A FREENESS THEOREM FOR NICHOLS ALGEBRAS

MATÍAS GRAÑA

Abstract. We prove a freeness result for Nichols algebras over certain Nichols subalgebras. This result allows in particular to classify pointed Hopf algebras of prime and prime squared index.

1. Introduction and Notation

1.1. Introduction. Let V be a vector space and c ∈ End(V ⊗ V) a solution of the braid equation

(c ⊗ id)(id ⊗c)(c ⊗ id) = (id ⊗c)(c ⊗ id)(id ⊗c) ∈ End(V ⊗ V ⊗ V).

Let AV and CV be respectively the tensor algebra and tensor coalgebra of V; they are both braided bialgebras. There exists a unique bialgebra map S : AV → CV extending the identity id : V → V in degree one. The image 𝔅(V) := Im(S) ⊂ CV is a braided bialgebra usually called quantum symmetric algebra. If c is rigid (see definition in 1.3.1 below), then we say that 𝔅(V) is a Nichols algebra. In such case it is known that 𝔅(V) is a braided Hopf algebra in a rigid braided category.

In this article we consider a rigid solution (V, c) of the braid equation and a subspace W ⊂ V such that c(W ⊗ W) = W ⊗ W. We identify the quantum symmetric algebra 𝔅(W) with a subalgebra of 𝔅(V). Under certain additional hypotheses it is a Nichols algebra, eventually in a different rigid braided category. Assume further that 𝔅(W) is finite dimensional. Our main result, Theorem 3.8, shows then that 𝔅(V) is free over 𝔅(W). More precisely, we describe a graded subalgebra K of 𝔅(V) (which is also a left coideal) such that 𝔅(V) ≃ K ⊗ 𝔅(W) as (K, 𝔅(W))-bimodules. Consequently, P_{𝔅(V)}(t) = P_{K}(t)P_{𝔅(W)}(t), (where P_{A}(t) = ∑_{i} dim A_{i}t^{i} is the Hilbert polynomial of A = ⊕_{i}A_{i}) and in particular the dimension of 𝔅(W) divides that of 𝔅(V). Moreover, assume that there is a decomposition V = ⊕_{i}V_{i} such that c(V_{i} ⊗ V_{j}) = V_{j} ⊗ V_{i} for all i, j and that there exists for each i a (possibly trivial) subspace W_{i} ⊆ V_{i} verifying the hypotheses of the first part. Then ∏_{i} P_{𝔅(W_{i})}(t) divides P_{𝔅(V)}(t).

It was recently proved independently by Scharfschwerdt [Sf] and Takeuchi [T3] that the Nichols– Zoeller theorem [NZ] holds in Yetter–Drinfeld categories. In general, however, our algebras 𝔅(V),

Date: September 1999.

This work was partially supported by CONICET, UBA, CONICOR and SeCyT-UNC.

2 MATÍAS GRAÑA

𝔅(W) lie in different braided categories, and their result does not apply in the present situation (moreover, our result applies also to the case when 𝔅(V) is infinite dimensional). We prove 3.8 using algebras of quantum differential operators AV ↪ End(𝔅(V))^{op}. To this end, we generalize the definition for the “abelian case” given, for instance, in [FG] (this algebra was considered also by Kashiwara and Lusztig). The author was not able to find this general notion in the literature.

As an application of our main result we classify, in 4.1 and 4.2, pointed Hopf algebras of index p and p^{2} (p a prime number). This classification generalizes results from [D] and [AS2]. Here, for a pointed Hopf algebra A we define the index as the ratio dim A/#G(A) =: [A : G(A)], where G(A) is the group of group-likes of A.

The results of this article can be combined with a parameterization of Nichols algebras of ranks 3 and 4 in _{kΓ}^{kΓ}YD (Γ a finite group) in order to give a classification of Nichols algebras of dimension < 32. This, together with the “Lifting procedure” of [AS1], is a first step for the classification of pointed Hopf algebras of index < 32. These and related problems are considered in [G2].

The article is organized as follows: in Section 2 we state and prove some first results on Quantum Symmetric algebras and Nichols algebras. For V ∈ _{kΓ}^{kΓ}YD, we define the subspaces that will play the rôle of W in the applications and study the behavior of the braiding with respect to these subspaces. We also prove in Section 2 a generalization of [AS1, 3.4-3.5] for general abelian braided categories. Section 3 is devoted to algebras of Quantum Differential Operators and to the main result. Finally, in Section 4 we classify all pointed Hopf algebras of prime index and all pointed Hopf algebras of prime squared index.

1.2. Acknowledgements. I thank N. Andruskiewitsch for his careful reading of the material and for the important suggestions he made. His contribution improved and made more clear the presentation of this work. I also thank him for the statement of 2.2.

I thank also the referee, who in particular pointed out the Remark 2.14, reducing the proof of 3.5.

