Φ_{I}^{+} where u_{ij}, u_{α} are non-zero scalars. This implies the first claim of the Theorem. The second follows in a similar way. □

Let 𝔅̂(V) be the braided Hopf algebra in _{Γ}^{Γ}YD generated by x_{1}, …, x_{θ} with relations (4.6), where the x_{i}’s are primitive. Let K(V) be the subalgebra of 𝔅̂(V) generated by x_{α}^{N_{I}}, α ∈ Φ_{I}^{+}, I ∈ X; it is a Yetter-Drinfeld submodule of 𝔅̂(V). Theorem 4.8. K(V) is a braided Hopf subalgebra in _{Γ}^{Γ}YD of 𝔅̂(V).

Proof. (a). As in the proof of Theorem 4.5 we first assume that the braiding is symmetric. If i ≠ j, then χ_{j}(g_{i})χ_{i}(g_{j}) = 1 and hence the corresponding Serre relation (4.6) says that x_{i}x_{j} = x_{j}x_{i}. Thus, we can easily reduce to the connected case. In such case, χ_{j}(g_{i}) = q^{d_{i}a_{ij}} as before and the Theorem is shown in [dCP].

(b). In the general case, we change the group as in the proof of Theorem 4.5. The isomorphism ψ : T (V) → T (V_{F}) respects the Serre relations up to non-zero scalars by Lemma 3.15. Also, it maps subcoalgebras stable under the action of the group to subcoalgebras by Lemma 3.10 (iii). We conclude from (a) that K(V) is a subcoalgebra of 𝔅̂(V). □

5. Linking datum and glueing of connected components

5.1. Linking datum. In this Section, we fix a finite abelian group Γ, a finite Cartan matrix (a_{ij})_{1≤i,j≤θ} and g_{1}, …, g_{θ} ∈ Γ, χ_{1}, …, χ_{θ} ∈ Γ̂ such that (1.1) and (1.2) hold. We preserve the conventions and hypotheses from Section 4. Definition 5.1. We say that two vertices i and j are linkable (or that i is linkable to j) if

i ≁ j, (5.2) g_{i}g_{j} ≠ 1 and (5.3) χ_{i}χ_{j} = 1. (5.4)

If i is linkable to j, then χ_{i}(g_{j})χ_{j}(g_{i}) = 1 by (5.2); it follows then from (5.4) that

(5.5)  χ_{j}(g_{j}) = χ_{i}(g_{i})^{−1}.

FINITE QUANTUM GROUPS OVER ABELIAN GROUPS OF PRIME EXPONENT 13

Lemma 5.6. Assume that i and k, resp. j and ℓ, are linkable. Then a_{ij} = a_{kℓ}, a_{ji} = a_{ℓk}. In particular, a vertex i can not be linkable to two different vertices j and h.

Proof. If a_{iℓ} ≠ 0 then a_{ij} = a_{ji} = 0 (otherwise j ∼ ℓ) and a_{kℓ} = a_{ℓk} = 0 (otherwise i ∼ k). If a_{jk} ≠ 0 then a_{ij} = a_{ji} = 0 (otherwise i ∼ k) and a_{kℓ} = a_{ℓk} = 0 (otherwise j ∼ ℓ). Assume that a_{iℓ} = 0 = a_{jk}. Then

χ_{i}(g_{i})^{a_{ij}} = χ_{i}(g_{j})χ_{j}(g_{i}) = χ_{k}^{−1}(g_{j})χ_{ℓ}^{−1}(g_{i}) = χ_{j}(g_{k})χ_{i}(g_{ℓ}) = χ_{ℓ}^{−1}(g_{k})χ_{k}^{−1}(g_{ℓ}) = χ_{k}(g_{k})^{−a_{kℓ}} = χ_{i}(g_{i})^{a_{kℓ}}.

Then N_{i} divides a_{ij} − a_{kℓ} and analogously, N_{k} divides a_{ij} − a_{kℓ}. So that a_{ij} = a_{kℓ} by the assumptions on the order of N_{i} and N_{k}; by symmetry, a_{ji} = a_{ℓk}. Assume that a vertex i is linkable to j and h. Then 2 = a_{ii} = a_{jh}, so j = h. □

A linking datum of finite Cartan type for Γ is a collection

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

where (a_{ij}) is a Cartan matrix of finite type, (g_{i}), (χ_{j}) are elements as above, and (λ_{ij})_{1≤i<j≤θ, i≁j} are elements in 𝕜 such that λ_{ij} is arbitrary if i and j are linkable but 0 otherwise. The elements λ_{ij} are called the linking elements of D. Given a linking datum, we say that two vertices i and j are linked if λ_{ij} ≠ 0.

This definition generalizes part of the definition of compatible datum in [AS1, Section 5]. We shall represent a linking datum by the Dynkin diagram of the Cartan matrix (a_{ij}) joining linked vertices by a dotted line. To have a complete picture we add the pair (g_{i}, χ_{i}) below the vertex i. Definition 5.7. Let us fix a decomposition Γ =< Y_{1} > ⊕ ⋯ ⊕ < Y_{s} >; let M_{h} denote the order of Y_{h}, 1 ≤ h ≤ s. Let D = D (Γ, (a_{ij})_{1≤i,j≤θ}, (g_{i})_{1≤i≤θ}, (χ_{j})_{1≤j≤θ}, (λ_{ij})_{1≤i<j≤θ, i≁j}) be a linking datum. We denote by 𝔲(D) the algebra presented by generators a_{1}, …, a_{θ}, y_{1}, …, y_{s} and relations

(5.8)  y_{h}^{M_{h}} = 1, y_{m}y_{h} = y_{h}y_{m},  for all 1 ≤ m, h ≤ s,

(1.4), (1.5), (1.6) and

(5.9)  a_{α}^{N_{I}} = 0,  for all α ∈ Φ_{I}^{+}, I ∈ X .

Remark 5.10. In the preceding definition, one could consider only linking data with λ_{ij} = 1 or 0. Indeed, one can replace the generator a_{i} by λ_{ij}^{−1}a_{i} whenever λ_{ij} ≠ 0 for some j which is unique by Lemma 5.6. The other relations do not change since they are homogeneous in the a_{i}’s. However, in the more general case where the relations (5.9) have a non-zero right side, one needs general linking data. Example 5.11. Here is a linking datum where all the connected components are points:

\begin{matrix}• & … & • \\ • & & • \\ • & … & • \end{matrix}

14 NICOLÁS ANDRUSKIEWITSCH AND HANS-JÜRGEN SCHNEIDER

Example 5.12. Let B := (b_{ij})_{1≤i,j≤R} be a finite Cartan matrix, 0 ≤ M ≤ R and q ∈ 𝕜 a root of unity of order N; we assume N is odd, and prime to 3 if B contains a component of type G_{2}. Let d_{1}, …, d_{R} be integers in {1, 2, 3} such that d_{i}b_{ij} = d_{j}b_{ji}. Let θ = R + M, B̃ := (b_{ij})_{1≤i,j≤M} and A = (a_{ij}) be the Cartan matrix

A = (\begin{matrix}B & 0 \\ 0 & B̃ \end{matrix}) .

