A CHARACTERIZATION OF QUANTUM GROUPS

NICOLÁS ANDRUSKIEWITSCH AND HANS-JÜRGEN SCHNEIDER

Abstract. We classify pointed Hopf algebras with finite Gelfand-Kirillov dimension, which are domains, whose groups of group-like elements are abelian, and whose infinitesimal braidings are positive.

Introduction

Since the appearance of quantum groups [KR, Sk, Dr, Ji], there were many attempts to define them intrinsically. Important descriptions of the so-called ”nilpotent” parts were given by Ringel [Ri], Lusztig [L2] and Rosso [Ro]. However, the question of finding an abstract characterization of the quantized enveloping algebras remained open.

In the main Theorem 5.2 of this paper, we classify all Hopf algebras over an algebraically closed field of characteristic 0 which are

• pointed, that is all their simple comodules are one-dimensional, and have an abelian group of group-like elements,

• domains of finite Gelfand-Kirillov dimension, and

• have positive infinitesimal braiding (see Section 1). The first two conditions are natural. The positivity condition should be related to the existence of a real involution.

In Theorem 4.2, we describe these Hopf algebras by generators and relations. They are natural generalizations of quantized enveloping algebras with positive parameter. To prove our main Theorem, we combine the lifting method for pointed Hopf algebras [AS1, AS2, AS4] with a characterization obtained by Rosso of the ”nilpotent part” of a quantized enveloping algebra in terms of finiteness of the Gelfand-Kirillov dimension [Ro].

Among the main differences between the new Hopf algebras and multiparametric quantized enveloping algebras, let us mention that we have one parameter of deformation for each connected component of the Dynkin diagram (this is explained as follows: in the ”classical limit”, one may have different scalar multiples of the Sklyanin brackets in the different connected components) and linking relations, see (4.12), generalizing the classical relations E_{i}F_{j} − F_{j}E_{i} = δ_{ij}(K_{i} − K_{i}^{−1}).

Note that we are not assuming that the Hopf algebras have an a-priori assigned Dynkin diagram as in [W, KW]; it comes from our hypothesis, and here is where we rely on Rosso’s result [Ro].

Version of January 9 , 2002.

This paper is available at www.mate.uncor.edu/andrus/articulos.html and www.mathematik.uni-muenchen.de/∼hanssch/Publications.html. This work was partially supported by AN-PCyT, Agencia Córdoba Ciencia, CONICET, the Graduiertenkolleg of the Math. Institut (Universität München)

and Secyt (UNC).

The article is organized as follows. Section 1 contains preliminary material. In Section 2, we collect some well-known facts about quantum groups, and give details of some proofs when they are not easily available in the literature. Section 3 contains a technical result on the coradical filtration of certain Hopf algebras, generalizing an idea of Takeuchi. In Section 4, we construct a new family of pointed Hopf algebras with generic braiding and establish the main basic properties of them. The approach is similar to [AS4] but instead of dimension arguments, we use the technical results on the coradical filtration obtained in Section 3; these results should be useful also for other classes of Hopf algebras. In Section 5, we prove our main Theorem. A key point is Lemma 5.1, which implies that a wide class of Hopf algebras with finite Gelfand-Kirillov dimension is generated by group-like and skew-primitive elements.

Acknowledgements. We thank N. Reshetikhin for reviving our interest in this question and J. Alev for interesting conversations. This paper was begun during visits to the MSRI, in the framework of the full-year Program on Noncommutative Algebra (August 1999 - May 2000). We thank the organizers for the kind invitation and the MSRI for the excellent working conditions. Part of the work of the first author was done during a visit to the University of Reims (October 2001 - January 2002); he is very grateful to J. Alev for his kind hospitality.

1. Preliminaries

Notation. Let 𝕜 be an algebraically closed field of characteristic 0. Our references are [M], [Sw] for Hopf algebras; [KL], for growth of algebras and Gelfand-Kirillov dimension; and [AS5] for pointed Hopf algebras. We use standard notation for Hopf algebras: Δ, S, ϵ, denote respectively the comultiplication, the antipode, the counit; we use a short version of Sweedler’s notation: Δ(x) = x_{(1)} ⊗ x_{(2)}, x in a coalgebra C.

Let H be a Hopf algebra with bijective antipode. We denote by G(H) be the group of grouplike elements of H; and by P_{g,h}(H) the space of g, h skew-primitive elements x of H, that is with Δ(x) = g ⊗ x + x ⊗ h, where g, h ∈ G(H); then P(H) := P_{1,1}(H). The braided category of Yetter-Drinfeld modules over H is denoted by _{H}^{H}YD, cf. the conventions of [AS5].

The adjoint representation ad of a Hopf algebra 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 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. If x ∈ P(R) then the braided adjoint representation of x is 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, ad_{(b#1)}(d#1) = (ad_{c}b(d))#1.

If Γ is an abelian group, we denote by Γ̂ the group of characters of Γ. If V is a 𝕜Γ-module (resp., 𝕜Γ-comodule), then we denote V^{χ} := {v ∈ V : h.v = χ(h)v, ∀h ∈ Γ}, χ ∈ Γ̂; resp., V_{g} := {v ∈ V : δ(v) = g ⊗ v}, g ∈ G. A Yetter-Drinfeld module over Γ is 𝕜Γ-module V which is also a 𝕜Γ-comodule, and such that each homogeneous component V_{g}, g ∈ Γ, is a 𝕜Γ-submodule. Thus, a vector space V provided with a direct sum decomposition V = ⊕_{g∈G,χ∈Γ̂}V_{g}^{χ} is a Yetter-Drinfeld module over H = 𝕜Γ.

A CHARACTERIZATION OF QUANTUM GROUPS 3

Braided vector spaces. A braided vector space (V, c) is a finite-dimensional vector space provided with an isomorphism c : V ⊗ V → V ⊗ V which is a solution of the braid equation, that is (c ⊗ id )(id ⊗c)(c ⊗ id ) = (id ⊗c)(c ⊗ id )(id ⊗c). Examples of braided vector spaces are Yetter-Drinfeld modules: if V ∈ _{H}^{H}YD, then c : V ⊗ V → V ⊗ V, c(v ⊗ w) = v_{(−1)}.w ⊗ v_{(0)}, is a solution of the braid equation. Definition 1.1. Let (V, c) be a finite-dimensional braided vector space. We shall say that the braiding c : V ⊗ V → V ⊗ V is diagonal if there exists a basis x_{1}, …, x_{θ} of V and non-zero scalars q_{ij} such that c(x_{i} ⊗ x_{j}) = q_{ij}x_{j} ⊗ x_{i}, 1 ≤ i, j ≤ θ. The matrix (q_{ij}) is called the matrix of the braiding.

Furthermore, we shall say that a diagonal braiding with matrix (q_{ij}) is indecomposable if for all i ≠ j, there exists a sequence i = i_{1}, i_{2}, …, i_{t} = j of elements of {1, …, θ} such that q_{i_{s},i_{s+1}}q_{i_{s+1},i_{s}} ≠ 1, 1 ≤ s ≤ t − 1. Otherwise, we say that the matrix is decomposable.

We attach a graph to a diagonal braiding in the following way. The vertices of the graph are the elements of {1, …, θ}, and there is an edge between i and j if they are different and q_{ij}q_{ji} ≠ 1. Thus, “indecomposable” means that the corresponding graph is connected. The components of the matrix are the principal submatrices corresponding to the connected components of the graph. If i and j are vertices in the same connected component, then we write i ∼ j. We shall denote by X the set of connected components of the matrix (q_{ij}). If I ∈ X, then V_{I} denotes the subspace of V spanned by x_{i}, i ∈ I.

We shall say that a braiding c is generic if it is diagonal with matrix (q_{ij}) where q_{ii} is not a root of 1, for any i.

Let 𝕜 = ℂ. We shall say that a braiding c is positive if it is generic with matrix (q_{ij}) where q_{ii} is a positive real number, for all i.

We shall say that a diagonal braiding c with matrix (q_{ij}) is of Cartan type if q_{ii} ≠ 1 for all i, and there are integers a_{ij} with a_{ii} = 2, 1 ≤ i ≤ θ, and 0 ≤ −a_{ij} < ord q_{ii} (which could be infinite), 1 ≤ i ≠ j ≤ θ, such that q_{ij}q_{ji} = q_{ii}^{a_{ij}} for all i and j. Since clearly a_{ij} = 0 implies that a_{ji} = 0 for all i ≠ j, (a_{ij}) is a generalized Cartan matrix. This generalizes the definition in [AS2, p. 4]. In this case, the braiding is indecomposable if and only if the corresponding Cartan matrix is indecomposable. We shall also denote by X the set of connected components of the Dynkin diagram corresponding to the matrix (a_{ij}); clearly, this agrees with the previous convention.

Let (V, c) be a braided vector space of Cartan type with generalized Cartan matrix (a_{ij}). We say that (V, c) is of DJ-type (or Drinfeld-Jimbo type) if there exist positive integers d_{1}, …, d_{θ} such that

For all i, j, d_{i}a_{ij} = d_{j}a_{ji} (thus (a_{ij}) is symmetrizable). (1.1)
For all I ∈ X , there exists q_{I} ∈ 𝕜, which is not a root of unity, such that q_{ij} = q_{I}^{d_{i}a_{ij}} for all i, j ∈ I. (1.2)

In particular, q_{ij} = 1 if i ≁ j.