1.3. Notation. We refer to [Mo] for notation on Hopf algebras. We fix k an algebraically closed field of characteristic 0. All vector spaces, algebras, tensor products, homs are considered over k unless explicitly stated. For comultiplication we use Sweedler’s notation without summation symbol (Δ(z) = z_{(1)} ⊗ z_{(2)}).

It is worth noting that most of the results of this article (in particular, the main Theorem 3.8) are true in positive characteristic (with the same proofs). The results in section 4 also hold, assuming char k does not divide dim A.

A FREENESS THEOREM FOR NICHOLS ALGEBRAS 3

We call a braided pair (or BP) a pair (V, c), where V is a vector space and c : V ⊗ V → V ⊗ V is a (bijective) solution of the braid equation. Throughout, V shall be finite dimensional. It is well known that there exists a braided category containing V as an object and such that c coincides with the braiding in the category, a construction due to [Ly] (see also [H, Sn]). Moreover, this category is abelian and has countable direct sums. Explicitly, one can take the category of comodules over the FRT bialgebra generated by (V, c) (see for instance [AA]).

When working in a non-symmetric braided category, one must distinguish between right and left duals. Since it is more convenient for our purposes, we shall use right duals, and hence consider evaluation and coevaluation maps as

e : V ⊗ V^{∗} → k, b : k → V^{∗} ⊗ V.

Let f : X ⊗ Y → Y ⊗ X be a map of finite dimensional vector spaces. It is custom to denote by f^{♭} : Y ⊗ X^{∗} → X^{∗} ⊗ Y the morphism defined by

⟨(id ⊗x), f^{♭}(y ⊗ α)⟩ = ⟨f(x ⊗ y), (α ⊗ id)⟩,

where we use the pairing ⟨(a ⊗ b), (β ⊗ α)⟩ = ⟨a, α⟩⟨b, β⟩.

Definition 1.3.1 ([Ly, Gu]). Let (V, c) be a braided pair such that V is finite dimensional. We say that c is rigid (or that (V, c) is an RBP) if c^{♭} is an isomorphism (it is proved in [LS] that this is equivalent to (c^{−1})^{♭} being an isomorphism).

When (V, c) is an RBP, we have isomorphisms

c_{V,V^{∗}} = (c^{−1})^{♭} : V ⊗ V^{∗} → V^{∗} ⊗ V,
c_{V^{∗},V} = (c^{♭})^{−1} : V^{∗} ⊗ V → V ⊗ V^{∗} and
c_{V^{∗},V^{∗}} = c^{∗} : V^{∗} ⊗ V^{∗} → V^{∗} ⊗ V^{∗},

where we use the pairing above for the definition of c^{∗}. Furthermore, in this case both V and V^{∗} can be considered to be objects of one and the same braided category, and the evaluation and coevaluation maps e : V ⊗ V^{∗} → k and b : k → V^{∗} ⊗ V are maps in the category. Hence, the condition on c to be rigid is equivalent to the condition on V to lie in a rigid braided category.

Let (V, c) be a BP. Since c is a solution of the braid equation, the Artin braid group 𝔹_{n} acts on V^{⊗n} giving braidings

c_{V^{⊗i},V^{⊗j}} : V^{⊗i} ⊗ V^{⊗j} → V^{⊗j} ⊗ V^{⊗i}.

4 MATÍAS GRAÑA

Let T V be the tensor space T V = ⊕_{n}V^{⊗n}. This makes (T V, c) into a braided pair. If (V, c) is an RBP then c gives an action of 𝔹_{n} on T^{n}(V ⊕ V^{∗}) and (T (V ⊕ V^{∗}), c) becomes a BP.

Let (V, c) be a BP. We denote respectively by AV, CV the tensor algebra and the tensor coalgebra of V. As vector spaces, they coincide with T V. The multiplication components of AV are simply the identity maps m_{i,j} = id : A^{i}V ⊗ A^{j}V → A^{i+j}V. Dually, the comultiplication components of CV are the identity maps Δ_{i,j} = id : C^{i+j}V → C^{i}V ⊗ C^{j}V. Both AV and CV are graded braided Hopf algebras. We denote the comultiplication components of AV and the multiplication components of CV by

S_{i,j} : A^{i+j}V → A^{i}V ⊗ A^{j}V,
T_{i,j} : C^{i}V ⊗ C^{j}V → C^{i+j}V,

(they are given by (i, j)-shuffles, we refer for instance to [AG] for the details). There exists a unique (graded) map of Hopf algebras S =: AV → CV such that S|_{V} = id : V → V. We denote by S^{n} the restriction of this map to A^{n}V.

Definition 1.3.2. Let (V, c) be a BP. We say that the image S(AV) ⊆ CV is a Quantum Symmetric Algebra, or QSA. It is a braided graded Hopf algebra and is determined by V. We denote it by 𝔅(V). We denote also by 𝔅^{i}(V) the homogeneous component of degree i, i.e. 𝔅(V) = ⊕_{i}𝔅^{i}(V). Notice that 𝔅(V) ≃ T V/ ker(S).

