POINTED HOPF ALGEBRAS

NICOLÁS ANDRUSKIEWITSCH AND HANS-JÜRGEN SCHNEIDER

Abstract. This a survey on pointed Hopf algebras over algebraically closed fields of characteristic

0. We propose to classify pointed Hopf algebras A by first determining the graded Hopf algebra gr A associated to the coradical filtration of A. The A_{0}-convariants elements form a braided Hopf algebra R in the category of Yetter-Drinfeld modules over the coradical A_{0} = 𝕜Γ, Γ the group of group-like elements of A, and gr A ≃ R#A_{0}. We call the braiding of the primitive elements of R the infinitesimal braiding of A. If this braiding is of Cartan type [AS2], then it is often possible to determine R, to show that R is generated as an algebra by its primitive elements and finally to compute all deformations or liftings, that is pointed Hopf algebras such that gr A ≃ R#𝕜Γ. In the last Chapter, as a concrete illustration of the method, we describe explicitly all finite-dimensional pointed Hopf algebras with abelian group G(A) and infinitesimal braiding of type A_{n} (up to some exceptional cases). In other words, we compute all the liftings of type A_{n}; this result is our main new contribution in this paper.

Contents

Introduction 2 1. Braided Hopf algebras 4 2. Nichols algebras 15 3. Types of Nichols algebras 21 4. Nichols algebras of Cartan type 31 5. Classification of pointed Hopf algebras by the lifting method 34 6. Pointed Hopf algebras of type A_{n} 42 References 62

2000 Mathematics Subject Classification. Primary: 17B37; Secondary: 16W30. Key words and phrases. Pointed Hopf algebras, finite quantum groups. This work was partially supported by ANPCyT, Agencia Córdoba Ciencia, CONICET, DAAD, the

Graduiertenkolleg of the Math. Institut (Universität München) and Secyt (UNC).

Introduction

A Hopf algebra A over a field 𝕜 is called pointed [Sw], [M1], if all its simple left or right comodules are one-dimensional. The coradical A_{0} of A is the sum of all its simple subcoalgebras. Thus A is pointed if and only if A_{0} is a group algebra.

We will always assume that the field 𝕜 is algebraically closed of characteristic 0 (although several results of the paper hold over arbitrary fields).

It is easy to see that A is pointed if it is generated as an algebra by group-like and skewprimitive elements. In particular, group algebras, universal enveloping algebras of Lie algebras and the q-deformations of the universal enveloping algebras of semisimple Lie algebras are all pointed.

An essential tool in the study of pointed Hopf algebras is the coradical filtration

A_{0} ⊂ A_{1} ⊂ ⋯ ⊂ A, ⋃_{n≥0} A_{n} = A

of A. It is dual to the filtration of an algebra by the powers of the Jacobson radical. For pointed Hopf algebras it is a Hopf algebra filtration, and the associated graded Hopf algebra gr A has a Hopf algebra projection onto A_{0} = 𝕜Γ, Γ = G(A) the group of all group-like elements of A. By a theorem of Radford [Ra], gr A is a biproduct

gr A ≅ R#𝕜Γ,

where R is a graded braided Hopf algebra in the category of left Yetter-Drinfeld modules over 𝕜Γ [AS2].

This decomposition is an analog of the theorem of Cartier–Kostant–Milnor–Moore on the semidirect product decomposition of a cocommutative Hopf algebra into an infinitesimal and a group algebra part.

The vector space V = P (R) of the primitive elements of R is a Yetter-Drinfeld submodule. We call its braiding

c : V ⊗ V → V ⊗ V

the infinitesimal braiding of A. The infinitesimal braiding is the key to the structure of pointed Hopf algebras.

The subalgebra 𝔅(V) of R generated by V is a braided Hopf subalgebra. As an algebra and coalgebra, 𝔅(V) only depends on the infinitesimal braiding of V. In his thesis [N] published in 1978, Nichols studied Hopf algebras of the form 𝔅(V)#𝕜Γ under the name of bialgebras of type one. We call 𝔅(V) the Nichols algebra of V. These Hopf algebras were found independently later by Woronowicz [Wo] and other authors.

Important examples of Nichols algebras come from quantum groups [Dr1]. If 𝔤 is a semisimple Lie algebra, U_{q}^{≥0}(𝔤), q not a root of unity, and the finite-dimensional Frobenius-Lusztig kernels 𝔲_{q}^{≥0}(𝔤), q a root of unity of order N, are both of the form 𝔅(V)#𝕜Γ with Γ = ℤ^{θ} resp. (ℤ/(N))^{θ}, θ ≥ 1. ([L3], [Ro1], [Sbg], and [L2], [Ro1], [Mu]). (Here, there are some technical conditions on N).

In general, the classification problem of pointed Hopf algebras has three parts:

(1) Structure of the Nichols algebras 𝔅(V).

(2) The lifting problem: Determine the structure of all pointed Hopf algebras A with G(A) = Γ such that gr A ≅ 𝔅(V)#𝕜Γ.

(3) Generation in degree one: Decide which Hopf algebras A are generated by group-like and skew-primitive elements, that is gr A is generated in degree one.

We conjecture that all finite-dimensional pointed Hopf algebras over an algebraically closed field of characteristic 0 are indeed generated by group-like and skew-primitive elements.

In the sequel, we describe the steps of this program in detail and explain the positive results obtained so far in this direction. It is not our intention to give a complete survey on all aspects of pointed Hopf algebras.

We will mainly report on recent progress in the classification of pointed Hopf algebras with abelian group of group-like elements.

If the group Γ is abelian, and V is a finite-dimensional Yetter-Drinfeld module, then the braiding is given by a family of scalars q_{ij} ∈ 𝕜, 1 ≤ i ≤ θ, in the form

c(x_{i} ⊗ x_{j}) = q_{ij}x_{j} ⊗ x_{i}, where x_{1}, …, x_{θ} is a basis of V.

Moreover there are elements g_{1}, …, g_{θ} ∈ Γ, and characters χ_{1}, …, χ_{θ} ∈ Γ̂ such that q_{ij} = χ_{j}(g_{i}). The group acts on x_{i} via the character χ_{i}, and x_{i} is a g_{i}-homogeneous element with respect to the coaction of Γ. We introduced braidings of Cartan type [AS2] where

q_{ij}q_{ji} = q_{ii}^{a_{ij}}, 1 ≤ i, j ≤ θ, and (a_{ij}) is a generalized Cartan matrix.

If (a_{ij}) is a Cartan matrix of finite type, then the algebras 𝔅(V) can be understood as twisting of the Frobenius-Lusztig kernels 𝔲^{≥0}(𝔤), 𝔤 a semisimple Lie algebra.

By deforming the quantum Serre relations for simple roots which lie in two different connected components of the Dynkin diagram, we define finite-dimensional pointed Hopf algebras 𝔲(D) in terms of a ”linking datum D of finite Cartan type“ [AS4]. They generalize the Frobenius-Lusztig kernels 𝔲(𝔤) and are liftings of 𝔅(V)#𝕜Γ.

In some cases linking data of finite Cartan type are general enough to obtain complete classification results.

For example, if Γ = (ℤ/(p))^{s}, p a prime > 17 and s ≥ 1, we have determined the structure of all finite-dimensional Hopf algebras A with G(A) ≃ Γ. They are all of the form 𝔲(D) [AS4].

Similar data allow a classification of infinite-dimensional pointed Hopf algebras A with abelian group G(A), without zero divisors and finite Gelfand-Kirillov dimension, in the case when the infinitesimal braiding is ”positive“ [AS5]. But the general case is more involved. We also have to deform the root vector relations of the 𝔲(𝔤)′s.

The structure of pointed Hopf algebras A with non-abelian group G(A) is widely unknown. One basic open problem is to decide which finite groups appear as groups of group-like elements of finite-dimensional pointed Hopf algebras which are link-indecomposable in the sense of [M2]. In our formulation, this problem is the main part of following question: given a finite group Γ, determine all Yetter-Drinfeld modules V over Γ such that 𝔅(V) is finite dimensional. On the one hand, there are a number of severe constraints on V [Gñ3]. See also the exposition in [A, 5.3.10]. On the other hand, it is very hard to prove the finiteness of the dimension, and in fact this was done only for a few examples [MiS], [FK], [FP] which are again related to root systems. The examples over the symmetric groups in [FK] were introduced to describe the cohomology ring of the flag variety. At this stage, the main difficulty is to decide when certain Nichols algebras over non-abelian groups, for example the symmetric groups 𝕊_{n}, are finite-dimensional.

The last Chapter can serve as a concrete illustration of the theory explained in this paper. We describe explicitly all finite-dimensional pointed Hopf algebras with abelian group G(A) and infinitesimal braiding of type A_{n} (up to some exceptional cases). The main results in this Chapter are new, and complete proofs are given. The only cases which were known before are the easy case A_{1} [AS1], and A_{2} [AS3].

The new relations concern the root vectors e_{i,j}, 1 ≤ i < j ≤ n + 1. The relations e_{i,j}^{N} = 0 in 𝔲_{q}^{≥0}(sl_{n}), q a root of unity of order N, are replaced by

e_{i,j}^{N} = u_{i,j} for a family u_{i,j} ∈ 𝕜Γ, 1 ≤ i < j ≤ n + 1,

depending on a family of free parameters in 𝕜.

To study the relations between a filtered object and its associated graded object is a basic technique in modern algebra. We would like to stress that finite dimensional pointed Hopf algebras enjoy a remarkable rigidity; it is seldom the case that one is able to describe precisely all the liftings of a graded object, as in this context.

Acknowledgements. We would like to thank Jacques Alev, Matías Graña and Eric Müller for suggestions and stimulating discussions.

Conventions. As said above, our ground field 𝕜 is algebraically closed field of characteristic 0. Throughout, “Hopf algebra” means “Hopf algebra with bijective antipode”.

We denote by τ : V ⊗ W → W ⊗ V the usual transposition, that is τ(v ⊗ w) = w ⊗ v.

We use Sweedler’s notation for the comultiplication and coaction; but, to avoid confusions, we use the following variant for the comultiplication of a braided Hopf algebra R: Δ_{R}(r) = r^{(1)} ⊗ r^{(2)}.

1. Braided Hopf algebras

1.1. Braided categories.

Braided Hopf algebras play a central rôle in this paper. Although we have tried to minimize the use of categorical language, we briefly and informally recall the notion of a braided category which is the appropriate setting for braided Hopf algebras.

Braided categories were introduced in [JS]. We refer to [Ka, Ch. XI, Ch. XIII] for a detailed exposition. There is a hierarchy of categories with a tensor product functor:

a). A monoidal or tensor category is a collection (C, ⊗, a, 𝕀, ℓ, r), where

• C is a category and ⊗ : C × C → C is a functor,

• 𝕀 is an object of C, and

• a : V ⊗ (W ⊗ U) → (V ⊗ W) ⊗ U, ℓ : V → V ⊗ 𝕀, r : V → 𝕀 ⊗ V are natural isomorphisms; such that the so-called “pentagon” and “triangle” axioms are satisfied, see [Ka, Ch. XI, (2.6) and (2.9)]. These axioms essentially express that the tensor product of a finite number of objects is well-defined, regardless of the place where parentheses are inserted; and that 𝕀 is a unit for the tensor product.

b). A braided (tensor) category is a collection (C, ⊗, a, 𝕀, ℓ, r, c), where

• (C, ⊗, a, 𝕀, ℓ, r) is a monoidal category and

• c_{V,W} : V ⊗ W → W ⊗ V is a natural isomorphism; such that the so-called “hexagon” axioms are satisfied, see [Ka, Ch. XIII, (1.3) and (1.4)]. A very important consequence of the axioms of braided category, is the following equality for any objects V, W, U:

(1.1) (c_{V,W} ⊗ id_{U})(id_{V} ⊗ c_{U,W})(c_{U,V} ⊗ id_{W}) = (id_{W} ⊗ c_{U,V})(c_{U,W} ⊗ id_{V})(id_{U} ⊗ c_{V,W}),

see [Ka, Ch. XIII, (1.8)]. For simplicity we have omitted the associativity morphisms.

c). A symmetric category is a braided category where c_{V,W} c_{W,V} = id_{W ⊗V} for all objects V, W. Symmetric categories have been studied since the pioneering work of Mac Lane.

d). A left dual of an object V of a monoidal category, is a collection (V^{∗}, ev_{V} , b_{V}), where V^{∗} is another object and ev_{V} : V^{∗} ⊗ V → 𝕀, b_{V} : 𝕀 → V ⊗ V^{∗} are morphisms such that the compositions

V ⟶ 𝕀 ⊗ V ⟶^{b_{V} ⊗ id_{V}} V ⊗ V^{∗} ⊗ V ⟶^{id_{V} ⊗ ev_{V}} V ⊗ 𝕀 ⟶ V

and

V^{∗} ⟶ V^{∗} ⊗ 𝕀 ⟶^{id_{V^{∗}} ⊗ b_{V}} V^{∗} ⊗ V ⊗ V^{∗} ⟶^{ev_{V} ⊗ id_{V^{∗}}} 𝕀 ⊗ V^{∗} ⟶ V^{∗}

are, respectively, the identity of V and V^{∗}. A braided category is rigid if any object V admits a left dual [Ka, Ch. XIV, Def. 2.1].

1.2. Braided vector spaces and Yetter-Drinfeld modules.

We begin with the fundamental

Definition 1.1. Let V be a vector space and c : V ⊗ V → V ⊗ V a linear isomorphism. Then (V, c) is called a braided vector space, if c is a solution of the braid equation, that is

(1.2) (c ⊗ id )(id ⊗c)(c ⊗ id ) = (id ⊗c)(c ⊗ id )(id ⊗c).

It is well-known that the braid equation is equivalent to the quantum Yang-Baxter equation:

R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12}. (1.3)

Here we use the standard notation: R_{13} : V ⊗V ⊗V → V ⊗V ⊗V is the map given by ∑_{j} r_{j}⊗id ⊗r^{j}, if R = ∑_{j} r_{j} ⊗ r^{j}. Similarly for R_{12}, R_{23}.

The equivalence between solutions of (1.2) and solutions of (1.3) is given by the equality c = τ ∘ R. For this reason, some authors call (1.2) the quantum Yang-Baxter equation.

An easy and for this paper important example is given by a family of non-zero scalars q_{ij} ∈ 𝕜, i, j ∈ I, where V is a vector space with basis x_{i}, i ∈ I. Then

c(x_{i} ⊗ x_{j}) = q_{ij}x_{j} ⊗ x_{i}, for all i, j ∈ I

is a solution of the braid equation.

Examples of braided vector spaces come from braided categories. In this article, we are mainly concerned with examples related to the notion of Yetter-Drinfeld modules.

Definition 1.2. Let H be a Hopf algebra. A (left) Yetter-Drinfeld module V over H is simultaneously a left H-module and a left H-comodule satisfying the compatibility condition

δ(h.v) = h_{(1)}v_{(−1)}Sh_{(3)} ⊗ h_{(2)}.v_{(0)}, v ∈ V, h ∈ H. (1.4)

We denote by _{H}^{H}YD the category of Yetter-Drinfeld modules over H; the morphisms in this category preserve both the action and the coaction by H. The category _{H}^{H}YD is a braided monoidal category; indeed the tensor product of two Yetter-Drinfeld modules is again a Yetter-Drinfeld module, with the usual tensor product module and comodule structure. The compatibility condition (1.4) is not difficult to verify.

For any two Yetter-Drinfeld-modules M and N, the braiding c_{M,N} : M ⊗ N → N ⊗ M is given by

c_{M,N}(m ⊗ n) = m_{(−1)}.n ⊗ m_{(0)}, m ∈ M, n ∈ N, (1.5)

The subcategory of _{H}^{H}YD consisting of finite dimensional Yetter-Drinfeld modules is rigid. Namely, if V ∈ _{H}^{H}YD is finite-dimensional, the dual V^{∗} = Hom(V, 𝕜) is in _{H}^{H}YD with the following action and coaction:

• (h · f)(v) = f(S(v)) for all h ∈ H, f ∈ V^{∗}, v ∈ V.

• If f ∈ V^{∗}, then δ(f) = f_{(−1)} ⊗ f_{(0)} is determined by the equation

f_{(−1)}f_{(0)}(v) = S^{−1}(v_{−1})f(v_{0}), v ∈ V.

Then the usual evaluation and coevaluation maps are morphisms in _{H}^{H}YD.

Let V, W be two finite-dimensional Yetter-Drinfeld modules over H. We shall consider the isomorphism Φ : W^{∗} ⊗ V^{∗} → (V ⊗ W)^{∗} given by

Φ(φ ⊗ ψ)(v ⊗ w) = ψ(v)φ(w), φ ∈ W^{∗}, ψ ∈ V^{∗}, v ∈ V, w ∈ W. (1.6)

Remark 1.3. We see that a Yetter-Drinfeld module is a braided vector space. Conversely, a braided vector space (V, c) can be realized as a Yetter-Drinfeld module over some Hopf algebra H if and only if c is rigid [Tk1]. If this is the case, it can be realized in many different ways.

We recall that a Hopf bimodule over a Hopf algebra H is simultaneously a bimodule and a bicomodule satisfying all possible compatibility conditions. The category _{H}^{H}M_{H}^{H} of all Hopf bimodules over H is a braided category. The category _{H}^{H}YD is equivalent, as a braided category, to the category of Hopf bimodules. This was essentially first observed in [Wo] and then independently in [AnDe, Appendix], [Sbg], [Ro1].

POINTED HOPF ALGEBRAS 7

If H is a finite dimensional Hopf algebra, then the category _{H}^{H}YD is equivalent to the category of modules over the double of H [Mj1]. The braiding in _{H}^{H}YD corresponds to the braiding given by the “canonical” R-matrix of the double. In particular, if H is a semisimple Hopf algebra then _{H}^{H}YD is a semisimple category. Indeed, it is known that the double of a semisimple Hopf algebra is again semisimple.

The case of Yetter-Drinfeld modules over group algebras is especially important for the applications to pointed Hopf algebras. If H = 𝕜Γ, where Γ is a group, then an H-comodule V is just a Γ-graded vector space: V = ⊕_{g∈Γ}V_{g}, where V_{g} = {v ∈ V | δ(v) = g ⊗ v}. We will write _{Γ}^{Γ}YD for the category of Yetter-Drinfeld modules over 𝕜Γ, and say that V ∈ _{Γ}^{Γ}YD is a Yetter-Drinfeld module over Γ (when the field is fixed).

Remark 1.4. Let Γ be a group, V a left 𝕜Γ-module, and a left 𝕜Γ-comodule with grading V = ⊕_{g∈Γ}V_{g}. We define a linear isomorphism c : V ⊗ V → V ⊗ V by

c(x ⊗ y) = gy ⊗ x, for all x ∈ V_{g}, g ∈ Γ, y ∈ V. (1.7)

Then a) V ∈ _{Γ}^{Γ}YD if and only if gV_{h} ⊂ V_{ghg^{−1}} for all g, h ∈ Γ.

b) If V ∈ _{Γ}^{Γ}YD, then (V, c) is a braided vector space.

c) Conversely, if V is a faithful Γ-module (that is, if for all g ∈ Γ, gv = v for all v ∈ V, implies g = 1), and if (V, c) is a braided vector space, then V ∈ _{Γ}^{Γ}YD.

Proof. a) is clear from the definition.

By applying both sides of the braid equation to elements of the form x ⊗ y ⊗ z, x ∈ V_{g}, y ∈ V_{h}, z ∈ V, it is easy to see that (V, c) is a braided vector space if and only if

c(gy ⊗ gz) = ghz ⊗ gy, for all g, h ∈ Γ, y ∈ V_{h}, z ∈ V. (1.8)

Let us write gy = ∑_{a∈Γ} x_{a}, where x_{a} ∈ V_{a} for all a ∈ Γ. Then c(gy ⊗ gz) = ∑_{a∈Γ} agz ⊗ x_{a}. Hence (1.8) means that agz = ghz, for all z ∈ V and a ∈ Γ such that the homogeneous component x_{a} is not zero. This proves b) and c).

Remark 1.5. If Γ is abelian, a Yetter-Drinfeld module over H = 𝕜Γ is nothing but a Γ-graded Γ-module.

Assume that Γ is abelian and furthermore that the action of Γ is diagonalizable (this is always the case if Γ is finite). That is, V = ⊕_{χ∈Γ̂}V^{χ}, where V^{χ} = {v ∈ V | gv = χ(g)v for all g ∈ Γ}. Then

V = ⊕_{g∈Γ,χ∈Γ̂}V_{g}^{χ}, (1.9) where V_{g}^{χ} = V^{χ} ∩ V_{g}. Conversely, any vector space with a decomposition (1.9) is a Yetter-Drinfeld module over Γ. The braiding is given by

c(x ⊗ y) = χ(g)y ⊗ x, for all x ∈ V_{g}, g ∈ Γ, y ∈ V^{χ}, χ ∈ Γ̂.

It is useful to characterize abstractly those braided vector spaces which come from Yetter-Drinfeld modules over groups or abelian groups. The first part of the following definition is due to M. Takeuchi.

Definition 1.6. Let (V, c) be a finite dimensional braided vector space.

• (V, c) is of group type if there exists a basis x_{1}, …, x_{θ} of V and elements g_{i}(x_{j}) ∈ V for all i, j such that

c(x_{i} ⊗ x_{j}) = g_{i}(x_{j}) ⊗ x_{i}, 1 ≤ i, j ≤ θ; (1.10)

necessarily g_{i} ∈ GL(V).

• (V, c) is of finite group type (resp. of abelian group type) if it is of group type and the subgroup of GL(V) generated by g_{1}, …, g_{θ} is finite (resp. abelian).

• (V, c) is of diagonal type if V has a basis x_{1}, …, x_{θ} such that

c(x_{i} ⊗ x_{j}) = q_{ij}x_{j} ⊗ x_{i}, 1 ≤ i, j ≤ θ, (1.11)

for some q_{ij} in 𝕜. The matrix (q_{ij}) is called the matrix of the braiding.

• If (V, c) is of diagonal type, then we say that it is indecomposable if for all i and 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 can also refer then to the components of the matrix.

If V ∈ _{Γ}^{Γ}YD is finite-dimensional with braiding c, then (V, c) is of group type by (1.4). Conversely, assume that (V, c) is a finite-dimensional braided vector space of group type. Let Γ be the subgroup of GL(V) generated by g_{1}, …, g_{θ}. Define a coaction by δ(x_{i}) = g_{i} ⊗ x_{i} for all i. Then V is a Yetter-Drinfeld module over Γ with braiding c by Remark 1.4 , c).

A braided vector space of diagonal type is clearly of abelian group type; it is of finite group type if the q_{ij}’s are roots of one.

1.3. Braided Hopf algebras.

The notion of “braided Hopf algebra” is one of the basic features of braided categories. We will deal in this paper only with braided Hopf algebras in categories of Yetter-Drinfeld modules, mainly over a group algebra.

Let H be a Hopf algebra. First, the tensor product in _{H}^{H}YD allows to define algebras and coalgebras in _{H}^{H}YD. Namely, an algebra in the category _{H}^{H}YD is an associative algebra (R, m), where m : R ⊗ R → R is the product, with unit u : 𝕜 → R, such that R is a Yetter-Drinfeld module over H and both m and u are morphisms in _{H}^{H}YD.

Similarly, a coalgebra in the category _{H}^{H}YD is a coassociative coalgebra (R, Δ), where Δ : R → R ⊗ R is the coproduct, with counit ε : R → 𝕜, such that R is a Yetter-Drinfeld module over H and both Δ and ε are morphisms in _{H}^{H}YD.

Let now R, S be two algebras in _{H}^{H}YD. Then the braiding c : S ⊗ R → R ⊗ S allows to provide the Yetter-Drinfeld module R ⊗ S with a ”twisted” version of algebra in _{H}^{H}YD. Namely, the product in R ⊗ S is m_{R⊗S} := (m_{R} ⊗ m_{S})(id ⊗c ⊗ id ):

R ⊗ S ⊗ R ⊗ S ⟶ R ⊗ S
id ⊗c⊗id ↓  ↓^{=}
R ⊗ R ⊗ S ⊗ S ⟶^{m_{R}⊗m_{S}} R ⊗ S.

We shall denote this algebra by R\underline{⊗}S. The difference with the usual tensor product algebra is the presence of the braiding c instead of the usual transposition τ.

Definition 1.7. A braided bialgebra in _{H}^{H}YD is a collection (R, m, u, Δ, ε), where

• (R, m, u) is an algebra in _{H}^{H}YD.

• (R, Δ, ε) is a coalgebra in _{H}^{H}YD.

• Δ : R → R\underline{⊗}R is a morphism of algebras.

• u : 𝕜 → R and ε : R → 𝕜 are morphisms of algebras.

We say that it is a braided Hopf algebra in _{H}^{H}YD if in addition:

• The identity is convolution invertible in End (R); its inverse is the antipode of R.

A graded braided Hopf algebra in _{H}^{H}YD is a braided Hopf algebra R in _{H}^{H}YD provided with a grading R = ⊕_{n≥0}R(n) of Yetter-Drinfeld modules, such that R is a graded algebra and a graded coalgebra.