Nichols algebras. Let V ∈ _{H}^{H}YD. A braided graded Hopf algebra R = ⊕_{n≥0}R(n) in _{H}^{H}YD is called a Nichols algebra of V if 𝕜 ≃ R(0) and V ≃ R(1) in _{H}^{H}YD, and

(1) P (R) = R(1),

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

The Nichols algebra of V exists and is unique up to isomorphisms; it will be denoted by 𝔅(V). It depends, as an algebra and coalgebra, only on the underlying braided vector space (V, c). The underlying algebra is called a quantum symmetric algebra in [Ro]. We shall identify V with the subspace of homogeneous elements of degree one in 𝔅(V). See [AS5] for more details and some historical references.

Given a braided vector space of any of the types in Definition 1.1, we will say that its Nichols algebra is of the same type.

Lemma 1.2. [AS2, Lemma 4.2]. Let V be a finite-dimensional Yetter-Drinfeld module over an ablian group. Let X = {I_{1}, …, I_{N}} be a numeration of the set of connected components. Then 𝔅(V) ≃ 𝔅(V_{I_{1}}) \underline{⊗} … \underline{⊗} 𝔅(V_{I_{N}}) as braided Hopf algebras with the braided tensor product algebra structure \underline{⊗} . □

Lifting method for pointed Hopf algebras. Recall that a Hopf algebra A is pointed if any irreducible A-comodule is one-dimensional. That is, if the coradical A_{0} equals the group algebra 𝕜G(A).

Let A be a pointed Hopf algebra let A_{0} = 𝕜G(A) ⊆ AA_{1} ⊆ … be the coradical filtration and let gr A = ⊕_{n≥0}gr A(n) be the associated graded coalgebra, which is a graded Hopf algebra [M]. The graded projection π : gr A → gr A(0) ≃ 𝕜G(A) is a Hopf algebra map and a retraction of the inclusion. Let R = {a ∈ A : (id ⊗π)Δ(a) = a ⊗ 1} be the algebra of coinvariants of π; R is a braided Hopf algebra in the category _{G(A)}^{G(A)}YD of Yetter-Drinfeld modules over 𝕜G(A) and gr A can be reconstructed from R and 𝕜G(A) as a bosonization: gr A ≃ R#𝕜G(A). Moreover, R = ⊕_{n≥0}R(n), where R(n) = gr A(n) ∩ R is a graded braided Hopf algebra. We then have several invariants of our initial pointed Hopf algebra A: The graded braided Hopf algebra R; it is called the diagram of A. The braided vector space (V, c), where V := R(1) = P (R) and c : V ⊗ V → V ⊗ V is the braiding in _{G(A)}^{G(A)}YD. It will be called the infinitesimal braiding of A. The dimension of V = P (R), called the rank of A, or of R. The subalgebra R′ of R generated by R(1), which is the Nichols algebra of V: R′ ≃ 𝔅(V). See [AS5] for more details.

2. Nichols algebras of Cartan type

Nichols algebras of diagonal type. In this section, (V, c) denotes a finite-dimensional braided vector space; we assume that the braiding c is diagonal with matrix (q_{ij}), with respect to a basis x_{1}, … x_{θ}.

Let Γ be the free abelian group of rank θ with basis g_{1}, …, g_{θ}. We define characters χ_{1}, …, χ_{θ} of Γ by

χ_{i}(g_{j}) = q_{ji}, 1 ≤ i, j ≤ θ.

We consider V as a Yetter-Drinfeld module over 𝕜Γ by defining x_{i} ∈ V_{g_{i}}^{χ_{i}}, for all i.

A CHARACTERIZATION OF QUANTUM GROUPS 5

We begin with some relations that hold in any Nichols algebra of diagonal type. Lemma 2.1. Let R be a braided Hopf algebra in _{𝕜Γ}^{𝕜Γ}YD, such that V ↪ P (R).

(a). If q_{ii} is a root of 1 of order N > 1 for some i ∈ {1, …, θ}, then x_{i}^{N} ∈ P (R).

(b). Let i ≠ j ∈ {1, …, θ} such that q_{ij}q_{ji} = q_{ii}^{1−r}, where r is an integer such that 0 ≤ r−1 < ord q_{ii} (which could be infinite). Then (ad x_{i})^{r}(x_{j}) is primitive in R.

Assume for the rest of the Lemma that R = 𝔅(V).

(c). In the situation of (a), resp. (b), x_{i}^{N} = 0, resp. ad x_{i}^{r}(x_{j}) = 0.

(d). If R = 𝔅(V) is an integral domain, then q_{ii} = 1 or it is not a root of 1, for all i.

(e). If i ≠ j, then ad_{c}(x_{i})^{r}(x_{j}) = 0 if and only if (r)!_{q_{ii}} ∏_{0≤k≤r−1} (1 − q_{ii}^{k}q_{ij}q_{ji}) = 0.

Proof. (a) and (b) are consequences of the quantum binomial formula, see e. g. [AS2, Appendix] for (b). Then (c) and (d) follow; the second statement in (c) is also a consequence of (e). Part (e) is from [Ro, Lemma 14]; it can also be shown using skew-derivations as in [AS5, Section 2.2]. □

We now recall a variation of a well-known result of Reshetikhin on twisting [Re]. Let (V̂, ĉ) be another braided vector space of the same dimension as V, such that the braiding ĉ is diagonal with matrix (q̂_{ij}) with respect to a basis x̂_{1}, … x̂_{θ}. We define characters χ̂_{1}, …, χ̂_{θ} of Γ by

χ̂_{i}(g_{j}) = q̂_{ji}, 1 ≤ i, j ≤ θ.

We consider V̂ as a Yetter-Drinfeld module over Γ by defining x̂_{i} ∈ V̂ χ̂_{i} ; g_{i}, for all i. Proposition 2.2. Assume that for all i, j, q_{ii} = q̂_{ii} and

(2.1)  q_{ij}q_{ji} = q̂_{ij}q̂_{ji}.

Then there exists an ℕ-graded isomorphism of 𝕜Γ-comodules ψ : 𝔅(V) → 𝔅(V̂) such that

(2.2)  ψ(x_{i}) = x̂_{i},  1 ≤ i ≤ θ.

Let σ : Γ × Γ → 𝕜^{×} be the unique bilinear form such that σ(g_{i}, g_{j}) = q̂_{ij}q_{ij}^{−1}, if i ≤ j, and is equal to 1 otherwise; σ is a group 2-cocycle and we have, for all g, h ∈ Γ,

(2.3)  ψ(xy) = σ(g, h)ψ(x)ψ(y),  x ∈ 𝔅(V)_{g}, y ∈ 𝔅(V)_{h};
(2.4)  ψ([x, y]_{c}) = σ(g, h)[ψ(x), ψ(y)]_{c},  x ∈ V_{g}^{χ}, y ∈ 𝔅(V)_{h}^{η}.

Proof. See [AS5, Prop. 3.9 and Remark 3.10]. □

Remark 2.3. In the situation of the proposition, we say that 𝔅(V) and 𝔅(V̂) are twist-equivalent; note that 𝔅(V) is twist-equivalent to a 𝔅(V̂) with q̂_{ij} = q̂_{ji} for all i and j, since all the q_{ij}q_{ji}’s have square roots in 𝕜. Lemma 2.4. Assume that the braiding with matrix (q_{ij}) is generic and of Cartan type with generalized Cartan matrix (a_{ij}). Then (a_{ij}) is symmetrizable.

Proof. By [K, Ex. 2.1], it is enough to show that a_{i_{1}i_{2}}a_{i_{2}i_{3}} … a_{i_{t−1}i_{t}}a_{i_{t}i_{1}} = a_{i_{2}i_{1}}a_{i_{3}i_{2}} … a_{i_{t}i_{t−1}}a_{i_{1}i_{t}} for all i_{1}, i_{2}, …, i_{t}. But

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

by substituting q_{i_{1}i_{1}}^{a_{i_{1}i_{2}}} = q_{i_{2}i_{2}}^{a_{i_{2}i_{1}}}, then q_{i_{2}i_{2}}^{a_{i_{2}i_{1}}} = q_{i_{3}i_{3}}^{a_{i_{3}i_{2}}} and so on. The claim follows because q_{i_{1}i_{1}} is not a root of one. □

The following result is due to Rosso, who sketched an argument in [Ro, Th. 2.1]. We include a proof for completeness.

Lemma 2.5. [Ro]. Let 𝕜 = ℂ. Assume that the braiding with matrix (q_{ij}) is positive and of Cartan type with generalized Cartan matrix (a_{ij}). Then (a_{ij}) is symmetrizable, with symmetrizing diagonal matrix (d_{i}); and there is a collection of positive numbers (q_{I})_{I∈X} such that (q_{ij}) is twist-equivalent to (q̂_{ij}), where

q̂_{ij} = q_{I}^{d_{i}a_{ij}} for all i, j ∈ I.

That is, the braiding associated to (q̂_{ij}) is of DJ-type.

Proof. We can assume that the braiding is indecomposable; write I = {1, …, θ}. By Remark 2.3, we can assume that q_{ij} = q_{ji}, for all i, j ∈ I. Given j ∈ I, there exists a sequence i_{1} = 1, i_{2}, …, i_{t} = j of elements in I, such that a_{i_{ℓ}i_{ℓ+1}} ≠ 0 for all ℓ, 1 ≤ ℓ < t. Then

q_{11}^{a_{i_{1}i_{2}}a_{i_{2}i_{3}}...a_{i_{t−1}i_{t}}} = q_{jj}^{a_{i_{2}i_{1}}a_{i_{3}i_{2}}...a_{i_{t}i_{t−1}}},

as in the proof of the previous Lemma. Since α_{j} := a_{i_{1}i_{2}}a_{i_{2}i_{3}} … a_{i_{t−1}i_{t}} and β_{j} := a_{i_{2}i_{1}}a_{i_{3}i_{2}} … a_{i_{t}i_{t−1}} are integers of the same sign, we can find b ∈ ℕ and a family (d_{i})_{i∈I} of positive integers such that \frac{α_{j}}{β_{j}} = \frac{d_{j}}{b}. Let q_{I} be the unique positive number such that q_{I}^{b} = q_{11}. Then q_{jj}^{β_{j}} = q_{11}^{α_{j}} = q_{11}^{β_{j} \frac{d_{j}}{b}} = q_{I}^{d_{j}β_{j}}.