Definition 1.3.3. If (V, c) is an RBP, then the quantum symmetric algebra generated by V will be called Nichols algebra and denoted also by 𝔅(V). In this case, since V belongs to a rigid braided category, then V^{∗} also does, and in particular it makes sense to consider 𝔅(V^{∗}), which is also a Nichols algebra.

Nichols algebras have some important properties that distinguish them from the general concept of QSA: they have a dual braided Hopf algebra (and in particular have non-zero spaces of integrals when they are finite dimensional). Moreover, finite dimensional Nichols algebras satisfy a Poincaré duality.

Let (V, c) be a BP. It is easy to see that the braiding commutes with the maps S^{n} (that is, (S^{n} ⊗ id)c_{V,V^{⊗n}} = c_{V,V^{⊗n}}(id ⊗S^{n})), whence c gives a braiding c_{𝔅(V)} : 𝔅(V) ⊗ 𝔅(V) → 𝔅(V) ⊗ 𝔅(V), and (𝔅(V), c_{𝔅(V)}) becomes a braided pair.

For a root of unity q, we denote by N(q) its order if q ≠ 1. If q = 1 then we define N(q) = ∞. We recall the definition of q-numbers: for n, m ∈ ℕ, n ≥ m, define

(n)_{q} = \frac{q^{n} − 1}{q − 1} = 1 + q + q^{2} + ⋯ + q^{n−1}

(n)_{q}^{!} = ∏_{i=1}^{n}(i)_{q} ( \begin{matrix}n \\ m \end{matrix} )_{q} = \frac{(n)_{q}^{!}}{(m)_{q}^{!} (n − m)_{q}^{!}}

Let H be a Hopf algebra with bijective antipode. We denote by _{H}^{H}YD the category of (left-left) Yetter–Drinfeld modules over H of arbitrary dimension. It is a braided category. The full subcategory of finite dimensional Yetter–Drinfeld modules is furthermore rigid. For an object V ∈ _{H}^{H}YD, we shall denote by δ : V → H ⊗V the H-coaction and by ⇀: H ⊗V → V the H-action. We sometimes denote the action by h(v) := h ⇀ v.

2. First results on Nichols algebras

Definition 2.1. Let (V, c) be a BP, V = ⊕_{i}V_{i}. We say that this direct sum is a decomposition of (V, c) (and that (V, c) is decomposable) if c|_{V_{i}⊗V_{j}} : V_{i} ⊗ V_{j} → V_{j} ⊗ V_{i}. In this case we denote by b_{ij} the restriction b_{ij} = c|_{V_{i}⊗V_{j}}. It is straightforward to see that if (V, c) is an RBP and V = ⊕_{i}V_{i} is a decomposition of V as a BP, then this decomposition induces a decomposition of V^{∗} = ⊕_{i}V_{i}^{∗} and taking the corresponding restrictions c_{i} = b_{ii} of c then each (V_{i}, c_{i}) becomes an RBP.

If Γ is a finite group, H = kΓ and V ∈ _{H}^{H}YD, then V can be decomposed as V = ⊕_{i}M(g_{i}, ρ_{i}), where ρ_{i} : Γ_{g_{i}} → Aut(Y_{i}) is an irreducible representation of the centralizer Γ_{g_{i}} of g_{i} (the definition of M(g, ρ) is given in 2.15 below). However, sometimes it is possible to further decompose some of the summands, or to give a different decomposition. These decompositions usually fail to lie in _{H}^{H}YD, but since (V, c) is an RBP any decomposition gives a direct sum of RBP’s. For instance, it is proved in [AG, Prop. 3.1.11] that if V is 2-dimensional then it can be decomposed as a sum of two 1-dimensional subspaces. We say that (V, c) comes from the abelian case if V can be decomposed as a sum of 1-dimensional subspaces.

When V = ⊕_{i}V_{i} is a decomposition of (V, c), then the knowledge of the QSA generated by the V_{i}’s gives information on the QSA generated by V. The next result, which is a generalization of [AS1, Lemma 3.4 and Prop. 3.5] for the non-abelian case, gives a first relation between 𝔅(V) and the 𝔅(V_{i})’s.

6 MATÍAS GRAÑA

Theorem 2.2. Let (V, c) be a braided pair and suppose it is decomposable as V = ⊕_{i=1}^{n}V_{i} such that 𝔅(V_{i}) is finite dimensional ∀i. Then dim 𝔅(V) ≥ ∏_{i=1}^{n} dim 𝔅(V_{i}). Furthermore, the equality holds if and only if b_{ij} = b_{ji}^{−1} ∀i ≠ j.