Remark 1.8. There is a non-categorical version of braided Hopf algebras, see [Tk1]. Any braided Hopf algebra in _{H}^{H}YD gives rise to a braided Hopf algebra in the sense of [Tk1] by forgetting the action and coaction, and preserving the multiplication, comultiplication and braiding. For the converse see [Tk1, Th. 5.7]. Analogously, one can define graded braided Hopf algebras in the spirit of [Tk1].

Let R be a finite-dimensional Hopf algebra in _{H}^{H}YD. The dual S = R^{∗} is a braided Hopf algebra in _{H}^{H}YD with multiplication Δ_{R}^{∗}Φ and comultiplication Φ^{−1}μ_{R}^{∗}, cf. (1.6); this is R^{∗bop} in the notation of [AG, Section 2].

In the same way, if R = ⊕_{n≥0}R(n) is a graded braided Hopf algebra in _{H}^{H}YD with finitedimensional homogeneous components, then the graded dual S = R^{∗} = ⊕_{n≥0}R(n)^{∗} is a graded braided Hopf algebra in _{H}^{H}YD.

1.4. Examples. The quantum binomial formula.

We shall provide many examples of braided Hopf algebras in Chapter 2. Here we discuss a very simple class of braided Hopf algebras.

We first recall the well-known quantum binomial formula. Let U and V be elements of an associative algebra over 𝕜[q], q an indeterminate, such that V U = qUV. Then

(U + V)^{n} = ∑_{1≤i≤n} \binom{n}{i}_{q}U^{i}V^{n−i}, if n ≥ 1. (1.12)

Here

\binom{n}{i}_{q} = \frac{(n)_{q}!}{(i)_{q}!(n − i)_{q}!}, where (n)_{q}! = ∏_{1≤i≤n}(i)_{q}, and (i)_{q} = ∑_{0≤j≤i−1} q^{j}.

By specialization, (1.12) holds for q ∈ 𝕜. In particular, if U and V are elements of an associative algebra over 𝕜, and q is a root of 1 of order dividing n, such that V U = qUV then

(U + V)^{n} = U^{n} + V^{n}. (1.13)

Example 1.9. Let (q_{ij})_{1≤i,j≤θ} be a matrix such that

(1.14)  q_{ij}q_{ji} = 1, 1 ≤ i, j ≤ θ, i ≠ j.

Let N_{i} be the order of q_{ii}, when this is finite.

Let R be the algebra presented by generators x_{1}, …, x_{θ} with relations

x_{i}^{N_{i}} = 0, if ord q_{ii} < ∞. (1.15)

x_{i}x_{j} = q_{ij}x_{j}x_{i}, 1 ≤ i < j ≤ θ. (1.16)

Given a group Γ and elements g_{1}, …, g_{θ} in the center of Γ, and characters χ_{1}, …, χ_{θ} of Γ, there exists a unique structure of Yetter-Drinfeld module over Γ on R, such that

x_{i} ∈ R_{g_{i}}^{χ_{i}}, 1 ≤ i ≤ θ.

Note that the braiding is determined by

c(x_{i} ⊗ x_{j}) = q_{ij} x_{j} ⊗ x_{i}, where q_{ij} = χ_{j}(g_{i}), 1 ≤ i, j ≤ θ.

Furthermore, R is a braided Hopf algebra with the comultiplication given by Δ(x_{i}) = x_{i}⊗1+1⊗x_{i}. To check that the comultiplication preserves (1.15) one uses (1.13); the verification for (1.16) is easy. We know [AS1] that dim R is infinite unless all the orders of q_{ii}’s are finite; in this last case, dim R = ∏_{1≤i≤θ} N_{i}. We also have P (R) = ⊕_{1≤i≤θ}𝕜x_{i}.

1.5. Biproducts, or bosonizations.

Let A, H be Hopf algebras and π : A → H and ι : H → A Hopf algebra homomorphisms. Assume that πι = id_{H}, so that π is surjective, and ι is injective. By analogy with elementary group theory, one would like to reconstruct A from H and the kernel of π as a semidirect product. However, the natural candidate for the kernel of π is the algebra of coinvariants

R := A^{co π} = {a ∈ A : (id ⊗π)Δ(a) = a ⊗ 1}

which is not, in general, a Hopf algebra. Instead, R is a braided Hopf algebra in _{H}^{H}YD with the following structure:

• The action · of H on R is the restriction of the adjoint action (composed with ι).

• The coaction is (π ⊗ id )Δ.

• R is a subalgebra of A.

• The comultiplication is Δ_{R}(r) = r_{(1)}ιπS(r_{(2)}) ⊗ r_{(3)}, for all r ∈ R.

Given a braided Hopf algebra R in _{H}^{H}YD, one can consider the bosonization or biproduct of R by H [Ra], [Mj2]. This is a usual Hopf algebra R#H, with underlying vector space R ⊗ H, whose multiplication and comultiplication are given by

\begin{aligned}(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)}. \end{aligned} (1.17)

The maps π : R#H → H and ι : H → R#H, π(r#h) = ϵ(r)h, ι(h) = 1#h, are Hopf algebra homomorphisms; we have R = {a ∈ R#H : (id ⊗π)Δ(a) = a ⊗ 1}.

Conversely, if A and H are Hopf algebras as above and R = A^{co π} then A ≃ R#H.

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

ϑ(ab) = a_{(1)}ϑ(b)ιπS(a_{(2)}), (1.18)

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

ϑ(ι(h)a) = h · ϑ(a). (1.19)

Notice also that ϑ induces a coalgebra isomorphism A/Aι(H)^{+} ≃ 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 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 defined by

ad_{c}x(y) = μ(μ ⊗ S_{R})(id ⊗c)(Δ_{R} ⊗ id )(x ⊗ y),

12 ANDRUSKIEWITSCH AND SCHNEIDER

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 ad_{c}x(y) = μ(id −c)(x ⊗ y) =: [x, y]_{c}. (1.20) For any x, y ∈ R, we call [x, y]_{c} := μ(id −c)(x ⊗ y) a braided commutator.

When A = R#H, then for all b, d ∈ R, ad_{(b#1)}(d#1) = (ad_{c}b(d))#1. (1.21) 1.6. Some properties of braided Hopf algebras.

In this Section, we first collect several useful facts about braided Hopf algebras in the category of Yetter-Drinfeld modules over an abelian group Γ. We begin with some identities on braided commutators.

In the following two Lemmata, R denotes a braided Hopf algebra in _{Γ}^{Γ}YD. Let a_{1}, a_{2}, ⋯ ∈ R be elements such that a_{i} ∈ R_{g_{i}}^{χ_{i}}, for some χ_{i} ∈ Γ̂, g_{i} ∈ Γ. Lemma 1.10. (a). [[a_{1}, a_{2}]_{c}, a_{3}]_{c} + χ_{2}(g_{1})a_{2}[a_{1}, a_{3}]_{c} = [a_{1}, [a_{2}, a_{3}]_{c}]_{c} + χ_{3}(g_{2})[a_{1}, a_{3}]_{c}a_{2}. (1.22)

(b). If [a_{1}, a_{2}]_{c} = 0 and [a_{1}, a_{3}]_{c} = 0 then [a_{1}, [a_{2}, a_{3}]_{c}]_{c} = 0.

(c). If [a_{1}, a_{3}]_{c} = 0 and [a_{2}, a_{3}]_{c} = 0 then [[a_{1}, a_{2}]_{c}, a_{3}]_{c} = 0.

(d). Assume that χ_{1}(g_{2})χ_{2}(g_{1})χ_{2}(g_{2}) = 1. Then

[[a_{1}, a_{2}]_{c}, a_{2}]_{c} = χ_{2}(g_{1})^{−1}χ_{1}(g_{2})[a_{2}, [a_{2}, a_{1}]_{c}]_{c} (1.23) Proof. Left to the reader.

The following technical Lemma will be used at a crucial point in Section 6.1. Lemma 1.11. Assume that χ_{2}(g_{2}) ≠ −1 and

(1.24)  χ_{1}(g_{2})χ_{2}(g_{1})χ_{2}(g_{2}) = 1,
(1.25)  χ_{2}(g_{3})χ_{3}(g_{2})χ_{2}(g_{2}) = 1.

If

(1.26)  [a_{2}, a_{2}]_{c}, a_{1}]_{c} = 0,
(1.27)  [[a_{2}, a_{2}]_{c}, a_{3}]_{c} = 0,
(1.28)  [a_{1}, a_{3}]_{c} = 0,

then [[[a_{1}, a_{2}]_{c}, a_{3}]_{c}, a_{2}]_{c} = 0. (1.29)

Proof. We compute:

[[[a_{1}, a_{2}]_{c}, a_{3}]_{c}, a_{2}]_{c} = a_{1}a_{2}a_{3}a_{2} − χ_{2}(g_{1}) a_{2}a_{1}a_{3}a_{2} − χ_{3}(g_{1})χ_{3}(g_{2}) a_{3}a_{1}a_{2}^{2}
+ χ_{3}(g_{1})χ_{3}(g_{2})χ_{2}(g_{1}) a_{3}a_{2}a_{1}a_{2} − χ_{2}(g_{1})χ_{2}(g_{2})χ_{2}(g_{3}) a_{2}a_{1}a_{2}a_{3}
+ χ_{2}(g_{1})^{2}χ_{2}(g_{2})χ_{2}(g_{3}) a_{2}^{2}a_{1}a_{3} + χ_{2}(g_{1})χ_{2}(g_{2})χ_{2}(g_{3})χ_{3}(g_{1})χ_{3}(g_{2}) a_{2}a_{3}a_{1}a_{2}
− χ_{2}(g_{1})^{2}χ_{2}(g_{2})χ_{2}(g_{3})χ_{3}(g_{1})χ_{3}(g_{2}) a_{2}a_{3}a_{2}a_{1}.

We index consecutively the terms in the right-hand side by roman numbers: (I), …, (V III). Then (II) + (V II) = 0, by (1.25) and (1.28). Now,

(I) = \frac{1}{χ_{3}(g_{2})(1 + χ_{2}(g_{2}))}a_{1}a_{2}^{2}a_{3} + \frac{χ_{2}(g_{2})χ_{3}(g_{2})}{1 + χ_{2}(g_{2})}a_{1}a_{3}a_{2}^{2}

= \frac{1}{χ_{3}(g_{2})(1 + χ_{2}(g_{2}))}a_{1}a_{2}^{2}a_{3} + \frac{χ_{2}(g_{2})χ_{3}(g_{2})χ_{3}(g_{1})}{1 + χ_{2}(g_{2})}a_{3}a_{1}a_{2}^{2}

= (I a) + (I b),

by (1.27) and (1.28). By the same equations (1.27) and (1.28), we also have

(V III) = −\frac{χ_{2}(g_{1})^{2}χ_{2}(g_{2})χ_{2}(g_{3})χ_{3}(g_{1})}{1 + χ_{2}(g_{2})}a_{2}^{2}a_{3}a_{1} − \frac{χ_{2}(g_{1})^{2}χ_{2}(g_{2})χ_{2}(g_{3})χ_{3}(g_{1})χ_{3}(g_{2})^{2}}{1 + χ_{2}(g_{2})} a_{3}a_{2}^{2}a_{1}

= −\frac{χ_{2}(g_{1})^{2}χ_{2}(g_{2})χ_{2}(g_{3})}{1 + χ_{2}(g_{2})}a_{2}^{2}a_{1}a_{3} − \frac{χ_{2}(g_{1})^{2}χ_{2}(g_{2})χ_{2}(g_{3})χ_{3}(g_{1})χ_{3}(g_{2})^{2}}{1 + χ_{2}(g_{2})} a_{3}a_{2}^{2}a_{1}

= (V III a) + (V III b).

We next use (1.26) to show that

(I a) + (V) + (V I) + (V III a) = 0,
(I b) + (III) + (IV) + (V III b) = 0.

In the course of the proof of these equalities, we need (1.24) and (1.25). This finishes the proof of (1.29).

Let H be a Hopf algebra. Then the existence of an integral for finite-dimensional braided Hopf algebras implies Lemma 1.12. Let R = ⨁_{n=0}^{N} R(n) be a finite-dimensional graded braided Hopf algebra in _{H}^{H}YD with R(N) ≠ 0. There exists λ ∈ R(N) which is a left integral on R and such that R(i) ⊗ R(N − i) → 𝕜, x ⊗ y ↦ λ(xy), is a non-degenerate pairing, for all 0 ≤ i ≤ N. In particular,

dimR(i) = dimR(N − i).

Proof. This is essentially due to Nichols [N, 1.5]. In this formulation, one needs the existence of non-zero integrals on R; this follows from [FMS]. See [AG, Prop. 3.2.2] for details.

1.7. The infinitesimal braiding of Hopf algebras whose coradical is a Hopf subalgebra.

For the convenience of the reader, we first recall in this Section some basic definitions from coalgebra theory.

Definition 1.13. Let C be a coalgebra.

• G(C) := {x ∈ C \ {0} | Δ(x) = x ⊗ x} is the set of all group-like elements of C.

• If g, h ∈ G(C), then x ∈ C is (g, h)-skew primitive if Δ(x) = x ⊗ h + g ⊗ x. The space of all (g, h)-skew primitive elements of C is denoted by P(C)_{g,h}. If C is a bialgebra or a braided bialgebra, and g = h = 1, then P (C) = P(C)_{1,1} is the space of primitive elements.

• The coradical of C is C_{0} := ∑ D, where D runs through all the simple subcoalgebras of C; it is the largest cosemisimple subcoalgebra of C. In particular, 𝕜G(C) ⊆ C_{0}.

• C is pointed if 𝕜G(C) = C_{0}.

• The coradical filtration of C is the ascending filtration C_{0} ⊆ C_{1} ⊆ ⋯ ⊆ C_{j} ⊆ C_{j+1} ⊆ …, defined by C_{j+1} := {x ∈ C | Δ(x) ∈ C_{j} ⊗ C + C ⊗ C_{0}}. This is a coalgebra filtration: ΔC_{j} ⊆ ∑_{0≤i≤j} C_{i} ⊗ C_{j−i}; and it is exhaustive: C = ⋃_{n≥0} C_{n}.

• A graded coalgebra is a coalgebra G provided with a grading G = ⊕_{n≥0}G(n) such that ΔG(j) ⊆ ∑_{0≤i≤j} G(i) ⊗ G(j − i) for all j ≥ 0.

• A coradically graded coalgebra [CM] is a graded coalgebra G = ⊕_{n≥0}G(n) such that its coradical filtration coincides with the standard ascending filtration arising from the grading: (gr C)_{n} = ⊕_{m≤n}gr C(m). A strictly graded coalgebra [Sw] is a coradically graded coalgebra G such that G(0) is one-dimensional.

• The graded coalgebra associated to the coalgebra filtration of C is gr C = ⊕_{n≥0}gr C(n), where gr C(n) := C_{n}/C_{n−1}, n > 0, gr C(0) := C_{0}. It is a coradically graded coalgebra.

We shall need a basic technical fact on pointed coalgebras.

Lemma 1.14. [M1, 5.3.3]. A morphism of pointed coalgebras which is injective in the first term of the coalgebra filtration, is injective.

Let now A be a Hopf algebra. We shall assume in what follows that the coradical A_{0} is not only a subcoalgebra but a Hopf subalgebra of A; this is the case if A is pointed.

To study the structure of A, we consider its coradical filtration; because of our assumption on A, it is also an algebra filtration [M1]. Therefore, the associated graded coalgebra gr A is a graded Hopf algebra. Furthermore, H := A_{0} ≃ gr A(0) is a Hopf subalgebra of gr A; and the projection π : gr A → gr A(0) with kernel ⊕_{n>0}gr A(n), is a Hopf algebra map and a retraction of the inclusion. We can then apply the general remarks of Section 1.5. Let R be the algebra of coinvariants of π; R is a braided Hopf algebra in _{H}^{H}YD and gr A can be reconstructed from R and H as a bosonization gr A ≃ R#H.

The braided Hopf algebra R is graded, since it inherits the gradation from gr A: R = ⊕_{n≥0}R(n), where R(n) = gr A(n) ∩ R. Furthermore, R is strictly graded; this means, (a). R(0) = 𝕜1 (hence the coradical is trivial, cf. [Sw, Chapter 11]).

(b). R(1) = P (R) (the space of primitive elements of R).

It is in general not true that a braided Hopf algebra R satisfying (a) and (b), also satisfies (c). R is generated as an algebra over 𝕜 by R(1).

A braided graded Hopf algebra satisfying (a), (b) and (c) is called a Nichols algebra. In the next chapter we will discuss this notion in detail. Notice that the subalgebra R′ of R generated by R(1), a Hopf subalgebra of R, is indeed a Nichols algebra.

Definition 1.15. The braiding

c : V ⊗ V → V ⊗ V,

of V := R(1) = P (R) is called the infinitesimal braiding of A.

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

The dimension of V = P (R) is called the rank of A.

2. Nichols algebras

Let H be a Hopf algebra. In this Chapter, we discuss a functor 𝔅 from the category _{H}^{H}YD to the category of braided Hopf algebras in _{H}^{H}YD; given a Yetter-Drinfeld module V, the braided Hopf algebra 𝔅(V) is called the Nichols algebra of V.

The structure of a Nichols algebra appeared first in the paper ”Bialgebras of type one“ [N] of Nichols and was rediscovered later by several authors. In our language, a bialgebra of type one is just a bosonization 𝔅(V)#H. Hence Nichols algebras are the H-coinvariant elements of bialgebras of type one, also called quantum symmetric algebras in [Ro2]. Several years after [N], Woronowicz defined Nichols algebras in his approach to ”quantum differential calculus” [Wo]; again, they appeared as the invariant part of his ”algebra of quantum differential forms”. Lusztig’s algebras 𝔣 [L3], defined by the non-degeneracy of a certain invariant bilinear form, are Nichols algebras. In fact Nichols algebras can always be defined by the non-degeneracy of an invariant bilinear form [AG]. The algebras 𝔅(V) are called bitensor algebras in [Sbg].

In a sense, Nichols algebras are similar to symmetric algebras; indeed, both notions coincide in the trivial braided category of vector spaces, or more generally in any symmetric category (e. g. in the category of super vector spaces). But when the braiding is not a symmetry, a Nichols algebra could have a much richer structure. We hope that this will be clarified in the examples. On the other hand, Nichols algebras are also similar to universal enveloping algebras. However, in spite of the efforts of several authors, it is not clear to us how to achieve a compact, functorial definition of a ”braided Lie algebra” from a Nichols algebra.

We believe that Nichols algebras are very interesting objects of an essentially new nature.

2.1. Definition of Nichols algebras.

We now present one of the main notions of this survey.

Definition 2.1. Let V be a Yetter-Drinfeld module over H. 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

P (R) = R(1), (2.1)

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

The dimension of V will be called the rank of R.

We need some preliminaries to show the existence and uniqueness of the Nichols algebra of V in _{H}^{H}YD.

Let V be a Yetter-Drinfeld module over H. Then the tensor algebra T (V) = ⨁_{n≥0} T (V)(n) of the vector space V admits a natural structure of a Yetter-Drinfeld module, since _{H}^{H}YD is a braided category. It is then an algebra in _{H}^{H}YD. There exists a unique algebra map Δ : T (V) → T (V)\underline{⊗}T (V) such that Δ(v) = v ⊗ 1 + 1 ⊗ v, for all v ∈ V. For example, if x, y ∈ V, then

Δ(xy) = 1 ⊗ xy + x ⊗ y + x_{(−1)} · y ⊗ x_{(0)} + yx ⊗ 1.

With this structure, T (V) is a graded braided Hopf algebra in _{H}^{H}YD with counit ε : T (V) → 𝕜, ε(v) = 0, if v ∈ V. To show the existence of the antipode, one notes that the coradical of the coalgebra T (V) is 𝕜, and uses a result of Takeuchi [M1, 5.2.10]. Hence all the braided bialgebra quotients of T (V) in _{H}^{H}YD are braided Hopf algebras in _{H}^{H}YD.

Let us consider the class 𝔖 of all I ⊂ T (V) such that

• I is a homogeneous ideal generated by homogeneous elements of degree ≥ 2,

• I is also a coideal, i. e. Δ(I) ⊂ I ⊗ T (V) + T (V) ⊗ I.

Note that we do not require that the ideals I are Yetter-Drinfeld submodules of T (V). Let then 𝔖̃ be the subset of 𝔖 consisting of all I ∈ 𝔖 which are Yetter-Drinfeld submodules of T (V). The ideals

I(V) = ∑_{I∈𝔖}I,  Ĩ(V) = ∑_{J∈𝔖̃}J

are the largest elements in 𝔖, respectively 𝔖̃.

POINTED HOPF ALGEBRAS 17

If I ∈ 𝔖 then R := T (V)/I = ⊕_{n≥0}R(n) is a graded algebra and a graded coalgebra with

R(0) = 𝕜, V ≃ R(1) ⊂ P (R).

If actually I ∈ 𝔖̃, then R is a graded braided Hopf algebra in _{H}^{H}YD.

We can show now existence and uniqueness of Nichols algebras.

Proposition 2.2. Let 𝔅(V) := T (V)/Ĩ(V). Then the following hold:

1. V = P (𝔅(V)), hence 𝔅(V) is a Nichols algebra of V.

2. I(V) = Ĩ(V).

3. Let R = ⊕_{n≥0}R(n) be a graded Hopf algebra in _{H}^{H}YD such that R(0) = 𝕜1 and R is generated as an algebra by V := R(1). Then there exists a surjective map of graded Hopf algebras R → 𝔅(V), which is an isomorphism of Yetter-Drinfeld modules in degree 1.

4. Let R = ⊕_{n≥0}R(n) be a Nichols algebra of V. Then R ≃ 𝔅(V) as braided Hopf algebras in _{H}^{H}YD.

5. Let R = ⨁_{n≥0} R(n) be a graded braided Hopf algebra in _{H}^{H}YD with R(0) = 𝕜1 and R(1) = P (R) = V. Then 𝔅(V) is isomorphic to the subalgebra 𝕜⟨V ⟩ of R generated by V.

Proof. 1. We have to show the equality V = P (𝔅(V)). Let us consider the inverse image X in T (V) of all homogeneous primitive elements of 𝔅(V) in degree n ≥ 2. Then X is a graded Yetter-Drinfeld submodule of T (V), and for all x ∈ X, Δ(x) ∈ x⊗1+1⊗x+T (V)⊗Ĩ(V)+Ĩ(V)⊗T (V). Hence the ideal generated by Ĩ(V) and X is in 𝔖̃, and X ⊂ Ĩ(V) by the maximality of Ĩ(V). Hence the image of X in 𝔅(V) is zero. This proves our claim since the primitive elements form a graded submodule.

2. We have to show that the surjective map 𝔅(V) → T (V)/I(V) is bijective. This follows from 1. and Lemma 1.14.

3. The kernel I of the canonical projection T (V) → R belongs to 𝔖̃; hence I ⊆ Ĩ(V).

4. follows again from Lemma 1.14, as in 2.

5. follows from 4.

If U is a braided subspace of V ∈ _{H}^{H}YD, that is a subspace such that c(U ⊗U) ⊂ U ⊗U, where c is the braiding of V, we can define 𝔅(U) := T (U)/I(U) with the obvious meaning of I(U). Then the description stated in Proposition 2.8 also applies to 𝔅(U).

Corollary 2.3. The assignment V ↦ 𝔅(V) is a functor from _{H}^{H}YD to the category of braided Hopf algebras in _{H}^{H}YD.

If U is a Yetter-Drinfeld submodule of V, or more generally if U is a braided subspace of V, then the canonical map 𝔅(U) → 𝔅(V) is injective.

18 ANDRUSKIEWITSCH AND SCHNEIDER

Proof. If ϕ : U → V is a morphism in _{H}^{H}YD, then T (ϕ) : T (U) → T (V) is a morphism of braided Hopf algebras. Since T (ϕ)(I(U)) is a coideal and a Yetter-Drinfeld submodule of T (V), the ideal generated by T (ϕ)(Ĩ(U)) is contained in Ĩ(V). Hence by Proposition 2.2, 𝔅 is a functor.

The second part of the claim follows from Proposition 2.2, 5.

The duality between conditions (2.1) and (2.2) in the definition of Nichols algebra, emphasized by Parts 3 and 5 of Proposition 2.2, is explicitly stated in the following

Lemma 2.4. Let R = ⊕_{n≥0}R(n) be a graded braided Hopf algebra in _{H}^{H}YD; suppose finitedimensional homogeneous components and R(0) = 𝕜1. Let S = ⊕_{n≥0}R(n)^{∗} be the graded dual of R. Then R(1) = P (R) if and only if S is generated as an algebra by S(1).

Proof. See for instance [AS2, Lemma 5.5].

Example 2.5. Let F be a field of positive characteristic p. Let S be the (usual) Hopf algebra F [x]/⟨x^{p^{2}}⟩ with x ∈ P (S). Then x^{p} ∈ P (S). Hence R = S^{∗} satisfies (2.2) but not (2.1).