Thus q_{jj} = q_{I}^{d_{j}}; and for all i, j ∈ I, q_{ij}q_{ji} = q_{I}^{d_{i}a_{ij}} = q_{I}^{d_{j}a_{ji}}, and the claim follows. □

Remark 2.6. The diagonal braiding with matrix

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

where q is not a root of one, is generic of Cartan type but not of DJ-type.

We now state a very elegant description of Nichols algebras used by Lusztig in a fundamental way [L2].

Proposition 2.7. Let (V, c) be as above and assume that q_{ij} = q_{ji} for all i, j. Let B_{1}, …, B_{θ} be non-zero elements in 𝕜. There is a unique bilinear form ( |) : T (V) × T (V) → 𝕜 such that (1|1) = 1

A CHARACTERIZATION OF QUANTUM GROUPS 7

and

(x_{j}|x_{j}) = δ_{ij}B_{i}, for all i, j; (2.5) (x|yy′) = (x_{(1)}|y)(x_{(2)}|y′), for all x, y, y′ ∈ T (V); (2.6) (xx′|y) = (x|y_{(1)})(x′|y_{(2)}), for all x, x′, y ∈ T (V). (2.7)

This form is symmetric and also satisfies

(2.8)  (x|y) = 0,  for all x ∈ T (V)_{g}, y ∈ T (V)_{h}, g ≠ h ∈ Γ.

The homogeneous components of T (V) with respect to its usual ℕ-grading are also orthogonal with respect to ( |).

The quotient T (V)/I(V), where I(V) = {x ∈ T (V) : (x|y) = 0∀y ∈ T (V)} is the radical of the form, is canonically isomorphic to the Nichols algebra of V. Thus, ( |) induces a non-degenerate bilinear form on 𝔅(V), which will be again be denoted by ( |).

Proof. The existence and uniqueness of the form, and the claims about symmetry and orthogonality, are proved exactly as in [L2, 1.2.3]. It follows from the properties of the form that I(V) is a Hopf ideal. We now check that T (V)/I(V) is the Nichols algebra of V; it is enough to verify that the primitive elements of T (V)/I(V) are in V. Let x be a primitive element in T (V)/I(V), homogeneous of degree n ≥ 2. Then (x|yy′) = 0 for all y, y′ homogeneous of degrees m, m′ ≥ 1 with m + m′ = n; thus x = 0. □

Nichols algebras arising from quantum groups. Let (a_{ij})_{1≤i,j≤θ} be a generalized symmetrizable Cartan matrix [K]; let (d_{1}, …, d_{θ}) be positive integers such that d_{i}a_{ij} = d_{j}a_{ji}. Let q ∈ 𝕜, q ≠ 0, 1, and not a root of 1. We assume that the braided vector space (V, c) is given by the matrix q_{ij} = q^{d_{i}a_{ij}}.

We now want to derive some precise information about the algebra 𝔅(V) mainly from [L2]. We need to consider vector spaces over the field of rational functions ℚ(v).

Let (W, d) denote a finite-dimensional braided vector space over ℚ(v); we assume that the braiding d is diagonal with matrix (v^{d_{i}a_{ij}}), with respect to a basis y_{1}, … y_{θ}. We define characters η_{1}, …, η_{θ} of Γ by

η_{i}(g_{j}) = v^{d_{i}a_{ij}}, 1 ≤ i, j ≤ θ.

We consider W as a Yetter-Drinfeld module over ℚ(v)Γ by defining y_{i} ∈ W_{g_{i}}^{η_{i}}, for all i.

Let v_{i} := v^{d_{i}}, 1 ≤ i ≤ θ. We now take B_{i} := (1 − v_{i}^{−2})^{−1} ∈ ℚ(v) as in [L2, 1.2.3]. By proposition 2.7, 𝔅(W) is Lusztig’s braided Hopf algebra f (with Cartan datum given by i.j = d_{i}a_{ij}, 1 ≤ i, j ≤ θ.

Let A := ℚ[v, v^{−1}]; let [n]_{i} := \frac{v_{i}^{n} − v_{i}^{−n}}{v_{i} − v_{i}^{−1}} , [r]_{i}! = [1]_{i}[2]_{i} … [r]_{i}. Let 𝔅(W)_{A} be the A-subalgebra of

𝔅(W) generated by all

y_{i}^{(r)} := \frac{y_{i}^{r}}{[r]_{i}!}, 1 ≤ i ≤ θ, r ≥ 0.

The canonical bilinear form ( |) : 𝔅(W) × 𝔅(W) → ℚ(v) does not restrict to an A-bilinear form 𝔅(W)_{A} × 𝔅(W)_{A} → A, since by [L2, 1.4.4],

(y_{i}^{(r)}|y_{i}^{(r)}) = vi^{\frac{r(r+1)}{2}} \frac{(v_{i} − v_{i}^{−1})^{−r}}{[r]_{i}!} .

Following an idea of Müller [Mu], we define 𝔅̃(W)_{A} as the A-subalgebra of 𝔅(W) generated by all

ỹ_{i} := (1 − v_{i}^{−2}) y_{i} = B_{i}^{−1}y_{i}, 1 ≤ i ≤ θ.

Then ( |) restricts to an A-bilinear form ( |) : 𝔅(W)_{A} × 𝔅̃(W)_{A} → A, by the argument in [Mu, Lemma 2.2 (a)].

Note that 𝔅(W)_{A} and 𝔅̃(W)_{A} inherit the Hopf algebra structure from 𝔅(W), since A is a principal ideal domain with quotient field ℚ(v).

Let W be the Weyl group of the generalized Cartan matrix (a_{ij}) [L2, 2.1.1]; then W is finite if and only if (a_{ij}) is of finite type. Let w ∈ W be an element with reduced expression w = s_{i_{1}}s_{i_{2}} … s_{i_{P}}, P ∈ ℕ. For any c = (c_{1}, …, c_{P}) ∈ ℕ^{P}, let

L(c) := y_{i_{1}}^{(c_{1})} T_{i_{1}} (y_{i_{2}}^{(c_{2})}) … T_{i_{1}} T_{i_{2}} … T_{i_{P −1}} (y_{i_{P}}^{(c_{P})}) ,

where T_{i_{ℓ}} are the ℚ(v)-algebra automorphisms of U named T′_{i_{ℓ},−1} in [L2, 37.1.3]. Note that L(c) = L(h, c, 0, 1), with h := (i_{1}, …, i_{P}), in [L2, 38.2.3]. By [L2, 41.1.3],

T_{i_{1}} T_{i_{2}} … T_{i_{ℓ−1}} (y_{i_{ℓ}}^{(r)}) ∈ 𝔅(W)_{A}, 1 ≤ ℓ < P, r ≥ 0,

where we have identified 𝔅(W) with U^{+} by [L2, 3.2.6], that is we identify E_{i} := θ_{i}^{+} in Lusztig’s notation with y_{i}. We denote

z_{ℓ} := T_{i_{1}} T_{i_{2}} … T_{i_{ℓ−1}} (y_{i_{ℓ}}) , 1 ≤ ℓ ≤ P.

Then z_{ℓ} ∈ 𝔅(W)_{A} and

z_{ℓ}^{r} = [r]_{i_{ℓ}}! T_{i_{1}} T_{i_{2}} … T_{i_{ℓ−1}} (y_{i_{ℓ}}^{(r)}) , 1 ≤ ℓ ≤ P, r ≥ 0.

Theorem 2.8. (Lusztig). For all c = (c_{1}, …, c_{P}), c′ ∈ ℕ^{P}

(2.9)  (L(c)|L(c′)) = δ_{c,c′} ∏_{1≤s≤P} ∏_{1≤t≤c_{s}} (1 − v_{i}^{−2t})^{−1} .

Proof. This follows from [L2, 38.2.3 and 1.4.4]. □

We regard 𝕜 as an A-algebra via the algebra map φ : A → 𝕜 given by φ(v) = q. We define

𝔅(W)_{𝕜} := 𝔅(W)_{A} ⊗_{A} 𝕜,  𝔅̃(W)_{𝕜} := 𝔅̃(W)_{A} ⊗_{A} 𝕜.

Then 𝔅(W)_{𝕜}, 𝔅̃(W)_{𝕜} are graded braided Hopf algebras in _{𝕜Γ}^{𝕜Γ}YD and by tensoring with 𝕜 over A, we get a bilinear form

( |) : 𝔅(W)_{𝕜} × 𝔅̃(W)_{𝕜} → 𝕜.

Define π : T (V) → 𝔅(W)_{𝕜} and π̃ : T (V) → 𝔅̃(W)_{𝕜} by π(x_{i}) := y_{i} ⊗ 1, π̃(x_{i}) := ỹ_{i} ⊗ 1, 1 ≤ i ≤ θ. Since π and π̃ induce isomorphisms of braided vector spaces of (V, c) with 𝔅(W)_{𝕜}(1) and 𝔅̃(W)_{𝕜}(1), the composition

( |) : T (V) × T (V) ⟶^{π×π̃} 𝔅(W)_{𝕜} × 𝔅̃(W)_{𝕜} ⟶^{( |)} 𝕜

is the canonical bilinear form of T (V) as in Proposition 2.7 with scalars B_{i} = 1, 1 ≤ i ≤ θ. These arguments allow to adapt many results of Lusztig to the case when q is not a root of 1.