Let Γ = (ℤ/(N))^{R}, g_{1}, …, g_{R} the canonical basis of Γ and χ_{1}, …, χ_{R} be the character given by χ_{i}(g_{j}) = q^{d_{i}b_{ij}}; let g_{R+j} = g_{j}, χ_{R+j} = χ_{j}^{−1}, 1 ≤ j ≤ M. Note that j and j + R are linkable, 1 ≤ j ≤ M. Finally, let λ_{j,j+R} = 1 if 1 ≤ j ≤ M and 0 otherwise; then (λ_{ij})_{1≤i<j≤θ} is a linking datum for Γ, (a_{ij}), g_{1}, …, g_{θ} and χ_{1}, …, χ_{θ}. The Hopf algebra 𝔲(D) with comultiplication determined by (1.9) is the parabolic part of a Frobenius-Lusztig kernel. Since the numeration of the Dynkin diagram is so far arbitrary, any such parabolic appears in this way. Example 5.13. Here are some exotic examples of linking data:

Take 4 copies of A_{3} and label the vertices such that {1, 2, 3}, {4, 5, 6}, {7, 8, 9} and {10, 11, 12} are the connected components. Then link 3 with 4, 6 with 7, 9 with 10 and 12 with 1. It is possible to realize this linking over ℤ/(N)^{12} for any odd N; the corresponding braiding will be symmetric in each component, that is, the corresponding subalgebra is the ”Borel” part of a Frobenius-Lusztig kernel. More examples arise considering more copies of more general components.

See [Di] for a combinatorial description of all linkings of Dynkin diagrams.

5.2. Altering the multiplication by a cocycle. The following variation of Drinfeld’s twisting was stated by Doi: if H is a Hopf algebra and σ : H × H → 𝕜 is an invertible 2-cocycle, so that

σ(x_{(1)}, y_{(1)})σ(x_{(2)}y_{(2)}, z) = σ(y_{(1)}, z_{(1)})σ(x, y_{(2)}z_{(2)}),
σ(1, 1) = 1,

for all x, y, z ∈ H, then H_{σ}– the same H but with the multiplication ._{σ} below– is again a Hopf algebra, where

x._{σ}y = σ(x_{(1)}, y_{(1)})x_{(2)}y_{(2)}σ^{−1}(x_{(3)}, y_{(3)}).

Lemma 5.14. [DT] Let U, B be Hopf algebras.

(a) Let τ : U ⊗ B → 𝕜 be a bilinear map such that for all u, v ∈ U, a, b ∈ B

(1) τ(uv, a) = τ(u, a_{(1)})τ(v, a_{(2)}),

(2) τ(u, ab) = τ(u_{(1)}, b)τ(u_{(2)}, a),

(3) τ(1, a) = ε(a),

(4) τ(u, 1) = ε(u). Let H be the tensor product Hopf algebra H = U ⊗ B and let σ : H ⊗ H → 𝕜 be the bilinear map σ(u ⊗ a, v ⊗ b) = ε(u)τ(v, a)ε(b), for all u, v ∈ U, a, b ∈ B. Then τ is convolution invertible with inverse given by τ^{−1}(v, a) = φ(Sv)(a) = φ(v)(S^{−1}a); σ is an invertible 2-cocycle– with inverse σ^{−1}(u ⊗ a, v ⊗ b) = ε(u)τ^{−1}(v, a)ε(b), for all u, v ∈ U, a, b ∈ B– and consequently H_{σ} is a Hopf algebra.

(b) Assume that B is finite dimensional and let φ : U → (B^{∗})^{cop} be a Hopf algebra homomorphism. Then τ : U ⊗ B → 𝕜, τ(v, a) = φ(v)(a), is invertible– with inverse given by τ^{−1}(v, a) = φ(Sv)(a) = φ(v)(S^{−1}a), and satisfies 1, 2, 3 and 4. Reciprocally, given such τ there is a unique such φ. □

The following result is probably known. We include it for completeness. Lemma 5.15. Let U, B and τ be as in the preceeding Lemma. Assume that

• U is generated as an algebra by skew-primitive elements u_{i}, i ∈ I and group-like elements g_{k}, k ∈ K, which in addition generate G(U) as a monoid;

• B is generated as an algebra by skew-primitive elements b_{j}, j ∈ J and group-like elements h_{ℓ},

ℓ ∈ L, which in addition generate G(B) as a monoid. Let A be an algebra and let α : U → A, β : B → A be algebra maps and let γ : (U ⊗ B)_{σ} → A be given by γ(u ⊗ b) = α(u)β(b) for all u ∈ U, b ∈ B. Then γ is an algebra map if and only if

(5.16)  τ(u_{(1)}, b_{(1)})α(u_{(2)})β(b_{(2)}) = β(b_{(1)})α(u_{(1)})τ(u_{(2)}, b_{(2)}),

whenever u, resp. b, belongs to the family u_{i}, i ∈ I or g_{k}, k ∈ K, resp. b_{j}, j ∈ J or h_{ℓ}, ℓ ∈ L.

Proof. (Sketch). Clearly, γ is an algebra map if and only if (5.16) holds for all u ∈ U, b ∈ B. It follows also easily that (5.16) holds when u = 1, or b = 1, or u ∈ G(U) and b ∈ G(B). Next, let u, v ∈ U and b, c ∈ B be arbitrary elements; one can then check that (5.16) holds for uv and bc if it holds for all the possibilities u_{(1)} and c_{(1)}; u_{(2)} and b_{(1)}; v_{(1)} and c_{(2)}; v_{(2)} and b_{(2)}. From this observation and the hypothesis the Lemma follows. □

5.3. Glueing of connected components. In this subsection, we fix a linking datum

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

of finite Cartan type. Let A = 𝔲(D) be the algebra defined in Definition 5.7. Theorem 5.17. (a) There exists a unique Hopf algebra structure on A determined by (1.9).

(b) The dimension of A is |Γ| ∏_{I∈X} N_{I}^{|Φ_{I}^{+}|}.

Proof. By induction on the number of connected components. Here is the first step:

Lemma 5.18. Theorem (5.17) is true if the Dynkin diagram corresponding to (a_{ij})_{1≤i,j≤θ} is connected.

Proof. Let V = ⊕_{1≤i≤θ}V_{g_{i}}^{χ_{i}} be a Yetter-Drinfeld module over Γ with dim V_{g_{i}}^{χ_{i}} = 1 and pick x_{i} ∈ V_{g_{i}}^{χ_{i}}−0. By Theorem 4.5 and the formulas for the biproduct, there exists a unique algebra map F : A → 𝔅(V)#𝕜Γ such that F(a_{i}) = x_{i}#1, F(y_{t}) = 1#y_{t}. Also, by Theorem 4.5 again, there are algebra maps G_{1} : 𝔅(V) → A, G_{2} : 𝕜Γ → A such that G_{1}(x_{i}) = a_{i}, G_{2}(y_{t}) = y_{t}. Let G : 𝔅(V)#𝕜Γ → A,

16 NICOLÁS ANDRUSKIEWITSCH AND HANS-JÜRGEN SCHNEIDER

G(x#u) = G_{1}(x)G_{2}(u), x ∈ 𝔅(V), u ∈ 𝕜Γ; then G is an algebra map by (1.4). It is clear now that F is an isomorphism with inverse G; thus A is a Hopf algebra and has the desired dimension by the dimension formula in Theorem 4.5. □

For the rest of this proof we assume: there exists θ̃ < θ such that i ∼ j, resp. i ≁ h, if 1 ≤ i ≤ θ̃ and 1 ≤ j ≤ θ̃, resp. θ̃ < h ≤ θ. Let J = {1, …, θ̃} ∈ X. Let Υ :=< Z_{1} > ⊕ ⋯ ⊕ < Zθ̃ >, where the order of Z_{i} is the least common multiple of ord g_{i} and ord χ_{i}, 1 ≤ i ≤ θ̃. Let η_{j} be the unique character of Υ such that η_{j}(Z_{i}) = χ_{j}(g_{i}), 1 ≤ i ≤ θ̃, 1 ≤ j ≤ θ̃. This is well defined because ord g_{i} divides ord Z_{i} for all i.