Proof. Let {v_{i}^{1}, …, v_{i}^{N_{i}}} be a set of homogeneous vectors of T (V_{i}) such that the set

{S^{d_{i}^{1}}(v_{i}^{1}), …, S^{d_{i}^{N_{i}}}(v_{i}^{N_{i}})}

is a basis of 𝔅(V_{i}), where d_{i}^{k} is the degree of v_{i}^{k}. The set

B = {v_{1}^{k_{1}} ⊗ ⋯ ⊗ v_{n}^{k_{n}} ∈ T (V_{1}) ⊗ ⋯ ⊗ T (V_{n}) | 1 ≤ k_{i} ≤ N_{i}}

is linearly independent in T (V), and has (∏_{i} dim 𝔅(V_{i})) elements. Note that the set

{S^{d_{1}^{k_{1}}}v_{1}^{k_{1}} ⊗ ⋯ ⊗ S^{d_{n}^{kn}}v_{n}^{k_{n}} | 1 ≤ k_{i} ≤ N_{i}}

is also linearly independent. It is enough for the first part to prove that B remains linearly independent after applying S to it. Then, suppose that a linear combination ∑ λ_{k_{1},... ,k_{n}}v_{1}^{k_{1}} ⊗ ⋯ ⊗ v_{n}^{k_{n}} lies in ker S. Now, the space T (V) can be decomposed in homogeneous components with respect of the V_{i}’s, and the hypothesis made on c (namely, that c behaves well with the decomposition V = ⊕_{i}V_{i}) guarantees that c is a homogeneous morphism. Hence, the linear combination of above can be treated for each homogeneous component, and we consequently may suppose that there exist d_{1}, …, d_{n} such that λ_{k_{1},... ,k_{n}} = 0 if d_{i}^{k_{i}} ≠ d_{i} for some i. Let m = d_{1} + ⋯ + d_{n}.

Now, by (2.8), for m = i + j we have

(S^{i} ⊗ S^{j}) ∘ S_{i,j} = S^{m} = T_{i,j} ∘ (S^{i} ⊗ S^{j}).

This implies that

\begin{aligned}0 & = S^{m}(∑ λ_{k_{1},... ,k_{n}}v_{1}^{k_{1}} ⊗ ⋯ ⊗ v_{n}^{k_{n}}) \\ & = T_{d_{1},... ,d_{n}}(S^{d_{1}} ⊗ ⋯ ⊗ S^{d_{n}}) (∑ λ_{k_{1},... ,k_{n}}v_{1}^{k_{1}} ⊗ ⋯ ⊗ v_{n}^{k_{n}}) \\ & = ∑_{x^{−1}∈Sh_{d_{1},... ,dn}} s(x)(S^{d_{1}} ⊗ ⋯ ⊗ S^{d_{n}}) (∑ λ_{k_{1},... ,k_{n}}v_{1}^{k_{1}} ⊗ ⋯ ⊗ v_{n}^{k_{n}}) , \end{aligned} (2.3)

where Sh_{d_{1},... ,d_{n}} ⊂ 𝕊^{m} is the shuffle. Furthermore, the homogeneous component with degrees (d_{1}, …, d_{n}) of CV can be decomposed as a direct sum taking all the possible orders for d_{i} tensorands of V_{i}. Explicitly, let

Z_{d_{1},... ,d_{n}} = {f : {1, …, m} → {1, …, n} | #(f^{−1}(i)) = d_{i} ∀1 ≤ i ≤ n}.

A FREENESS THEOREM FOR NICHOLS ALGEBRAS 7

The decomposition just stated is nothing but

(the (d_{1}, …, d_{n}) component of CV) = ⨁_{f∈Z_{d_{1},... ,dn}} V_{f(1)} ⊗ V_{f(2)} ⊗ ⋯ ⊗ V_{f(m)} = ⨁_{f∈Z_{d_{1},... ,dn}} V_{f}.

It is clear that there exists a bijection γ : (Sh_{d_{1},... ,d_{n}})^{−1} → Z_{d_{1},... ,d_{n}} such that for x ∈ (Sh_{d_{1},... ,d_{n}})^{−1} the image of s(x)|_{V_{1}^{⊗d_{1}}⊗···⊗V_{n}^{⊗dn}} lies in V_{γ(x)}. Hence (2.3) implies, for each x ∈ (Sh_{d_{1},... ,d_{n}})^{−1}, that

0 = s(x)(S^{d_{1}} ⊗ ⋯ ⊗ S^{d_{n}}) (∑ λ_{k_{1},... ,k_{n}}v_{1}^{k_{1}} ⊗ ⋯ ⊗ v_{n}^{k_{n}}) ,

but since s(x) is an isomorphism for each x ∈ 𝕊_{n}, this implies that

0 = (S^{d_{1}} ⊗ ⋯ ⊗ S^{d_{n}}) (∑ λ_{k_{1},... ,k_{n}}v_{1}^{k_{1}} ⊗ ⋯ ⊗ v_{n}^{k_{n}}) = ∑ λ_{k_{1},... ,k_{n}}(S^{d_{1}}v_{1}^{k_{1}} ⊗ ⋯ ⊗ S^{d_{n}}v_{n}^{k_{n}}),

which in turn implies that λ_{k_{1},... ,k_{n}} = 0 ∀k_{1}, …, k_{n}. This proves the first part.