Theorem 2.9. [Ro, Theorem 15]; [L2, Section 37]. Let (V, c) be a braided vector space of DJ-type. Then 𝔅(V) ≃ 𝕜⟨x_{1}, …, x_{θ}|ad_{c}(x_{i})^{1−a_{ij}} = 0, 1 ≤ i ≠ j ≤ θ⟩. □

The following Theorem is part of the folklore of quantum groups. Theorem 2.10. Let (V, c) be a braided vector space of DJ-type, with generalized Cartan matrix (a_{ij}).

(i). If the Gelfand-Kirillov dimension of 𝔅(V) is finite, then (a_{ij}) is a finite Cartan matrix.

(ii). If (a_{ij}) is a finite Cartan matrix, then the Gelfand-Kirillov dimension of 𝔅(V) is finite and equal to the number of positive roots.

Proof. (i). We can assume that the braiding is connected. Let w ∈ W be an element with reduced expression w = s_{i_{1}}s_{i_{2}} … s_{i_{P}}, P ∈ ℕ. We keep the notation above. For all ℓ, 1 ≤ ℓ ≤ P, we choose an element t_{ℓ} ∈ T (V) with π(t_{ℓ}) = z_{ℓ} ⊗ 1, and set b_{ℓ} := canonical image of t_{ℓ} in 𝔅(V). We claim that the ordered monomials b_{1}^{c_{1}}b_{2}^{c_{2}} … b_{P}^{c_{P}}, c_{1}, …, c_{P} ≥ 0, are linearly independent.

To prove the claim, we choose a_{c} ∈ A such that a_{c}L(c) ∈ 𝔅̃(W)_{A} for all c = (c_{1}, …, c_{P}) ∈ ℕ^{P}. If L̃(c) := a_{c}L(c) ∏_{1≤s≤P} ∏_{1≤t≤c_{s}} (1 − v_{i}^{−2t}) ∈ 𝔅̃(W)_{A}, then

(2.10)  (L(c)|L̃(c′)) = δ_{c,c′}a_{c},

for all c, c′ ∈ ℕ^{P}, by (2.9). It follows that π(t_{1}^{c_{1}}t_{2}^{c_{2}} … t_{P}^{c_{P}}) = α_{c}L(c) ⊗ 1 ∈ 𝔅̃(W)_{𝕜}, where α_{c} = φ (∏_{1≤s≤P} ([c_{s}]_{i_{s}}!)^{−1}) is a non-zero scalar in 𝕜. Choose elements ℓ̃(c) ∈ T (V) with π̃(ℓ̃(c)) = L̃(c) ⊗ 1, c ∈ ℕ^{P}. Then for all c, c′ ∈ ℕ^{P}, we conclude that (t_{1}^{c_{1}}t_{2}^{c_{2}} … t_{P}^{c_{P}} |ℓ̃(c′)) = 0, if c ≠ c′, and (t_{1}^{c_{1}}t_{2}^{c_{2}} … t_{P}^{c_{P}} |ℓ̃(c)) ≠ 0. Since the form ( | ) factorizes over 𝔅(V) × 𝔅(V), the elements b_{1}^{c_{1}}b_{2}^{c_{2}} … b_{P}^{c_{P}}, c_{1}, …, c_{P} ≥ 0, are linearly independent.

The following statement is well-known:

Let A be a k-algebra, where k is a field. Let a_{1}, …, a_{P} ∈ A such that the ordered monomials a_{1}^{c_{1}}a_{2}^{c_{2}} … a_{P}^{c_{P}}, c_{1}, …, c_{P} ≥ 0, are linearly independent. Then the Gelfand-Kirillov dimension of A is ≥ P.

We conclude that the Gelfand-Kirillov dimension of 𝔅(V) is ≥ P. If the Cartan matrix (a_{ij}) is not finite, then there are elements in the Weyl group of arbitrary lenght, and (i) follows.

(ii). Now the Weyl group is finite, and we take the longest element w_{0}. Let us consider the natural map φ : 𝕜⟨x_{1}, …, x_{θ}|ad_{c}(x_{i})^{1−a_{ij}}, 1 ≤ i ≠ j ≤ θ⟩ → 𝔅(V). Since the elements b_{1}^{c_{1}}b_{2}^{c_{2}} … b_{P}^{c_{P}}, c_{1}, …, c_{P} ≥ 0, are the image of the PBW-basis of 𝕜⟨x_{1}, …, x_{θ}|ad_{c}(x_{i})^{1−a_{ij}}, 1 ≤ i ≠ j ≤ θ⟩, see e. g. [DCK, Proposition 1.7], the claim follows. □

10 NICOLÁS ANDRUSKIEWITSCH AND HANS-JÜRGEN SCHNEIDER

Rosso’s characterization of Nichols algebras. We recall some important results of Rosso. Theorem 2.11. [Ro, Lemma 19] Let (V, c) be a braided vector space of diagonal type. If 𝔅(V) has finite Gelfand-Kirillov dimension, then for all i ≠ j, there exists r > 0 such that ad_{c}(x_{i})^{r}(x_{j}) = 0. □

Corollary 2.12. Let (V, c) be a braided vector space of diagonal type with indecomposable matrix. Assume that 𝔅(V) has finite Gelfand-Kirillov dimension.

(a). If there exists i such that q_{ii} = 1, then θ = 1.

(b). If the braiding is generic then it is of Cartan type.

Proof. This follows from Theorem 2.11 and Lemma 2.1 (e). □

Theorem 2.13. [Ro, Theorem 21] Let (V, c) be a finite-dimensional braided vector space with positive braiding. Then the following are equivalent:

(a). 𝔅(V) has finite Gelfand-Kirillov dimension.

(b). (V, c) is twist-equivalent to a braiding of DJ-type with finite Cartan matrix.

Proof. We can assume that the matrix (q_{ij}) is indecomposable. (b) ⟹ (a) follows from Theorem 2.10 (ii). (a) ⟹ (b). (V, c) is of Cartan type by Corollary 2.12 (b) with Cartan matrix (a_{ij}). We know that (a_{ij}) is symmetrizable by Lemma 2.4 and that (V, c) is twist-equivalent to a braiding of DJ-type by Lemma 2.5. By Theorem 2.10 (i), the Cartan matrix (a_{ij}) is finite. □

Remark 2.14. See [AS2] for the analogous problem of characterizing finite-dimensional braided vector spaces with diagonal braiding such that 𝔅(V) has finite dimension.

3. Coradically graded coalgebras

In this Section we prove a general criterion to determine the coradical filtration of certain Hopf algebras. We generalize a method of Takeuchi [T], who computed the coradical filtration of U_{q}(g) in this way; see also [Mu]. We first extend the definition of coradically graded coalgebras [CM].

Let T ≥ 1 be a natural number. If i = (i_{1}, …, i_{T}) ∈ ℕ^{T}, then we set |i| = i_{1} + ⋯ + i_{T}. Definition 3.1. An ℕ^{T} -graded coalgebra is a coalgebra C provided with an ℕ^{T} -grading C = ⊕_{i∈ℕ^{T}}C(i) such that ΔC(i) ⊂ ⊕_{j}C(j) ⊗ C(i − j). An ℕ^{T} -graded coalgebra C is coradically graded if the n-th term of the coradical filtration is

C_{n} = ⊕_{i∈ℕ^{T} ,|i|≤n}C(i), ∀n ∈ ℕ.

We denote by π_{i} : C → C(i) the projection associated to the grading.

An ℕ^{T} -graded coalgebra C is strictly coradically graded if C(0) = C_{0}, the coradical of C, and Δ_{i,j} : C(i + j) → C(i) ⊗ C(j), Δ_{i,j} = (π_{i} ⊗ π_{j}) ∘ Δ, is injective, for all i, j ∈ ℕ^{T}. Lemma 3.2. (a). Let C be a strictly coradically ℕ^{T} -graded coalgebra and let D be a strictly coradically ℕ^{S}-graded coalgebra. Then C ⊗ D is strictly coradically ℕ^{T +S}-graded with respect to the tensor product grading.

(b). If C is strictly coradically ℕ^{T} -graded, then it is coradically graded.

A CHARACTERIZATION OF QUANTUM GROUPS 11

(c). If C is coradically ℕ-graded, then it is strictly coradically graded.

Proof. (a) follows from the definition. We prove (b) by induction on n, the case n = 0 being part of the hypothesis. Assume n > 0. If c ∈ ⊕_{|i|≤n}C(i), then Δ(c) ∈ C(0)⊗C(n)+⊕_{|j|<n}C(n − j)⊗C(j) ⊂ C_{0} ⊗ C + C ⊗ C_{n−1}; hence c ∈ C_{n} (the argument does not need the second hypothesis). Conversely, let c ∈ C_{n} ∩ C(i) with |i| > n. By the recursive hypothesis, Δ(c) ∈ C(0) ⊗ C + ⊕_{|j|<n}C(i − j) ⊗ C(j). If e ∈ N^{T} has |e| = 1, then Δ_{e,i−e}(c) = 0, thus c = 0.

(c). See for example [MiS, Lemma 2.3]. □ We now consider the following situation. Let U be a Hopf algebra, H a Hopf subalgebra and

N_{1}, …, N_{T} subalgebras of U, such that the multiplication induces a linear isomorphism

μ : N_{1} ⊗ ⋯ ⊗ N_{T} ⊗ H → U.