• B := 𝔲(D_{1}), with D_{1} = D (Γ, (a_{ij})_{θ̃<i,j≤θ}, (g_{i})_{θ̃<i≤θ}, (χ_{j})_{θ̃<j≤θ}, (λ_{ij})_{θ̃<i<j≤θ,i≁j}), with genera-

tors b_{θ̃+1}, …, b_{θ} (instead of the a_{i}’s) and y_{1}, …, y_{s};

• U := 𝔲(D_{2}), with D_{2} = D (Υ, (a_{ij})_{1≤i,j≤θ̃}, (Z_{i})_{1≤i≤θ̃}, (η_{j})_{1≤j≤θ̃}, (λ_{ij})_{1≤i<j≤θ̃, i≁j}), with gener-

ators u_{1}, …, uθ̃ (instead of the a_{i}’s) and z_{1}, …, zθ̃.

Note that the linking datum of U is empty since J is connected. By the induction hypothesis, dim B = |Γ| ∏_{I∈X ,I≠J} N_{I}^{|Φ_{I}^{+}|} and dim U = |Υ|N_{J}^{|Φ_{J}^{+}|}.

Lemma 5.19. (a). For each i, 1 ≤ i ≤ θ̃, there exists a unique character γ_{i} : B → 𝕜 such that

(5.20)  γ_{i}(y_{k}) = χ_{i}(y_{k}),  γ_{i}(b_{j}) = 0,

1 ≤ k ≤ s, θ̃+ 1 ≤ j ≤ θ.

(b). Let (λ̃_{ij})_{1≤i<j≤θ, i≁j} be an arbitrary linking datum. For each i, 1 ≤ i ≤ θ̃, there exists a unique (ε, γ_{i})–derivation δ_{i} : B → 𝕜 such that

(5.21)  δ_{i}(y_{k}) = 0,  δ_{i}(b_{j}) = λ̃_{ij},

1 ≤ k ≤ s, θ̃+ 1 ≤ j ≤ θ.

(c). There exists a unique Hopf algebra map φ : U → (B^{∗})^{cop} such that

(5.22)  φ(z_{i}) = γ_{i},  φ(u_{i}) = δ_{i},

1 ≤ i ≤ θ̃.

Proof. (a). We have to show that γ_{i} preserves the relations (5.8), (1.4), (1.5), (1.6), (5.9). This is clear for (5.8), (1.4), (1.5), (5.9). We check (1.6): let θ̃+ 1 ≤ j, h ≤ θ such that λ_{jh} ≠ 0. Then

χ_{i}(g_{j}g_{h}) = χ_{j}χ_{h}(g_{i})^{−1} = 1,

by (5.2) and (5.4). So that relations (1.6) hold and (a) is proven.

(b). This is equivalent to: there exists an algebra map T : B → M_{2}(k) such that

(5.23)  T (y_{k}) = (\begin{matrix}1 & 0 \\ 0 & γ_{i}(y_{k}) \end{matrix}) ,  T (b_{j}) = (\begin{matrix}0 & λ̃_{ij} \\ 0 & 0 \end{matrix}) ,

1 ≤ k ≤ s, θ̃+1 ≤ j ≤ θ. Then T is of the form T (a) = (\begin{matrix}ϵ(a) & δ_{i}(a) \\ 0 & γ_{i}(a) \end{matrix}) and δ_{i} is the desired derivation.

So, we need to show that the relations (5.8), (1.4), (1.5), (1.6), (5.9) hold for the matrices in (5.23). This is evident for (5.8). For (1.4) it amounts to λ̃_{ij}χ_{i}(y_{k})^{−1} = λ̃_{ij}χ_{j}(y_{k}), which follows from (5.4) when λ̃_{ij} ≠ 0. For (1.5) and (5.9) the argument is clear. Finally, the left hand side of (1.6) for j < h, is 0, whereas the right-hand side also vanishes since λ_{jh}γ_{i}(g_{j})γ_{i}(g_{h}) = λ_{jh}χ_{j}(g_{i})^{−1}χ_{h}(g_{i})^{−1} = λ_{jh} by (5.4) again.

(c). It is enough to verify that δ_{i}, γ_{i} satisfy the defining relations (5.8), (1.4), (1.5), (1.6), (5.9) for U. Indeed, this will automatically imply that φ is a Hopf algebra map. Note that (1.6) are empty since the Dynkin diagram of U is connected. For (5.8), it is enough to verify that the equalities hold when applied to the generators b_{θ̃+1}, …, b_{θ}, y_{1}, …, y_{s} since both sides are algebra maps. This is now not difficult; for instance (γ_{m}γ_{h})(b_{j}) = γ_{m}(g_{j})γ_{h}(b_{j}) + γ_{m}(b_{j}) = 0 = (γ_{h}γ_{m})(b_{j}). The first relations in (5.8) for Υ hold since ord γ_{i} divides ord Z_{i} for all i. For (1.4) we need again to verify only on generators, since both sides are skew-derivations; this verification is in turn straightforward. The left-hand side of the Serre relations (1.5) is a skew-derivation by [AS2, Lemma A.1]; again we are reduced to see that (ad δ_{i})^{1−a_{ij}}δ_{j}(b_{h}) = 0 = (ad δ_{i})^{1−a_{ij}}δ_{j}(y_{t}), θ̃ + 1 ≤ h ≤ θ, 1 ≤ t ≤ s. Write (ad δ_{i})^{1−a_{ij}}δ_{j} = δ_{i}(ad δ_{i})^{−a_{ij}}δ_{j} − q̂((ad δ_{i})^{−a_{ij}}δ_{j}) δ_{i}, where q̂ is a root of 1. Then (ad δ_{i})^{1−a_{ij}}δ_{j}(y_{t}) = δ_{i}(y_{t})(ad δ_{i})^{−a_{ij}}δ_{j}(y_{t}) − q̂(ad δ_{i})^{−a_{ij}}δ_{j}(y_{t})δ_{i}(y_{t}) = 0. Similarly,

(ad δ_{i})^{1−a_{ij}}δ_{j}(b_{h}) = δ_{i}(b_{h})(ad δ_{i})^{−a_{ij}}δ_{j}(1) − q̂(ad δ_{i})^{−a_{ij}}δ_{j}(g_{h})δ_{i}(b_{h}) = 0,

since (ad δ_{i})^{−a_{ij}}δ_{j} is a homogeneous polynomial in δ_{i}, δ_{j} of positive degree. Finally, relations (5.9) follow from the next Lemma. □

Lemma 5.24. Let B be a finite dimensional pointed Hopf algebra generated as an algebra by grouplike elements and a family b_{j}, j ∈ J, of (h_{j}, 1)-primitives, for some h_{j} ∈ G(B). Let Û be the algebra presented by generators u_{1}, …, uθ̃ and z_{1}, …, zθ̃ with exactly the same relations as for U except for (5.9); it is a Hopf algebra via (1.9). Let N = N_{J}. Assume there exists a Hopf algebra map ϕ : Û → (B^{∗})^{cop} such that γ_{i} := ϕ(z_{i}) and δ_{i} := ϕ(u_{i}) satisfy

(5.25)  γ_{i}(b_{j}) = 0,  δ_{i}(g) = 0,  j ∈ J , g ∈ G(B),

for all 1 ≤ i ≤ θ̃. Then ϕ(u_{α}^{N}) = 0 for all α ∈ Φ_{J}^{+}.