For the second part, suppose there exist i, j, i ≠ j such that b_{ji}b_{ij} ≠ id, and let v ∈ V_{i}, w ∈ V_{j} such that b_{ji}b_{ij}(v ⊗ w) ≠ (v ⊗ w). Consider the element z ∈ T (V) given by z = (id −b_{ij})(v ⊗ w). Since the linear span of the set S(B) = {S(y), y ∈ B} is a subspace of 𝔅(V) with dimension ∏_{l} dim 𝔅(V_{l}), it is sufficient to prove that S(z) does not lie in this linear span. Note that since deg z = 2, S(z) = (id +c)(z). Suppose that

S(z) = S (∑ λ_{k_{1},... ,k_{n}}v_{1}^{k_{1}} ⊗ ⋯ ⊗ v_{n}^{k_{n}}) .

Looking at the homogeneous components, we must have λ_{k_{1},... ,k_{n}} = 0 if (d_{1}^{k_{1}}, …, d_{n}^{k_{n}}) ≠ e_{i} + e_{j}, where e_{l} = (0, …, 0, 1, 0, …, 0) with the 1 in the l-coordinate. In other words, if i < j there exist u_{1}, …, u_{s} ∈ V_{i}, u′_{1}, …, u′_{s} ∈ V_{j} and λ_{1}, …, λ_{s} ∈ k such that

S(z) = S(∑_{t} λ_{t}u_{t} ⊗ u′_{t})

(if i > j we have S(z) = S(∑_{t} λ_{t}u′_{t} ⊗ u_{t}) and it is analogous). Now, since z = (1 − c)(v ⊗ w) and S(z) = (1 + c)(z), we have

(1 − c^{2})(v ⊗ w) = (1 + c)(∑_{t} λ_{t}u_{t} ⊗ u′_{t}),

but the component in V_{j} ⊗ V_{i} in the left hand side vanish, and hence we have c(∑_{t} λ_{t}u_{t} ⊗ u′_{t}) = 0. Since c is an isomorphism, we must have ∑_{t} λ_{t}u_{t} ⊗u′_{t} = 0, but this implies that the V_{i} ⊗V_{j} component in the right hand side vanish, whence c^{2}(v ⊗ w) = (v ⊗ w), a contradiction.

It only remains to prove that if b_{ij}b_{ji} = id ∀i ≠ j then S(B) is a basis of 𝔅(V). This is the same as saying that B generates T (V) modulo ker S. Let {w_{i}^{1}, …, w_{i}^{M_{i}}} be a basis of V_{i}. Let

B̃ = {w_{a_{1}}^{r_{1}} ⊗ ⋯ ⊗ w_{a_{m}}^{r_{m}} | m ∈ ℕ, 1 ≤ r_{t} ≤ M_{a_{t}} and a_{1} ≤ a_{2} ≤ … ≤ a_{m}}.

8 MATÍAS GRAÑA

Since, modulo ker(S), the elements of B̃ are linear combinations of the elements of B, it remains to prove that B̃ generates T (V) modulo ker(S). To see this, it suffices to prove that if i < j and 1 ≤ r_{i} ≤ M_{i}, 1 ≤ r_{j} ≤ M_{j}, then w_{j}^{r_{j}} ⊗ w_{i}^{r_{i}} can be expressed in terms of elements in B̃. Let z = (1 − c)(w_{j}^{r_{j}} ⊗ w_{i}^{r_{i}}). Since b_{ij}b_{ji} = id, we have S(z) = (1 + c)(z) = 0, which gives the relation

w_{j}^{r_{j}} ⊗ w_{i}^{r_{i}} = b_{ji}(w_{j}^{r_{j}} ⊗ w_{i}^{r_{i}}) mod ker(S).

Now, b_{ji}(w_{j}^{r_{j}} ⊗ w_{i}^{r_{i}}) ∈ V_{i} ⊗ V_{j}, and it can be written as a linear combination of elements in B̃.

From now on, we concentrate on Nichols algebras. Let (V, c) be an RBP. Since V and V^{∗} belong to a braided category C, we have that V^{⊗n} and (V^{∗})^{⊗m} also belong to C, and it is easy to see that the map S : AV → CV is a map in the category, whence both 𝔅(V) and 𝔅(V^{∗}) lie in C, and in particular they lie in one and the same braided category.