We assume that the following condition holds:

For all l, 1 ≤ l ≤ T, N_{l}H is a Hopf subalgebra of U. Furthermore, the projection π_{l} : N_{l}H → H, defined by π_{l}(nh) = ε(n)h, n ∈ N_{l}, h ∈ H, is a Hopf algebra map, and N_{l} = (N_{l}H)^{coπ_{l}}.

Then N_{l} is a braided Hopf algebra in _{H}^{H}YD and N_{l}H ≃ N_{l}#H. Let Δ_{l} be the comultiplication of the braided Hopf algebra N_{l} and let j_{l} : N_{l}H ⊗ N_{l} → N_{l}H ⊗ N_{l} be the map given by j_{l}(rh ⊗ s) = rs_{(−1)}h ⊗ s_{(0)}; we know that

(3.1)  Δ(s) = j_{l}Δ_{l}(s), for all s ∈ N_{l}.

Lemma 3.3. The map j : N_{1} ⊗ N_{1} ⊗ N_{2} ⊗ N_{2} ⊗ ⋯ ⊗ N_{T} ⊗ N_{T} ⊗ H ⊗ H → U ⊗ U given by

(3.2) j(u_{1} ⊗ v_{1} ⊗ u_{2} ⊗ v_{2} ⊗ ⋯ ⊗ u_{T} ⊗ v_{T} ⊗ h ⊗ k) = j_{1}(u_{1} ⊗ v_{1})j_{2}(u_{2} ⊗ v_{2}) … j_{T} (u_{T} ⊗ v_{T})(h ⊗ k)

(product in U ⊗ U), is a linear isomorphism, and the following diagram commutes:

(3.3)  \begin{matrix}N_{1} ⊗ ⋯ ⊗ N_{T} ⊗ H & ⟶^{μ} & U \\ ^{Δ_{1}⊗Δ_{2}⊗···⊗Δ_{H}} ↓ & & ↓^{Δ} \\ N_{1} ⊗ N_{1} ⊗ N_{2} ⊗ N_{2} ⊗ ⋯ ⊗ N_{T} ⊗ N_{T} ⊗ H ⊗ H & ⟶^{j} & U ⊗ U. \end{matrix}

Proof. The commutativity of (3.3) follows from (3.1) since Δ is multiplicative. We prove now that j is a linear isomorphism. Let z = u_{1} ⊗ v_{1} ⊗ u_{2} ⊗ v_{2} ⊗ ⋯ ⊗ u_{T} ⊗ v_{T} ⊗ h ⊗ k ∈ N_{1} ⊗ N_{1} ⊗ N_{2} ⊗ N_{2} ⊗ ⋯ ⊗ N_{T} ⊗ N_{T} ⊗ H ⊗ H. Then

j(z) = u_{1}(v_{1})_{(−1)}u_{2}(v_{2})_{(−1)} … u_{T} (v_{T})_{(−1)}h ⊗ (v_{1})_{(0)}(v_{2})_{(0)} … (v_{T})_{(0)}k

= u_{1} (ad (v_{1})_{(−T)}u_{2}) (ad ((v_{1})_{(−T +1)}(v_{2})_{(−T +1)})u_{3}) … (ad ((v_{1})_{(−2)}(v_{2})_{(−2)} … (v_{T −1})_{(−2)})u_{T}) ×

× (v_{1})_{(−1)}(v_{2})_{(−1)} … (v_{T})_{(−1)}h ⊗ (v_{1})_{(0)}(v_{2})_{(0)} … (v_{T}))_{(0)}k

= (μ ⊗ μ) (u_{1} ⊗ (v_{1})_{(−2)} · (u_{2} ⊗ (v_{2})_{(−2)} · (⋯ ⊗ (v_{T −1})_{(−2)} · u_{T} )) ⊗ (v_{1})_{(−1)}(v_{2})_{(−1)} … (v_{T})_{(−1)}h

⊗(v_{1})_{(0)} ⊗ (v_{2})_{(0)} ⊗ ⋯ ⊗ (v_{T})_{(0)} ⊗ k) .

Here, in the first equality we applied the rule (v_{i})_{(−1)}u_{j} = (ad (v_{i})_{(−2)}u_{j})(v_{i})_{(−1)} for i < j systematically; in the second equality, we use · for the tensor product of copies of the adjoint representation. We now apply successively several isomorphisms to this expression. First, if M is any left H-comodule

12 NICOLÁS ANDRUSKIEWITSCH AND HANS-JÜRGEN SCHNEIDER

and N is any left H-module, the map ϕ_{N,M} : N ⊗M → N ⊗M, ϕ_{M}(n⊗m) = m_{(−1)}·n⊗m_{(0)} is an isomorphism with inverse ϕ_{M}^{−1}(n⊗m) = S^{−1}(m_{(−1)})·n⊗m_{(0)}. We apply id_{N_{1}⊗···⊗N_{T}} ⊗ϕ_{H,N_{1}⊗···⊗N_{T}}^{−1} ⊗id_{H} to (μ ⊗ μ)^{−1}j(z) and get

u_{1} ⊗ (v_{1})_{(−1)} · (u_{2} ⊗ (v_{2})_{(−1)} · (⋯ ⊗ (v_{T −1})_{(−1)} · u_{T} )) ⊗ h ⊗ (v_{1})_{(0)} ⊗ (v_{2})_{(0)} ⊗ ⋯ ⊗ (v_{T})_{(0)} ⊗ k; to this expression, we apply id_{N_{1}} ⊗ϕ_{N_{2}⊗···⊗N_{T} ⊗H,N_{1}}^{−1} ⊗ id_{N_{2}⊗···⊗N_{T} ⊗H} and get

u_{1} ⊗ u_{2} ⊗ (v_{2})_{(−1)} · (⋯ ⊗ (v_{T −1})_{(−1)} · u_{T} ) ⊗ h ⊗ v_{1} ⊗ (v_{2})_{(0)} ⊗ ⋯ ⊗ (v_{T})_{(0)} ⊗ k;

iterating this procedure, we obtain z. This shows that j is bijective. □

Theorem 3.4. We keep the notations above. We assume that N_{l} = ⊕_{i∈ℕ}N_{l}(i) is a coradically graded Hopf algebra in _{H}^{H}YD, for all l, 1 ≤ l ≤ T, and that H is cosemisimple. Let U(i) = μ (N_{1}(i_{1}) ⊗ ⋯ ⊗ N_{T} (i_{T}) ⊗ H), for all i = (i_{1}, …, i_{T}) ∈ ℕ^{T}. Then U = ⊕_{i∈ℕ^{T}}U(i) is a coradically ℕ^{T} -graded coalgebra.

Proof. The tensor product coalgebra N_{1} ⊗ ⋯ ⊗ N_{T} ⊗ H is a strictly coradically ℕ^{T} -graded coalgebra by Lemma 3.2 (a) and (c) since each N_{l} is coradically graded and H is cosemisimple. Since each N_{l}(i) is a Yetter-Drinfeld submodule of N_{l}, the map j in Lemma 3.1 is homogeneous. Hence, it follows from Lemma 3.3 that U is strictly coradically ℕ^{T} -graded coalgebra. Then U is coradically ℕ^{T} -graded by Lemma 3.2 (b). □

4. A family of pointed Hopf algebras

In this Section, we fix • a free abelian group Γ of finite rank s,

• a finite Cartan matrix (a_{ij}) ∈ ℤ^{θ×θ} [K]; we denote by (d_{1}, …, d_{θ}) a diagonal matrix of positive integers such that d_{i}a_{ij} = d_{j}a_{ji}, which is minimal with this property;

• a family (q_{I})_{I∈X} of elements in 𝕜 which are not roots of 1;

• elements g_{1}, …, g_{θ} ∈ Γ, characters χ_{1}, …, χ_{θ} ∈ Γ̂ such that ⟨χ_{i}, g_{i}⟩ = q_{I}^{d_{i}} for all i, and

(4.1) ⟨χ_{j}, g_{i}⟩⟨χ_{i}, g_{j}⟩ = q_{I}^{d_{i}a_{ij}}, for all 1 ≤ i, j ≤ θ, i ∈ I.

Definition 4.1. [AS4]. We say that two vertices i and j are linkable (or that i is linkable to j) if i ≁ j, (4.2) g_{i}g_{j} ≠ 1 and (4.3) χ_{i}χ_{j} = ε. (4.4)

Here, ε denotes the trivial representation of Γ. One can easily see, cf. [AS4], that: (4.5) If i is linkable to j, then χ_{i}(g_{j}) = χ_{j}(g_{i})^{−1} = χ_{i}(g_{i}) = χ_{j}(g_{j})^{−1}. (4.6) If i and k, resp. j and ℓ, are linkable, then a_{ij} = a_{kℓ}, a_{ji} = a_{ℓk}. (4.7) A vertex i can not be linkable to two different vertices j and h.

A linking datum for Γ, (a_{ij}), (q_{I})_{I∈X}, g_{1}, …, g_{θ} and χ_{1}, …, χ_{θ} is a collection (λ_{ij})_{1≤i<j≤θ,i≁j} of elements in {0, 1} such that λ_{ij} is arbitrary if i and j are linkable but 0 otherwise. Given a linking datum, we say that two vertices i and j are linked if λ_{ij} ≠ 0.

The collection D = D((a_{ij}), (q_{I}), (g_{i}), (χ_{i}), (λ_{ij})), where (λ_{ij}) is a linking datum, will be called a generic datum of finite Cartan type for Γ. If 𝕜 = ℂ, a generic datum of finite Cartan type will be called positive if q_{I} > 0, for all I ∈ X.