Proof. There exists a Hopf algebra projection ϖ : Û → 𝕜Υ such that ϖ(u_{i}) = 0 and ϖ(z_{i}) = z_{i} for all i. Let K be the subalgebra of Û generated by u_{α}^{N}, α ∈ Φ_{J}^{+}, and z_{i}^{N}, 1 ≤ i ≤ θ̃. We claim that

(5.26)  ϕ(u) = ϕ(ϖ(u))

for all u ∈ K. Clearly, this implies the Lemma. By Theorem 4.8, we know that K is a Hopf subalgebra of Û. We have to prove that ϕ(u)(b) = ϕ(ϖ(u))(b) for b a monomial in the group-likes of B and the b_{j}’s. We do this by induction on the length of the monomial.

18 NICOLÁS ANDRUSKIEWITSCH AND HANS-JÜRGEN SCHNEIDER

We first check the case of length 1. Here we show more generally that

ϕ(u)(g) = ϕ(ϖ(u))(g) and ϕ(u)(b_{j}) = ϕ(ϖ(u))(b_{j})

for all g ∈ G(B), j ∈ J and u ∈ G(Û) or u of the form u_{i_{1}} … u_{i_{t}}z, with z group-like, 1 ≤ i_{1}, …, i_{t} ≤ θ̃, and t ≥ 2. Note that each element in K is a linear combination of such u’s since N ≥ 2. The case when u is a group-like is clear. Let u = u_{i_{1}} … u_{i_{t}}z, with z group-like, 1 ≤ i_{1}, …, i_{t} ≤ θ̃, and t ≥ 2. Then ϕ(u) = δ_{i_{1}} … δ_{i_{t}}ϕ(z), and

ϕ(u)(g) = δ_{i_{1}}(g) … δ_{i_{t}}(g)ϕ(z)(g) = 0,
ϕ(u)(b_{j}) = ∑_{1≤r≤t+1} δ_{i_{1}}(g_{j}) … δ_{i_{r−1}}(g_{j})δ_{i_{r}}(b_{j})δ_{i_{r+1}}(1) … ϕ(z)(1) = 0,

where we used (5.25) and t ≥ 2.

Assume then that b = cd where c and d are monomials satisfying the claim. Since ϕ and ϖ are Hopf algebra maps, we have ϕ(u)(cd) = ϕ(u_{(2)})(c)ϕ(u_{(1)})(d) = ϕ(ϖ(u_{(2)}))(c)ϕ(ϖ(u_{(1)}))(d) = ϕ(ϖ(u))(cd). □

We are ready now to conclude the proof of the Theorem. Consider the cocycle σ : (U ⊗ B) ⊗ (U ⊗ B) → 𝕜 obtained as in Lemma 5.14 from the map φ constructed in Lemma 5.19. Consider the Hopf algebra (U ⊗B)_{σ}; it has dimension |Υ||Γ| ∏_{I∈X} N_{I}^{|Φ_{I}^{+}|}. We claim that the group-like elements z_{i} ⊗g_{i}^{−1} are central in (U ⊗ B)_{σ} for all i. By definition of (U ⊗ B)_{σ}, we have to show for all u ∈ U, b ∈ B and 1 ≤ i ≤ θ̃

(5.27)  φ(z_{i})(b_{(1)})uz_{i} ⊗ b_{(2)}g_{i}^{−1}φ(z_{i}^{−1})(b_{(3)}) = φ(u_{(1)})(g_{i}^{−1})z_{i}u_{(2)} ⊗ g_{i}^{−1}bφ(Su_{(3)})(g_{i}^{−1}).

Since u⊗b = (u⊗1)(1⊗b) in (U ⊗B)_{σ} for all u ∈ U, b ∈ B, it is enough to check (5.27) on generators of U and B. This in turn follows easily from the definitions.

Let Ã be the quotient of (U ⊗ B)_{σ} by the central Hopf subalgebra 𝕜[z_{i} ⊗ g_{i}^{−1} : 1 ≤ i ≤ θ̃] with quotient map π. Then dim Ã = |Γ| ∏_{I∈X} N_{I}^{|Φ_{I}^{+}|} by a result of the second author [Mo, Th. 3.3.1]. Next we claim the existence of a surjective algebra map F : A → Ã such that

F(a_{i}) = π(u_{i} ⊗ 1), F(a_{j}) = π(1 ⊗ b_{j}), F(y_{k}) = π(1 ⊗ y_{k}),

for 1 ≤ i ≤ θ̃, θ̃ + 1 ≤ j ≤ θ, 1 ≤ k ≤ s. Again we have to verify the relations (5.8), (1.4), (1.5), (1.6), (5.9). Up to (1.6) these relations already hold in (U ⊗ B)_{σ}. For (1.6), it is enough to show that π(u_{i} ⊗ 1)π(1 ⊗ b_{j}) − χ_{j}(g_{i})π(1 ⊗ b_{j})π(u_{i} ⊗ 1) = λ_{ij} (1 − π(1 ⊗ g_{i}g_{j})) , 1 ≤ i ≤ θ̃, θ̃+ 1 ≤ j ≤ θ. A tedious computation shows that the left-hand side is equal to χ_{j}(g_{i})λ̃_{ij} (π(z_{i} ⊗ g_{j}) − 1). Since π(z_{i}⊗g_{i}^{−1}) = 1, we have π(z_{i}⊗g_{j}) = π(1⊗g_{i}g_{j}). Hence the claim follows if we choose λ̃_{ij} = −χ_{i}(g_{j})λ_{ij} for all 1 ≤ i ≤ θ̃, θ̃+ 1 ≤ j ≤ θ.

On the other hand, we have algebra maps G_{1} : U → A, G_{2} : B → A given by G_{1}(u_{i}) = a_{i}, G_{1}(z_{i}) = g_{i}, G_{2}(b_{j}) = a_{j} G_{2}(y_{k}) = y_{k}, 1 ≤ i ≤ θ̃, θ̃ + 1 ≤ j ≤ θ, 1 ≤ k ≤ s. Here we use that ord g_{i} divides ord Z_{i} for all i. Let G : U ⊗ B → A be defined by G(u ⊗ b) = G_{1}(u)G_{2}(b)

FINITE QUANTUM GROUPS OVER ABELIAN GROUPS OF PRIME EXPONENT 19

for all u ∈ U, b ∈ B. We claim that G is an algebra map. By Lemma 5.15, we have to verify φ(u_{(1)})(b_{(1)})G_{1}(u_{(2)})G_{2}(b_{(2)}) = G_{2}(b_{(1)})G_{1}(u_{(1)})φ(u_{(2)})(b_{(2)}), for all generators. This is a straightforward task; for the case u_{i} and b_{j} we need again the condition λ̃_{ij} = −χ_{i}(g_{j})λ_{ij}.

Since clearly G factorizes through Ã, F is an isomorphism and the Theorem follows. □

Remark 5.28. Let us consider a Hopf algebra defined as in Definition 5.7 but replacing (5.9) by (1.6) only for simple α, and demanding (5.9) for non-simple α. Then it is possible to prove the analogue of Theorem 5.17. This gives in particular a new proof of [AS1, Proposition 5.2] and generalizes [AS3, Theorem 3.6]. Indeed, the connected components of the Dynkin diagrams in loc. cit. are of type A_{1}. This result was also found independently by A. Masuoka [Ma].

6. Lifting of relations

In this Section, we assume the situation described in Section 4. To lift the Serre relations, we need the following Lemma. Lemma 6.1. Let 1 ≤ i ≠ j ≤ θ and let I be the connected component containing i.

(a). If i ∼ j, assume that N_{I} ≠ 3; if i ∼ j and I is of type B_{n}, C_{n} or F_{4} assume further N_{I} ≠ 5. Then there exists no ℓ, 1 ≤ ℓ ≤ θ, such that g_{i}^{1−a_{ij}}g_{j} = g_{ℓ}, χ_{i}^{1−a_{ij}}χ_{j} = χ_{ℓ}.