We shall use the bilinear form (|) : T^{n}V ⊗ T^{n}V^{∗} → k given by

(z_{1}z_{2} ⋯ z_{n}|w_{1}w_{2} ⋯ w_{n}) = ∏_{1≤i≤n}⟨z_{i}, w_{n+1−i}⟩ = ⟨z_{1}, w_{n}⟩ ⋯ ⟨z_{n}, w_{1}⟩, (2.4)

where z_{1}, z_{2}, ⋯, z_{n} ∈ T^{n}(V), w_{1}, w_{2}, ⋯, w_{n} ∈ T^{n}(V^{∗}) and ⟨·, ·⟩ is the evaluation map. We define also (z|w) = 0 if z ∈ T^{n}(V), w ∈ T^{m}(V^{∗}) and n ≠ m. It is easy to see that c_{V^{∗},V^{∗}} is the transpose of c_{V,V} when one identifies (V ⊗ V)^{∗} ≃ V^{∗} ⊗ V^{∗} with this form. Furthermore, as proved in [AG, Prop. 3.2.20], this bilinear form satisfies for z ∈ T^{n}(V), w ∈ T^{n}(V^{∗}),

(S^{n}z|w) = (z|S^{n}w), (2.5)

from where S^{n} : T^{n}V^{∗} → T^{n}V^{∗} is the transpose of S^{n} : T^{n}V → T^{n}V with respect to the pairing (|). We thus have proved:

Lemma 2.6. Let (V, c) be an RBP. Then 𝔅^{n}(V^{∗}) ≃ (𝔅^{n}(V))^{∗} for all n ≥ 0. If 𝔅(V) is finite dimensional then 𝔅(V^{∗}) ≃ (𝔅(V))^{∗} and (𝔅(V), c_{𝔅(V),𝔅(V)}) is an RBP.

Definition 2.7. Let (V, c) be an RBP. For y ∈ V^{∗}, we denote by ∂_{y} the operator

∂_{y} = (id ⊗y) ∘ Δ^{i−1,1} : 𝔅^{i}(V) → 𝔅^{i−1}(V).

If B = {x_{1}, …, x_{n}} is a basis of V, let {∂_{1}, …, ∂_{n}} be the basis of V^{∗} dual to B. We denote also by ∂_{j}, j = 1, …, n the operator ∂_{∂_{j}}. We warn that for the case V ∈ _{H}^{H}YD, usually these are not morphisms in _{H}^{H}YD.

Because of the coassociativity of AV, we have for i_{1}, i_{2}, i_{3} ∈ ℕ

(2.8) (S_{i_{1},i_{2}} ⊗ id)S_{i_{1}+i_{2},i_{3}} = S_{i_{1},i_{2},i_{3}} = (id ⊗S_{i_{2},i_{3}})S_{i_{1},i_{2}+i_{3}},

which has the following consequence:

Proposition 2.9. Let (V, c) be an RBP. Consider the map ∂ : V^{∗} → End 𝔅(V) given in 2.7, and denote by ¯∂ : T (V^{∗}) → (End 𝔅(V))^{op} the (unique) algebra map extending ∂ to the tensor algebra. Then ¯∂ factors through 𝔅(V^{∗}) as

¯∂ = T (V^{∗}) ↠^{π} 𝔅(V^{∗}) ↪^{ι} (End 𝔅(V))^{op}.

Proof. We use the bilinear form (|) of 2.4. Notice that if z ∈ T^{n}(V), z′ ∈ T^{m}(V), w ∈ T^{n}(V^{∗}), w′ ∈ T^{m}(V^{∗}), then

(z · z′|w′ · w) = (z′|w′)(z|w). (2.10)

For w ∈ T^{n}(V^{∗}), z ∈ T^{m}(V) we take the pairing z ⊗ w ↦ (S^{n}z|w). By definition this pairing gives a functional w ∈ (𝔅(V))^{∗}, and we define the operator h(w) ∈ End 𝔅(V) by h(w)(x) = w ⇀ x, i.e. h(w)(x) = x_{(1)}w(x_{(2)}). In other terms, if z ∈ T^{m}(V) then S(z) ∈ 𝔅^{m}(V) and we have, using (2.5),

(2.11) h(w)(S(z)) = S(z_{(1)})w(S(z_{(2)})) = (S(z_{(1)}))(z_{(2)}|S^{n}w) = (S^{m−n} ⊗ ( · |S^{n}w))S_{m−n,n}(z).

We prove now that h = ¯∂. In degree 1 this is just the definition of h, so it is enough to prove that h is multiplicative. Let w_{i} ∈ T^{n_{i}}(V^{∗}) for i = 1, 2 and z ∈ T^{n_{1}+n_{2}+n_{3}}(V). Then

(h(w_{1}) ·_{op} h(w_{2})) (pz) = h(w_{2})(pz_{(1)})w_{1}(z_{(2)}) = pz_{(1)}w_{2}(z_{(2)})w_{1}(z_{(3)})
= S^{n_{3}}(z_{(1)})(z_{(2)}|S^{n_{2}}w_{2})(z_{(3)}|S^{n_{1}}w_{1}) by (2.11)
= (S^{n_{3}} ⊗ w_{2} ⊗ w_{1})(id ⊗S^{n_{2}} ⊗ S^{n_{1}})(S_{n_{3},n_{2},n_{1}}(z))
= (S^{n_{3}} ⊗ w_{1} · w_{2})(id ⊗S^{n_{2}+n_{1}})(S_{n_{3},n_{2}+n_{1}}(z)) by (2.10)
= (S^{n_{3}} ⊗ S^{n_{2}+n_{1}}(w_{1} · w_{2}))(S_{n_{3},n_{2}+n_{1}}z)
= h(w_{1} · w_{2})(pz) by (2.11).