Let D′ be a generic datum of finite Cartan type over a free abelian group Γ′ of finite rank, formed by (a′_{ij}) ∈ ℤ^{θ′×θ′}, (q′_{I})_{I∈X′}, g′_{1}, …, g′_{θ}, χ′_{1}, …, χ′_{θ} and a linking datum (λ′_{ij})_{1≤i<j≤θ,i≁j}. The data D and D′ are called isomorphic if θ = θ′, and if there exist a group isomorphism φ : Γ → Γ′, a permutation σ ∈ 𝕊_{θ}, and elements 0 ≠ α_{i} ∈ 𝕜, for all 1 ≤ i ≤ θ such that

• φ(g_{i}) = g′_{σ(i)}, for all 1 ≤ i ≤ θ;

• χ_{i} = χ′_{σ(i)}φ, for all 1 ≤ i, j ≤ θ;

• λ_{ij} = \begin{cases}α_{i}α_{j}λ′_{σ(i)σ(j)}, & if σ(i) < σ(j) \\ −α_{i}α_{j}χ_{j}(g_{i})λ′_{σ(j)σ(i)}, & if σ(i) > σ(j) \end{cases} , for all 1 ≤ i < j ≤ θ, i ≁ j.

In this case the triple (φ, σ, (α_{i})) will be called an isomorphism from D to D′.

Note that then a_{ij} = a′_{σ(i)σ(j)}, for all 1 ≤ i, j ≤ θ. Thus σ is an isomorphism of the corresponding Dynkin diagrams. Indeed, for all i, j, χ_{j}(g_{i}) = χ′_{σ(i)}(g′_{σ(i)}), hence χ_{j}(g_{i})χ_{i}(g_{j}) = χ_{i}(g_{i})^{a_{ij}} = χ_{i}(g_{i})^{a′_{σ(i)σ(j)}}, and a_{ij} = a′_{σ(i)σ(j)}, since χ_{i}(g_{i}) is not a root of 1.

Let Isom(D, D′) be the set of all isomorphisms from D to D′.

To state the next theorem, we follow the conventions of [AS4, Section 4]. We assume that the Cartan matrix is a matrix of blocks corresponding to the connected components; that is, for each I ∈ X, there exist c_{I}, d_{I} such that I = {j : c_{I} ≤ j ≤ d_{I}}.

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. Note that

(4.8)  V_{g_{i}}^{χ_{i}} ≇ V_{g_{j}}^{χ_{j}} in _{𝕜Γ}^{𝕜Γ}YD, for all 1 ≤ i, j ≤ θ, i ≠ j.

For suppose that V_{g_{i}}^{χ_{i}} ≅ V_{g_{j}}^{χ_{j}}, and i ≠ j. Then g_{i} = g_{j}, χ_{i} = χ_{j} and

χ_{i}(g_{i})^{2} = χ_{i}(g_{j})χ_{j}(g_{i}) = χ_{i}(g_{i})^{a_{ij}}.

Thus a_{ij} = 2, since χ_{i}(g_{i}) is not a root of one; but this is a contradiction.

Lusztig defined root vectors X_{α}, α ∈ Φ^{+} [L2], in the case of braidings of DJ-type; these are the elements b_{1}, …, b_{P} in the proof of Theorem 2.10; they can be expressed as iterated braided commutators. As in [AS4], this definition can be extended to generic braidings of finite Cartan type, first in the tensor algebra T (V), and then in suitable quotients.

We fix a ℤ-basis Y_{h}, 1 ≤ h ≤ s of Γ.

Theorem 4.2. Let D = D((a_{ij}), (q_{I}), (g_{i}), (χ_{i}), (λ_{ij})) be a generic datum of finite Cartan type for the free abelian group Γ of finite rank. Let U(D) be the algebra presented by generators a_{1}, …, a_{θ}, y_{1}^{±1}, …, y_{s}^{±1} and relations

(4.9)  y_{m}^{±1}y_{h}^{±1} = y_{h}^{±1}y_{m}^{±1}, y_{m}^{±1}y_{m}^{∓1} = 1,  1 ≤ m, h ≤ s,
(4.10)  y_{h}a_{j} = χ_{j}(Y_{h})a_{j}y_{h},  1 ≤ h ≤ s, 1 ≤ j ≤ θ,
(4.11)  (ad a_{i})^{1−a_{ij}}a_{j} = 0,  1 ≤ i ≠ j ≤ θ, i ∼ j,
(4.12)  a_{i}a_{j} − χ_{j}(g_{i})a_{j}a_{i} = λ_{ij}(1 − g_{i}g_{j}),  1 ≤ i < j ≤ θ, i ≁ j;

then U(D) is a pointed Hopf algebra with structure determined by

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

Furthermore, U(D) has a PBW-basis given by monomials in the root vectors b_{1} := a_{β_{1}}, …, b_{P} := a_{β_{P}}. The coradical filtration of U(D) is given by

(4.14)  U(D)_{N} = span of a_{i_{1}}a_{i_{2}} … a_{i_{s}}y : s ≤ ℕ, y ∈ G(H).

There is an isomorphism of graded Hopf algebras ψ : 𝔅(V)#𝕜Γ → gr U(D), given by x_{i}#1 ↦ \overline{a_{i}}, 1 ≤ i ≤ θ, 1#Y_{h} ↦ y_{h}, 1 ≤ h ≤ s.

U(D) has finite Gelfand-Kirillov dimension and is a domain.

In (4.12) and (4.13), the elements g_{i}, 1 ≤ i ≤ θ, must be read as the word in the generators y_{h}, 1 ≤ h ≤ s given by y_{1}^{t_{i,1}} ⋯ y_{s}^{t_{i,s}}, if g_{i} = Y_{1}^{t_{i,1}} ⋯ Y_{s}^{t_{i,s}}, where t_{i,1}, ⋯, t_{i,s} are natural numbers.

The relations (4.11) are the quantum Serre relations. By (4.10) and (4.13), (ad a_{i})(a_{j}) = a_{i}a_{j} − χ_{j}(g_{i})a_{j}a_{i} = (ad_{c}a_{i})(a_{j}). Here, ad_{c} is the braided adjoint representation in the tensor algebra of V. Hence the left hand side of (4.11) should be more formally written as

(ad_{c}a_{i})^{1−aij}(a_{j}) = ∑_{l=0}^{1−a_{ij}}(−1)^{l}\binom{1 − a_{ij}}{l}_{q_{ii}}qii^{\frac{l(l−1)}{2}} q_{ij}^{l}a_{i}^{1−a_{ij}−l}a_{j}a_{i}^{l},

where q_{ij} := χ_{j}(g_{i}) for all i, j.

Proof. Step I. It is not difficult to see that Δ is well-defined by (4.13), using Lemma 2.1 (b) for the quantum Serre relations.

Step II. As in [AS4, Th. 4.5], we deduce from Theorem 2.10, via Proposition 2.2, that 𝔅(V) ≃ 𝕜⟨x_{1}, …, x_{θ}|ad_{c}(x_{i})^{1−a_{ij}} = 0, 1 ≤ i ≠ j ≤ θ⟩.

Step III. We now prove the statement about the PBW-basis. We argue exactly as in the proof of [AS4, Th. 5.17]; we proceed by induction on the number of connected components. If the Dynkin diagram is connected, the claim follows from Step II as in [AS4, Lemma 5.18]. For the inductive step, one repeats the argument in [AS4, Lemma 5.19]. We assume there exists θ̂ < θ such that J = {1, …, θ̂} ∈ X. Let Υ :=< Z_{1} > ⊕ ⋯ ⊕ < Zθ̂ >, be a free abelian group of rank θ̂. Let η_{j} be the unique character of Υ such that η_{j}(Z_{i}) = χ_{j}(g_{i}), 1 ≤ i, j ≤ θ̂.

Let D_{1} be the generic datum over Γ given by (a_{ij})_{θ̂<i,j≤θ}, (g_{i})_{θ̂<i≤θ}, (χ_{j})_{θ̂<j≤θ}, (λ_{ij})_{θ̂<i<j≤θ,i≁j}. Let B := U(D_{1}), with generators b_{θ̂+1}, …, b_{θ} (instead of the a_{i}’s) and y_{1}, …, y_{s}.

Let D_{2} be the generic datum over Υ given by (a_{ij})_{1≤i,j≤θ̂}, (Z_{i})_{1≤i≤θ̂}, (η_{j})_{1≤j≤θ̂}, with empty linking datum. Let U := U(D_{2}) with generators u_{1}, …, uθ̂ (instead of the a_{i}’s) and z_{1}, …, zθ̂.

Then the analogue of [AS4, Lemma 5.19] holds replacing the dual of B by the Hopf dual. Notice that the argument in loc. cit. works since algebra maps and skew-derivations are in the Hopf dual. The proof is actually easier because there are less relations to check. Finally, we form the Hopf algebra (U ⊗ B)_{σ} as in loc. cit., and consider the quotient Ã of (U ⊗ B)_{σ} by the central Hopf subalgebra 𝕜[(z_{i} ⊗ g_{i}^{−1}) : 1 ≤ i ≤ θ̂]. The same argument as in loc. cit. shows that Ã ≃ U(D) as Hopf algebras. On the other hand, the monomials b_{1}^{c_{1}}b_{2}^{c_{2}} … b_{P}^{c_{P}} y, c_{j} ∈ ℕ, 1 ≤ j ≤ P, y ∈ Γ × Υ form a basis of (U ⊗ B)_{σ}. Since Γ × Υ splits as the product of Γ and the group generated by (z_{i} ⊗ g_{i}^{−1}), 1 ≤ i ≤ θ̂, we conclude that the images of the monomials b_{1}^{c_{1}}b_{2}^{c_{2}} … b_{P}^{c_{P}} y, c_{j} ∈ ℕ, 1 ≤ j ≤ P, y ∈ Γ form a basis of Ã.