Example 2.6. Let S = 𝕜[X] = ⊕_{n≥0}S(n) be a polynomial algebra in one variable. We consider S as a braided Hopf algebra in _{H}^{H}YD, where H = 𝕜Γ, Γ an infinite cyclic group with generator g, with action, coaction and comultiplication given by

δ(X^{n}) = g^{n} ⊗ X^{n}, g · X = qX, Δ(X) = X ⊗ 1 + 1 ⊗ X.

Here q ∈ 𝕜 is a root of 1 of order N. That is, S is a so-called quantum line. Then S satisfies (2.2) but not (2.1) since X^{N} is also primitive. Hence the graded dual R = S^{d} = ⊕_{n≥0}S(n)^{∗} is a braided Hopf algebra satisfying (2.1) but not (2.2).

However, in characteristic 0 we do not know any finite dimensional example of a braided Hopf algebra satisfying (2.1) but not (2.2).

Conjecture 2.7. [AS2, Conjecture 1.4] Any finite dimensional braided Hopf algebra in _{H}^{H}YD satisfying (2.1) also satisfies (2.2). (Recall that the base field 𝕜 has characteristic zero.)

The compact description of 𝔅(V) in Lemma 2.2 shows that it depends only on the algebra and coalgebra structure of T (V). Since the comultiplication of the tensor algebra was defined using the ”twisted” multiplication of T (V)\underline{⊗}T (V), we see that 𝔅(V) depends as an algebra and coalgebra only on the braiding of V. The explicit formula for the comultiplication of T (V) leads to the following alternative description of 𝔅(V).

2.2. The braid group.

Let us recall that the braid group 𝔹_{n} is presented by generators σ_{1}, …, σ_{n−1} with relations

σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1}, 1 ≤ i ≤ n − 2,
σ_{i}σ_{j} = σ_{j}σ_{i}, , 1 ≤ i, j ≤ n − 2, |i − j| > 1.

Here are some basic well-known facts about the braid group.

There is a natural projection π : 𝔹_{n} → 𝕊_{n} sending σ_{i} to the transposition τ_{i} := (i, i + 1) for all i. The projection π admits a set-theoretical section s : 𝕊_{n} → 𝔹_{n} determined by

s(τ_{i}) = σ_{i}, 1 ≤ i ≤ n − 1,
s(τω) = s(τ)s(ω), if ℓ(τω) = ℓ(τ) + ℓ(ω).

Here ℓ denotes the lenght of an element of 𝕊_{n} with respect to the set of generators τ_{1}, …, τ_{n−1}. The map s is called the Matsumoto section. In other words, if ω = τ_{i_{1}} … τ_{i_{M}} is a reduced expression of ω ∈ 𝕊_{n}, then s(ω) = σ_{i_{1}} … σ_{i_{M}}.

Let q ∈ 𝕜, q ≠ 0. The quotient of the group algebra 𝕜(𝔹_{n}) by the two-sided ideal generated by the relations

(σ_{i} − q)(σ_{i} + 1), 1 ≤ i ≤ n − 1,

is the so-called Hecke algebra of type A_{n}, denoted by H_{q}(n).

Using the section s, the following distinguished elements of the group algebra 𝕜𝔹_{n} are defined:

𝔖_{n} := ∑_{σ∈𝕊_{n}} s(σ), 𝔖_{i,j} := ∑_{σ∈X_{i,j}} s(σ);

here X_{i,j} ⊂ 𝕊_{n} is the set of all (i, j)-shuffles. The element 𝔖_{n} is called the quantum symmetrizer.

Given a braided vector space (V, c), there are representations of the braid groups ρ_{n} : 𝔹_{n} → Aut (V^{⊗n}) for any n ≥ 0, given by

ρ_{n}(σ_{i}) = id ⊗ ⋯ ⊗ id ⊗c ⊗ id ⊗ ⋯ ⊗ id ,

where c acts in the tensor product of the i and i + 1 copies of V. By abuse of notation, we shall denote by 𝔖_{n}, 𝔖_{i,j} also the corresponding endomorphisms ρ(𝔖_{n}), ρ(𝔖_{i,j}) of V^{⊗n} = T^{n}(V).

If C = ⨁_{n≥0} C(n) is a graded coalgebra with comultiplication Δ, we denote by Δ_{i,j} : C(i+j) → C(i) ⊗ C(j), i, j ≥ 0, the (i, j)-graded component of the map Δ.

Proposition 2.8. Let V ∈ _{H}^{H}YD. Then

Δ_{i,j} = 𝔖_{i,j}, (2.3)

𝔅(V) = ⨁_{n≥0} T^{n}(V)/ker(𝔖_{n}). (2.4)

Proof. See for instance [Sbg].

This description of the relation of 𝔅(V) does not mean that the relations are known. In general it is very hard to compute the kernels of the maps 𝔖_{n} in concrete terms. For any braided vector space (V, c), we may define 𝔅(V) by (2.4).

2.3. Invariance under twisting.

Twisting is a method to construct new Hopf algebras by ”deforming” the comultiplication; originally due to Drinfeld [Dr2], it was adapted to Hopf algebras in [Re].

Let A be a Hopf algebra and F ∈ A ⊗ A be an invertible element. Let Δ_{F} := F ΔF^{−1} : A → A ⊗ A; it is again an algebra map. If

(2.5)  (1 ⊗ F)(id ⊗Δ)(F) = (F ⊗ 1)(Δ ⊗ id )(F),
(2.6)  (id ⊗ε)(F) = (ε ⊗ id )(F) = 1,

then A_{F} (the same algebra, but with comultiplication Δ_{F}) is again a Hopf algebra. We shall say that A_{F} is obtained from A via twisting by F; F is a cocycle in a suitable sense.

There is a dual version of the twisting operation, which amounts to a twist of the multiplication [DT]. Let A be a Hopf algebra and let σ : A × A → 𝕜 be an invertible 2-cocycle^{1}, that is

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 ∈ A. Then A_{σ} – the same A 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)}).

Assume now that H is a Hopf algebra, R is a braided Hopf algebra in _{H}^{H}YD, and A = R#H. Let π : A → H and ι : H → A be the canonical projection and injection. Let σ : H × H → 𝕜 be an invertible 2-cocycle, and define σ_{π} : A × A → 𝕜 by

σ_{π} := σ(π ⊗ π);

σ_{π} is an invertible 2-cocycle, with inverse (σ^{−1})_{π}. The maps π : A_{σ_{π}} → H_{σ}, ι : H_{σ} → A_{σ_{π}} are still Hopf algebra maps. Because the comultiplication is not changed, the space of coinvariants of π is R; this is a subalgebra of A_{σ_{π}} that we denote R_{σ}; the multiplication in R_{σ} is given by

x._{σ}y = σ(x_{(−1)}, y_{(−1)})x_{(0)}y_{(0)}, x, y ∈ R = R_{σ}. (2.7)

Equation (2.7) follows easily using (1.17). Clearly, R_{σ} is a Yetter-Drinfeld Hopf algebra in _{H_{σ}}^{H_{σ}}YD. The coaction of H_{σ} on R_{σ} is the same as the coaction of H on R, since the comultiplication was not altered. The explicit formula for the action of H_{σ} on R_{σ} can be written down; we shall do this only in the setting we are interested in.

Let H = 𝕜Γ be a group algebra; an invertible cocycle σ : H × H → 𝕜 is uniquely determined by its restriction σ : Γ × Γ → 𝕜^{×}, a group cocycle with respect to the trivial action.

1 Here ”invertible” means that the associated linear map σ : A ⊗ A → 𝕜 is invertible with respect to the convolution product.

POINTED HOPF ALGEBRAS 21

Lemma 2.9. Let Γ be an abelian group and let R be a braided Hopf algebra in _{Γ}^{Γ}YD. Let σ : Γ × Γ → 𝕜^{×} be a cocycle. Let S be the subalgebra of R generated by P (R). In the case y ∈ S_{h}^{η}, for some h ∈ Γ and η ∈ Γ̂, the action of H = H_{σ} on R_{σ} is

g ⇀_{σ} y = σ(g, h)σ^{−1}(h, g)η(g)y, g ∈ Γ. (2.8)

Hence, the braiding c_{σ} in R_{σ} is given in this case by

c_{σ}(x ⊗ y) = σ(g, h)σ^{−1}(h, g)η(g) y ⊗ x, x ∈ R_{g}, g ∈ Γ. (2.9)

Therefore, for such x and y, we have

[x, y]_{c_{σ}} = σ(g, h)[x, y]_{c}. (2.10)

Proof. To prove (2.8), it is enough to assume y ∈ P (R)_{h}^{η}.

Let A = R#H; in A_{σ_{π}}, we have

g._{σ}y = σ(g, π(y)) g σ^{−1}(g, 1) + σ(g, h) gy σ^{−1}(g, 1) + σ(g, h) gh σ^{−1}(g, π(y))
= σ(g, h)gy;
y._{σ}g = σ(π(y), g) g σ^{−1}(1, g) + σ(h, g) yg σ^{−1}(1, g) + σ(h, g) hg σ^{−1}(π(y), g)
= σ(h, g)yg;

hence

g._{σ}y = σ(g, h)gy = σ(g, h)η(g)yg = σ(g, h)σ^{−1}(h, g)η(g)y._{σ}g,

which is equivalent to (2.8). Now (2.9) follows at once, and (2.10) follows from (2.7) and (2.9):

[x, y]_{c_{σ}} = x._{σ}y − ._{σ}c_{σ}(x ⊗ y) = σ(g, h)xy − σ(g, h)σ^{−1}(h, g)η(g) σ(h, g)yx = σ(g, h)[x, y]_{c}.

The proof of the following Lemma is clear, since the comultiplication of a Hopf algebra is not changed by twisting.

Lemma 2.10. Let H be a Hopf algebra and let R be a braided Hopf algebra in _{H}^{H}YD. Let σ : H × H → 𝕜 be an invertible 2-cocycle. Then R is a Nichols algebra if and only if R_{σ} is a Nichols algebra in _{H_{σ}}^{H_{σ}}YD.

3. Types of Nichols algebras

We now discuss several examples of Nichols algebras. We are interested in explicit presentations, e. g. by generators and relations, of 𝔅(V), for braided vector spaces in suitable classes, for instance, those of group type. We would also like to determine when 𝔅(V) has finite dimension, or polynomial growth.

22 ANDRUSKIEWITSCH AND SCHNEIDER

3.1. Symmetries and braidings of Hecke type. We begin with the simplest class of braided vector spaces.

Example 3.1. Let τ : V ⊗ V → V ⊗ V be the usual transposition; the braided vector space (V, τ) can be realized as a Yetter-Drinfeld module over any Hopf algebra H, with trivial action and coaction. Then 𝔅(V) = Sym (V), the symmetric algebra of V.

The braided vector space (V, −τ), which can be realized e. g. in _{ℤ}^{ℤ}YD, has 𝔅(V) = Λ(V), the exterior algebra of V.

Example 3.2. Let V = ⊕_{i∈ℤ/2}V (i) be a super vector space and let c : V ⊗ V → V ⊗ V be the supersymmetry:

c(v ⊗ w) = (−1)^{i.j}w ⊗ v v ∈ V (i), w ∈ V (j).

Clearly, V can be realized as a Yetter-Drinfeld module over ℤ/2. Then 𝔅(V) ≃ Sym (V (0)) ⊗ Λ(V (1)), the super-symmetric algebra of V.

The simple form of 𝔅(V) in these examples can be explained in the following context.

Definition 3.3. We say that a braided vector space (V, c) is of Hecke-type with label q ∈ 𝕜, q ≠ 0, if

(c − q)(c + 1) = 0.

In this case, the representation of the braid group ρ_{n} : 𝔹_{n} → Aut (V^{⊗n}) factorizes through the Hecke algebra H_{q}(n), for all n ≥ 0; cf. Section 2.2.

If q = 1, one says that c is a symmetry. Then ρ_{n} factorizes through the symmetric group 𝕊_{n}, for all n ≥ 0. The categorical version of symmetries is that of symmetric categories, see Section 1.1.

Proposition 3.4. Let (V, c) be a braided vector space of Hecke-type with label q, wich is either 1 or not a root of 1. Then 𝔅(V) is a quadratic algebra; that is, the ideal I(V) is generated by I(V)(2) = Ker 𝔖_{2}.

Moreover, 𝔅(V) is a Koszul algebra and its Koszul dual is the Nichols algebra 𝔅(V^{∗}) corresponding to the braided vector space (V^{∗}, q^{−1}c^{t}).

A nice exposition on Koszul algebras is [BGS, Chapter 2].

Proof. The argument for the first claim is taken from [AA, Prop. 3.3.1]. The image of the quantum symmetrizer 𝔖_{n} in the Hecke algebra H_{q}(n) is [n]_{q}!M_{ε}, where M_{ε} satsfies the following properties:

M_{ε}^{2} = M_{ε}, M_{ε}c_{i} = c_{i}M_{ε} = qM_{ε}, 1 ≤ i ≤ n − 1.

See for instance [HKW]. Now, we have to show that Ker 𝔖_{n} = T^{n}(V) ∩ I, where I is the ideal generated by Ker 𝔖_{2} = Ker (c + 1) = Im (c − q); but clearly T^{n}(V) ∩ I = ∑_{i} I^{n,i}, where

I^{n,i} = T^{i−1}(V) ⊗ Im (c − q) ⊗ T^{n−i−1}(V) = Im (c_{i} − q).

POINTED HOPF ALGEBRAS 23

It follows that T^{n}(V) ∩ I ⊆ Ker 𝔖_{n}, a fact that we already know from the general theory. But moreover, T^{n}(V) ∩ I is a H_{q}(n)-submodule of T^{n}(V) since

c_{j}(c_{i} − q) = (c_{j} − q)(c_{i} − q) + q(c_{i} − q).

This computation also shows that the action of H_{q}(n) on the quotient module T^{n}(V)/T^{n}(V) ∩ I is via the character that sends σ_{i} to q; hence M_{ε} acts on T^{n}(V)/T^{n}(V) ∩ I by an automorphism, and a fortiori T^{n}(V) ∩ I ⊇ Ker 𝔖_{n}. Having shown the first claim, the second claim is essentially a result from [Gu, Wa]; see also the exposition in [AA, Sections 3.3 and 3.4].

Example 3.5. Let q ∈ 𝕜^{×}, q is not a root of 1. The braided vector space (V, qτ) can be realized in _{ℤ}^{ℤ}YD. It can be shown that 𝔅(V) = T (V), the tensor algebra of V, for all q in an open set. Problem: Determine this open set.

It would be interesting to know whether other conditions on the minimal polynomial of a braiding have consequences on the structure of the corresponding Nichols algebra. The first candidate should be a braiding of BMW-type.

3.2. Braidings of diagonal type.

In this Section, (V, c) denotes a finite dimensional braided vector space of diagonal type; that is, V has a basis x_{1}, …, x_{θ} such that (1.11) holds for some non-zero q_{ij} in 𝕜. Our first goal is to determine polynomial relations on the generators x_{1}, …, x_{θ} that should hold in 𝔅(V). We look at polynomial expressions in these generators which are homogeneous of degree ≥ 2, and give rise to primitive elements in any braided Hopf algebra containing V inside its primitive elements. For related material, see [Kh].

Lemma 3.6. Let R be a braided Hopf algebra in _{H}^{H}YD, for some Hopf algebra H, such that V ⊆ P (R) as braided vector spaces. (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 1 ≤ i, j ≤ θ, i ≠ j, such that q_{ij}q_{ji} = q_{ii}^{r}, where 0 ≤ −r < ord q_{ii} (which could be infinite).

Then (ad_{c}x_{i})^{1−r}(x_{j}) is primitive in R.

Proof. (a) and (b) are consequences of the quantum binomial formula, see e. g. [AS2, Appendix] for (b).

We apply these first remarks to 𝔅(V) and see how conditions on the Nichols algebra induce conditions on the braiding.

Lemma 3.7. Let R = 𝔅(V). (a). If q_{ii} is a root of 1 of order N > 1 then x_{i}^{N} = 0. In particular, if 𝔅(V) is an integral domain,

then q_{hh} = 1 or it is not a root of 1, for all h. (b). If i ≠ j, then (ad_{c}x_{i})^{r}(x_{j}) = (r)!_{q_{ii}} ∏_{0≤k≤r} (1 − q_{ii}^{k}q_{ij}q_{ji}) x_{i}^{r}x_{j}.

24 ANDRUSKIEWITSCH AND SCHNEIDER

(c). If i ≠ j and q_{ij}q_{ji} = q_{ii}^{r}, where 0 ≤ −r < ord q_{ii} (which could be infinite), then (ad_{c}x_{i})^{1−r}(x_{j}) = 0.

(d). If 𝔅(V) has finite growth, then for all i ≠ j, there exists r_{ij} > 0 such that (ad_{c}x_{i})^{r_{ij}}(x_{j}) = 0.

(e). If 𝔅(V) is a domain of finite growth and there exists k such that q_{kk} = 1, then q_{ii} = 1, for all 1 ≤ i ≤ θ, and q_{ij}q_{ji} = 1 for all 1 ≤ i, j ≤ θ, i ≠ j.

(f). If 𝔅(V) has finite growth and i ≠ j, then min{r > 0 | (ad_{c}x_{i})^{r}(x_{j}) = 0} = min ({ord N_{ii}} ⋃{k > 0 | q_{ij}q_{ji}q_{ii}^{k} = 1}) .

Proof. Parts (a) and (c) follow from Lemma 3.6. Part (b) is apparently well-known and appears in [Ro2] without proof. Part(d) is [Ro2, Lemma 20]. The rest follows from these parts.

We now discuss how the twisting operation, cf. Section 2.3, affects Nichols algebras of diagonal type.

Definition 3.8. We shall say that two braided vector spaces (V, c) and (W, d) of diagonal type, with matrices (q_{ij}) and (q̂_{ij}), are twist-equivalent if dim V = dim W and, for all i, j, q_{ii} = q̂_{ii} and

q_{ij}q_{ji} = q̂_{ij}q̂_{ji}. (3.1)

Proposition 3.9. Let (V, c) and (W, d) be two twist-equivalent braided vector spaces of diagonal type, with matrices (q_{ij}) and (q̂_{ij}); say with respect to basis x_{1}, … x_{θ}, resp. x̂_{1}, … x̂_{θ}. Then there exists a linear isomorphism ψ : 𝔅(V) → 𝔅(W) such that

ψ(x_{i}) = x̂_{i}, 1 ≤ i ≤ θ. (3.2)

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

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

We consider V, W as Yetter-Drinfeld modules over Γ by declaring x_{i} ∈ V_{g_{i}}^{χ_{i}}, x̂_{i} ∈ V χ̂_{i} ; g_{i}. Hence, 𝔅(V), 𝔅(W) are braided Hopf algebras in _{Γ}^{Γ}YD.

Let σ : Γ × Γ → 𝕜^{×} be the unique bilinear form such that

σ(g_{i}, g_{j}) = \{ \begin{matrix}q̂_{ij}q_{ij}^{−1},  i ≤ j, \\ 1,  i > j; \end{matrix} (3.3)

it is a group cocycle. We claim that φ : W → 𝔅(V)_{σ}(1), φ(x̂_{i}) = x_{i}, 1 ≤ i ≤ θ, is an isomorphism in _{Γ}^{Γ}YD. It clearly preserves the coaction; for the action, we assume i ≤ j and compute

g_{j} ·_{σ} x_{i} = σ(g_{j}, g_{i})σ^{−1}(g_{i}, g_{j})χ_{i}(g_{j})x_{i}
= (q̂_{ij})^{−1}q_{ij}q_{ji}x_{i} = q̂_{ji}x_{i},

and also

g_{i} ·_{σ} x_{j} = σ(g_{i}, g_{j})σ^{−1}(g_{j}, g_{i})χ_{j}(g_{i})x_{j}
= q̂_{ij}q_{ij}^{−1}q_{ij}x_{j} = q̂_{ij}x_{j},

where we have used (2.8) and the hypothesis (3.1). This proves the claim. By Lemma 3.6, φ extends to an isomorphism φ : 𝔅(W) → 𝔅(V)_{σ}; ψ = φ^{−1} is the map we are looking for.

Remarks 3.10. (i). The map ψ defined in the proof is much more than just linear; by (2.7) and (2.10), we have

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

(ii). A braided vector space (V, c) of diagonal type, with matrix (q_{ij}), is twist-equivalent to (W, d), with a symmetric matrix (q̂_{ij}).

3.3. Braidings of diagonal type but not Cartan.

In the next Chapter, we shall concentrate on braidings of Cartan type. There are a few examples of Nichols algebras 𝔅(V) of finite group type and rank 2, which are not of Cartan type, but where we know that the dimension is finite. We now list them all, following [N, Gñ3]. The braided vector space is necessarily of diagonal type; we shall give the matrix Q of the braiding, the constraints on their entries and the dimension d of 𝔅(V). Below, ω, resp. ζ, denotes an arbitrary primitive third root of 1, resp. different from 1.

(3.6) (\begin{matrix}q_{11} & q_{12} \\ q_{21} & −1 \end{matrix}); q_{11}^{−1} = q_{12}q_{21} ≠ 1;  d = 4 ord(q_{12}q_{21}).

(3.7) (\begin{matrix}q_{11} & q_{12} \\ q_{21} & ω \end{matrix}); q_{11}^{−1} = q_{12}q_{21} ≠ ±1, ω^{−1};  d = 9 ord(q_{11}) ord(q_{12}q_{21}ω).

(3.8) (\begin{matrix}−1 & q_{12} \\ q_{21} & ω \end{matrix}); q_{12}q_{21} = −1;  d = 108.

(3.9) (\begin{matrix}−1 & q_{12} \\ q_{21} & ω \end{matrix}); q_{12}q_{21} = ω;  d = 72.

(3.10) (\begin{matrix}−1 & q_{12} \\ q_{21} & ω \end{matrix}); q_{12}q_{21} = −ω;  d = 36.

(3.11) (\begin{matrix}−1 & q_{12} \\ q_{21} & ζ \end{matrix}); q_{12}q_{21} = ζ^{−2};  d = 4 ord(ζ) ord(−ζ^{−1}).

3.4. Braidings of finite non-abelian group type.

We begin with a class of examples studied in [MiS].

Let Γ be a group and T ⊂ Γ a subset such that for all g ∈ Γ, t ∈ T, gtg^{−1} ∈ T. Thus T is a union of conjugacy classes of Γ. Let ϕ : Γ × T → 𝕜 \ {0} be a function such that for all g, h ∈ Γ and t ∈ T,

(3.12) ϕ(1, t) = 1,
(3.13) ϕ(gh, t) = ϕ(g, hth^{−1})ϕ(h, t).

We can then define a Yetter-Drinfeld module V = V (Γ, T, ϕ) over Γ with 𝕜-basis x_{t}, t ∈ T, and action and coaction of W given by

(3.14) gx_{t} = ϕ(g, t)x_{gtg^{−1}},
(3.15) δ(x_{t}) = t ⊗ x_{t}

for all g ∈ Γ, t ∈ T.

Conversely, if the function ϕ defines a Yetter-Drinfeld module on the vector space V by (3.14), (3.15), then ϕ satisfies (3.12), (3.13).

Note that the braiding c of V (Γ, T, ϕ) is determined by

c(x_{s} ⊗ x_{t}) = ϕ(s, t)x_{sts_{−1}} ⊗ x_{t} for all s, t ∈ T,

hence by the values of ϕ on T × T.

The main examples come from the theory of Coxeter groups ([BL, Chapitre IV]). Let S be a subset of the group W of elements of order 2. For all s, s′ ∈ S let m(s, s′) be the order of ss′. (W, S) is called a Coxeter system and W a Coxeter group if W is generated by S with defining relations (ss′)^{m(s,s′)} = 1 for all s, s′ ∈ S such that m(s, s′) is finite.

Let (W, S) be a Coxeter system. For any g ∈ W there is a sequence (s_{1}, …, s_{q}) of elements in S with g = s_{1} · · · · · s_{q}. If q is minimal among all such representations, then q = l(g) is called the length of g, and (s_{1}, …, s_{q}) is a reduced representation of g.

Definition 3.11. Let (W, S) be a Coxeter system, and T = {gsg^{−1} | g ∈ W, s ∈ S}. Define ϕ : W × T → 𝕜 \ {0} by

ϕ(g, t) = (−1)^{l(g)} for all g ∈ W, t ∈ T. (3.16)

This ϕ satisfies (3.12) and (3.13). Thus we have associated to each Coxeter group the Yetter-Drinfeld module V (W, T, ϕ) ∈ _{W}^{W} YD.

The functions ϕ satisfying (3.12), (3.13) can be constructed up to a diagonal change of the basis from characters of the centralizers of elements in the conjugacy classes. This is a special case of the description of the simple modules in _{Γ}^{Γ}YD (see [W] and also [L4]); the equivalent classification of the simple Hopf bimodules over Γ was obtained in [DPR] (over 𝕜) and then in [Ci] (over any field).