(b). Assume that i ∼ j and N_{I} ≠ 3. If I is of type B_{n}, C_{n} or F_{4}, resp. G_{2}, assume further that N_{I} ≠ 5, resp. N_{I} ≠ 7. Then χ_{i}^{1−a_{ij}}χ_{j} ≠ ε.

Proof. (a). Assume that g_{i}^{1−a_{ij}}g_{j} = g_{ℓ}, χ_{i}^{1−a_{ij}}χ_{j} = χ_{ℓ} for some ℓ. Substituting g_{ℓ} and χ_{ℓ} in ⟨χ_{ℓ}, g_{i}⟩⟨χ_{i}, g_{ℓ}⟩ = q_{i}^{a_{iℓ}} and using ⟨χ_{j}, g_{i}⟩⟨χ_{i}, g_{j}⟩ = q_{i}^{a_{ij}} we conclude that

(6.2)  N_{i} divides 2 − a_{ij} − a_{iℓ}.

Changing the rôles of i and j we obtain in the same way

(6.3)  N_{j} divides a_{ji}(1 − a_{ij}) − a_{jℓ} + 2.

First assume that i ≁ j. In particular, a_{ji} = 0 = a_{ij} and a_{iℓ} = 0 or a_{jℓ} = 0. If a_{iℓ} = 0, resp. a_{jℓ} = 0, then we get from (6.2), resp. (6.3), that N_{i} = 2, resp. N_{j} = 2, which is not possible.

Next assume that i ∼ j. If j = ℓ then N_{i} divides 2(1 − a_{ij}) by (6.2). The only possibility is a_{ij} = −2 and N_{i} = 3; but this was excluded in the hypothesis. If i = ℓ then N_{i} divides −a_{ij} by (6.2) and N_{j} divides a_{ji}a_{ij} − 2 by (6.3); but this contradicts our general assumptions on the N_{i}’s.

Finally, if i ≠ ℓ and j ≠ ℓ then a_{iℓ} ≠ −3 and a_{jℓ} ≠ −3. We discuss the different possible values of a_{ij}. If a_{ij} = 0 or −1, by (6.2) and since N_{i} is odd we see that N_{i} = 3 or 5, cases excluded by hypothesis. If a_{ij} = −2 then a_{ji} = −1. By (6.3), N_{j} divides −1 − a_{jℓ}; this discards everything except a_{jℓ} = −1. But in this last case, a_{iℓ} = 0 and N_{i} divides 4 by (6.2), a contradiction. Finally, a_{ij} = −3 is impossible by analogous arguments.

(b). Assume that χ_{i}^{1−a_{ij}}χ_{j} = ε. We consider first the case a_{ij} ≠ 0. Evaluating at g_{i}, we get q_{i}^{1−a_{ij}}χ_{j}(g_{i}) = 1; hence q_{i} = χ_{i}(g_{j}). Evaluating at g_{j}, we get then q_{j} = q_{i}^{a_{ij}−1}. Since q_{j}^{a_{ji}} = q_{i}^{a_{ij}} we

20 NICOLÁS ANDRUSKIEWITSCH AND HANS-JÜRGEN SCHNEIDER

finally obtain

(6.4)  N_{i} divides a_{ij}a_{ji} − a_{ij} − a_{ji}.

The possible values of a_{ij}a_{ji} − a_{ij} − a_{ji} are 3, 5 or 7, where 5, resp. 7, is only possible if I is of type B_{n}, C_{n} or F_{4}, resp. G_{2}. This contradicts the hypothesis.

We consider finally the case a_{ij} = 0; so that χ_{i}χ_{j} = ε. Since I is connected, there is a sequence i = i_{1}, i_{2}, …, i_{t} = j of elements in I such that a_{i_{ℓ}i_{ℓ+1}} ≠ 0, 2 for all ℓ, 1 ≤ ℓ < t. Then

q_{i}^{a_{i_{1}i_{2}}a_{i_{2}i_{3}}...a_{i_{t−1}i_{t}}} = q_{i_{2}}^{a_{i_{2}i_{1}}a_{i_{2}i_{3}}...a_{i_{t−1}i_{t}}} = ⋯ = q_{j}^{a_{i_{2}i_{1}}a_{i_{3}i_{2}}...a_{i_{t}i_{t−1}}},

by substituting q_{i}^{a_{i_{1}i_{2}}} = q_{i_{2}}^{a_{i_{2}i_{1}}}, then q_{i_{2}}^{a_{i_{2}i_{1}}} = q_{i_{3}}^{a_{i_{3}i_{2}}} and so on. Note that q_{i} = q_{j}^{−1} since a_{ij} = 0 and χ_{i}χ_{j} = ε. Hence

(6.5)  N_{i} divides a_{i_{1}i_{2}}a_{i_{2}i_{3}} … a_{i_{t−1}i_{t}} + a_{i_{2}i_{1}}a_{i_{3}i_{2}} … a_{i_{t}i_{t−1}}.

The possible values of the sum in (6.5) are ±2 or ±3. Hence (6.5) contradicts our assumptions in (b). □

Let now A be a pointed Hopf algebra with G(A) ≃ Γ, not necessarily finite dimensional. Let R be the diagram of A (see Section 2.1). We assume there is an isomorphism P(R) ≅ V in _{Γ}^{Γ}YD. Then

⊕\begin{matrix}g,h∈Γ \\ ε≠χ∈Γ̂ \end{matrix}P_{g,h}(A)^{χ} ⟶^{≅} A_{1}/A_{0} ⟵^{≅} V #𝕜Γ

(see [AS1, Lemma 5.4]). Let a_{i} ∈ P_{g_{i},1}(A)^{χ_{i}}, 1 ≤ i ≤ θ, such that \overline{a_{i}} is mapped onto x_{i} for all i. Then we know from [AS2, Lemma 5.4] that for all g ∈ Γ, χ ∈ Γ̂ with χ ≠ ε:

(6.6)  P_{g,1}(A)^{χ} ≠ 0 ⟺ there is some 1 ≤ ℓ ≤ θ : g = g_{ℓ}, χ = χ_{ℓ};
(6.7)  P_{g,1}(A)^{ε} = 𝕜(1 − g).

Recall that we have fixed Γ, (a_{ij})_{1≤i,j≤θ}, (g_{i})_{1≤i≤θ}, (χ_{j})_{1≤j≤θ} as in the situation of Section 4. Theorem 6.8. Let A and a_{1}, …, a_{θ} be as above.

(a). There is a linking datum D (Γ, (a_{ij})_{1≤i,j≤θ}, (g_{i})_{1≤i≤θ}, (χ_{j})_{1≤j≤θ}, (λ_{ij})_{1≤i<j≤θ, i≁j}) with linking elements (λ_{ij})_{1≤i<j≤θ, i≁j} such that (1.6) holds.

(b). Let I ∈ X. Assume that N_{I} ≠ 3. If I is of type B_{n}, C_{n} or F_{4}, resp. G_{2}, assume further that N_{I} ≠ 5, resp. N_{I} ≠ 7. Then the quantum Serre relations (1.5) hold for all i ≠ j ∈ I.

Proof. It is known that (ad a_{i})^{1−a_{ij}}a_{j} ∈ P_{g_{i}^{1−a_{ij}} g_{j},1}(A)^{χ_{i}^{1−a_{ij}} χ_{j}}, see for instance [AS2, Appendix]. Part (b) of the Theorem then follows from Lemma 6.1, (6.6) and (6.7).