Step IV. The claim about the GK-dimension follows from the previous step. The claim about the coradical filtration also follows from the previous step, together with Theorem 3.4. Indeed, Theorem 3.4 applies since Nichols algebras are coradically graded by definition.

Step V. The existence of ψ follows from the definition of Nichols algebras, since by the statement about the coradical filtration we have a monomorphism of Yetter-Drinfeld modules V → U(D)_{1}/U(D)_{0}. Since the restriction of ψ to the first term of the coradical filtration (𝔅(V)#𝕜Γ)_{1} is injective, ψ is injective [M, Th. 5.3.1]. It is surjective by the PBW-basis claim. Therefore ψ is bijective.

Step VI. We finally prove that U(D) is a domain. Here we follow [DCK, Corollary 1.8]. We introduce an ℕ^{P +1}-filtration on U(D) by the degree defined by

deg (b_{1}^{c_{1}}b_{2}^{c_{2}} … b_{P}^{c_{P}} y) = (c_{1}, c_{2}, …, c_{P} , ∑_{1≤j≤P} c_{j} ht β_{j}), , c_{j} ∈ ℕ, 1 ≤ j ≤ P, y ∈ Γ;

here ht β is the height of the root β as in [DCK]. We claim that this is an algebra filtration. If α = e_{1}α_{1} + ⋯ + e_{θ}α_{θ}, we set g_{α} = g_{1}^{e_{1}} … g_{θ}^{e_{θ}}, χ_{α} = χ_{1}^{e_{1}} … χ_{θ}^{e_{θ}}. Recall that b_{k} = a_{β_{k}}, 1 ≤ k ≤ P. To

16 NICOLÁS ANDRUSKIEWITSCH AND HANS-JÜRGEN SCHNEIDER

prove the claim one has to verify that for all k > l

(4.15)  b_{k}b_{l} − χ_{β_{l}}(g_{β_{k}})b_{l}b_{k} = ∑_{c∈ℕ^{P}} ρ_{c} b_{1}^{c_{1}}b_{2}^{c_{2}} … b_{P}^{c_{P}} ,

where ρ_{c} ∈ 𝕜, and ρ_{c} = 0 unless deg (b_{1}^{c_{1}}b_{2}^{c_{2}} … b_{P}^{c_{P}}) < deg (b_{l}b_{k}).

If β_{l} and β_{k} have support in the same connected component, then (4.15) follows from the formula of Levendorskii and Soibelman [DCK, Lemma 1.7] using Lemma 2.2. Note that here we are using the special order of the set of positive roots, in which the roots with support on the first component are smaller than the roots in the second, and so on.

If β_{l} and β_{k} have support in different connected components, then (4.15) follows from the linking relations since the monomials in any root vector are homogeneous.

As in [DCK, Corollary 1.8], it is not difficult to verify that the associated graded ring is a domain, which implies that U(D) is a domain. □

To describe the isomorphisms between Hopf algebras U(D) and U(D′), we first formulate a Lemma which is needed in this general form in the proof of the main theorem 5.2. Lemma 4.3. Let Γ be a finitely generated abelian group, A a pointed Hopf algebra with coradical A_{0} = 𝕜Γ, V ∈ _{𝕜Γ}^{𝕜Γ}YD with 𝕜-basis x_{i} ∈ V_{g_{i}}^{χ_{i}}, g_{i} ∈ Γ, χ_{i} ∈ Γ̂, 1 ≤ i ≤ θ.

Assume that grA ≅ 𝔅(V)#𝕜Γ as graded Hopf algebras, and

(4.16)  χ_{i} ≠ ε, for all 1 ≤ i ≤ θ.

Then the first term of the coradical filtration of A is

A_{1} = A_{0} ⊕ ⨁_{g∈Γ,1≤i≤θ} P_{gg_{i},g}(A)^{χ_{i}}.

If (g_{i}, χ_{i}) ≠ (g_{j}, χ_{j}) for all i ≠ j, then the vector spaces P_{gg_{i},g}(A)^{χ_{i}} are one-dimensional for all g ∈ Γ, 1 ≤ i ≤ θ.

Proof. By assumption, A_{1}/A_{0} ≅ V #𝕜Γ, and the elements x_{i}#g ∈ P_{gg_{i},g}(𝔅(V)#𝕜Γ)^{χ_{i}}, 1 ≤ i ≤ θ, g ∈ Γ, x ∈ P_{gg_{i},g}(A)^{χ_{i}} form a 𝕜-basis of V #𝕜Γ. Hence

A_{1}/A_{0} ≅ ⨁_{1≤i≤θ}(A_{1}/A_{0})^{χ_{i}}.

Let h_{1}, ⋯, h_{s} be generators of Γ. Fix 1 ≤ i ≤ θ. Then for any a ∈ A_{1} with ā ∈ (A_{1}/A_{0})^{χ_{i}}, and 1 ≤ l ≤ s, the adjoint action of h_{l} on a is given by h_{l} · a = χ_{i}(h_{l})a + v_{l}, for some v_{l} ∈ A_{0}. Since Γ is abelian, the vector space spanned by a, v_{1}, ⋯, v_{s} is Γ-stable containing a. This shows that A_{1} is locally finite under the adjoint action of Γ. We claim that A_{1} is completely reducible as Γ-module. Indeed, let U be any locally finite Γ-module, let χ ∈ Γ̂, and let U^{(χ)} = {u ∈ U : ∃s > 0 such that (g − χ(g))^{s}(u) = 0∀g ∈ Γ}. Then U = ⊕_{χ∈Γ̂}U^{(χ)} (see for instance [D, Th. 1.3.19]). Now, A_{0} ⊂ (A_{1})^{ε} ⊂ (A_{1})^{(ε)}, but by (4.16), they are all three equal. The claim follows.

Hence A_{1} = A_{0} ⊕ ⨁_{1≤i≤θ}(A_{1})^{χ_{i}}. By the theorem of Taft and Wilson [M, Theorem 5.4.1], A_{1} = A_{0} + (⨁_{g,h∈Γ} P_{g,h}(A)). Hence

A_{1} = A_{0} ⊕ ⨁_{g,h∈Γ,ε≠χ∈Γ̂} P_{g,h}(A)^{χ},

and the lemma follows. □

Note that (4.16) holds for generic braidings, since in this case no χ_{i}(g_{i}) is a root of 1.

If A, B are Hopf algebras, we denote the set of all Hopf algebra isomorphisms from A to B by Isom(A, B). Theorem 4.4. Let D and D′ be generic data of finite Cartan type for Γ. Then the Hopf algebras U(D) and U(D′) are isomorphic if and only if D is isomorphic to D′.

More precisely, let a_{1}, ⋯, a_{θ} resp. a′_{1}, ⋯, a′_{θ} be the simple root vectors in U(D) resp. U(D′) of Theorem 4.2, and let g_{1}, ⋯, g_{θ} resp. g′_{1}, ⋯, g′_{θ} be the group-like elements in D resp. D′ . Then the map

Isom(U(D), U(D′)) → Isom(D, D′),

given by ϕ ↦ (φ, σ, (α_{i})), where φ(g) = ϕ(g), φ(g_{i}) = g′_{σ(i)}, ϕ(a_{i}) = α_{i}a′_{σ(i)}, for all g ∈ Γ, 1 ≤ i ≤ θ, is bijective.

Proof. Let V resp. V′ be the Yetter-Drinfeld module of the infinitesimal braiding of A := U(D) resp. of A′ := U(D). Let ϕ : A → A′ be an isomorphism of Hopf algebras. Then ϕ induces isomorphisms A_{0} → A′_{0} and A_{1} → A′_{1}. Hence ϕ defines an isomorphism of groups φ : Γ → Γ, and for all g, h ∈ Γ, χ ∈ Γ̂, a linear isomorphism

P_{g,h}(A)^{χ} ≅ P_{φ(g),φ(h)}(A′)^{χφ^{−1}}.

By Theorem 4.2, the assumptions of Lemma 4.3 are satisfied for A, V and A′, V′. Then it follows from Lemma 4.3 and (4.8) that there is a uniquely determined permutation σ ∈ 𝕊_{θ} such that ϕ induces an isomorphism

P_{g_{i},1}(A)^{χ_{i}} ≅ P_{g′_{σ(i)},1}(A′)^{χ′_{σ(i)}}, with φ(g_{i}) = g′_{σ(i)}, χ_{i}φ^{−1} = χ′_{σ(i)}, for all 1 ≤ i ≤ θ.

Moreover, since for all i, P_{g_{i},1}(A)^{χ_{i}} and P_{g′_{σ(i)},1}(A)^{χ′_{σ(i)}} are one-dimensional with basis a_{i} and a′_{i}, there are non-zero scalars α_{i} ∈ 𝕜 with ϕ(a_{i}) = α_{i}a′_{σ(i)}, for all 1 ≤ i ≤ θ.

Then the elements ϕ(a_{i}), 1 ≤ i ≤ θ, satisfy the Serre relations (4.11), and they satisfy (4.12) if and only if the triple (φ, σ, (α_{i})) is an isomorphism of generic data. Thus the map Isom(U(D), U(D′)) → Isom(D, D′) in the theorem is well-defined and injective. Surjectivity of this map follows from the description of the Hopf algebras U(D) and U(D′) in Theorem 4.2. □

The main reason why the proof of the preceding theorem works is the knowledge of the coradical filtration. The same ideas allow to determine all Hopf subalgebras of U(D).