Let t be an element in Γ. We denote by O_{t} and Γ^{t} the conjugacy class and the centralizer of t in Γ. Let U be any left 𝕜Γ^{t}-module. It is easy to see that the induced representation V = 𝕜Γ ⊗_{𝕜Γ^{t}} U is a Yetter-Drinfeld module over Γ with the induced action of Γ and the coaction

δ : V → 𝕜Γ ⊗ V, δ(g ⊗ u) = gtg^{−1} ⊗ g ⊗ u for all g ∈ Γ, u ∈ U.

We will denote this Yetter-Drinfeld module over Γ by M(t, U).

Assume that Γ is finite. Then V = M(t, U) is a simple Yetter-Drinfeld module if U is a simple representation of Γ^{t}, and each simple module in _{Γ}^{Γ}YD has this form. If we take from each conjugacy class one element t and non-isomorphic simple Γ^{t}-modules, any two of these simple Yetter-Drinfeld modules are non-isomorphic.

Let s_{i}, 1 ≤ i ≤ θ, be a complete system of representatives of the residue classes of Γ^{t}. We define t_{i} = s_{i}ts_{i}^{−1} for all 1 ≤ i ≤ θ. Thus

Γ/Γ^{t} → O_{t}, s_{i}Γ^{t} ↦ t_{i}, 1 ≤ i ≤ θ,

is bijective, and as a vector space, V = ⨁_{1≤i≤θ} s_{i} ⊗ U. For all g ∈ Γ and 1 ≤ i ≤ θ, there is a uniquely determined 1 ≤ j ≤ θ with s_{j}^{−1}gs_{i} ∈ Γ^{t}, and the action of g on s_{i} ⊗ u, u ∈ U, is given by

gs_{i} ⊗ u = s_{j} ⊗ (s_{j}^{−1}gs_{i})u.

In particular, if U is a one-dimensional Γ^{t}-module with basis u and action hu = χ(g)u for all h ∈ Γ^{t} defined by the character χ : Γ^{t} → 𝕜 \ {0}, then V has a basis x_{i} = s_{i} ⊗ u, 1 ≤ i ≤ θ, and the action and coaction of Γ are given by

gx_{i} = χ(s_{j}^{−1}gs_{i})x_{j} and δ(x_{i}) = x_{j},

if s_{j}^{−1}gs_{i} ∈ Γ^{t}. Note that gt_{i}g^{−1} = t_{j}. Hence the module we have constructed is V (Γ, T, ϕ), where T is the conjugacy class of t, and ϕ is given by ϕ(g, t_{i}) = χ(s_{j}^{−1}gs_{i}).

We now construct another example of a function ϕ satisfying (3.12), (3.13).

Definition 3.12. Let T be the set of all transpositions in the symmetric group 𝕊_{n}. Define ϕ : 𝕊_{n} × T → 𝕜 \ {0} for all g ∈ 𝕊_{n}, 1 ≤ i < j ≤ n, by

χ(g, (ij)) = \begin{cases}1 & , if g(i) < g(j), \\ −1 & , if g(i) > g(j). \end{cases} (3.17)

Let t = (12). The centralizer of t in 𝕊_{n} is ⟨(34), (45), …, (n − 1, n)⟩ ∪ ⟨(34), (45), …, (n − 1, n)⟩(12). Let χ be the character of (𝕊_{n})^{t} with χ((ij)) = 1 for all 3 ≤ i < j ≤ n, and χ((12)) = −1. Then the function ϕ definied by (3.17) is given by the character χ as described above.

Up to base change we have found all functions ϕ satisfying (3.12), (3.13) for Γ = 𝕊_{n}, where T is the conjugacy class of all transpositions, and ϕ(t, t) = −1 for all t =∈ T. The case ϕ(t, t) = 1 for some t ∈ T would lead to a Nichols algebra 𝔅(V) of infinite dimension.

To determine the structure of 𝔅(V) for the Yetter-Drinfeld modules defined by the functions ϕ in (3.16) and (3.17) seems to be a fundamental and very hard combinatorial problem. Only a few partial results are known [MiS], [FK], [FP].

We want to describe some of the relations of 𝔅(V). Let us first recall an important tool introduced by Nichols [N, 3.3] to deal with 𝔅(V) over group algebras 𝕜Γ without knowing the relations explicitly (see [MiS, 2.4]). Let V ∈ _{Γ}^{Γ}YD be of finite dimension θ. We choose a basis x_{i} ∈ V_{g_{i}} with g_{i} ∈ Γ, 1 ≤ i ≤ θ, of Γ-homogeneous elements. Let I ∈ 𝔖 and R = T (V)/I (see Section 2.1). Then R is a graded Hopf algebra in _{Γ}^{Γ}YD with R(0) = 𝕜1 and R(1) = V. For all 1 ≤ i ≤ θ let σ_{i} : R → R be the algebra automorphism given by the action of g_{i}.

Proposition 3.13. 1) For all 1 ≤ i ≤ θ, there exists a uniquely determined (id, σ_{i})-derivation D_{i} : R → R with D_{i}(x_{j}) = δ_{i,j} (Kronecker δ) for all j.

2) I = I(V), that is R = 𝔅(V), if and only if ⋂_{i=1}^{θ} ker(D_{i}) = 𝕜1.

These skew derivations are very useful to find relations of 𝔅(V). We consider some special cases.

Example 3.14. Let W = 𝕊_{n}, n ≥ 2, and T = {(ij) | 1 ≤ i < j ≤ n} the set of all transpositions. Define χ by (3.16) and let V = V (W, T, ϕ). Then the following relations hold in 𝔅(V) for all 1 ≤ i < j ≤ n, 1 ≤ k < l ≤ n:

(3.18)  x_{(ij)}^{2} = 0.
(3.19)  If {i, j} ∩ {k, l} = ∅, then  x_{(ij)}x_{(kl)} + x_{(kl)}x_{(ij)} = 0.
(3.20)  If i < j < k, then  x_{(ij)}x_{(jk)} + x_{(jk)}x_{(ik)} + x_{(ik)}x_{(ij)} = 0,
x_{(jk)}x_{(ij)} + x_{(ik)}x_{(jk)} + x_{(ij)}x_{(ik)} = 0.

Example 3.15. Let W = 𝕊_{n}, n ≥ 2, and T = {(ij) | 1 ≤ i < j ≤ n} the set of all transpositions. Define ϕ by (3.17) and let V = V (W, T, ϕ). Then the following relations hold in 𝔅(V) for all 1 ≤ i < j ≤ n, 1 ≤ k < l ≤ n:

(3.21)  x_{(ij)}^{2} = 0.
(3.22)  If {i, j} ∩ {k, l} = ∅, then x_{(ij)}x_{(kl)} − x_{(kl)}x_{(ij)} = 0.
(3.23)  If i < j < k, then x_{(ij)}x_{(jk)} − x_{(jk)}x_{(ik)} − x_{(ik)}x_{(ij)} = 0,
x_{(jk)}x_{(ij)} − x_{(ik)}x_{(jk)} − x_{(ij)}x_{(ik)} = 0.

The algebras 𝔅̃(V) generated by all x_{(ij)}, 1 ≤ i < j ≤ n, with the quadratic relations in the examples 3.14 resp. 3.15 are braided Hopf algebras in the category of Yetter-Drinfeld modules over 𝕊_{n}. 𝔅̃(V) in example 3.15 is the algebra E_{n} introduced by Fomin and Kirillov in [FK] to describe the cohomology ring of the flag variety. We believe that indeed the quadratic relations in the examples 3.14 and 3.15 are defining relations for 𝔅(V), that is 𝔅̃(V) = 𝔅(V) in these cases.

It was noted in [MiS] that the conjecture in [FK] about the ”Poincaré-duality“ of the dimensions of the homogeneous components of the algebras E_{n} (in case they are finite-dimensional) follows from the braided Hopf algebra structure as a special case of Lemma 1.12.

Another result about the algebras E_{n} by Fomin and Procesi [FP] says that E_{n+1} is a free module over E_{n}, and P_{E_{n}} divides P_{E_{n+1}}, where P_{A} denotes the Hilbert series of a graded algebra A. The proof in [FP] used the relations in Example 3.15.

This result is in fact a special case of a very general splitting theorem for braided Hopf algebras in [MiS, Theorem3.2] which is an application of the fundamental theorem for Hopf modules in the braided situation. This splitting theorem generalizes the main result of [Gñ2].

In [MiS] some partial results are obtained about the structure of the Nichols algebras of Coxeter groups. In particular

Theorem 3.16. [MiS] Let (W, S) be a Coxeter system, T the set of all W -conjugates of elements in S, ϕ defined by (3.16), V = V (W, T, ϕ) and R = 𝔅(V). For all g ∈ W, choose a reduced representation g = s_{1} ⋯ s_{q}, s_{1}, ⋯, s_{q} ∈ S, of g, and define

x_{g} = x_{s_{1}} ⋯ x_{s_{q}}.

Then the subalgebra of R generated by all x_{s}, s ∈ S has the 𝕜-basis x_{g}, g ∈ W. For all g ∈ W, the g-homogeneous component R_{g} of R is isomorphic to R_{1}.

If R is finite-dimensional, then W is finite and dim(R) = ord(W)dim(R_{1}).

This theorem holds for more general functions ϕ, in particular for S_{n} and ϕ defined in (3.17).

Let (W, S) be a Coxeter system and V = V (W, T, ϕ) as in Theorem [MiS]. Then 𝔅(V) was computed in [MiS] in the following cases:

• W = 𝕊_{3}, S = {(12), (23)}: The relations of 𝔅(V) are the quadratic relations in Example 3.14, and dim𝔅(V) = 12.

• W = 𝕊_{4}, S = {(12), (23), (34)}: The relations of 𝔅(V) are the quadratic relations in Example 3.14, and dim𝔅(V) = 24 · 24.

• W = D_{4}, the dihedral group of order 8, S = {t, t′}, where t, t′ are generators of D_{4} of order 2 such that tt′ is of order 4. There are quadratic relations and relations of order 4 defining 𝔅(V), and dim𝔅(V) = 64.

In all three cases the integral, which is the longest non-zero word in the generators x_{t}, can be described in terms of the longest element in the Coxeter group. In all the other cases it is not known whether 𝔅(V) is finite-dimensional.

In [FK] it is shown that

• dim(E_{3}) = 12.

• dim(E_{4}) = 24 · 24.

• dim(E_{5}) is finite by using a computer program. Again, for the other cases n > 5 it is not known whether E_{n} is finite-dimensional.

In [Gñ3, 5.3.2] another example of a finite-dimensional Nichols algebra of a braided vector space (V, c) of finite group type is given with dim(V) = 4 and dim(𝔅(V)) = 72. The defining relations of 𝔅(V) are quadratic and of order 6.

By a result of Montgomery [M2], any pointed Hopf algebra B can be decomposed as a crossed product

B ≃ A#_{σ}𝕜G, σ a 2-cocycle

of A, its link-indecomposable component containing 1 (a Hopf subalgebra) and a group algebra 𝕜G. However, the structure of such link-indecomposable Hopf algebras A, in particular in the case when A is finite-dimensional and the group of its group-like elements G(A) is non-abelian, is not known. To define link-indecomposable pointed Hopf algebras, we recall the definition of the quiver of A in [M2]. The vertices of the quiver of A are the elements of the group G(A); for g, h ∈ G(A), there exists an arrow from h to g if P_{g,h}(A) is non-trivial, that is if 𝕜(g − h) ⫋ P_{g,h}(A). The Hopf algebra A is called link-indecomposable, if its quiver is connected as an undirected graph.

Definition 3.17. Let Γ be a finite group and V ∈ _{Γ}^{Γ}YD. V is called link-indecomposable if the group Γ is generated by the elements g with V_{g} ≠ 0.

By [MiS, 4.2], V ∈ _{Γ}^{Γ}YD is link-indecomposable if and only if the Hopf algebra 𝔅(V)#𝕜Γ is link-indecomposable.

Thus by the examples constructed above, there are link-indecomposable, finite-dimensional pointed Hopf algebras A with G(A) isomorphic to S_{n}, 3 ≤ n ≤ 5, or to D_{4}.

Question 3.18. Which finite groups are isomorphic to G(A) for some finite-dimensional, linkindecomposable pointed Hopf algebra A? Are there finite groups which do not occur in this form?

Finally, let us come back to the simple Yetter-Drinfeld modules V = M(t, U) ∈ _{Γ}^{Γ}YD, where t ∈ Γ and U is a simple left Γ^{t}-module of dimension > 1. In this case, strong restrictions are known for 𝔅(V) to be finite-dimensional. By Schur’s lemma, t acts as a scalar q on U.

Proposition 3.19. [Gñ3, 3.1] Assume that dim 𝔅(V) is finite. If dim U ≥ 3, then q = −1; and if dim U = 2, then q = −1 or q is a root of unity of order three.

In the proof of Proposition 3.19, a result of Lusztig on braidings of Cartan type (see [AS2, Theorem 3.1]) is used. In a similar way Graña showed

Proposition 3.20. [Gñ3, 3.2] Let Γ be a finite group of odd order, and V ∈ _{Γ}^{Γ}YD. Assume that 𝔅(V) is finite-dimensional. Then the multiplicity of any simple Yetter-Drinfeld module over Γ as a direct summand in V is at most 2.

In particular, up to isomorphism there are only finitely many Yetter-Drinfeld modules V ∈ _{Γ}^{Γ}YD such that 𝔅(V) is finite-dimensional.

The second statement in Proposition 3.20 was a conjecture in a preliminary version of [AS2].

3.5. Braidings of (infinite) group type.

We briefly mention Nichols algebras over a free abelian group of finite rank with a braiding which is not diagonal.

Example 3.21. Let Γ = ⟨g⟩ be a free group in one generator. Let V(t, 2) be the Yetter-Drinfeld module of dimension 2 such that V(t, 2) = V(t, 2)_{g} and the action of g on V(t, 2) is given, in a basis x_{1}, x_{2}, by

g · x_{1} = tx_{1}, g · x_{2} = tx_{2} + x_{1}.

Here t ∈ 𝕜^{×}. Then:

(a). If t is not a root of 1, then 𝔅(V(t, 2)) = T (V(t, 2)).

(b). If t = 1, then 𝔅(V(1, 2)) = 𝕜 < x_{1}, x_{2}|x_{1}x_{2} = x_{2}x_{1} +x_{1}^{2} >; this is the well-known Jordanian quantum plane.

Example 3.22. More generally, if t ∈ 𝕜^{×}, let V(t, θ) be the Yetter-Drinfeld module of dimension θ ≥ 2 such that V(t, θ) = V(t, θ)_{g} and the action of g on V(t, θ) is given, in a basis x_{1}, …, x_{θ}, by

g · x_{1} = tx_{1}, g · x_{j} = tx_{j} + x_{j−1}, 2 ≤ j ≤ θ.

Note there is an inclusion of Yetter-Drinfeld modules V(t, 2) ↪ V(t, θ); hence, if t is not a root of 1, 𝔅(V(t, θ)) has exponential growth.

Question 3.23. Compute 𝔅(V(1, θ)); does it have finite growth?

4. Nichols algebras of Cartan type

We now discuss fundamental examples of Nichols algebras of diagonal type that come from the theory of quantum groups.

We first need to fix some notation. Let A = (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 g be the Kac-Moody algebra corresponding to the Cartan matrix A. Let X be the set of connected components of the Dynkin diagram corresponding to it. For each I ∈ X, we let 𝔤_{I} be the Kac-Moody Lie algebra corresponding to the generalized Cartan matrix (a_{ij})_{i,j∈I} and 𝔫_{I} be the Lie subalgebra of 𝔤_{I} spanned by all its positive roots. We omit the subindex I when I = {1, …, θ}. 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^{+}.

Example 4.1. Let q ∈ 𝕜, q ≠ 0, and consider the braided vector space (𝕍, c), where 𝕍 is a vector space with a basis x_{1}, …, x_{θ} and the braiding c is given by

c(x_{i} ⊗ x_{j}) = q^{d_{i}a_{ij}}x_{j} ⊗ x_{i}, (4.1)

Theorem 4.2. [L3] Let (𝕍, c) be a braided vector space with braiding matrix (4.1). If q is not a root of 1, then

𝔅(𝕍) = 𝕜⟨x_{1}, …, x_{θ}|ad_{c}(x_{i})^{1−a_{ij}}x_{j} = 0, i ≠ j⟩.

The Theorem says that 𝔅(𝕍) is the well-known ”positive part” U_{q}^{+}(g) of the quantum enveloping algebra of g.

To state the following important Theorem, we recall the definition of braided commutators (1.20). Lusztig defined root vectors X_{α} ∈ 𝔅(𝕍), α ∈ Φ^{+} [L2]. One can see from [L1, L2] 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}. See also [Ri].

Theorem 4.3. [L1, L2, L3, Ro1, Mu]. Let (𝕍, c) be a braided vector space with braiding matrix (4.1). Assume that q is a root of 1 of odd order N; and that 3 does not divide N if there exists I ∈ X of type G_{2}.

The algebra 𝔅(𝕍) is finite dimensional if and only if (a_{ij}) is a finite Cartan matrix.

If this happens, then 𝔅(𝕍) can be presented by generators X_{i}, 1 ≤ i ≤ θ, and relations

(4.2)  ad_{c}(X_{i})^{1−a_{ij}}(X_{j}) = 0,  i ≠ j,
(4.3)  X_{α}^{N} = 0,  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}} , 0 ≤ h_{j} ≤ N − 1, 1 ≤ j ≤ P.

The Theorem says that 𝔅(𝕍) is the well-known ”positive part” 𝔲_{q}^{+}(g) of the so-called Frobenius-Lusztig kernel of g.

Motivated by the preceding Theorems and results, we introduced the following notion in [AS2] (see also [FG]).

Definition 4.4. Let (V, c) a braided vector space of diagonal type with basis x_{1}, …, x_{θ}, and matrix (q_{ij}), that is

c(x_{i} ⊗ x_{j}) = q_{i,j}x_{j} ⊗ x_{i}, for all 1 ≤ i, j ≤ θ.

We shall say that (V, c) is of Cartan type if for all i, j, q_{ij} is a root of unity, q_{ii} ≠ 1 and there exists a_{ij} ∈ ℤ such that

q_{ij}q_{ji} = q_{ii}^{a_{ij}}, for all 1 ≤ i, j ≤ θ. (4.4)

The integers a_{ij} are uniquely determined when chosen in the following way:

a_{ii} = 2; (4.5) − ord q_{ii} < a_{ij} ≤ 0, i ≠ j. (4.6)

It follows that (a_{ij}) is a generalized Cartan matrix (GCM) in the sense of the book [K]. We shall adapt the terminology from generalized Cartan matrices and Dynkin diagrams to braidings of Cartan type. For instance, we shall say that (V, c) is of finite Cartan type if it is of Cartan type and the corresponding GCM is actually of finite type, i. e. a Cartan matrix associated to a finite dimensional semisimple Lie algebra. We shall say that a Yetter-Drinfeld module V is of Cartan type if the matrix (q_{ij}) as above is of Cartan type.

To formulate our first main result, we need one more definition from [AS2] for braidings which are close to the braidings of the Frobenius-Lusztig kernels.

Definition 4.5. Let (V, c) be a braided vector space of Cartan type with Cartan matrix (a_{ij}). We say that (V, c) is of FL-type if there exist positive integers d_{1}, …, d_{θ} such that

(4.7) For all i, j, d_{i}a_{ij} = d_{j}a_{ji} (thus (a_{ij}) is symmetrizable). (4.8) There exists q ∈ 𝕜 such that q_{ij} = q^{d_{i}a_{ij}} for all i, j.

We call (V, c) locally of FL-type if any principal 2 × 2 submatrix of (q_{ij}) defines a braiding of FL-type.

We now fix for each α ∈ Φ^{+} such a representation of X_{α} as an iterated braided commutator. For a general braided vector space (V, c) of finite Cartan type, 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 (q_{ij}).

Theorem 4.6. [AS2, AS4]. Let (V, c) be a braided vector space of Cartan type. We also assume that q_{ij} has odd order for all i, j.

34 ANDRUSKIEWITSCH AND SCHNEIDER

(i). Assume that (V, c) is locally of FL-type and that, for all i, the order of q_{ii} is relatively prime to 3 whenever a_{ij} = −3 for some j, and is different from 3, 5, 7, 11, 13, 17. If 𝔅(V) is finite dimensional, then (V, c) is of finite Cartan type.

(ii). If (V, c) is of finite Cartan type, then 𝔅(V) is finite dimensional, and if moreover 3 does not divide the order of q_{ii} for all i in a connected component of the Dynkin diagram of type G_{2}, then

dim 𝔅(V) = ∏_{I∈X} N_{I}^{dim 𝔫_{I}} ,

where N_{I} = ord(q_{ii}) for all i ∈ I and I ∈ X. The Nichols algebra 𝔅(V) is presented by generators x_{i}, 1 ≤ i ≤ θ, and relations

(4.9)  ad_{c}(x_{i})^{1−a_{ij}}(x_{j}) = 0,  i ≠ j,
(4.10)  x_{α}^{N_{I}} = 0,  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}} , 0 ≤ h_{j} ≤ N_{I} − 1, if β_{j} ∈ I, 1 ≤ j ≤ P.

Let 𝔅̂(V) be the braided Hopf algebra in _{Γ}^{Γ}YD generated by x_{1}, …, x_{θ} with relations (4.9), 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 4.7. [AS4] K(V) is a braided Hopf subalgebra in _{Γ}^{Γ}YD of 𝔅̂(V).

5. Classification of pointed Hopf algebras by the lifting method 5.1. Lifting of Cartan type.

We propose to subdivide the classification problem for finite-dimensional pointed Hopf algebras into the following problems: (a). Determine all braided vector spaces V of group type such that 𝔅(V) is finite dimensional. (b). Given a finite group Γ, determine all realizations of braided vector spaces V as in (a) as

Yetter-Drinfeld modules over Γ. (c). The lifting problem: For 𝔅(V) as in (a), compute all Hopf algebras A such that gr A ≃

𝔅(V)#H . (d). Investigate whether any finite dimensional pointed Hopf algebra is generated as an algebra

by its group-like and skew-primitive elements. Problem (a) was discussed in Chapters 3 and 4. We have seen the very important class of

braidings of finite Cartan type and some isolated examples where the Nichols algebra is finitedimensional. But the general case of problem (a) seems to require completely new ideas.

Problem (b) is of computational nature. For braidings of finite Cartan type with Cartan matrix (a_{ij})_{1≤i,j≤θ} and an abelian group Γ we have to compute elements g_{1}, …, g_{θ} ∈ Γ and characters χ_{1}, …, χ_{θ} ∈ Γ̂ such that

χ_{i}(g_{j})χ_{j}(g_{i}) = χ_{i}(g_{i})^{a_{ij}}, for all 1 ≤ i, j ≤ θ. (5.1)

To find these elements one has to solve a system of quadratic congruences in several unknowns. In many cases they do not exist. In particular, if θ > 2(ordΓ)^{2}, then the braiding cannot be realized over the group Γ. We refer to [AS2, Section 8] for details.

Problem (d) is the subject of Section 5.4.

We will now discuss the lifting problem (c).

The coradical filtration 𝕜Γ = A_{0} ⊂ A_{1} ⊂ … of a pointed Hopf algebra A is stable under the adjoint action of the group. For abelian groups Γ and finite-dimensional Hopf algebras, the following stronger result holds. It is the starting point of the lifting procedure, and we will use it several times.

If M is a 𝕜Γ-module, we denote by M^{χ} = {m ∈ M | gm = χ(g)m for all g ∈ Γ}, χ ∈ Γ̂, the isotypic component of type χ.

Lemma 5.1. Let A be a finite-dimensional Hopf algebra with abelian group G(A) = Γ and diagram R. Let V = R(1) ∈ _{Γ}^{Γ}YD with basis x_{i} ∈ V_{g_{i}}^{χ_{i}}, g_{i} ∈ Γ, χ_{i} ∈ Γ̂, 1 ≤ i ≤ θ.

(a). The isotypic component of trivial type of A_{1} is A_{0}. Therefore, A_{1} = A_{0} ⊕ (⊕_{χ≠ε}(A_{1})^{χ}) and

⊕_{χ≠ε}(A_{1})^{χ} ⟶^{≃} A_{1}/A_{0} ⟵^{≃} V #𝕜Γ. (5.2)

(b). For all g ∈ Γ, χ ∈ Γ̂ with χ ≠ ε,

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

Proof. (a) follows from [AS1, Lemma 3.1] and implies (b). See [AS1, Lemma 5.3].

We assume that A is a finite-dimensional pointed Hopf algebra with abelian group G(A) = Γ, and that

grA ≃ 𝔅(V)#𝕜Γ,

where V ∈ _{Γ}^{Γ}YD is a given Yetter-Drinfeld module with basis x_{i} ∈ V_{g_{i}}^{χ_{i}}, g_{1} …, g_{θ} ∈ Γ, χ_{1}, …, χ_{θ} ∈ Γ̂, 1 ≤ i ≤ θ.

We first lift the basis elements x_{i}. Using (5.2), we choose a_{i} ∈ P(A)_{g_{i},1}^{χ_{i}} such that the canonical image of a_{i} in A_{1}/A_{0} is x_{i} (which we identify with x_{i}#1), 1 ≤ i ≤ θ. Since the elements x_{i} together with Γ generate 𝔅(V)#𝕜Γ, it follows from a standard argument that a_{1}, …, a_{θ} and the elements in Γ generate A as an algebra.