To prove part (a), let us assume that i ≁ j. By Lemma 6.1, (6.6) and (6.7) again, a_{i}a_{j}−χ_{j}(g_{i})a_{j}a_{i} = λ_{ij}(1 − g_{i}g_{j}), for some λ_{ij} ∈ 𝕜. We can choose λ_{ij} = 0 when g_{i}g_{j} = 1 or else if χ_{i}χ_{j} ≠ ε. That is, λ_{ij} is a linking datum for (a_{ij}), g_{1}, …, g_{θ} and χ_{1}, …, χ_{θ}; and (1.6) hold. □

Lemma 6.9. Let A and a_{1}, …, a_{θ} be as above. Assume further that • the hypothesis from Theorem 6.8 part (b) holds for all I ∈ X.

FINITE QUANTUM GROUPS OVER ABELIAN GROUPS OF PRIME EXPONENT 21

• A is finite dimensional.

• g_{i}^{N_{i}} = 1, 1 ≤ i ≤ θ. Then the relations (5.9) hold in A.

Proof. Let us fix I ∈ X. Let Û be the algebra presented by generators â_{i}, i ∈ I, y_{1}, …, y_{s} and relations (5.8), (1.4), (1.5) and (1.6); it is a Hopf algebra via (1.9). Let N = N_{I} and let K be the subalgebra of Û generated by â_{α}^{N}, α ∈ Φ_{I}^{+}, and g_{i}^{N}, i ∈ I. By Theorem 4.8, we know that K is a Hopf subalgebra of Û. Note that K is a graded Hopf algebra with trivial coradical. By the choice of the a_{i}’s in A and Theorem 6.8, we see there is a well-defined Hopf algebra map Û → A such that â_{i} ↦ a_{i}, i ∈ I. The image of K under this map is a finite dimensional pointed Hopf algebra; it has a trivial coradical by [Mo] and therefore it is trivial. This implies the Lemma. □

Theorem 6.10. Let A be as above and assume that • the hypothesis from Theorem 6.8 part (b) holds for all I ∈ X.

• gr A ≃ 𝔅(V)#𝕜Γ, hence A is finite dimensional.

• g_{i}^{N_{i}} = 1, 1 ≤ i ≤ θ. Then there exists a linking datum D = D (Γ, (a_{ij})_{1≤i,j≤θ}, (g_{i})_{1≤i≤θ}, (χ_{j})_{1≤j≤θ}, (λ_{ij})_{1≤i<j≤θ, i≁j}) such that

A ≃ 𝔲(D).

Proof. By Theorem 6.8 and Lemma 6.9, there exist elements (λ_{ij})_{1≤i<j≤θ, i≁j} such that

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

is a linking datum, and a surjective Hopf algebra map J : A → A, where A = 𝔲(D). But dim A = dim gr A = |Γ| dim 𝔅(V) = dim A by Theorems 4.5 and 5.17; hence J is an isomorphism. □

7. Hopf algebras generated in degree one

In this Section, Γ is a finite abelian group. Let V ∈_{Γ}^{Γ} YD be of Cartan type with braiding (b_{ij}) and Cartan matrix (a_{ij}). As in [AS2] we say that the braiding (b_{ij}) satisfies the relative primeness condition if for all i, j, (a_{ij}) is 0 or relatively prime to the order of b_{ii}.

The next lemma follows from [AS2]. We will apply it in the case of 2 × 2 (hence symmetrizable) Cartan matrices. Lemma 7.1. Let V ∈_{Γ}^{Γ} YD be of symmetrizable Cartan type with braiding (b_{ij})_{1≤i,j≤θ}. Assume that for all 1 ≤ i, j ≤ θ, the order of b_{ij} is odd, and that (b_{ij})_{1≤i,j≤θ} satisfies the relative primeness condition. If 𝔅(V) is finite-dimensional, then V is of finite Cartan type.

Proof. By [AS2, Lemma 4.1] we can realize the braiding over a suitable finite abelian group Γ̃ and twist with a 2-cocycle F such that the resulting braiding (b_{ij}^{F}) is symmetric with elements of odd order, has the same diagonal elements and is of Cartan type with the same Cartan matrix (a_{ij}) as V. We can now conclude from [AS2, Lemma 4.3] that (b_{ij}^{F}) is of FL-type (see [AS2]). Let Ṽ^{F} be the Yetter-Drinfeld module over Γ̃ with braiding (b_{ij}^{F}). Since 𝔅(V) and 𝔅(Ṽ^{F}) have the same dimension, 𝔅(Ṽ^{F}) is finite-dimensional. Then (a_{ij}) is of finite Cartan type by [AS2, Theorem 3.1]. □

Lemma 7.2. Let S = ⊕_{n≥0}S(n) be a finite-dimensional graded Hopf algebra in _{Γ}^{Γ}YD such that S(0) = 𝕜1. Assume that S(1) is of finite Cartan type with basis (x_{i})_{1≤i≤θ}, braiding (b_{ij})_{1≤i,j≤θ} and Cartan matrix (a_{ij})_{1≤i,j≤θ} as in (3.7) . For all 1 ≤ l ≤ θ, let q_{l} = b_{ll} and N_{l} = ord(q_{l}).

Let 1 ≤ i, j ≤ θ, i ≠ j, and assume that N_{i}, N_{j} and ord(b_{ij}) are odd, and N_{i} is not divisible by 3 and > 7.

(a) Assume i ∼ j and let I be the connected component containing i, j. If the type of I is B_{n}, C_{n} or F_{4}, assume that N_{i} is not divisible by 5. If the type is G_{2}, assume that N_{i} is not divisible by 5 or 7. Then (ad_{c}x_{i})^{1−a_{ij}}x_{j} = 0.

(b) Assume i ≁ j and q_{i}q_{j} = 1 or ord(q_{i}q_{j}) = ord(q_{i}). Then x_{i}x_{j} − b_{ij}x_{j}x_{i} = 0.

Proof. Define z_{1} := x_{i}, z_{2} := (ad_{c}x_{i})^{1−a_{ij}}x_{j}. In both cases we have to show z_{2} = 0. We assume that z_{2} is not 0. Let g_{i} ∈ Γ, χ_{i} ∈ Γ̂, 1 ≤ i, j ≤ θ, with b_{ij} = χ_{j}(g_{i}) for all i, j. Then action and coaction on z_{1}, z_{2} are given by δ(z_{1}) = g_{i} ⊗ z_{1}, δ(z_{2}) = g_{i}^{1−a_{ij}}g_{j} ⊗ z_{2} and h · z_{1} = χ_{i}(h)z_{1}, h · z_{2} = (χ_{i}^{1−a_{ij}}χ_{j})(h)z_{2} for all h ∈ Γ. The elements z_{1}, z_{2} are linearly independent since they are non-zero and of different degree. The braiding (B_{kl})_{1≤k,l≤2} of the 2-dimensional Yetter-Drinfeld module with basis z_{1}, z_{2} is given by

B_{11} = χ_{i}(g_{i}) = q_{i},  B_{12} = (χ_{i}^{1−a_{ij}}χ_{j})(g_{i}) = q_{i}b_{ji}^{−1},
B_{21} = χ_{i}(g_{i}^{1−a_{ij}}g_{j}) = q_{i}^{1−a_{ij}}b_{ji},  B_{22} = (χ_{i}^{1−a_{ij}}χ_{j})(g_{i}^{1−a_{ij}}g_{j}) = q_{i}^{1−a_{ij}}q_{j}.

Then B_{12}B_{21} = q_{i}^{2−a_{ij}}. We claim that (B_{kl}) is of Cartan type and satisfies the relative primeness condition, that is there are integers A_{12}, A_{21} such that

(7.3)  q_{i}^{2−a_{ij}} = q_{i}^{A_{12}} ,and A_{12} is relatively prime to N_{i},
(7.4)  q_{i}^{2−a_{ij}} = (q_{i}^{1−a_{ij}}q_{j})^{A_{21}} , and A_{21} is relatively prime to ord(q_{i}^{1−a_{ij}}q_{j}).