The factorizability through 𝔅(V^{∗}) is now a consequence of (2.11). The injectivity of ι follows immediately from the non-degeneracy of (|).

As a first consequence of this result, we have Corollary 2.12. Let ∂_{0} ∈ V^{∗} be such that c(∂_{0} ⊗ ∂_{0}) = q∂_{0} ⊗ ∂_{0}, where q ≠ 1 is a root of unity of order N. Then (∂_{0})^{N} = 0 as an operator of 𝔅(V).

Definition 2.13. We shall denote by [,] the action given by ¯∂, i.e. for S^{n}z ∈ 𝔅^{n}(V), S^{m}w ∈ 𝔅^{m}(V^{∗}) (n ≥ m),

[S^{n}z, S^{m}w] = S^{n−m}(z_{(1)})(S^{m}z_{(2)}|w) ∈ 𝔅^{n−m}(V).

We characterize now the RBP’s (V, c) arising in Yetter–Drinfeld categories over group algebras. The following remark is due to the referee.

Remark 2.14. Let V be a finite dimensional module in _{kΓ}^{kΓ}YD, where Γ is a group. As a kΓ-comodule, V can be decomposed as V = ⊕_{g∈Γ}V^{g}, where δ(x) = g ⊗ x ∀x ∈ V^{g}. Then, for x ∈ V^{g}, y ∈ V we have c(x ⊗ y) = g ⇀ y ⊗ x ∈ V ⊗ x. Thus V has a basis {x_{1}, …, x_{n}} such that c(x_{i} ⊗ V) = V ⊗ x_{i}. Conversely, let (V, c) be a BP with a basis {x_{1}, …, x_{n}} such that c(x_{i} ⊗V) = V ⊗x_{i}. Let g_{i} ∈ Aut(V) be defined by

c(x_{i} ⊗ y) = g_{i}(y) ⊗ x_{i},

and let Γ be the subgroup of Aut(V) generated by {g_{1}, …, g_{n}}. Then V is an object of _{kΓ}^{kΓ}YD. In particular, it is an RBP.

Proof. The comodule structure is determined by δ(x_{i}) = g_{i} ⊗ x_{i}. The module structure is given by g_{i} ⇀ y = g_{i}(y). We must prove the YD-compatibility condition between ⇀ and δ, which reads as δ(g ⇀ y) = gy_{(−1)}g^{−1} ⊗ gy_{(0)}, with g ∈ Γ, δ(y) = y_{(−1)} ⊗ y_{(0)}. It is sufficient to prove it for y = x_{i} and g = g_{j}, which is equivalent to prove that g_{j}(x_{i}) ∈ V^{g_{j}g_{i}g_{j}^{−1}}. Let g_{j}x_{i} = ∑_{k} c_{k}x_{k}. The braid equation applied to x_{j} ⊗ x_{i} ⊗ y implies that

∑_{k} g_{j}g_{i}(y) ⊗ c_{k}x_{k} ⊗ x_{j} = ∑_{k} g_{k}g_{j}(y) ⊗ c_{k}x_{k} ⊗ x_{j},

which tells that c_{k} = 0 if g_{j}g_{i} ≠ g_{k}g_{j}.

Remark 2.15. Let V ∈ _{kΓ}^{kΓ}YD, Γ a finite group. Then V = ⊕_{i}M(g_{i}, ρ_{i}), where ρ_{i} : Γ_{g_{i}} → Aut(Y_{i}) is an irreducible representation of Γ_{g_{i}}, and

M(g_{i}, ρ_{i}) = Ind_{Γ_{g_{i}}}^{Γ} ρ_{i} = kΓ ⊗_{kΓ_{g_{i}}} Y_{i},

with comodule structure δ(h ⊗ y) = hg_{i}h^{−1} ⊗ (h ⊗ y). Let {x_{1}^{i}, …, x_{r_{i}}^{i}} be a basis of Y_{i} and let {h_{1}^{i}, …, h_{s_{i}}^{i}} be a set of representatives of left coclasses in Γ/Γ_{g_{i}}. Then M(g_{i}, ρ_{i}) has basis {z_{jl}^{i} = h_{j}^{i} ⊗ x_{l}^{i}, 1 ≤ j ≤ s_{i}, 1 ≤ l ≤ r_{i}}, and consequently {z_{jl}^{i}}_{i,j,l} is a basis of V. Let {∂_{jl}^{i}}_{i,j,l} be