Our aim is to find relations between the a′_{i}s and the elements in Γ which define a quotient Hopf algebra of the correct dimension dim𝔅(V) · ord(Γ). The idea is to ”lift“ the relations between the x′_{i}s and the elements in Γ in 𝔅(V)#𝕜Γ.

We now assume moreover that V is of finite Cartan type with Cartan matrix (a_{ij}) with respect to the basis x_{1}, …, x_{θ}, that is (5.1) holds. We also assume

(5.5) ord(χ_{j}(g_{i})) is odd for all i, j, (5.6) N_{i} = ord(χ_{i}(g_{i})) is prime to 3 for all i ∈ I, I ∈ X of type G_{2}.

We fix a presentation Γ = ⟨y_{1}⟩ ⊕ ⋯ ⊕ ⟨y_{σ}⟩, and denote by M_{ℓ} the order of y_{ℓ}, 1 ≤ ℓ ≤ σ. Then Theorem 4.6 and formulas (1.17) imply that 𝔅(V)#𝕜Γ can be presented by generators h_{ℓ}, 1 ≤ ℓ ≤ σ, and x_{i}, 1 ≤ i ≤ θ with defining relations

h_{ℓ}^{M_{ℓ}} = 1, 1 ≤ ℓ ≤ σ; (5.7) h_{ℓ}h_{t} = h_{t}h_{ℓ}, 1 ≤ t < ℓ ≤ σ; (5.8) h_{ℓ}x_{i} = χ_{i}(y_{ℓ})x_{i}h_{ℓ}, 1 ≤ ℓ ≤ σ, 1 ≤ i ≤ θ; (5.9)

x_{α}^{N_{I}} = 0, α ∈ Φ_{I}^{+}, I ∈ X; (5.10)

ad (x_{i})^{1−a_{ij}}(x_{j}) = 0, i ≠ j, (5.11)

and where the Hopf algebra structure is determined by

Δ(h_{ℓ}) = h_{ℓ} ⊗ h_{ℓ}, 1 ≤ ℓ ≤ σ; (5.12) Δ(x_{i}) = x_{i} ⊗ 1 + g_{i} ⊗ x_{i}, 1 ≤ i ≤ θ. (5.13)

Thus A is generated by the elements a_{i}, 1 ≤ i ≤ θ, and h_{l}, 1 ≤ l ≤ σ. By our previous choice, relations (5.7), (5.8), (5.9) and (5.12), (5.13) all hold in A with the x′_{i}s replaced by the a′_{i}s.

The remaining problem is to lift the quantum Serre relations (5.11) and the root vector relations (5.10). We will do this in the next two Sections.

5.2. Lifting the quantum Serre relations.

We divide the problem into two cases.

• Lifting of the “quantum Serre relations” x_{i}x_{j} − χ_{j}(g_{i})x_{j}x_{i} = 0, when i ≠ j are in different components of the Dynkin diagram.

• Lifting of the “quantum Serre relations” ad_{c}(x_{i})^{1−a_{ij}}(x_{j}) = 0, when i ≠ j are in the same

component of the Dynkin diagram. The first case is settled in the next Lemma from [AS4].

Lemma 5.2. Assume that 1 ≤ i, j ≤ θ, i < j and i ≁ j. Then

a_{i}a_{j} − χ_{j}(g_{i})a_{j}a_{i} = λ_{ij}(1 − g_{i}g_{j}), (5.14)

where λ_{ij} is a scalar in 𝕜 which can be chosen such that

(5.15) λ_{ij} is arbitrary if g_{i}g_{j} ≠ 1 and χ_{i}χ_{j} = ε, but 0 otherwise.

POINTED HOPF ALGEBRAS 37

Proof. It is easy to check that a_{i}a_{j} − χ_{j}(g_{i})a_{j}a_{i} ∈ P(A)_{g_{i}g_{j},1}^{χ_{i}χ_{j}}. Suppose that χ_{i}χ_{j} ≠ ε and a_{i}a_{j} − χ_{j}(g_{i})a_{j}a_{i} ≠ 0. Then by (5.3), χ_{i}χ_{j} = χ_{l} and g_{i}g_{j} = g_{l} for some 1 ≤ l ≤ θ.

Substituting g_{l} and χ_{l} in χ_{i}(g_{l})χ_{l}(g_{i}) = χ_{i}(g_{i})^{a_{il}} and using χ_{i}(g_{j})χ_{j}(g_{i}) = 1 (since a_{ij} = 0, since i and j lie in different components), we get χ_{i}(g_{i})^{2} = χ_{i}(g_{i})^{a_{il}}.

Thus we have shown that a_{il} ≡ 2 mod N_{i}, and in the same way a_{jl} ≡ 2 mod N_{j}. Since i ≁ j, a_{il} or a_{jl} must be 0, and we obtain the contradiction N_{i} = 2 or N_{i} = 2.

Therefore χ_{i}χ_{j} = ε, and the claim follows from (5.4), or a_{i}a_{j} − χ_{j}(g_{i})a_{j}a_{i} = 0, and the claim is trivial.

Lemma 5.2 motivates the following notion.

Definition 5.3. [AS4] We say that two vertices i and j are linkable (or that i is linkable to j) if

i ≁ j, (5.16) g_{i}g_{j} ≠ 1 and (5.17) χ_{i}χ_{j} = ε. (5.18)

The following elementary properties are easily verified: (5.19) If i is linkable to j, then χ_{i}(g_{j})χ_{j}(g_{i}) = 1, χ_{j}(g_{j}) = χ_{i}(g_{i})^{−1}. (5.20) If i and k, resp. j and ℓ, are linkable, then a_{ij} = a_{kℓ}, a_{ji} = a_{ℓk}. (5.21) A vertex i can not be linkable to two different vertices j and h. A linking datum is a collection (λ_{ij})_{1≤i<j≤θ, i≁j} of elements in 𝕜 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 notion of a linking datum encodes the information about lifting of relations in the first case.

Definition 5.4. The collection D formed by a finite Cartan matrix (a_{ij}), and g_{1}, …, g_{θ} ∈ Γ, χ_{1}, …, χ_{θ} ∈ Γ̂ satisfying (5.1), (5.5) and (5.6), and a linking datum (λ_{ij})_{1≤i<j≤θ, i≁j} will be called a linking datum of finite Cartan type for Γ. We define the Yetter-Drinfeld module V ∈ _{Γ}^{Γ}YD of D as the vector space with basis x_{1}, …, x_{θ} with x_{i} ∈ V_{g_{i}}^{χ_{i}} for all i.

If D is a linking datum of finite Cartan type for Γ, we define the Hopf algebra 𝔲(D) by generators a_{i}, 1 ≤ i ≤ θ, and h_{l}, 1 ≤ l ≤ σ and the relations (5.7),(5.8),(5.9),(5.10), the quantum Serre relations (5.11) for i ≠ j and i ∼ j, (5.12),(5.13) with the x_{i}’s replaced by the a_{i}’s, and the lifted quantum Serre relations (5.14).

In the definition of 𝔲(D) we could always assume that the linking datum contains only elements λ_{ij} ∈ {0, 1} (by multiplying the generators a_{i} with non-zero scalars).

Theorem 5.5. [AS4] Let Γ be a finite abelian group and D a linking datum of finite Cartan type for Γ with Yetter-Drinfeld module V. Then 𝔲(D) is a finite-dimensional pointed Hopf algebra with gr𝔲(D) ≃ 𝔅(V)#𝕜Γ.

The proof of the Theorem is by induction on the number of irreducible components of the Dynkin diagram. In the induction step a new Hopf algebra is constructed by twisting the multiplication of the tensor product of two Hopf algebras by a 2-cocycle. The 2-cocycle is defined in terms of the linking datum.

Note that the Frobenius-Lusztig kernel 𝔲_{q}(𝔤) of a semisimple Lie algebra 𝔤 is a special case of 𝔲(D). Here the Dynkin diagram of D is the disjoint union of two copies of the Dynkin diagram of 𝔤, and corresponding points are linked pairwise. But many other linkings are possible, for example 4 copies of A_{3} linked in a circle [AS4, 5.13]. See [D] for a combinatorical description of all linkings of Dynkin diagrams.

Let us now turn to the second case. Luckily it turns out that (up to some small order exceptions) in the second case the Serre relations simply hold in the lifted situation without any change.

Theorem 5.6. [AS4, Theorem 6.8]. 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 hold for all i, j ∈ I, i ≠ j, i ∼ j.

5.3. Lifting the root vector relations.

Assume first that the root α is simple and corresponds to a vertex i. It is not difficult to see, using the quantum binomial formula, that a_{i}^{N_{i}} is a (g_{i}^{N_{i}}, 1)-skew-primitive. By Lemma 5.1, we have

a_{i}^{N_{i}} = μ_{i} (1 − g_{i}^{N_{i}}) , (5.22)

for some scalar μ_{i}; this scalar can be chosen so that

(5.23) μ_{i} is arbitrary if g_{i}^{N_{i}} ≠ 1 and χ_{i}^{N_{i}} = 1 but 0 otherwise.

Now, if the root α is not simple then a_{i}^{N_{i}} is not necessarily a skew-primitive, but a skew-primitive “modulo root vectors of shorter lenght”.

In general, we define the root vector a_{α} for α ∈ Φ_{I}^{+}, I ∈ X, by replacing the x_{i} by a_{i} in the formal expression for x_{α} as a braided commutator in the simple root vectors. Then a_{α}^{N_{I}}, α ∈ I, should be an element u_{α} in the group algebra of the subgroup generated by the N_{I}-th powers of the elements in Γ.

Finally, the Hopf algebra generated by a_{i}, 1 ≤ i ≤ θ, and h_{l}, 1 ≤ l ≤ σ with the relations (5.7),(5.8),(5.9) (with a_{i} instead of x_{i}),

• the lifted root vector relations a_{α}^{N_{I}} = u_{α}, α ∈ Φ_{I}^{+}, I ∈ X,

• the quantum Serre relations (5.11) for i ≠ j and i ∼ j (with a_{i} instead of x_{i}),

• the lifted quantum Serre relations (5.14), should have the correct dimension dim(𝔅(V)) · ord(Γ).

We carried out all the steps of this program in the following cases:

(a) All connected components of the Dynkin diagram are of type A_{1} [AS1].

(b) The Dynkin diagram is of type A_{2}, and N > 3 is odd [AS3].

(c) The Dynkin diagram is arbitrary, but we assume g_{i}^{N_{i}} = 1 for all i [AS4].

(d) The Dynkin diagram is of type A_{n}, any n ≥ 2, and N > 3 , see Section 6 of this paper. The cases A_{2}, N = 3 and B_{2}, N odd and ≠ 5, were recently done in [BDR]. Here N denotes the common order of χ_{i}(g_{i}) for all i when the Dynkin diagram is connected.

5.4. Generation in degree one.

Let us now discuss step (d) of the Lifting method.

It is not difficult to show that our conjecture 2.7 about Nichols algebras, in the setting of H = 𝕜Γ, is equivalent to

Conjecture 5.7. [AS3]. Any pointed finite dimensional Hopf algebra over 𝕜 is generated by group-like and skew-primitive elements.

We have seen in Section 2.1 that the corresponding conjecture is false when the Hopf algebra is infinite-dimensional or when the Hopf algebra is finite-dimensional and the characteristic of the field is > 0. A strong indication that the conjecture is true is given by:

Theorem 5.8. [AS4]. Let A be a finite-dimensional pointed Hopf algebra with coradical 𝕜Γ and diagram R, that is

gr A ≃ R#𝕜Γ.

Assume that R(1) is a Yetter-Drinfeld module of finite Cartan type with braiding (q_{ij})_{1≤i,j≤θ}. For all i, let q_{i} = q_{ii}, N_{i} = ord(q_{i}). Assume that ord(q_{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. If i and j belong to different components, 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.

Let us discuss the idea of the proof of Theorem 5.8. At one decisive point, we use our previous results about braidings of Cartan type of rank 2.

Let S be the graded dual of R. By the duality principle in Lemma 2.4, S is generated in degree one since P (R) = R(1). Our problem is to show that R is generated in degree one, that is S is a Nichols algebra.

Since S is generated in degree one, there is a surjection of graded braided Hopf algebras S → 𝔅(V), where V = S(1) has the same braiding as R(1). But we know the defining relations of 𝔅(V), since it is of finite Cartan type. So have to show that these relations also hold in S.

40 ANDRUSKIEWITSCH AND SCHNEIDER

In the case of a quantum Serre relation ad_{c}(x_{i})^{1−a_{ij}}(x_{j}) = 0, i ≠ j, we consider the Yetter-Drinfeld submodule W of S generated by x_{i} and ad_{c}(x_{i})^{1−a_{ij}}(x_{j}) and assume that ad_{c}(x_{i})^{1−a_{ij}}(x_{j}) ≠ 0. The assumptions (1) and (2) of the Theorem guarantee that W also is of Cartan type, but not of finite Cartan type. Thus ad_{c}(x_{i})^{1−a_{ij}}(x_{j}) = 0 in S.

Since the quantum Serre relations hold in S, the root vector relations follow automatically from the next Lemma which is a consequence of Theorem 4.7.

Lemma 5.9. [AS4, 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 this Section. 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, α ∈ Φ_{I}^{+}, I ∈ X ,

hold in S.

5.5. Applications.

As a special case of the theory explained above we obtain a complete answer to the classification problem in a significant case.

Theorem 5.10. [AS4] Let p be a prime > 17, s ≥ 1, and Γ = (ℤ/(p))^{s}. Up to isomorphism there are only finitely many finite-dimensional pointed Hopf algebras A with G(A) ≃ Γ. They all have the form

A ≃ 𝔲(D), where D is a linking datum of finite Cartan type for Γ.

If we really want to write down all these Hopf algebras we still have to solve the following serious problems:

• Determine all Yetter-Drinfeld modules V over Γ = (ℤ/(p))^{s} of finite Cartan type.

• Determine all the possible linkings for the modules V over (ℤ/(p))^{s} in (a).

By [AS2, Proposition 8.3], dimV ≤ 2s\frac{p − 1}{p − 2}, for all the possible V in (a). This proves the finiteness statement in Theorem 5.10.

Note that we have a precise information about the dimension of the Hopf algebras in 5.10: dim 𝔲(D) = p^{s|ϕ^{+}|},

where |ϕ^{+}| is the number of the positive roots of the root system of rank θ ≤ 2s\frac{p − 1}{p − 2} of the Cartan matrix of D.

POINTED HOPF ALGEBRAS 41

For arbitrary finite abelian groups Γ, there usually are infinitely many non-isomorphic pointed Hopf algebras of the same finite dimension. The first examples were found in 1997 independently in [AS1], [BDG], [G]. Now it is very easy to construct lots of examples by lifting. Using [AS3, Lemma 1.2] it is possible to decide when two liftings are non-isomorphic.

But we have a bound on the dimension of A:

Theorem 5.11. [AS4] For any finite (not necessarily abelian) group Γ of odd order there is a natural number n(Γ) such that

dimA ≤ n(Γ)

for any finite-dimensional pointed Hopf algebra A with G(A) = 𝕜Γ.

Remark 5.12. As a corollary of the Theorem and its proof, we get the complete classification of all finite dimensional pointed Hopf algebras with coradical of prime dimension p, p ≠ 2, 5, 7. By [AS2, Theorem 1.3], the only possibilities for the Cartan matrix of D with Γ of odd prime order p are

(a) A_{1} and A_{1} × A_{1},

(b) A_{2}, if p = 3 or p ≡ 1 mod 3,

(c) B_{2}, if p ≡ 1 mod 4,

(d) G_{2}, if p ≡ 1 mod 3,

(e) A_{2} × A_{1} and A_{2} × A_{1}, if p = 3. The Nichols algebras over ℤ/(p) for these Cartan matrices are listed in [AS2, Theorem 1.3]. Hence we obtain from Theorem 5.10 for p ≠ 2, 5, 7 the bosonizations of the Nichols algebras, the liftings in case (a), that is quantum lines and quantum planes [AS1], and the liftings of type A_{2} [AS3] in case (b).

This result was also obtained by Musson [Mus], using the lifting method and [AS2].

The case p = 2 was already done in [N]. In this case the dimension of the pointed Hopf algebras with 2-dimensional coradical is not bounded.

Let us mention briefly some classification results for Hopf algebras of special order which can be obtained by the methods we have described. Let p > 2 be a prime. Then all pointed Hopf algebras A of dimension p^{n}, 1 ≤ n ≤ 5 are known. If the dimension is p or p^{2}, then A is a group algebra or a Taft Hopf algebra. The cases of dimension p^{3} and p^{4} were treated in [AS1] and [AS3], and the classification of dimension p^{5} follows from [AS4] and [Gñ1]. Independently and by other methods, the case p^{3} was also solved in [CD] and [SvO].

See [A] for a discussion of what is known on classification of finite dimensional Hopf algebras.

To formulate a classification result for infinite-dimensional Hopf algebras, we now assume that 𝕜 is the field of complex numbers and we introduce a notion from [AS5]. The collection D formed by a free abelian group Γ of finite rank, a finite Cartan matrix (a_{ij})_{1≤i,j≤θ}, g_{1}, …, g_{θ} ∈ Γ, χ_{1}, …, χ_{θ} ∈ Γ̂, and a linking datum (λ_{ij})_{1≤i<j≤θ, i≁j}, will be called a positive datum if

χ_{i}(g_{j})χ_{j}(g_{i}) = χ_{i}(g_{i})^{d_{i}a_{ij}}, and χ_{i}(g_{i}) > 0, for all 1 ≤ i, j, ≤ θ,

where d_{1}, …, d_{θ} are integers between 1 and 3 such that d_{i}a_{ij} = d_{j}a_{ji}.

If D is a positive datum we define the Hopf algebra U(D) by generators a_{i}, 1 ≤ i ≤ θ, and h_{l}^{±}, 1 ≤ l ≤ σ and the relations h_{m}^{±}h_{l}^{±} = h_{l}^{±}h_{l}^{±}, h_{l}^{±}h_{l}^{∓} = 1, for all 1 ≤ l, m ≤ σ, defining the free abelian group of rank σ, and (5.9), the quantum Serre relations (5.11) for i ≠ j and i ∼ j, (5.12),(5.13) (with a_{i} instead of x_{i}), and the lifted quantum Serre relations (5.14).

If (V, c) is a finite-dimensional braided vector space, we will say that the braiding is positive if it is diagonal, and all the entries of the matrix (q_{ij}) of the braiding are positive, and q_{ii} ≠ 1 for all i.

The next theorem follows from a result of Rosso [Ro2, Theorem 21] and the theory described in the previous Sections.

Theorem 5.13. [AS5] Let A be a pointed Hopf algebra with abelian group Γ = G(A) and diagram R. Assume that R(1) has finite dimension θ and positive braiding. Then the following are equivalent:

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

(b). The group Γ is free abelian of finite rank, and

A ≃ U(D), where D is a positive datum for Γ.

It is likely that the positivity assumption on the infinitesimal braiding in the last theorem is related to the existence of a compact involution.

6. Pointed Hopf algebras of type A_{n}

In this Chapter, we develop from scratch, i. e. without using Lusztig’s results, the classification of all finite dimensional pointed Hopf algebras whose infinitesimal braiding is of type A_{n}. The main results of this Chapter are new.

6.1. Nichols algebras of type A_{n}.

Let N be an integer, N > 2, and let q be a root of 1 of order N. For the case N = 2, see [AnDa].

Let q_{ij}, 1 ≤ i, j ≤ n, be roots of 1 such that

q_{ii} = q, q_{ij}q_{ji} = \begin{cases}q^{−1}, & if |i − j| = 1, \\ 1, & if |i − j| ≥ 2. \end{cases} (6.1)

for all 1 ≤ i, j ≤ n. For convenience, we denote

B_{p,r}^{i,j} := ∏_{i≤ℓ≤j−1, p≤h≤r−1} q_{ℓ,h},

for any 1 ≤ i < j ≤ n + 1, 1 ≤ p < r ≤ n + 1. Then we have the following identities, whenever i < s < j, p < t < r:

(6.2)  B_{p,r}^{i,s}B_{p,r}^{s,j} = ∏_{i≤ℓ≤s−1,p≤h≤r−1} q_{ℓ,h} ∏_{s≤ℓ≤j−1,p≤h≤r−1} q_{ℓ,h} = B_{p,r}^{i,j};
(6.3)  B_{p,t}^{i,j}B_{t,r}^{i,j} = B_{p,r}^{i,j};

also,

(6.4)  B_{j,j+1}^{i,j}B_{i,j}^{j,j+1} = ∏_{i≤ℓ≤j−1} q_{ℓ,j} ∏_{i≤h≤j−1} q_{j,h} = q^{−1};
(6.5)  B_{i,j}^{i,j} = q.

We consider in this Section a vector space V = V_{n} with a basis x_{1}, …, x_{n} and braiding determined by:

c(x_{i} ⊗ x_{j}) = q_{ij} x_{j} ⊗ x_{i}, 1 ≤ i, j ≤ n;

that is, V is of type A_{n}.

Remark 6.1. Let Γ be a group, g_{1}, …, g_{n} in the center of Γ, and χ_{1}, …, χ_{n} in Γ̂ such that

q_{ij} = ⟨χ_{j}, g_{i}⟩, 1 ≤ i, j ≤ n.

Then V can be realized as a Yetter-Drinfeld module over Γ by declaring

x_{i} ∈ V_{g_{i}}^{χ_{i}}, 1 ≤ i ≤ n. (6.6)

For example, we could consider Γ = (ℤ/P)^{n}, where P is divisible by the orders of all the q_{ij}’s; and take g_{1}, …, g_{n} as the canonical basis of Γ.

We shall consider a braided Hopf algebra R provided with an inclusion of braided vector spaces V → P (R). We identify the elements x_{1}, …, x_{n} with their images in R. Distinguished examples of such R are the tensor algebra T (V) and the Nichols algebra 𝔅(V). Additional hypotheses on R will be stated when needed.

We introduce the family (e_{ij})_{1≤i<j≤n+1} of elements of R as follows:

e_{i,i+1} := x_{i}; (6.7)

e_{i,j} := [e_{i,j−1}, e_{j−1,j}]_{c}, 1 ≤ i < j ≤ n + 1, j − i ≥ 2. (6.8)

The braiding between elements of this family is given by:

c(e_{i,j} ⊗ e_{p,r}) = B_{p,r}^{i,j}e_{p,r} ⊗ e_{i,j}, 1 ≤ i < j ≤ n + 1, 1 ≤ p < r ≤ n + 1. (6.9)

In particular,

e_{i,j} = e_{i,j−1}e_{j−1,j} − B_{j−1,j}^{i,j−1}e_{j−1,j}e_{i,j−1}.

Remark 6.2. When V is realized as a Yetter-Drinfeld module over Γ as in Remark 6.1, we have e_{i,j} ∈ R_{g_{i,j}}^{χ_{i,j}}, where

χ_{i,j} = ∏_{i≤ℓ≤j−1} χ_{ℓ},  g_{i,j} = ∏_{i≤ℓ≤j−1} g_{ℓ},  1 ≤ i < j ≤ n + 1. (6.10)

Lemma 6.3. (a). If R is finite dimensional or R ≃ 𝔅(V), then

e_{i,i+1}^{N} = 0, if 1 ≤ i ≤ n. (6.11)

(b). Assume that R ≃ 𝔅(V). Then

(6.12) [e_{i,i+1}, e_{p,p+1}]_{c} = 0, that is e_{i,i+1}e_{p,p+1} = q_{ip}e_{p,p+1}e_{i,i+1}, if 1 ≤ i < p ≤ n, p − i ≥ 2.

(c). Assume that R ≃ 𝔅(V). Then

(6.13)  [e_{i,i+1}, [e_{i,i+1}, e_{i+1,i+2}]_{c}]_{c} = 0, if 1 ≤ i < n;
(6.14)  [e_{i+1,i+2}, [e_{i+1,i+2}, e_{i,i+1}]_{c}]_{c} = 0, if 1 ≤ i < n;

that is

(6.15)  e_{i,i+1}e_{i,i+2} = B_{i,i+2}^{i,i+1}e_{i,i+2}e_{i,i+1},
(6.16)  e_{i,i+2}e_{i+1,i+2} = B_{i+1,i+2}^{i,i+2}e_{i+1,i+2}e_{i,i+2}.

Proof. (a). This follows from Lemma 3.6 (a), use c(e_{i,i+1}^{N} ⊗ e_{i,i+1}^{N}) = e_{i,i+1}^{N} ⊗ e_{i,i+1}^{N} in the finite dimensional case.

(b) and (c). By Lemma 3.6 (b), the elements

[e_{i,i+1}, e_{p,p+1}]_{c}, [e_{i,i+1}, [e_{i,i+1}, e_{i+1,i+2}]_{c}]_{c} and [e_{i+1,i+2}, [e_{i+1,i+2}, e_{i,i+1}]_{c}]_{c}

are primitive. Since they are homogeneous of degree 2, respectively of degree 3, they should be 0. To derive (6.16) from (6.14), use (1.23).