In both cases 2 − a_{ij} is relatively prime to N_{i}, because of the hypothesis on N_{i}. This shows (7.3).

We now prove (7.4) in case (a). Then N_{I} = N_{i} = N_{j}, and it suffices to find an integer A_{21} relatively prime to N_{i} with q_{i}^{2−a_{ij}} = (q_{i}^{1−a_{ij}}q_{j})^{A_{21}}.

First assume that a_{ij} ≠ 0. Since a_{ji} ≠ 0 is relatively prime to N_{i}, it is enough to consider the a_{ji}-th power of (7.4). Since q_{i}^{a_{ij}} = q_{j}^{a_{ji}} by the Cartan condition for (b_{ij}), we have to solve (2 − a_{ij})a_{ji} ≡ ((1 − a_{ij})a_{ji} + a_{ij})A_{21} mod N_{i}. Since (a_{ij}) is of finite Cartan type, the possible values of (2 − a_{ij})a_{ji} are -3, -4, -5, -6, -9 (-4, -6 resp. -5, -9 only occur if the type is B_{n}, C_{n} or F_{4} resp. G_{2}); the possible values of ((1 − a_{ij})a_{ji} + a_{ij}) are -3, -5, -7, (-5, resp. -7 only occur if the type is B_{n}, C_{n} or F_{4} resp. G_{2}). Hence (2 − a_{ij})a_{ji} and ((1 − a_{ij})a_{ji} + a_{ij}) are relatively prime to N_{i} by assumption, and the claim follows. (Note that a + b is never 0).

FINITE QUANTUM GROUPS OVER ABELIAN GROUPS OF PRIME EXPONENT 23

If a_{ij} = 0, then by connectedness there is a sequence i = i_{1}, i_{2}, …, i_{t} = j of elements in I such that a_{i_{ℓ}i_{ℓ+1}} ≠ 0, 2 for all ℓ, 1 ≤ ℓ < t. Then as in the proof of Lemma 6.1 (b),

q_{i}^{a} = q_{j}^{b}, where a = a_{i_{1}i_{2}}a_{i_{2}i_{3}} … a_{i_{t−1}i_{t}}, and b = a_{i_{2}i_{1}}a_{i_{2}i_{3}} … a_{i_{t−1}i_{t}}.

Since the possible values of a, b are 1, 2, -1, -2, the b-th power of (7.4) leads to the congruence 2b ≡ (b + a)A_{21} mod N_{i}, and the claim again follows.

Assume case (b), in particular a_{ij} = 0. If q_{i}q_{j} = 1, we get a contradiction since the algebra generated by z_{1}, z_{2} is finite-dimensional, hence B_{22} ≠ 1 by [AS1, Lemma 3.1]. If ord(q_{i}q_{j}) = ord(q_{i}), (7.4) is solvable since N_{i} is odd.

Thus we have shown that (B_{kl}) is of Cartan type and satisfies the relative primeness condition. Hence (B_{kl}) is of finite Cartan type by Lemma 7.1. In both cases A_{12} = 2 − a_{ij} − N_{i} is a solution of (7.3), and −N_{i} < A_{12} ≤ 0. Hence the possible values of A_{12} are 0, -1, -2, -3, and we see that N_{i} ≤ 8. This contradicts our assumption, and we have shown the Serre relation z_{2} = 0. □

Lemma 7.5. Let S = ⊕_{n≥0}S(n) be a finite-dimensional graded Hopf algebra in _{Γ}^{Γ}YD such that S(0) = 𝕜1. Assume that V = S(1) is of Cartan type with basis (x_{i})_{1≤i,j≤θ} as described in the beginning of Section 4. Assume the Serre relations

(ad_{c}x_{i})^{1−a_{ij}}x_{j} = 0 for all 1 ≤ i, j ≤ θ, i ≠ j and i ∼ j.

Then the root vector relations

x_{α}^{N_{I}} = 0, for all α ∈ Φ_{I}^{+}, I ∈ X ,

hold in S.

Proof. We fix a connected component I ∈ X. Let V_{I} be the Yetter-Drinfeld submodule of V with basis x_{i}, i ∈ I, and 𝔅̂(V_{I}) the quotient of T (V_{I}) modulo the Serre relations of all elements x_{i}, x_{j} with i ≠ j in I. Let N_{I} = N. The map Ψ : T (V_{I}) ⊂ T (V) → S factorizes over 𝔅̂(V_{I}), since the Serre relations hold in S. By Theorem 4.8 the subalgebra K(V_{I}) of 𝔅̂(V) generated by the powers of the root vectors x_{α}^{N_{I}}, α ∈ Φ_{I}^{+}, is a braided Hopf subalgebra. As a coalgebra, K(V_{I}) is pointed and has trivial coradical. Hence K := Ψ(K(V_{I})) is a finite-dimensional pointed and graded Hopf subalgebra of S in _{Γ}^{Γ}YD with trivial coradical. We have to show the root vector relation x_{α}^{N_{I}} = 0, α ∈ Φ_{I}^{+} in S, or equivalently that K is one-dimensional, that is P(K) = 0.

Assume P(K) ≠ 0. Since P(K) is in _{Γ}^{Γ}YD, there are g ∈ Γ, χ ∈ Γ̂ with P(K)_{g}^{χ} ≠ 0. By [AS1, Lemma 3.1], we conclude χ(g) ≠ 1. But this is a contradiction, since for all g ∈ Γ, χ ∈ Γ̂, K_{g}^{χ} ≠ 0 implies χ(g) = 1. For, K is the 𝕜-span of all monomials

Ψ(x_{β_{1}}^{N}) ⋯ Ψ(x_{β_{m}}^{N}), m ≥ 1, β_{1}, …, β_{m} ∈ Φ_{I}^{+}.

For any β ∈ Φ_{I}^{+} there are natural numbers b_{i}^{β}, 1 ≤ i ≤ θ, such that β = ∑_{i=1}^{θ} b_{i}^{β}α_{i}, where α_{1}, …, α_{θ} are the simple roots. By (4.1), x_{β} ∈ T (V_{I})_{g_{β}}^{χ_{β}}, where g_{β} = ∏_{i∈I} g_{i}^{b_{i}^{β}}, χ_{β} = ∏_{i∈I} χ_{i}^{b_{i}^{β}}. Hence for all

24 NICOLÁS ANDRUSKIEWITSCH AND HANS-JÜRGEN SCHNEIDER

β_{1}, …, β_{m} ∈ Φ_{I}^{+},

Ψ(x_{β_{1}}^{N}) ⋯ Ψ(x_{β_{m}}^{N}) ∈ K_{g}^{χ}, where χ = χ_{β_{1}}^{N} ⋯ χ_{β_{m}}^{N}, g = g_{β_{1}}^{N} ⋯ g_{β_{m}}^{N}.

It remains to show that χ(g) = 1.