18 NICOLÁS ANDRUSKIEWITSCH AND HANS-JÜRGEN SCHNEIDER

5. Pointed Hopf algebras with generic braidings

We are going to show that the class of Hopf algebras described in the previous section has an intrinsic description.

The following key Lemma implies that pointed Hopf algebras belonging to a natural class are generated by group-like and skew primitive elements. Lemma 5.1. (a). Let S = ⊕_{n∈ℕ}S(n) be a graded braided Hopf algebra such that S(0) = ℂ.1, V := S(1) is finite-dimensional and generates S as an algebra. Assume that S has finite Gelfand-Kirillov dimension and that V has positive braiding. Then S is a Nichols algebra.

(b). Let R be as S in (a), except that we assume P (R) = R(1) instead of generation in degree 1. Then R is a Nichols algebra.

Proof. (a). 𝔅(V) has finite Gelfand-Kirillov dimension since it is a quotient of S. Assume first that the matrix is indecomposable. We can then apply Theorem 2.13; let (a_{ij}), (d_{1}, …, d_{θ}) and q such that q_{ij}q_{ji} = q^{d_{i}a_{ij}} for all i ≠ j and q_{ii} = q^{d_{ii}}.

Let i ≠ j. We claim that z_{2} = ad_{c}(x_{i})^{1−a_{ij}}(x_{j}) = 0 in S. Indeed, let z_{1} = x_{i}, suppose that z_{2} ≠ 0 and consider the two-dimensional subspace W of S generated by the primitive elements z_{1} and z_{2}.

We claim that 𝔅(W) has finite Gelfand-Kirillov dimension. For, let T be the subalgebra of S generated by W; then the graded Hopf algebra gr (T #ℂΓ) has finite Gelfand-Kirillov dimension, and contains 𝔅(W).

Also, the braiding of W is given by the matrix:

(\begin{matrix}q_{ii} & q_{ii}^{1−a_{ij}}q_{ij} \\ q_{ii}^{1−a_{ij}}q_{ji} & q_{ii}^{(1−a_{ij})^{2}}q_{ij}^{1−a_{ij}}q_{ji}^{1−a_{ij}}q_{jj} \end{matrix}) = (\begin{matrix}q^{d_{i}} & q^{d_{i}(1−a_{ij})}q_{ij} \\ q^{d_{i}(1−a_{ij})}q_{ji} & q^{d_{i}−d_{i}a_{ij}+d_{j}} \end{matrix})

By Theorem 2.11 and Lemma 2.1, there exists k ≥ 0 such that 1 = q^{d_{i}k+2d_{i}(1−a_{ij})}q_{ij}q_{ji}, hence 0 = d_{i}k + 2d_{i}(1 − a_{ij}) + d_{i}a_{ij} = d_{i}(k + 2 − a_{ij}), a contradiction. This shows that z_{2} = 0.

Therefore, we have an epimorphism of braided graded Hopf algebras 𝔅(V) → S, by Step II of Theorem 4.2, which is the identity in degree 1. Hence 𝔅(V) ≃ S.

Assume now that the matrix is decomposable. Let i, j belong to different components; in particular q_{ij}q_{ji} = 1. We claim that x_{i}x_{j} = q_{ij}x_{j}x_{i}. If not, let z_{1} := x_{i} and z_{2} := x_{i}x_{j} − q_{ij}x_{j}x_{i} is primitive by Lemma 2.1 (b). Let z_{1} = x_{i} and consider as before the subspace W of S generated by z_{1} and z_{2}. As before, 𝔅(W) has finite Gelfand-Kirillov dimension. Now the braiding of W is given by the matrix:

(\begin{matrix}q̃_{11} & q̃_{12} \\ q̃_{21} & q̃_{22} \end{matrix}) := (\begin{matrix}q_{ii} & q_{ii}q_{ij} \\ q_{ii}q_{ji} & q_{ii}q_{jj} \end{matrix}) .

By Theorem 2.11 again, there exists k ≥ 0 such that 1 = q_{ii}^{k+2}q_{ij}q_{ji} = q_{ii}^{k+2}, a contradiction. This concludes the proof of (a).

Finally, (a) and (b) are equivalent by [AS2, Lemma 5.5] and the definition of finite Gelfand-Kirillov dimension. □

A CHARACTERIZATION OF QUANTUM GROUPS 19

Theorem 5.2. Let A be a pointed Hopf algebra with abelian coradical and diagram R. Assume that R(1) is finite-dimensional with positive braiding. Then the following are equivalent:

(a). A is a domain with finite Gelfand-Kirillov dimension.

(b). The group Γ := G(A) is free abelian of finite rank, and there exist a positive datum D for Γ such that A ≃ U(D) as Hopf algebras.

Proof. (b) ⟹ (a): this is Theorem 4.2.

(a) ⟹ (b). By [KL, 6.5], gr A has finite GK-dimension; hence both R and 𝕜Γ also have. It is clear then that Γ should be a free abelian group of finite rank, say s. Consider the diagram R of A. From Theorem 2.13 and Lemma 5.1, we conclude the existence of the finite Cartan matrix (a_{ij}), the family (q_{I})_{I∈X}, and the elements g_{1}, …, g_{θ} ∈ Γ, χ_{1}, …, χ_{θ} ∈ Γ̂ satisfying (4.1), and such that no q_{I} is a root of 1 (in fact q_{I} > 0 and not 1 for all I), R = 𝔅(V) where V ∈ _{𝕜Γ}^{𝕜Γ}YD has a basis x_{i} ∈ V_{g_{i}}^{χ_{i}}, 1 ≤ i ≤ θ.

Since grA ≅ 𝔅(V)#𝕜Γ as graded Hopf algebras, and χ_{i}(g_{i}) ≠ 1 for all 1 ≤ i ≤ θ, it follows from Lemma 4.3 that the first term of the coradical filtration of A is

(5.1)  A_{1} = A_{0} ⊕ ⨁_{g∈Γ,1≤i≤θ} P_{gg_{i},g}(A)^{χ_{i}}.

We can then 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}. Let y_{1}, ⋯, y_{s} be free generators of G(A). It is clear that relations (4.9) and (4.10) hold.

Let i ≠ j. We claim:

(i). 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}.

(ii). If i ∼ j, then χ_{i}^{1−a_{ij}}χ_{j} ≠ ε.

We prove (i). Assume that g_{i}^{1−a_{ij}}g_{j} = g_{ℓ}, χ_{i}^{1−a_{ij}}χ_{j} = χ_{ℓ} for some ℓ. Then

q_{I}^{d_{i}a_{iℓ}} = ⟨χ_{ℓ}, g_{i}⟩⟨χ_{i}, g_{ℓ}⟩ = q_{I}^{2d_{i}(1−a_{ij})}⟨χ_{j}, g_{i}⟩⟨χ_{i}, g_{j}⟩ = q_{I}^{d_{i}(2−a_{ij})};

we conclude that 2 = a_{ij} + a_{iℓ}. The only possibility is a_{ij} = 0 and l = i. Then g_{j} = 1 which is impossible.

We prove (ii). Assume that χ_{i}^{1−a_{ij}}χ_{j} = ε, i ≠ j, i ∼ j. Evaluating at g_{i}, we get 1 = q_{ii}^{1−a_{ij}}q_{ij} = q_{ii}(q_{ij}q_{ji})^{−1}q_{ij} = q_{ii}q_{ji}^{−1}, so that q_{ii} = q_{ji}. Evaluating at g_{j}, we get 1 = q_{ji}^{1−a_{ij}}q_{jj}; hence q_{jj} = q_{ii}^{a_{ij}−1}. Hence 0 = d_{i}(1 − a_{ij}) + d_{j}; this is a contradiction.

If i ≠ j, then (ad a_{i})^{1−a_{ij}}a_{j} ∈ P_{g_{i}^{1−a_{ij}} g_{j},1}(H)^{χ_{i}^{1−a_{ij}} χ_{j}} by Lemma 2.1 (b). If i ∼ j, taking into account (5.1), (i) and (ii), we see that the quantum Serre relations (4.11) hold in A.

Finally, assume that i ≁ j; if 0 ≠ (ad a_{i})a_{j} ∈ P_{g_{i}g_{j},1}(A)^{χ_{i}χ_{j}}, then χ_{i}χ_{j} = ε by (5.1) and (i). So that a_{i}a_{j} − χ_{j}(g_{i})a_{j}a_{i} = λ_{ij}(1 − g_{i}g_{j}) for some λ_{ij} ∈ 𝕜, where λ_{ij} = 0 when χ_{i}χ_{j} ≠ ε. But we can also choose λ_{ij} = 0 when g_{i}g_{j} = 1. By (4.7), we can rescale a generator a_{i} with λ_{ij} ≠ 0 to have λ_{ij} = 1. Hence, (λ_{ij}) is a linking datum for (a_{ij}), g_{1}, …, g_{θ} and χ_{1}, …, χ_{θ}; and (4.12) holds.

We have found a positive datum D for Γ and constructed a homomorphism of Hopf algebras φ : U(D) → A. Now gr φ : gr U(D) → gr A is an isomorphism by Theorem 4.2; indeed gr φ is surjective and the restriction of gr φ to the first term of the coradical filtration is injective; thus gr φ is injective [M, Th. 5.3.1]. Hence φ is is an isomorphism. □

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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