Lemma 6.4. Assume that (6.12) holds in R. Then

[e_{i,j}, e_{p,r}]_{c} = 0, if 1 ≤ i < j < p < r ≤ n + 1. (6.17) [e_{p,r}, e_{i,j}]_{c} = 0, if 1 ≤ i < j < p < r ≤ n + 1. (6.18) [e_{i,p}, e_{p,j}]_{c} = e_{i,j}, if 1 ≤ i < p < j ≤ n + 1. (6.19)

Proof. (6.17). For j = i + 1 and r = p + 1, this is (6.12); the general case follows recursively using (1.22). (6.18) follows from (6.17), since B_{p,r}^{i,j}B_{i,j}^{p,r} = 1 in this case.

POINTED HOPF ALGEBRAS 45

(6.19). By induction on j − p; if p = j − 1 then (6.19) is just (6.8). For p < j, we have

e_{i,j+1} = [e_{i,j}, e_{j,j+1}]_{c} = [[e_{i,p}, e_{p,j}]_{c}, e_{j,j+1}]_{c} = [e_{i,p}, [e_{p,j}, e_{j,j+1}]_{c}]_{c} = [e_{i,p}, e_{p,j+1}]_{c}

by (1.22), since [e_{i,p}, e_{j,j+1}]_{c} = 0 by (6.17).

Lemma 6.5. Assume that (6.12) holds in R. Then for any 1 ≤ i < j ≤ n + 1,

Δ(e_{i,j}) = e_{i,j} ⊗ 1 + 1 ⊗ e_{i,j} + (1 − q^{−1}) ∑_{i<p<j} e_{i,p} ⊗ e_{p,j}. (6.20)

Proof. We proceed by induction on j − i. If j − i = 1, the formula just tells that the x_{i}’s are primitive. For the inductive step, we compute

Δ(e_{i,j}e_{j,j+1}) = (e_{i,j} ⊗ 1 + 1 ⊗ e_{i,j} + (1 − q^{−1}) ∑_{i<p<j} e_{i,p} ⊗ e_{p,j}) (e_{j,j+1} ⊗ 1 + 1 ⊗ e_{j,j+1})

= e_{i,j}e_{j,j+1} ⊗ 1 + B_{j,j+1}^{i,j}e_{j,j+1} ⊗ e_{i,j} + (1 − q^{−1}) ∑_{i<p<j} B_{j,j+1}^{p,j}e_{i,p}e_{j,j+1} ⊗ e_{p,j}

+ e_{i,j} ⊗ e_{j,j+1} + 1 ⊗ e_{i,j}e_{j,j+1} + (1 − q^{−1}) ∑_{i<p<j} e_{i,p} ⊗ e_{p,j}e_{j,j+1};

and also

Δ(e_{j,j+1}e_{i,j}) = (e_{j,j+1} ⊗ 1 + 1 ⊗ e_{j,j+1}) (e_{i,j} ⊗ 1 + 1 ⊗ e_{i,j} + (1 − q^{−1}) ∑_{i<p<j} e_{i,p} ⊗ e_{p,j})

= e_{j,j+1}e_{i,j} ⊗ 1 + e_{j,j+1} ⊗ e_{i,j} + (1 − q^{−1}) ∑_{i<p<j} e_{j,j+1}e_{i,p} ⊗ e_{p,j}

+ B_{i,j}^{j,j+1}e_{i,j} ⊗ e_{j,j+1} + 1 ⊗ e_{j,j+1}e_{i,j} + (1 − q^{−1}) ∑_{i<p<j}(B_{j,j+1}^{i,p})^{−1}e_{i,p} ⊗ e_{j,j+1}e_{p,j}.

Hence

Δ(e_{i,j+1}) = e_{i,j+1} ⊗ 1 + 1 ⊗ e_{i,j+1} + (1 − B_{j,j+1}^{ij}B_{ij}^{j,j+1})e_{i,j} ⊗ e_{j,j+1}

+ (1 − q^{−1}) ∑_{i<p<j} (B_{j,j+1}^{p,j}B_{j,j+1}^{i,p} − B_{j,j+1}^{i,j}) e_{j,j+1}e_{i,p} ⊗ e_{p,j}

+ (1 − q^{−1}) ∑_{i<p<j} e_{i,p} ⊗ (e_{p,j}e_{j,j+1} − B_{j,j+1}^{i,j}(B_{j,j+1}^{i,p})^{−1}e_{j,j+1}e_{p,j})

= e_{i,j+1} ⊗ 1 + 1 ⊗ e_{i,j+1} + (1 − q^{−1}) ∑_{i<p<j+1} e_{i,p} ⊗ e_{p,j+1};

by (6.4), (6.17) and the hypothesis.

Remark 6.6. Let Γ be a group with g_{1}, …, g_{n} in the center of Γ, χ_{1}, …, χ_{n} in Γ̂, as in 6.1. Let R be a braided Hopf algebra in _{Γ}^{Γ}YD such that (6.12) holds in R. It follows from (6.20) and the reconstruction formulas for the bosonization (1.17) that

Δ_{R#𝕜Γ}(e_{i,j}) = e_{i,j} ⊗ 1 + g_{i,j} ⊗ e_{i,j} + (1 − q^{−1}) ∑_{i<p<j} e_{i,p}g_{p,j} ⊗ e_{p,j}. (6.21)

Lemma 6.7. Assume that (6.12), (6.13), (6.14) hold in R. Then

[e_{i,j}, e_{p,r}]_{c} = 0, if 1 ≤ i < p < r < j ≤ n + 1; (6.22) [e_{i,j}, e_{i,r}]_{c} = 0, if 1 ≤ i < j < r ≤ n + 1; (6.23) [e_{i,j}, e_{p,j}]_{c} = 0, if 1 ≤ i < p < j ≤ n + 1. (6.24)

Proof. (a). We prove (6.22) by induction on j − i. If j − i = 3, then

[e_{i,i+3}, e_{i+1,i+2}]_{c} = [[e_{i,i+2}, e_{i+2,i+3}]_{c}, e_{i+1,i+2}]_{c} = [[[e_{i,i+1}, e_{i+1,i+2}]_{c}, e_{i+2,i+3}]_{c}, e_{i+1,i+2}]_{c} = 0,

by Lemma 1.11. If j − i > 3 we argue by induction on r − p. If r − p = 1, then there exists an index h such that either i < h < p < r = p + 1 < j or i < p < r = p + 1 < h < j. In the first case, by (6.19), we have

[e_{i,j}, e_{p,r}]_{c} = [[e_{i,h}, e_{h,j}]_{c}, e_{p,p+1}]_{c} = 0;

the last equality follows from Lemma 1.10 (c), because of (6.17) and the induction hypothesis. In the second case, we have

[e_{i,j}, e_{p,r}]_{c} = [[e_{i,h}, e_{h,j}]_{c}, e_{p,p+1}]_{c} = 0;

the last equality follows from Lemma 1.10 (c), because of the induction hypothesis and (6.18). Finally, if r − p > 1 then

[e_{i,j}, e_{p,r}]_{c} = [e_{i,j}, [e_{p,r−1}, e_{r−1,r}]_{c}]_{c} = 0

by Lemma 1.10 (b) and the induction hypothesis.

(b). We prove (6.23) by induction on r − i. If r − i = 2, then the claimed equality is just (6.13). If r − i > 2 we argue by induction on j − i. If j − i = 1 we have

[e_{i,i+1}, e_{i,r}]_{c} = [e_{i,i+1}, [e_{i,r−1}, e_{r−1,r}]_{c}]_{c} = [[e_{i,i+1}, e_{i,r−1}]_{c}, e_{r,r−1}]_{c} = 0

by (1.22), since [e_{i,i+1}, e_{r−1,r}]_{c} = 0 by (6.17). If j − i > 2, we have

[e_{i,j}, e_{i,r}]_{c} = [[e_{i,j−1}, e_{j−1,j}]_{c}, e_{i,r}]_{c} = 0

by Lemma 1.10 (c), because of the induction hypothesis and (6.22).

The proof of (6.24) is analogous to the proof of (6.23), using (6.14) instead of (6.13).

Lemma 6.8. Assume that (6.12), (6.13), (6.14) hold in R. Then

[e_{i,j}, e_{p,r}]_{c} = B_{jr}^{pj}(q − 1)e_{ir}e_{pj}, if 1 ≤ i < p < j < r ≤ n + 1. (6.25)

Proof. We compute:

[e_{i,j}, e_{p,r}]_{c} = [[e_{i,p}, e_{p,j}]_{c}, e_{p,r}]_{c}
= [e_{i,p}, [e_{p,j}, e_{p,r}]_{c}]_{c} + B_{p,r}^{p,j}[e_{i,p}, e_{p,r}]_{c}e_{p,j} − B_{p,j}^{i,p}e_{p,j}[e_{i,p}, e_{p,r}]_{c}
= B_{p,r}^{p,j}e_{i,r}e_{p,j} − B_{p,j}^{i,p}e_{p,j}e_{i,r}
= (B_{p,r}^{p,j} − B_{p,j}^{i,p}(B_{p,j}^{i,r})^{−1}) e_{i,r}e_{p,j}
= (B_{p,r}^{p,j} − (B_{p,j}^{p,r})^{−1}) e_{i,r}e_{p,j}
= B_{jr}^{pj}(q − 1)e_{ir}e_{pj}.

Here, the first equality is by (6.19); the second, by Lemma 1.10 (a); the third, by (6.23) and by (6.19); the fourth, by (6.22).

Lemma 6.9. Assume that (6.12), (6.13) and (6.14) hold in R. For any 1 ≤ i < j ≤ n + 1 we have

Δ(e_{i,j}^{N}) = e_{i,j}^{N} ⊗ 1 + 1 ⊗ e_{i,j}^{N} + (1 − q^{−1})^{N} ∑_{i<p<j} (B_{i,p}^{p,j})^{N(N−1)/2} e_{i,p}^{N} ⊗ e_{p,j}^{N}. (6.26)

Proof. By (6.20), and using several times the quantum binomial formula (1.13), we have

Δ(e_{i,j}^{N}) = (e_{i,j} ⊗ 1 + (1 − q^{−1}) ∑_{i<p<j} e_{i,p} ⊗ e_{p,j})^{N} + 1 ⊗ e_{i,j}^{N}

= e_{i,j}^{N} ⊗ 1 + (1 − q^{−1})^{N} ( ∑_{i<p<j} e_{i,p} ⊗ e_{p,j})^{N} + 1 ⊗ e_{i,j}^{N}

= e_{i,j}^{N} ⊗ 1 + (1 − q^{−1})^{N} ∑_{i<p<j} (e_{i,p} ⊗ e_{p,j})^{N} + 1 ⊗ e_{i,j}^{N}

= e_{i,j}^{N} ⊗ 1 + (1 − q^{−1})^{N} ∑_{i<p<j} (B_{i,p}^{p,j})^{N(N−1)/2} e_{i,p}^{N} ⊗ e_{p,j}^{N} + 1 ⊗ e_{i,j}^{N}.

Here, in the first equality we use that (1⊗e_{i,j})(e_{i,j}⊗1) = q(e_{i,j}⊗1)(1⊗e_{i,j}) and (1⊗e_{i,j})(e_{i,p}⊗e_{p,j}) = q(e_{i,p}⊗e_{p,j})(1⊗e_{i,j}), this last by (6.24); in the second, we use (e_{i,p}⊗e_{p,j})(e_{i,j} ⊗1) = q(e_{i,j} ⊗1)(e_{i,p}⊗ e_{p,j}), which follows from (6.23); the third, that (e_{i,p} ⊗ e_{p,j})(e_{i,s} ⊗ e_{s,j}) = q^{2}(e_{i,s} ⊗ e_{s,j})(e_{i,p} ⊗ e_{p,j}) for p < s, which follows from (6.23) and (6.24); the fourth, from (e_{i,p} ⊗ e_{p,j})^{h} = (B_{i,p}^{p,j})^{h(h−1)/2} e_{i,p}^{h} ⊗ e_{p,j}^{h}.

Remark 6.10. Let Γ be a group with g_{1}, …, g_{n} in the center of Γ, χ_{1}, …, χ_{n} in Γ̂, as in 6.1. Let R be a braided Hopf algebra in _{Γ}^{Γ}YD such that (6.12), (6.13) and (6.14) hold in R. By (6.26) and the reconstruction formulas (1.17), we have

Δ_{R#𝕜Γ}(e_{i,j}^{N}) = e_{i,j}^{N} ⊗ 1 + g_{i,j}^{N} ⊗ e_{i,j}^{N} + (1 − q^{−1})^{N} ∑_{i<p<j} (B_{i,p}^{p,j})^{N(N−1)/2} e_{i,p}^{N}g_{p,j}^{N} ⊗ e_{p,j}^{N}. (6.27)

Lemma 6.11. Assume that R = 𝔅(V). Then

e_{i,j}^{N} = 0, 1 ≤ i < j ≤ n + 1. (6.28)

Proof. This follows from Lemma 6.9 by induction on j − i, the case j − i = 1 being Lemma 6.3 (c).

Lemma 6.12. Assume that (6.12), (6.13), (6.14), (6.28), hold in R. Assume, furthermore, that R is generated as an algebra by the elements x_{1}, …, x_{n}. Then the algebra R is spanned as a vector space by the elements

e_{1,2}^{ϵ_{1,2}}e_{1,3}^{ϵ_{1,3}} … e_{1,n+1}^{ϵ_{1,n+1}}e_{2,3}^{ϵ_{2,3}} … e_{2,n+1}^{ϵ_{2,n+1}} … e_{n,n+1}^{ϵ_{n,n+1}}, with ϵ_{i,j} ∈ {0, 1, …, N − 1}. (6.29)

Proof. We order the family (e_{ij}) by

e_{1,2} ≺ e_{1,3} ≺ … e_{1,n+1} ≺ e_{2,3} ≺ … e_{2,n+1} ≺ … e_{n,n+1};

this induces an ordering in the monomials (6.29). If M is an ordered monomial, we set σ(M) := e_{r,s} if e_{r,s} is the first element appearing in M. Let B be the subspace generated by the monomials in (6.29). We show by induction on the length that, for any ordered monomial M and for any i, e_{i,i+1}M ∈ B and it is 0 or a combination of monomials N with σ(N) ⪰ min{e_{i,i+1}, σ(M)}, length of N ≤ length of M + 1. The statement is evident if the length of M is 0; so that assume that the length is positive. Write M = e_{p,q}M′ where e_{p,q} ⪯ M′. We have several cases:

If i < p or i = p and i + 1 < q, e_{i,i+1} ≺ e_{p,q} and we are done.

If i = p and i + 1 = q, then the claim is clear.

If p < q < i then e_{i,i+1}e_{p,q} = B_{p,q}^{i,i+1} e_{p,q}e_{i,i+1} by (6.18); hence e_{i,i+1}M = e_{i,i+1}e_{p,q}M′ = B_{p,q}^{i,i+1} e_{p,q}e_{i,i+1}M′; by the inductive hypothesis and the fact that e_{p,q} ⪯ min{e_{i,i+1}, σ(M′)}, the claim follows.

If p < i = q then e_{i,i+1}e_{p,i} = (B_{i,i+1}^{p,i})^{−1}(e_{p,i}e_{i,i+1} − e_{p,i+1}) by (6.19); again, the inductive hypothesis and e_{p,q} ⪯ min{e_{i,i+1}, σ(M′)} imply that e_{p,i}e_{i,i+1}M′ has the form we want. To see that e_{p,i+1}M” satisfies the claim when e_{p,i} = σ(M′), we use e_{p,i+1}e_{p,i} = (B_{p,i+1}^{p,i})_{−1}e_{p,i}e_{p,i+1} by (6.23).

If p < i < q then e_{i,i+1}e_{p,q} = (B_{i,i+1}^{p,q})^{−1}e_{p,q}e_{i,i+1} by (6.22) or (6.24); we then argue as in the two preceding cases.

POINTED HOPF ALGEBRAS 49

Therefore, B = R since it is a left ideal containing 1.

We shall say that the elements e_{1,2}, e_{1,3}, …, e_{1,n+1}e_{2,3} … e_{2,n+1} … e_{n,n+1}, in this order, form a PBW-basis for R if the monomials (6.29) form a basis of R. Then we have:

Theorem 6.13. The elements e_{1,2}, e_{1,3}, …, e_{1,n+1}e_{2,3} … e_{2,n+1} … e_{n,n+1}, in this order, form a PBW basis for 𝔅(V_{n}). In particular,

dim 𝔅(V_{n}) = N^{\frac{n(n+1)}{2}} .

Proof. We proceed by induction on n. The case n = 1 is clear, see [AS1, Section 3] for details. We assume the statement for n − 1. We consider V_{n} as a Yetter-Drinfeld module over Γ = (ℤ/P)^{n}, as explained in Remark 6.1. Let Z_{n} = 𝔅(V_{n})#𝕜Γ. Let i_{n} : V_{n−1} → V_{n} be given by x_{i} ↦ x_{i} and p_{n} : V_{n} → V_{n−1} by x_{i} ↦ x_{i}, 1 ≤ i ≤ n − 1 and x_{n} ↦ 0. The splitting of Yetter-Drinfeld modules id_{V_{n−1}} = p_{n}i_{n} gives rise to a splitting of Hopf algebras id_{Z_{n−1}} = π_{n}ι_{n}, where ι_{n} : Z_{n−1} → Z_{n} and π_{n} : Z_{n} → Z_{n−1} are respectively induced by i_{n}, p_{n}. Let

R_{n} = Z_{n}^{co π_{n}} = {z ∈ Z_{n} : (id ⊗π_{n})Δ(z) = z ⊗ 1}.

Then R_{n} is a braided Hopf algebra in the category _{Z_{n−1}}^{Z_{n−1}}YD; we shall denote by c_{R_{n}} the corresponding braiding of R_{n}. We have Z_{n} ≃ R_{n}#Z_{n−1} and in particular dim Z_{n} = dim R_{n} dim Z_{n−1}.

For simplicity, we denote h_{i} = e_{i,n+1}, 1 ≤ i ≤ n. We have h_{i}h_{j} = B_{j,n+1}^{i,n+1}h_{j}h_{i}, for i < j, by (6.24). We claim that h_{1}, …, h_{n} are linearly independent primitive elements of the braided Hopf algebra R_{n}.

Indeed, it follows from (6.8) that π_{n}(h_{i}) = 0; by (6.20), we conclude that h_{i} ∈ R_{n}. We prove by induction on j = n + 1 − i that h_{i} is a primitive element of R_{n}, the case j = 1 being clear. Assume the statement for j. Now

x_{i−1} ⇀ h_{i} = x_{i−1}h_{i} + g_{i−1}h_{i}S(x_{i−1}) = x_{i−1}h_{i} − g_{i−1}h_{i}g_{i−1}^{−1}x_{i−1}
= x_{i−1}h_{i} − B_{i,n+1}^{i−1,i}h_{i}x_{i−1} = [x_{i−1}, h_{i}]_{c} = h_{i−1}.

So

Δ_{R_{n}}(h_{i−1}) = Δ_{R_{n}}(x_{i−1} ⇀ h_{i}) = x_{i−1} ⇀ Δ(h_{i})
= g_{i−1} ⇀ 1 ⊗ x_{i−1} ⇀ h_{i} + x_{i−1} ⇀ h_{i} ⊗ 1 = h_{i−1} ⊗ 1 + 1 ⊗ h_{i−1}.

We prove also by induction on j = n + 1 − i that h_{i} ≠ 0 using (6.20) and the induction hypothesis on Z_{n−1}. Since h_{i} is homogeneous of degree j (with respect to the grading of Z_{n}), we conclude that h_{1}, …, h_{n} are linearly independent.

We next claim that c_{R_{n}}(h_{i} ⊗ h_{j}) = B_{j,n+1}^{i,n+1}h_{j} ⊗ h_{i}, for any i > j.

By (6.27), the coaction of Z_{n−1} on R_{n} satisfies

δ(h_{i}) = g_{i,n+1} ⊗ e_{i,n+1} + (1 − q^{−1}) ∑_{i<p<n+1} e_{i,p}g_{p,n+1} ⊗ e_{p,n+1}.

If j < i, we compute the action on R_{n}:

e_{i,p} ⇀ h_{j} = e_{i,p}h_{j} + g_{i,p}h_{j}S(e_{i,n+1}) + (1 − q^{−1}) ∑_{i<t<p} e_{i,t}g_{t,p}h_{j}S(e_{t,p})
= (B_{i,p}^{j,n+1})^{−1}h_{j}e_{i,p} + B_{j,n+1}^{i,p}h_{j}g_{i,p}S(e_{i,n+1})

+ h_{j}(1 − q^{−1}) ∑_{i<t<p} B_{j,n+1}^{t,p}(B_{i,t}^{j,n+1})^{−1}e_{i,t}g_{t,p}S(e_{t,p})

= B_{j,n+1}^{i,p}h_{j}e_{i,p}_{(1)}S(e_{i,p}_{(2)})
= 0,

by (6.24). Thus

c_{R_{n}}(h_{i} ⊗ h_{j}) = g_{i,n+1} ⇀ h_{j} ⊗ h_{i} = B_{j,n+1}^{i,n+1}h_{j} ⊗ h_{i}.

We next claim that the dimension of the subalgebra of R_{n} spanned by h_{1}, …, h_{n} is ≥ N^{n}. We already know that

Δ(h_{j}^{m_{j}}) = ∑_{0≤i_{j}≤m_{j}} \binom{m_{j}}{i_{j}}_{q}h_{j}^{i_{j}} ⊗ h_{j}^{m_{j}−i_{j}}, m_{j} ≤ N.

Let us denote m = (m_{1}, …, m_{j}, …, m_{n}), 1 = (1, …, 1, …, 1), N = (N, …, N). We consider the partial order i ≤ m, if i_{j} ≤ m_{j}, j = 1, …, n. We set h^{m} := h_{n}^{m_{n}} … h_{j}^{m_{j}} … h_{1}^{m_{1}}. ¿From the preceding claim, we deduce that

Δ(h^{m}) = h^{m} ⊗ 1 + 1 ⊗ h^{m} + ∑_{0≤i≤m, 0≠i≠m} c_{m,i}h^{i} ⊗ h^{m−i}, m ≤ N − 1;

where c_{m,i} ≠ 0 for all i. We then argue recursively as in the proof of [AS1, Lemma 3.3] to conclude that the elements h^{m}, m ≤ N−1, are linearly independent; hence the dimension of the subalgebra of R_{n} spanned by h_{1}, …, h_{n} is ≥ N^{n}, as claimed.

We can now finish the proof of the Theorem. Since dim Z_{n} ≤ N^{\frac{n(n+1)}{2}} by Lemma 6.12 and dim Z_{n−1} = N^{\frac{n(n−1)}{2}} by the induction hypothesis, we have dim R_{n} ≤ N^{n}. By what we have just seen, this dimension is exactly N^{n}. Therefore, dim Z_{n} = N^{\frac{n(n+1)}{2}} ; in presence of Lemma 6.12, this implies the Theorem.

Theorem 6.14. The Nichols algebra 𝔅(V) can be presented by generators e_{i,i+1}, 1 ≤ i ≤ n, and relations (6.12), (6.13), (6.14) and (6.28).

Proof. Let 𝔅′ be the algebra presented by generators e_{i,i+1}, 1 ≤ i ≤ n, and relations (6.12), (6.13), (6.14) and (6.28). We claim that 𝔅′ is is a braided Hopf algebra with the e_{i,i+1}’s primitive. Indeed, the claim follows without difficulty; use Lemma 6.9 for relations (6.28).

By Lemma 6.12, we see that the monomials (6.29) span 𝔅′ as vector space, and in particular that dim 𝔅′ ≤ N^{\frac{n(n+1)}{2}} . By Lemmas 6.3 and 6.11, there is a surjective algebra map ψ : 𝔅′ → 𝔅(V). By Theorem 6.13, ψ is an isomorphism.

6.2. Lifting of Nichols algebras of type A_{n}.

We fix in this Section a finite abelian group Γ such that our braided vector space V can be realized in _{Γ}^{Γ}YD, as in Remark 6.1. That is, we have g_{1}, …, g_{n} in Γ, χ_{1}, …, χ_{n} in Γ̂, such that q_{ij} = ⟨χ_{j}, g_{i}⟩ for all i,j, and V can be realized as a Yetter-Drinfeld module over Γ by (6.6).

We also fix a finite dimensional pointed Hopf algebra A such that G(A) is isomorphic to Γ, and the infinitesimal braiding of A is isomorphic to V as Yetter-Drinfeld module over Γ. That is, gr A ≃ R#𝕜Γ, and the subalgebra R′ of R generated by R(1) is isomorphic to 𝔅(V). We choose elements a_{i} ∈ (A_{1})_{g_{i}}^{χ_{i}} such that π(a_{i}) = x_{i}, 1 ≤ i ≤ n.