Let α, β ∈ Φ_{I}^{+}. Since the braiding is of Cartan type,

χ_{α}(g_{β})χ_{β}(g_{α}) = ∏_{i,j} χ_{i}^{b_{i}^{α}}(g_{j}^{b_{j}^{β}}) ∏_{i,j} χ_{j}^{b_{j}^{β}}(g_{i}^{b_{i}^{α}}) = ∏_{i,j}(χ_{i}(g_{j})χ_{j}(g_{i}))^{b_{i}^{α}b_{j}^{β}} = ∏_{i,j} χ_{i}(g_{i})^{a_{ij}b_{i}^{α}b_{j}^{β}},

and

χ_{α}(g_{α}) = ∏_{i} χ_{i}(g_{i})^{b_{i}^{α}b_{i}^{α}} ∏_{i<j}(χ_{i}(g_{j})χ_{j}(g_{i}))^{b_{i}^{α}b_{j}^{α}} = ∏_{i} χ_{i}(g_{i})^{b_{i}^{α}b_{i}^{α}} ∏_{i<j} χ_{i}(g_{i})^{a_{ij}b_{i}^{α}b_{j}^{α}}.

Hence, since all the χ_{i}(g_{i}) have order N,

χ_{α}^{N}(g_{β}^{N})χ_{β}^{N}(g_{α}^{N}) = 1, χ_{α}^{N}(g_{α}^{N}) = 1.

Therefore we obtain

χ(g) = ∏_{i,j} χ_{β_{i}}^{N}(g_{β_{j}}^{N}) = ∏_{i} χ_{β_{i}}^{N}(g_{β_{i}}^{N}) ∏_{i<j} χ_{β_{i}}^{N}(g_{β_{j}}^{N})χ_{β_{j}}^{N}(g_{β_{i}}^{N}) = 1.

□

Theorem 7.6. Let A be a finite-dimensional pointed Hopf algebra with coradical 𝕜Γ, and let R be the diagram of A, that is

gr A ≃ R#𝕜Γ,

and R = ⊕_{n≥0}R(n) is a graded braided Hopf algebra in _{Γ}^{Γ}YD with R(0) = 𝕜1, R(1) = P(R).

Assume that R(1) is a Yetter-Drinfeld module of finite Cartan type with braiding (b_{ij})_{1≤i,j≤θ}. For all i, let q_{i} = b_{ii}, N_{i} = ord(q_{i}). Assume that ord(b_{ij}) is odd and N_{i} is not divisible by 3 and > 7 for all 1 ≤ i, j ≤ θ.

(1) For any 1 ≤ i ≤ θ contained in a connected component of type B_{n}, C_{n} or F_{4} resp. G_{2}, assume that N_{i} is not divisible by 5 resp. by 5 or 7.

(2) For any 1 ≤ i, j ≤ θ and i ≁ j assume q_{i}q_{j} = 1 or ord(q_{i}q_{j}) = N_{i}.

Then R is generated as an algebra by R(1), that is A is generated by skew-primitive and group-like elements.

Proof. Let S := R^{∗} be the dual Hopf algebra of R in the braided sense (see for example [AG, Section 2]). S = ⊕_{n≥0}S(n) is a graded braided Hopf algebra in _{Γ}^{Γ}YD with S(0) = 𝕜1, S(n) = R(n)^{∗}, for all n ≥ 0. By assumption there are h_{i} ∈ Γ, η_{i} ∈ Γ̂, 1 ≤ i, j ≤ θ, with b_{ij} = η_{j}(h_{i}) for all i, j, and a basis (y_{i}) of R(1) with y_{i} ∈ R(1)_{h_{i}}^{η_{i}} for all i. Let (x_{i}) in V := S(1) = R(1)^{∗} be the dual basis of (y_{i}). Then x_{i} ∈ V_{g_{i}}^{χ_{i}} with χ_{i} = η_{i}^{−1}, g_{i} = h_{i}^{−1} and b_{ij} = χ_{j}(g_{i}) = η_{j}(h_{i}) for all 1 ≤ i, j ≤ θ. Thus V is a Yetter-Drinfeld module over Γ with the same braiding as R(1). By [AS2, Lemma 5.5], R is generated by R(1) if and only if S(1) = P(S). Hence by duality, S is generated by S(1), since R(1) = P(R). It

FINITE QUANTUM GROUPS OVER ABELIAN GROUPS OF PRIME EXPONENT 25

is easy to see that V = S(1) ⊂ P(S). Hence there are canonical surjections of graded braided Hopf algebras

T (V) → S → 𝔅(V).

Here T (V) is the tensor algebra, the elements x_{i} are primitive and of degree one, and both maps are the identity on V. The kernel I of the first map is a homogeneous ideal generated by elements of degree ≥ 2, a coideal and stable under the action and coaction of Γ. Since 𝔅(V) = T (V)/J, where J is the largest ideal with the same properties as I, there is a canonical surjection S → 𝔅(V).

The x_{i} satisfy the Serre relations (4.6) by Lemma 7.2, and then the root vector relations (4.7) by Lemma 7.5. Therefore it follows from the description of 𝔅(V) in Theorem 4.5 that S = 𝔅(V). This means S(1) = P(S), hence by duality that R is generated by R(1). □

A special case of the last Theorem together with a main result in [AS2] allows to prove the following Corollary 7.7. Let p > 17 be a prime number. Then any finite-dimensional pointed Hopf algebra with coradical 𝕜(ℤ/(p))^{s} for some natural number s is generated by group-like and skew-primitive elements.

Proof. Let A be a finite-dimensional pointed Hopf algebra with coradical 𝕜(ℤ/(p))^{s} and let R be the diagram of A. Then R(1) is a Yetter-Drinfeld module of finite Cartan type by [AS2, Corollary 1.2]. Hence the claim follows from Theorem 7.6. □

Let us state explicitly another Corollary of the Theorem. Corollary 7.8. Under the hypothesis of Theorem 7.6, if the Dynkin diagram attached to the pointed Hopf algebra is connected, then A is generated by group-like and skew-primitive elements. □

In principle, the idea behind the proof of Theorem 7.6 is as follows. Let A be a finite-dimensional pointed Hopf algebra with coradical 𝕜Γ, Γ any finite group. Let R be the diagram of A, and S := R^{∗} the dual braided Hopf algebra. Consider the diagram R̃ of the bosonization S#𝕜Γ. Then P(S) is naturally embedded in P(R̃) (and this embedding is in fact an isomorphism). Moreover, dim(P(R)) ≤ dim(P(S)) ≤ dim(P(R̃)), and dim(P(R)) = dim(P(R̃)) if and only if S(1) = P(S) or R = 𝔅(P(R)).

Corollary 7.7 can also be seen as a direct consequence of Section 6 and [AS2]: By [AS2] P(R̃) is of finite Cartan type. Then the result follows from Theorem 6.8 and 6.9 applied to A = S#𝕜Γ.

The next theorem is another application of this principle. It shows that only very special dimensions are possible for finite-dimensional pointed Hopf algebras. Theorem 7.9. For any finite group Γ of odd order there is a natural number n(Γ) such that the dimension of any finite-dimensional pointed Hopf algebra with coradical 𝕜Γ is ≤ n(Γ).

Proof. Let A be a finite-dimensional pointed Hopf algebra with coradical 𝕜Γ and diagram R and R̃ as defined above. Since R and R̃ are braided Hopf algebras over Γ of the same dimension, and dim(P(R)) ≤ dim(P(R̃)), we can iterate this process and after finitely many steps we obtain a graded braided Hopf algebra T over Γ with dim(R) = dim(T) and T = 𝔅(P(T)). By a result of Graña [Gñ3] using [AS2, Theorem 3.1] which follows from [L3], the number of isomorphism classes of Yetter-Drinfeld modules V over the fixed group Γ with finite-dimensional 𝔅(V) is finite. Thus we can take for n(Γ) the product of the largest such dimension with the order of Γ. □

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Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, (5000) Ciudad Universitaria, Córdoba, Argentina

  E-mail address: andrus@mate.uncor.edu

Mathematisches Institut, Universität München, Theresienstraße 39, D-80333 München, Germany

  E-mail address: hanssch@rz.mathematik.uni-muenchen.de