We shall consider, more generally, Hopf algebras H provided with

• a group isomorphism Γ → G(H);

• elements a_{1}, …, a_{n} in P(H)_{g_{i},1}^{χ_{i}}. Further hypotheses on H will be stated when needed. The examples of such H we are thinking of are the Hopf algebra A, and any bosonization R#𝕜Γ, where R is any braided Hopf algebra in _{Γ}^{Γ}YD provided with a monomorphism of Yetter-Drinfeld modules V → P (R); so that a_{i} := x_{i}#1, 1 ≤ i ≤ n. This includes notably the Hopf algebras T (V)#𝕜Γ, 𝔅̂(V)#𝕜Γ, 𝔅(V)#𝕜Γ.

Here 𝔅̂(V) be the braided Hopf algebra in _{Γ}^{Γ}YD generated by x_{1}, …, x_{θ} with relations (6.12), (6.13) and (6.14).

We introduce inductively the following elements of H:

E_{i,i+1} := a_{i}; (6.30)

E_{i,j} := ad (E_{i,j−1})(E_{j−1,j}), 1 ≤ i < j ≤ n + 1, j − i ≥ 2. (6.31)

Assume that H = R#𝕜Γ as above. Then, by the relations between braided commutators and the adjoint (1.21), the relations (6.12), (6.13) and (6.14) translate respectively to

(6.32)  ad E_{i,i+1}(E_{p,p+1}) = 0;  1 ≤ i < p ≤ n,  p − i ≥ 2;
(6.33)  (ad E_{i,i+1})^{2}(E_{i+1,i+2}) = 0,  1 ≤ i < n;
(6.34)  (ad E_{i+1,i+2})^{2}(E_{i,i+1}) = 0,  1 ≤ i < n.

Remark 6.15. Relations (6.32), (6.33) and (6.34) can be considered, more generally, in any H as above. If these relations hold in H, then we have a Hopf algebra map π_{H} : 𝔅̂(V)#𝕜Γ → H. On the other hand, we know by Remark 6.10 that the comultiplication of the elements E_{ij}^{N} is given by (6.27). Hence, the same formula is valid in H, provided that relations (6.32), (6.33) and (6.34) hold in it. In particular, the subalgebra of H generated by the elements E_{ij}^{N}, g_{i,j}^{N}, 1 ≤ i < j ≤ n+1, is a Hopf subalgebra of H.

Lemma 6.16. Relations (6.32), (6.33) and (6.34) hold in A if N > 3.

Proof. This is a particular case of Theorem 5.6; we include the proof for completeness. We know, by Lemma 2.10, that

ad E_{i,i+1}(E_{p,p+1}) ∈ P_{g_{i}g_{p},1}(A)^{χ_{i}χ_{p}},  1 ≤ i < p ≤ n,  p − i ≥ 2,
(ad E_{i,i+1})^{2}(E_{p,p+1}) ∈ P_{g_{i}^{2}g_{p},1}(A)^{χ_{i}^{2}χ_{p}},  1 ≤ i, p ≤ n,  |p − i| = 1.

Assume that ad E_{i,i+1}(E_{p,p+1}) ≠ 0, and χ_{i}χ_{p} ≠ ε, where 1 ≤ i < p ≤ n, p − i ≥ 2. By Lemma 5.1, there exists ℓ, 1 ≤ ℓ ≤ n, such that g_{i}g_{p} = g_{ℓ}, χ_{i}χ_{p} = χ_{ℓ}. But then

q = χ_{ℓ}(g_{ℓ}) = χ_{i}(g_{i})χ_{i}(g_{p})χ_{p}(g_{i})χ_{p}(g_{p}) = q^{2}.

Hence q = 1, a contradiction.

Assume next that ad E_{i,i+1}^{2}(E_{p,p+1}) ≠ 0, |p − i| = 1. and χ_{i}^{2}χ_{p} ≠ ε. By Lemma 5.1 , there exists ℓ, 1 ≤ ℓ ≤ n, such that g_{i}^{2}g_{p} = g_{ℓ}, χ_{i}^{2}χ_{p} = χ_{ℓ}. But then

q = χ_{ℓ}(g_{ℓ}) = χ_{i}(g_{i})^{4}χ_{i}(g_{p})^{2}χ_{p}(g_{i})^{2}χ_{p}(g_{p}) = q^{3}.

Hence q = ±1, a contradiction (we assumed N > 2).

It remains to exclude the cases χ_{i}χ_{p} = ε, |p − i| ≥ 2, and χ_{i}^{2}χ_{p} = ε, |p − i| = 1. The first case leads to the contradiction N = 3. In the second case it follows from the connectivity of A_{n} that N would divide 2 which is also impossible.

Lemma 6.17. If H = A, then E_{i,j}^{N} ∈ 𝕜Γ^{N}, for any 1 ≤ i < j ≤ n + 1.

Proof. We first show that E_{i,j}^{N} ∈ 𝕜Γ, 1 ≤ i < j ≤ n + 1. (For our further purposes, this is what we really need).

Let i < j. We claim that there exists no ℓ, 1 ≤ ℓ ≤ n, such that g_{i,j}^{N} = g_{ℓ}, χ_{i,j}^{N} = χ_{ℓ}. Indeed, otherwise we would have

q = χ_{ℓ}(g_{ℓ}) = χ_{i,j}(g_{i,j})^{N^{2}} = q^{N^{2}} = 1.

By Lemma 6.16 and Remark 6.15, we have

Δ(E_{i,j}^{N}) = E_{i,j}^{N} ⊗ 1 + g_{i,j}^{N} ⊗ E_{i,j}^{N} + (1 − q^{−1})^{N} ∑_{i<p<j} (B_{i,p}^{p,j})^{N(N−1)/2} E_{i,p}^{N}g_{p,j}^{N} ⊗ E_{p,j}^{N}. (6.35)

We proceed by induction on j − i. If j − i = 1, then, by Lemma 5.1 , either E_{i,i+1}^{N} ∈ 𝕜Γ or E_{i,i+1}^{N} ∈ P_{g_{i}^{N} ,1}(A)^{χ_{i}^{N}} and χ_{i}^{N} ≠ ε, hence g_{i}^{N} = g_{l}, χ_{i}^{N} = χ_{l} for some l; but this last possibility contradicts the claim above. Assume then that j − i > 1. By the recursion hypothesis, Δ(E_{i,j}^{N}) = E_{i,j}^{N} ⊗ 1 + g_{i,j}^{N} ⊗ E_{i,j}^{N} + u, for some u ∈ 𝕜Γ ⊗ 𝕜Γ. In particular, we see that E_{i,j}^{N} ∈ (A_{1})^{χ_{i}^{N}}. Then,

POINTED HOPF ALGEBRAS 53

by Lemma 5.1, either χ_{i}^{N} = ε and hence E_{i,i+1}^{N} ∈ 𝕜Γ, or else χ_{i}^{N} ≠ ε, which implies u = 0 and E_{i,i+1}^{N} ∈ P_{g_{i}^{N} ,1}(A)^{χ_{i}^{N}}. Again, this last possibility contradicts the claim above.

Finally, let C be the subalgebra of A generated by the elements E_{ij}^{N}, g_{i,j}^{N}, 1 ≤ i < j ≤ n + 1, which is a Hopf subalgebra of H. Since E_{i,j}^{N} ∈ 𝕜Γ ∩ C, we conclude that E_{i,j}^{N} ∈ C_{0} = 𝕜Γ^{N}.

To solve the lifting problem, we see from Lemma 6.17 that we first have to answer a combinatorial question in the group algebra of an abelian group. To simplify the notation we define

h_{ij} = g_{i,j}^{N}, C_{i,p}^{j} = (1 − q^{−1})^{N} (B_{i,p}^{p,j})^{N(N−1)/2} .

We are looking for families (u_{ij})_{1≤i<j≤n+1} of elements in 𝕜Γ such that

Δ(u_{ij}) = u_{ij} ⊗ 1 + h_{i,j} ⊗ u_{ij} + ∑_{i<p<j} C_{i,p}^{j}u_{i,p} h_{p,j} ⊗ u_{p,j}. (6.36)

The coefficients C_{i,p}^{j} satisfy the rule

C_{is}^{j}C_{st}^{j} = C_{is}^{t}C_{it}^{j}, for all 1 ≤ i < s < t < j ≤ n + 1. (6.37)

This follows from (6.2) and (6.3) since B_{is}^{sj}B_{st}^{tj} = B_{is}^{st}B_{is}^{tj}B_{st}^{tj} = B_{it}^{st}B_{it}^{tj}.

Theorem 6.18. Let Γ be a finite abelian group and h_{ij} ∈ Γ, 1 ≤ i < j ≤ n + 1, a family of elements such that

h_{ij} = h_{i,p} h_{p,j}, if i < p < j. (6.38)

Let C_{i,p}^{j} ∈ 𝕜^{×}, 1 ≤ i < p < j ≤ n + 1, be a family of elements satisfying (6.37). Then the solutions (u_{ij})_{1≤i<j≤n+1} of (6.36), u_{ij} ∈ 𝕜Γ for all i < j, have the form (u_{ij}(γ))_{1≤i<j≤n+1} where γ = (γ_{ij})_{1≤i<j≤n+1} is an arbitrary family of scalars γ_{ij} ∈ 𝕜 such that

(6.39) for all 1 ≤ i < j ≤ n + 1, γ_{ij} = 0 if h_{ij} = 1,

and where the elements u_{ij}(γ) are defined by induction on j − i by

u_{ij}(γ) = γ_{ij}(1 − h_{ij}) + ∑_{i<p<j} C_{ip}^{j}γ_{ip}u_{pj}(γ) for all 1 ≤ i < j ≤ n + 1. (6.40)

Proof. We proceed by induction on k. We claim that the solutions u_{ij} ∈ 𝕜Γ, 1 ≤ i < j ≤ n + 1, j − i ≤ k, of (6.36) for all i < j with j − i ≤ k are given by arbitrary families of scalars γ_{ij}, 1 ≤ i < j ≤ n + 1, j − i ≤ k such that

u_{ij} = γ_{ij}(1 − h_{ij}) + ∑_{i<p<j} C_{ip}^{j}γ_{ip}u_{pj} for all 1 ≤ i < j ≤ n + 1 with j − i ≤ k.

54 ANDRUSKIEWITSCH AND SCHNEIDER

Suppose k = 1. For any 1 ≤ i < n, j = i + 1, u_{i,i+1} is a solution of (6.36) if and only if u_{i,i+1} is (h_{i,i+1}, 1)-primitive in 𝕜Γ, that is u_{i,i+1} = γ_{i,i+1}(1 − h_{i,i+1}) for some γ_{i,i+1} ∈ 𝕜. We may assume that γ_{i,i+1} = 0, if h_{i,i+1} = 1.

For the induction step, let k > 2. We assume that γ_{ab} ∈ 𝕜, 1 ≤ a < b ≤ n + 1, b − a ≤ k − 1, is a family of scalars with γ_{ab} = 0, if h_{ab} = 1, and that the family u_{ab} ∈ 𝕜Γ, 1 ≤ a < b ≤ n + 1, b − a ≤ k − 1, defined inductively by the γ_{ab} by (6.40) is a solution of (6.36). Let 1 ≤ i < j ≤ n + 1, and j − i = k. We have to show that

Δ(u_{ij}) = u_{ij} ⊗ 1 + h_{ij} ⊗ u_{ij} + ∑_{i<p<j} C_{ip}^{j}u_{ip}h_{pj} ⊗ u_{pj} (6.41)

is equivalent to

u_{ij} = γ_{ij}(1 − h_{ij}) + ∑_{i<p<j} C_{ip}^{j}γ_{ip}u_{pj} for some γ_{ij} ∈ 𝕜. (6.42)

We then may define γ_{ij} = 0 if h_{ij} = 1.

We denote

z_{ij} := u_{ij} − ∑_{i<p<j} C_{ip}^{j}γ_{ip}u_{pj}.

Then (6.42) is equivalent to

Δ(z_{ij}) = z_{ij} ⊗ 1 + h_{ij} ⊗ z_{ij}. (6.43)

For all i < p < j we have Δ(u_{pj}) = u_{pj} ⊗ 1 + h_{pj} ⊗ u_{pj} + ∑_{p<s<j} C_{ps}^{j}u_{ps}h_{sj} ⊗ u_{sj}, since j − p < k. Using this formula for Δ(u_{pj}) we compute

Δ(z_{ij}) − z_{ij} ⊗ 1 − h_{ij} ⊗ z_{ij}

= Δ(u_{ij}) − ∑_{i<p<j} C_{ip}^{j}γ_{ip}Δ(u_{pj}) − z_{ij} ⊗ 1 − h_{ij} ⊗ z_{ij}

= Δ(u_{ij}) − ∑_{i<p<j} C_{ip}^{j}γ_{ip}(u_{pj} ⊗ 1 + h_{pj} ⊗ u_{pj} + ∑_{p<s<j} C_{ps}^{j}u_{ps}h_{sj} ⊗ u_{sj})

− (u_{ij} − ∑_{i<p<j} C_{ip}^{j}γ_{ip}u_{pj}) ⊗ 1 − h_{ij} ⊗ (u_{ij} − ∑_{i<p<j} C_{ip}^{j}γ_{ip}u_{pj})

= Δ(u_{ij}) − u_{ij} ⊗ 1 − h_{ij} ⊗ u_{ij}

+ ∑_{i<p<j} C_{ip}^{j}γ_{ip}h_{ij} ⊗ u_{pj} − ∑_{i<p<j} C_{ip}^{j}γ_{ip}h_{pj} ⊗ u_{pj} − ∑_{i<p<s<j} C_{ip}^{j}C_{ps}^{j}γ_{ip}u_{ps}h_{sj} ⊗ u_{sj}.

POINTED HOPF ALGEBRAS 55

Therefore, (6.41) and (6.42) are equivalent if and only if the identity

∑_{i<p<j} C_{ip}^{j}γ_{ip}h_{ij} ⊗ u_{pj} − ∑_{i<p<j} C_{ip}^{j}γ_{ip}h_{pj} ⊗ u_{pj} − ∑_{i<p<s<j} C_{ip}^{j}C_{ps}^{j}γ_{ip}u_{ps}h_{sj} ⊗ u_{sj} (6.44)

= − ∑_{i<p<j} C_{ip}^{j}u_{ip}h_{pj} ⊗ u_{pj}

holds.

To prove (6.44) we use (6.40) for all i < p, where i < p < j, that is u_{ip} = γ_{ip}(1 − h_{ip}) + ∑_{i<s<p} C_{is}^{p}γ_{is}u_{sp}. Then

∑_{i<p<j} C_{ip}^{j}h_{pj}u_{ip} ⊗ u_{pj} + ∑_{i<p<j} C_{ip}^{j}γ_{ip}h_{ij} ⊗ u_{pj} − ∑_{i<p<j} C_{ip}^{j}γ_{ip}h_{pj} ⊗ u_{pj}

= ∑_{i<p<j}(C_{ip}^{j}h_{pj}(γ_{ip}(1 − h_{ip}) + ∑_{i<s<p} C_{is}^{p}γ_{is}u_{sp}) + C_{ip}^{j}γ_{ip}(h_{ij} − h_{pj})) ⊗ u_{pj}

= ∑_{i<p<j} C_{ip}^{j}h_{pj} ∑_{i<s<p} C_{is}^{p}γ_{is}u_{sp} ⊗ u_{pj}, since h_{pj}(1 − h_{ip}) = h_{pj} − h_{ij} by (6.38),

= ∑_{i<s<p<j} C_{is}^{j}C_{sp}^{j}γ_{is}u_{sp}h_{pj} ⊗ u_{pj}, since C_{ip}^{j}C_{is}^{p} = C_{is}^{j}C_{sp}^{j} by (6.37).

This proves (6.44) by interchanging s and p.

Remarks 6.19. 1) Let γ = (γ_{ij})_{1≤1qi<j≤n+1} be an arbitrary family of scalars. Then it is easy to see that the family u_{ij}(γ) ∈ 𝕜Γ, 1 ≤ i < j ≤ n + 1, can be defined explicitly as follows:

u_{ij}(γ) = ∑_{i≤p<j} ϕ_{ip}^{j}(γ)(1 − h_{pj}) for all i < j,

where

ϕ_{ip}^{j}(γ) = ∑_{i=i_{1}<···<i_{k}=p,k≥1} C_{i_{1},i_{2}}^{j} … C_{i_{k−1},i_{k}}γ_{i_{1},i_{2}} … γ_{i_{k−1},i_{k}}γ_{pj} for all i ≤ p < j

is a polynomial of degree p in the free variables (γ_{ij})_{1≤i<j≤n+1}.

2) Let γ = (γ_{ij})_{1≤i<j≤n+1} and γ̃ = (γ̃_{ij})_{1≤i<j≤n+1} be families of scalars in 𝕜 satisfying (6.39). Assume that for all i < j, u_{ij}(γ) = u_{ij}(γ̃). Then γ = γ̃. This follows easily be induction on j − i from (6.40).

Lemma 6.20. Assume the situation of Theorem 6.18. Let γ = (γ_{ij})_{1≤i<j≤n+1} be a family of scalars in 𝕜 satisfying (6.39) and define u_{ij} = u_{ij}(γ) for all 1 ≤ i < j ≤ n + 1 by (6.40).

1) The following are equivalent: (a) For all i < j, u_{ij} = 0 if χ_{ij}^{N} ≠ ε.

56 ANDRUSKIEWITSCH AND SCHNEIDER

(b) For all i < j, γ_{ij} = 0 if χ_{ij}^{N} ≠ ε.

2) Assume that h_{ij} = g_{ij}^{N} for all i < j. Then the following are equivalent: (a) For all i < j, u_{ij} = 0 if χ_{ij}^{N}(g_{l}) ≠ 1 for some 1 ≤ l ≤ n.

(b) For all i < j, γ_{ij} = 0 if χ_{ij}^{N}(g_{l}) ≠ 1 for some 1 ≤ l ≤ n.

(c) The elements u_{ij}, 1 ≤ i < j ≤ n + 1, are central in 𝔅̂(V)#𝕜Γ.

Proof. 1) follows by induction on j − i.

Suppose j = i + 1. Then u_{i,i+1} = γ_{i,i+1}(1 − h_{i,i+1}). If h_{i,i+1} = 1, then both u_{i,i+1} and γ_{i,i+1} are 0. If h_{i,i+1} ≠ 1, then u_{i,i+1} = 0 if and only if γ_{i,i+1} = 0.

The induction step follows in the same way from (6.40), since for all i < p < j, if χ_{ij}^{N} ≠ ε, then χ_{ip}^{N} ≠ ε or χ_{pj}^{N} ≠ ε, hence by induction γ_{ip} = 0 or u_{pj} = 0, and u_{ij} = γ_{ij}(1 − h_{ij}).

2) Suppose that for all i < p < j, u_{pj} is central in 𝔅̂(V)#𝕜Γ, and let 1 ≤ l ≤ n. Then

h_{ij}x_{l} = x_{l}χ_{l}(h_{ij})h_{ij},

and we obtain from (6.40)

u_{ij}x_{l} = x_{l}γ_{ij}(1 − χ_{l}(h_{ij})h_{ij}) + x_{l} ∑_{i<p<j} C_{ip}^{j}γ_{ip}u_{pj}.

Hence u_{ij} is central in 𝔅̂(V)#𝕜Γ if and only if γ_{ij} = γ_{ij}χ_{l}(h_{ij}) for all 1 ≤ l ≤ n. Since the braiding is of type A_{n} and the order of q = χ_{l}(g_{l}) is N,

χ_{l}(h_{ij}) = χ_{l}(g_{ij}^{N}) = χ_{ij}^{−N}(g_{l}),

and the equivalence of (b) and (c) follows by induction on j − i. The equivalence of (a) and (b) is shown as in 1).

6.3. Classification of pointed Hopf algebras of type A_{n}.

Using the previous results we will now determine exactly which finite-dimensional pointed Hopf algebras are of type A_{n}. We will find a big new class of deformations of 𝔲_{q}^{≥0}(sl_{n}).

As before, we fix a natural number n, a finite abelian group Γ, an integer N > 2, a root of unity q of order N, g_{1}, …, g_{n} ∈ Γ, χ_{1}, …, χ_{n} ∈ Γ̂ such that q_{ij} = χ_{j}(g_{i}) for all i, j satisfy (6.1), and V ∈ _{Γ}^{Γ}YD with basis x_{i} ∈ V_{g_{i}}^{χ_{i}}, 1 ≤ i ≤ n.

We recall that 𝔅̂(V) is the braided Hopf algebra in _{Γ}^{Γ}YD generated by x_{1}, …, x_{n} with the quantum Serre relations (6.12), (6.13) and (6.14).

In 𝔅̂(V) we consider the iterated braided commutators e_{i,j}, 1 ≤ i < j ≤ n+1 defined inductively by (6.8) beginning with e_{i,i+1} = x_{i} for all i.

Let 𝔸 be the set of all families (a_{i,j})_{1≤i<j≤n+1} of integers a_{i,j} ≥ 0 for all 1 ≤ i < j ≤ n + 1. For any a ∈ 𝔸 we define

e^{a} := (e_{1,2})^{a_{1,2}}(e_{1,3})^{a_{1,3}} … (e_{n,n+1})^{a_{n,n+1}},

where the order in the product is the lexicographic order of the index pairs. We begin with the PBW-theorem for 𝔅̂(V).

Theorem 6.21. The elements e^{a}, a ∈ 𝔸, form a basis of the 𝕜-vector space 𝔅̂(V).

Proof. This can be shown similarly to the proof of Theorem 6.13. For general finite Cartan type the theorem can be derived from the PBW-basis of U_{q}(𝔤) (see [L3]) by changing the group and twisting as described in [AS4, Section 4.2].

The following commutation rule for the elements e_{ij}^{N} is crucial.

Lemma 6.22. For all 1 ≤ i < j ≤ n + 1, 1 ≤ s < t ≤ n + 1,

[e_{i,j}, e_{s,t}^{N}]_{c} = 0, that is e_{i,j}e_{s,t}^{N} = χ_{s,t}^{N}(g_{i,j})e_{s,t}^{N}e_{i,j}.

Proof. Since e_{i,j} is a linear combination of elements of the form x_{i_{1}} … x_{i_{k}} with k = j − i and g_{i_{1}} … g_{i_{k}} = g_{ij}, it is enough to consider the case when j = i + 1.

To show [e_{i,i+1}, e_{s,t}^{N}]_{c} = 0, we will distinguish several cases.

First assume that (i, i + 1) < (s, t). If i + 1 < s resp. i = s and i + 1 < t, then [e_{i,i+1}, e_{s,t}]_{c} = 0 by (6.17) resp. (6.23), and the claim follows.

If i + 1 = s, we denote x = e_{i+1,t}, y = e_{i,i+1}, z = e_{it} and α = χ_{i+1,t}(g_{i}), β = χ_{i+1,t}(g_{i,t}). Then

yx = αxy + z, by (6.19), and zx = βxz, by (6.24).

Moreover, α = χ_{i+1,t}(g_{i}) ≠ β = χ_{i+1,t}(g_{i,t}) = χ_{i+1,t}(g_{i})χ_{i+1,t}(g_{i+1,t}), and α^{N} = β^{N}, since χ_{i+1,t}(g_{i+1,t}) = q by (6.5). Therefore it follows from [AS4, Lemma 3.4] that yx^{N} = α^{N}x^{N}y, which was to be shown.

The claim is clear if i = s, and i + 1 = t, since χ_{i}^{N}(g_{i}) = 1.

It remains to consider the case when (i, i + 1) > (s, t). If s < i and t = i + 1, resp. s < i and i+1 < t, resp. s = i and t < i+1, then e_{s,t}e_{i,i+1} = χ_{i}(g_{s,t})e_{i,i+1}e_{s,t} by (6.24), resp.(6.22),resp.(6.23). Hence in all cases, e_{i,i+1}e_{s,t}^{N} = χ_{i}^{−N}(g_{s,t})e_{s,t}^{N}e_{i,i+1}. This proves the claim [e_{i,i+1}, e_{s,t}^{N}]_{c} = 0, since χ_{i}^{−N}(g_{s,t}) = χ_{s,t}^{N}(g_{i}).

We want to compute the dimension of certain quotient algebras of 𝔅(V)#𝕜Γ. Since this part of the theory works for any finite Cartan type, we now consider more generally a left 𝕜Γ-module algebra R over any abelian group Γ and assume that there are integers P and N_{i} > 1, elements y_{i} ∈ R, h_{i} ∈ Γ, η_{i} ∈ Γ̂, 1 ≤ i ≤ P, such that g · y_{i} = η_{i}(g)y_{i}, for all g ∈ Γ, 1 ≤ i ≤ P. (6.45)

y_{i}y_{j}^{N_{j}} = η_{j}^{N_{j}}(h_{i})y_{j}^{N_{j}}y_{i} for all 1 ≤ i, j ≤ P. (6.46) (6.47) The elements y_{1}^{a_{1}} … y_{P}^{a_{P}} , a_{1}, …, a_{P} ≥ 0, form a 𝕜 − basis of R.

Let 𝕃 be the set of all l = (l_{i})_{1≤i≤P} ∈ ℕ^{P} such that 0 ≤ l_{i} < N_{i} for all 1 ≤ i ≤ P. For a = (a_{i})_{1≤i≤P} ∈ ℕ^{P}, we define

y^{a} = y_{1}^{a_{1}} … y_{P}^{a_{P}} , and aN = (a_{i}N_{i})_{1≤i≤P} .

Then by (6.46), (6.47), the elements

y^{l}y^{aN}, l ∈ 𝕃, a ∈ ℕ^{P} ,

form a 𝕜-basis of R.

In the application to 𝔅(V)#𝕜Γ, P is the number of positive roots, and the y_{i} play the role of the root vectors e_{i,j}.

To simplify the notation in the smash product algebra R#𝕜Γ, we identify r ∈ R with r#1 and v ∈ 𝕜Γ with 1#v. For 1 ≤ i ≤ P, let η̃_{i} : 𝕜Γ → 𝕜Γ be the algebra map defined by η̃_{i}(g) = η_{i}(g)g for all g ∈ Γ. Then

vy_{i} = y_{i}η̃_{i}(v) for all v ∈ 𝕜Γ.

We fix a family u_{i}, 1 ≤ i ≤ P, of elements in 𝕜Γ, and denote

u^{a} := ∏_{1≤i≤P} u_{i}^{a_{i}}, if a = (a_{i})_{1≤i≤P} ∈ ℕ^{P} .

Let M be a free right 𝕜Γ-module with basis m(l), l ∈ 𝕃. We then define a right 𝕜Γ-linear map

φ : R#𝕜Γ → M by φ(y^{l}y^{aN}) := m(l)u^{a} for all l ∈ 𝕃, a ∈ ℕ^{P} .

Lemma 6.23. Assume that

• u_{i} is central in R#𝕜Γ, for all 1 ≤ i ≤ P, and

• u_{i} = 0 if η_{i}^{N_{i}}(h_{j}) ≠ 1 for some 1 ≤ j ≤ P.

Then the kernel of φ is a right ideal of R#𝕜Γ containing y_{i}^{N_{i}} − u_{i} for all 1 ≤ i ≤ P.

Proof. By definition, φ(y_{i}^{N_{i}}) = m(0)u_{i} = φ(u_{i}).

To show that the kernel of φ is a right ideal, let

z = ∑_{l∈𝕃,a∈ℕ^{P}} y^{l}y^{aN}v_{l,a}, where v_{l,a} ∈ 𝕜Γ, for all l ∈ 𝕃, a ∈ ℕ^{P} ,

be an element with φ(z) = 0. Then φ(z) = ∑_{l,a} m(l)u^{a}v_{l,a} = 0, hence

∑_{a∈ℕ^{P}} u^{a}v_{l,a} = 0, for all l ∈ 𝕃.

Fix 1 ≤ i ≤ P. We have to show that φ(zy_{i}) = 0.

For any l ∈ 𝕃, we have the basis representation

y^{l}y_{i} = ∑_{t∈𝕃,b∈ℕ^{P}} α_{t,b}^{l}y^{t}y^{bN}, where α_{t,b}^{l} ∈ 𝕜 for all t ∈ 𝕃, b ∈ ℕ^{P} .

POINTED HOPF ALGEBRAS 59

Since u^{a} is central in R#𝕜Γ, u^{a} = η̃_{i}(u^{a}) for all a ∈ ℕ^{P} . (6.48) For any a = (a_{i})_{1≤i≤P} ∈ ℕ^{P} and any family (g_{i})_{1≤i≤P} of elements in Γ we define η^{aN}((g_{i})) = ∏_{i} η_{i}^{a_{i}N_{i}}(g_{i}). Then by (6.46), for all a, b ∈ ℕ^{P}, y^{aN}y_{i} = y_{i}y^{aN}η^{aN}(g^{a}), and y^{bN}y^{aN} = y^{(a+b)N}η^{aN}(g^{b}), (6.49) for some families of elements g^{a}, g^{b} in Γ.

By a reformulation of our assumption, (6.50) u^{a}η^{aN}((g_{i})) = u^{a} for any a ∈ ℕ^{P} and family(g_{i}) in Γ. Using (6.49) we now can compute

zy_{i} = ∑_{l,a} y^{l}y^{aN}v_{l,a}y_{i} = ∑_{l,a} y^{l}y^{aN}y_{i}η̃_{i}(v_{l,a})

= ∑_{l,a} y^{l}y_{i}y^{aN}η^{aN}(g^{a})η̃_{i}(v_{l,a})

= ∑_{l,a} ∑_{t,b} α_{t,b}^{l}y^{t}y^{bN}e^{aN}η^{aN}(g^{a})η̃_{i}(v_{l,a})

= ∑_{l,a} ∑_{t,b} α_{t,b}^{l}y^{t}y^{(a+b)N}η^{aN}(g^{a})η^{aN}(g^{b})η̃_{i}(v_{l,a}).

Therefore

φ(zy_{i}) = ∑_{t} m(t) ∑_{l,a,b} α_{t,b}^{l}u^{a+b}η^{aN}(g^{a})η^{aN}(g^{b})η̃_{i}(v_{l,a})

= ∑_{t} m(t) ∑_{l,a,b} α_{t,b}^{l}u^{a+b}η̃_{i}(v_{l,a}), by (6.50),

= ∑_{t} m(t) ∑_{b,l} α_{t,b}^{l}u^{b} ∑_{a} u^{a}η̃_{i}(v_{l,a})

= ∑_{t} m(t) ∑_{b,l} α_{t,b}^{l}u^{b}η̃_{i}(∑_{a} u^{a}v_{l,a}), by (6.48),

= 0, since ∑_{a} u^{a}v_{l,a} = 0.

Theorem 6.24. Let u_{i}, 1 ≤ i ≤ P, be a family of elements in 𝕜Γ, and I the ideal in R#𝕜Γ generated by all y_{i}^{N_{i}} − u_{i}, 1 ≤ i ≤ P. Let A = (R#𝕜Γ)/I be the quotient algebra. Then the following are equivalent:

1) The residue classes of y^{l}g, l ∈ 𝕃, g ∈ Γ, form a 𝕜-basis of A.

2) u_{i} is central in R#𝕜Γ for all 1 ≤ i ≤ P, and u_{i} = 0 if η_{i}^{N_{i}} ≠ ε.

Proof. 1) ⇒ 2) : For all i and g ∈ Γ, gy_{i}^{N_{i}} = η_{i}^{N_{i}}(g)y_{i}^{N_{i}}g, hence u_{i}g = gu_{i} ≡ η_{i}^{N_{i}}(g)u_{i}g mod I. Since by assumption, 𝕜Γ is a subspace of A, we conclude that u_{i} = η_{i}^{N_{i}}(g)u_{i}, and u_{i} = 0 if η_{i}^{N_{i}} ≠ ε.

Similarly, for all 1 ≤ i, j ≤ n, y_{i}y_{j}^{N_{j}} = η_{j}^{N_{j}}(h_{i})y_{j}^{N_{j}}y_{i} by (6.46), hence y_{i}u_{j} ≡ η_{j}^{N_{j}}(h_{i})u_{j}y_{i} mod I. Since we already know that u_{i} = 0 if η_{j}^{N_{j}} ≠ ε, we see that y_{i}u_{j} ≡ u_{j}y_{i} mod I. On the other hand u_{j}y_{i} = y_{i}η̃_{i}(u_{j}). Then our assumption in 1) implies that η̃_{i}(u_{j}) = u_{j}. In other words, u_{j} is central in R#𝕜Γ.

2) ⇒ 1): Let J be the right ideal of R#𝕜Γ generated by all y_{i}^{N_{i}} − u_{i}, 1 ≤ i ≤ P. For any 1 ≤ i ≤ P and g ∈ Γ,

g(y_{i}^{N_{i}} − u_{i}) = y_{i}^{N_{i}}gη_{i}^{N_{i}}(g) − gu_{i} = (y_{i}^{N_{i}} − u_{i})η_{i}^{N_{i}}(g)g,

since gu_{i} = u_{i}η_{i}^{N_{i}}(g)g by 2).

And for all 1 ≤ i, j ≤ P,

y_{i}(y_{j}^{N_{j}} − u_{j}) = η_{j}^{N_{j}}(h_{i})y_{j}^{N_{j}}y_{i} − y_{i}u_{j} = (y_{j}^{N_{j}} − u_{j})η_{j}^{N_{j}}(h_{i})y_{i},

since by 2) y_{i}u_{j} = u_{j}y_{i} = u_{j}η_{j}^{N_{j}}(h_{i})y_{i}.

This proves J = I.

It is clear that the images of all y^{l}g, l ∈ 𝕃, g ∈ Γ, generate the vector space A. To show linear independency, suppose

∑_{l∈𝕃,g∈Γ}α_{l,g}y^{l}g ∈ I, with α_{l,g} ∈ 𝕜 for all l ∈ 𝕃, g ∈ Γ.

Since I = J, we obtain from Lemma 6.23 that φ(I) = 0. Therefore,

0 = φ( ∑_{l∈𝕃,g∈Γ}α_{l,g}y^{l}g) = ∑_{l∈𝕃,g∈Γ}α_{l,g}m(l)g,

hence α_{l,g} = 0 for all l, g.

We come back to A_{n}. Our main result in this Chapter is

Theorem 6.25. (i). Let γ = (γ_{i,j})_{1≤i<j≤n+1} be any family of scalars in 𝕜 such that for any i < j, γ_{i,j} = 0 if g_{i,j}^{N} = 1 or χ_{i,j}^{N} ≠ ε. Define u_{i,j} = u_{i,j}(γ) ∈ 𝕜Γ, 1 ≤ i < j ≤ n + 1, by (6.40). Then

A_{γ} := (𝔅̂(V)#𝕜Γ)/(e_{i,j}^{N} − u_{i,j} | 1 ≤ i < j ≤ n + 1)

is a pointed Hopf algebra of dimension N^{\frac{n(n+1)}{2}} ord(Γ) with grA_{γ} ≃ 𝔅(V)#𝕜Γ.

(ii). Conversely, if A is a finite-dimensional pointed Hopf algebra with grA ≃ 𝔅(V)#𝕜Γ, and N > 3, or if the infinitesimal braiding of A is of type A_{n} with N > 7 and not divisible by 3, then A is isomorphic to a Hopf algebra A_{γ} in (i).

Proof. (i). By Lemma 6.20, the elements u_{i,j} are central in Û := 𝔅̂(V)#𝕜Γ, and u_{i,j} = 0 if g_{i,j}^{N} = 1 or χ_{i,j}^{N} ≠ ε. Hence the residue classes of the elements e^{l}g, l ∈ 𝔸, 0 ≤ l_{i,j} < N for all 1 ≤ i < j ≤ n + 1, g ∈ Γ, form a basis of A_{γ} by Theorem 6.24. By Theorem 6.18, the u_{i,j} satisfy (6.36). The ideal I of Û generated by all e_{i,j}^{N} − u_{i,j} is a biideal, since

Δ(e_{i,j}^{N} − u_{i,j}) =(e_{i,j}^{N} − u_{i,j}) ⊗ 1 + g_{i,j}^{N} ⊗ (e_{i,j}^{N} − u_{i,j})

+ ∑_{i<p<j} C_{i,p}^{j}((e_{i,p}^{N} − u_{i,p})g_{p,j}^{N} ⊗ e_{p,j}^{N} + u_{i,p}g_{p,j}^{N} ⊗ (e_{p,j}^{N} − u_{p,j})) ∈ I ⊗ Û + Û ⊗ I,

by (6.35) and (6.36).

Since A_{γ} is generated by group-like and skew-primitive elements, and the group-like elements form a group, A_{γ} is a Hopf algebra.

For all 1 ≤ i ≤ n, let a_{i} ∈ gr(A_{γ})(1) be the residue class of x_{i} ∈ (A_{γ})_{1}. Define root vectors a_{i,j} ∈ gr(A_{γ}), 1 ≤ i < j ≤ N + 1 inductively as in (6.30) and (6.31). Then a_{i,j}^{N} = 0 in gr(A_{γ}) since e_{i,j}^{N} = 0 in A_{γ}. Therefore, by Theorem 6.14, there is a surjective Hopf algebra map

𝔅(V)#𝕜Γ → gr(A_{γ}) mapping x_{i}#g onto a_{i}g, 1 ≤ i ≤ n, g ∈ Γ.

This map is an isomorphism, since dim(gr(A_{γ})) = dim(A_{γ}) = N^{\frac{n(n+1)}{2}} |Γ| = dim(𝔅(V)#𝕜Γ) by Theorem 6.13.

(ii). As in Section 6.2, we choose elements a_{i} ∈ (A_{1})_{g_{i}}^{χ_{i}} such that π(a_{i}) = x_{i}, 1 ≤ i ≤ n. By assumption resp. by Theorem 6.16, there is a Hopf algebra map

ϕ : 𝔅̂(V)#𝕜Γ → A, ϕ(x_{i}#g) = a_{i}g, 1 ≤ i ≤ n, g ∈ Γ.

By Theorem 5.8, A is generated in degree one, hence ϕ is surjective. We define the root vector E_{i,j} ∈ A, 1 ≤ i < j ≤ n + 1, by (6.30), (6.31). By Lemma 6.17, E_{i,j}^{N} =: u_{i,j} ∈ 𝕜Γ for all 1 ≤ i < j ≤ n + 1. Then for all g ∈ Γ and i < j, gE_{i,j}^{N} = χ_{i,j}^{N}(g)E_{i,j}^{N}g, hence gu_{i,j} = χ_{i,j}^{N}(g)u_{i,j}g, and u_{i,j} = χ_{i,j}^{N}(g)u_{i,j}. By (6.35) and Theorem 6.18 we therefore know that u_{i,j} = u_{i,j}(γ) for all i < j, for some family γ = (γ_{i,j})_{1≤i≤J≤n+1} of scalars in 𝕜 such that for all i < j, γ_{i,j} = 0 if g_{i,j}^{N} = 1 or χ_{i,j}^{N} = ε. Hence ϕ indices a surjective Hopf algebra map A_{γ} → A which is an isomorphism since dim(A_{γ}) = N^{\frac{n(n+1)}{2}} ord(Γ) = dim(A) by 1).

Remark 6.26. Up to isomorphism, A_{γ} does not change if we replace each x_{i} by a non-zero scalar multiple of itself. Hence in the definition of A_{γ} we may always assume that

γ_{i,i+1} = 0 or 1 for all 1 ≤ i ≤ n.

We close the paper with a very special case of Theorem 6.25. We obtain a large class of nonisomorphic Hopf algebras which have exactly the same infinitesimal braiding as 𝔲_{q}^{≥0}(sl_{n}). Here q has order N, but the group is ∏_{i=1}^{n} ℤ/(Nm_{i}) and not (ℤ/(N))^{n} as for 𝔲_{q}^{≥0}(sl_{n}).

Example 6.27. Let N be > 2, q a root of unity of order N, and m_{1}, …, m_{n} integers > 1 such that m_{i} ≠ m_{j} for all i ≠ j. Let Γ be the commutative group generated by g_{1}, …, g_{n} with relations g_{i}^{Nm_{i}} = 1, 1 ≤ i ≤ n. Define χ_{1}, …, χ_{n} ∈ Γ̂ by

χ_{j}(g_{i}) = q^{a_{ij}}, where a_{ii} = 2 for all i, a_{ij} = −1 if |i − j| = 1, a_{ij} = 0 if |i − j| ≥ 2.

Then χ_{i,j}^{N} = ε and g_{i,j}^{N} ≠ 1 for all i < j. Thus for any family γ = (γ_{i,j})_{1≤i<j≤n+1} of scalars in 𝕜, A_{γ} in Theorem 6.25 has infinitesimal braiding of type A_{n}.

Moreover, if γ, γ̃ are arbitrary such families with γ_{i,i+1} = 1 = γ̃_{i,i+1} for all 1 ≤ i ≤ n, then

A_{γ} ≇ Aγ̃, if γ ≠ γ̃.

Proof. We let x̃_{i} and ẽ_{i,j} denote the elements of Aγ̃ corresponding to x_{i} and e_{i,j} in A_{γ} as above, for all i and i < j. Suppose ϕ : A_{γ} → Aγ̃ is a Hopf algebra isomorphism. By Lemma [AS3, Lemma 1.2] there are a permutation σ ∈ 𝕊_{n} and non-zero scalars α_{1}, …, α_{n} ∈ 𝕜 such that ϕ(g_{i}) = g_{σ(i)} and ϕ(x_{i}) = α_{i}x̃_{σ(i)} for all i. Since ord(g_{i}) = m_{i}N ≠ m_{j}N = ord(g_{j}) for all i ≠ j, σ must be the identity, and ϕ induces the identity on Γ by restriction. In particular, 1 − g_{i}^{N} = ϕ(x_{i}^{N}) = α_{i}^{N} x̃_{i}^{N} = α_{i}^{N}(1 − g_{i}^{N}), and α_{i}^{N} = 1 for all i. Therefore we obtain for all i < j,

u_{i,j}(γ) = ϕ(e_{i,j}^{N}) = α_{i}^{N}α_{i+1}^{N} … α_{j−1}^{N}ẽ_{i,j}^{N} = u_{i,j}(γ̃),

and by Remark 6.19, 2), γ = γ̃.

References

[AA] A. Abella and N. Andruskiewitsch, Compact quantum groups arising from the FRT construction, Bol. Acad. Ciencias (Córdoba) 63 (1999), 15-44.

[A] N. Andruskiewitsch, About finite dimensional Hopf algebras, Contemp. Math., to appear.

[AnDa] N. Andruskiewitsch and S. Dăscălescu, On quantum groups at −1, preprint (2000).

[AnDe] N. Andruskiewitsch and J. Devoto, Extensions of Hopf algebras, Algebra i Analiz 7 (1995), pp. 17–52.

[AG] N. Andruskiewitsch and M. Graña, Braided Hopf algebras over non-abelian groups, Bol. Acad. Ciencias (Córdoba) 63 (1999), 45-78.

[AS1] N. Andruskiewitsch and H.-J. Schneider, Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of order p^{3}, J. Algebra 209 (1998), 658–691.

[AS2] , Finite quantum groups and Cartan matrices, Adv. Math. 154 (2000), 1–45.

[AS3] , Lifting of Nichols algebras of type A_{2} and Pointed Hopf Algebras of order p^{4}, in ”Hopf algebras and quantum groups”, eds. S. Caeneppel and F. van Oystaeyen, M. Dekker, pp. 1–16.

[AS4] , Finite quantum groups over abelian groups of prime exponent, Ann. Sci. Ec. Norm. Super., to appear.

[AS5] , A chacterization of quantum groups, in preparation.

[BDG] M. Beattie, S. Dăscălescu, and L. Grünenfelder, On the number of types of finite-dimensional Hopf algebras, Inventiones Math. 136 (1999), pp. 1–7.

[BDR] M. Beattie, S. Dăscălescu, and S. Raianu, Lifting of Nichols algebras of type B_{2}, preprint 2001.

[BGS] A. Beilinson, V. Ginsburg and W. Sörgel, Koszul duality patterns in representation theory, J. of Amer. Math. Soc. 9, pp. 473–526.

[B] N. Bourbaki, Commutative algebra. Chapters 1-7, Springer-Verlag (1989).

[BL] N. Bourbaki, Groupes et algèbres de Lie. Chap. IV, V, VI, Hermann, Paris, 1968.

[CD] S. Caenepeel and S. Dăscălescu, Pointed Hopf algebras of dimension p^{3}, J. Algebra 209 (1998), pp.622–634.

[CM] W. Chin and I. Musson, The coradical filtration for quantized universal enveloping algebras, J. London Math. Soc. 53 (1996), pp. 50–67.

[Ci] C. Cibils, Tensor products of Hopf bimodules over a group algebra, Proc. A. M. S. 125 (1997), pp. 1315–1321.

[D] D. Didt, Linkable Dynkin diagrams, Work in progress.

[DPR] R. Dijkgraaf, V. Pasquier and P. Roche, Quasi Hopf algebras, group cohomology and orbifold models, Nuclear Phys. B Proc. Suppl. 18B (1991), pp. 60–72.

[DT] Y. Doi and M. Takeuchi, Multiplication alteration by two-cocycles. The quantum version, Commun. Algebra 22, No.14, (1994), 5715-5732.

[Dr1] V. Drinfeld, Quantum groups, Proceedings of the ICM Berkeley 1986, Amer .Math. Soc.

[Dr2] , Quasi-Hopf algebras, Leningrad Math. J. 1 (1990), pp. 1419–1457.

[FMS] D. Fischman, S. Montgomery, H.-J. Schneider, Frobenius extensions of subalgebras of Hopf algebras, Trans. Amer. Math. Soc. 349 (1997), 4857-4895.

[FK] S. Fomin and K. N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus, Progr. Math. 172, Birkhauser, (1999), pp. 146–182.

[FP] S. Fomin and C. Procesi, Fibered quadratic Hopf algebras related to Schubert calculus, J. Alg. 230 (2000), 174–183.

[FG] C. Fronsdal and A. Galindo, The Ideals of Free Differential Algebras, J. Algebra, 222 (1999), pp. 708-746.

[G] S. Gelaki, On pointed Hopf algebras and Kaplansky’s tenth conjecture, J. Algebra 209 (1998), pp. 635–657. [Gñ1] M. Graña, On Pointed Hopf algebras of dimension p^{5}, Glasgow Math. J. 42 (2000), 405–419. [Gñ2] , A freeness theorem for Nichols algebras, J. Algebra 231 (2000), pp. 235-257. [Gñ3] , On Nichols algebras of low dimension, Contemp. Math. 267 (2000), pp. 111–134.

[Gu] D. Gurevich, Algebraic aspects of the quantum Yang–Baxter equation, Leningrad J. M. 2 (1991), pp. 801–828.

[HKW] P. de la Harpe, M. Kervaire and C. Weber, On the Jones polynomial, L’Ens. Math. 32 (1987), pp. 271–335.

[JS] Joyal, A. & Street, R., Braided Tensor Categories, Adv. Math. 102 (1993), pp. 20–78.

[K] V. Kac, Infinite dimensional Lie algebras, Cambridge Univ. Press, Third edition, 1995.

[Ka] C. Kassel, Quantum groups, Springer-Verlag (1995).

[Kh] V. Kharchenko, An Existence Condition for Multilinear Quantum Operations, J. Alg. 217 (1999), pp. 188-228.

[L1] G. Lusztig, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 257–296.

[L2] , Quantum groups at roots of 1, Geom. Dedicata 35 (1990), 89–114.

[L3] , Introduction to quantum groups, Birkhäuser, 1993.

[L4] , Exotic Fourier transform, Duke Math. J. 73 (1994), 227–241.

[Mj1] S. Majid, Foundations of Quantum Group Theory. Cambridge Univ. Press, 1995.

[Mj2] , Crossed products by braided groups and bosonization, J.Algebra 163 (1994), pp. 165–190.

[MiS] A. Milinski and H-J. Schneider, Pointed Indecomposable Hopf Algebras over Coxeter Groups, Contemp. Math. 267 (2000), pp. 215–236.

[M1] S. Montgomery, Hopf algebras and their actions on rings, CBMS Lecture Notes 82, Amer. Math. Soc., 1993.

[M2] , Indecomposable coalgebras, simple comodules, and pointed Hopf algebras, Proc. AMS, 123 (1995), pp. 2343–2351.

[Mu] E. Müller, Some topics on Frobenius-Lusztig kernels, I, J. Algebra 206 (1998), 624–658.

[Mus] I. Musson, Finite Quantum Groups and Pointed Hopf Algebras, preprint (1999).

[N] W.D. Nichols, Bialgebras of type one, Commun. Alg. 6 (1978), 1521–1552.

[Ra] D. Radford, Hopf algebras with projection, J. Algebra 92 (1985), 322–347.

[Re] N. Reshetikhin, Multiparameter quantum groups and twisted quasitriangular Hopf algebras, Lett. Math. Phys. 20, (1990), pp. 331–335.

[Ri] C. Ringel, PBW-bases of quantum groups, J. Reine Angew. Math. 470 (1996), pp. 51–88.

[Ro1] M. Rosso, Groupes quantiques et algebres de battage quantiques, C.R.A.S. (Paris) 320 (1995), 145–148

[Ro2] , Quantum groups and quantum shuffles, Inventiones Math. 133 (1998), 399–416.

[Sbg] P. Schauenburg, A characterization of the Borel-like subalgebras of quantum enveloping algebras, Comm. in Algebra 24 (1996), pp. 2811–2823.

[SvO] D. Stefan and F. van Oystaeyen, Hochschild cohomology and coradical filtration of pointed Hopf algebras, J. Algebra 210 (1998), pp. 535–556.

[Sw] M.E. Sweedler, Hopf algebras, Benjamin, New York, 1969.

[Tk1] M. Takeuchi, Survey of braided Hopf algebras, Contemp. Math. 267 (2000), pp. 301–324.

[Wa] M. Wambst, Complexes de Koszul quantiques, Ann. Inst. Fourier (Grenoble) 43 (1993), pp. 1089–1156.

[W] S. Witherspoon, The representation ring of the quantum double of a finite group, J. Alg. 179 (1996), 305–329.

[Wo] S. L. Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys. 122 (1989), pp. 125–170.

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