A FREENESS THEOREM FOR NICHOLS ALGEBRAS 11

the dual basis. Let t_{j}^{i} = h_{j}^{i}g_{i}(h_{j}^{i})^{−1} (thus, the conjugacy class of g_{i} is {t_{1}^{i}, …, t_{s_{i}}^{i}}), and note that δ(z_{jl}^{i}) = t_{j}^{i} ⊗ z_{jl}^{i}. Let W_{jl}^{′i} = span{z_{j_{2}l_{2}}^{i_{2}} | (i, j, l) ≠ (i_{2}, j_{2}, l_{2})}. Then W_{jl}^{′i} is a kΓ-subcomodule of V. For further use, we compute:

(id ⊗∂_{j_{1}l_{1}}^{i_{1}})c(z_{j_{2}l_{2}}^{i_{2}} ⊗ z_{j_{3}l_{3}}^{i_{3}}) = (id ⊗∂_{j_{1}l_{1}}^{i_{1}})(t_{j_{2}}^{i_{2}} ⇀ z_{j_{3}l_{3}}^{i_{3}} ⊗ z_{j_{2}l_{2}}^{i_{2}}) = δ_{i_{1},i_{2}}δ_{j_{1},j_{2}}δ_{l_{1},l_{2}}(t_{j_{1}}^{i_{1}} ⇀ z_{j_{3}l_{3}}^{i_{3}}).

Moreover,

(∂_{j_{1}l_{1}}^{i_{1}} ⊗ id)c(z_{j_{1}l_{1}}^{i_{1}} ⊗ z_{j_{2}l_{2}}^{i_{2}}) = ∂_{j_{1}l_{1}}^{i_{1}}(t_{j_{1}}^{i_{1}} ⇀ z_{j_{2}l_{2}}^{i_{2}})z_{j_{1}l_{1}}^{i_{1}} = δ_{i_{1},i_{2}}∂_{j_{1}l_{1}}^{i_{1}}(t_{j_{1}}^{i_{1}} ⇀ z_{j_{2}l_{2}}^{i_{1}})z_{j_{1}l_{1}}^{i_{1}}. (2.16)

Now, we have δ(t_{j_{1}}^{i_{1}} ⇀ z_{j_{2}l_{2}}^{i_{1}}) = (t_{j_{1}}^{i_{1}}t_{j_{2}}^{i_{1}}(t_{j_{1}}^{i_{1}})^{−1}) ⊗ t_{j_{1}}^{i_{1}} ⇀ z_{j_{2}l_{2}}^{i_{1}}, whence (2.16) vanishes if t_{j_{1}}^{i_{1}}t_{j_{2}}^{i_{1}}(t_{j_{1}}^{i_{1}})^{−1} ≠ t_{j_{1}}^{i_{1}}, i.e. if t_{j_{2}}^{i_{1}} ≠ t_{j_{1}}^{i_{1}}, but for the definition of t_{j}^{i} and h_{j}^{i} this is equivalent to j_{1} ≠ j_{2}. Moreover, for y ∈ Y_{i_{1}}, t_{j_{1}}^{i_{1}} ⇀ (h_{j_{1}}^{i_{1}} ⊗ y) = h_{j_{1}}^{i_{1}}g_{i_{1}}(h_{j_{1}}^{i_{1}})^{−1}h_{j_{1}}^{i_{1}} ⊗ y = h_{j_{1}}^{i_{1}} ⊗ ρ_{i_{1}}(g_{i_{1}})(y) = q_{i_{1}}h_{j_{1}}^{i_{1}} ⊗ y, from where (2.16) vanishes if l_{1} ≠ l_{2}. Then, we have proved

• c(z_{jl}^{i} ⊗ z_{jl}^{i}) = q_{i}z_{jl}^{i} ⊗ z_{jl}^{i},

• c(z_{jl}^{i} ⊗ W_{jl}^{′i}) = W_{jl}^{′i} ⊗ z_{jl}^{i},

• δ(W_{jl}^{′i}) ⊆ kΓ ⊗ W_{jl}^{′i}.

We close this section with a useful bound for dimension of Nichols algebras:

Lemma 2.17. Let V ∈ _{kΓ}^{kΓ}YD, Γ finite, V finite dimensional. Then by 2.15, we have a basis {x_{1}, …, x_{n}} of V such that c(x_{i} ⊗ V) = V ⊗ x_{i} and c(x_{i} ⊗ x_{i}) = q_{i}x_{i} ⊗ x_{i}, q_{i} a root of unity. Let N_{i} = N(q_{i}). Then ∏_{i} N_{i} ≤ dim(𝔅(V)).

Proof. Let {∂_{1}, …, ∂_{n}} be the basis of V^{∗} dual to {x_{1}, …, x_{n}}. Note first that x_{i}^{N_{i}−1} ≠ 0 because 𝔅(kx_{i}) ↪ 𝔅(V), or simply by direct computation, since ∂_{i}(x_{i}^{j}) = (j)_{q_{i}} x_{i}^{j−1} and thence (∂_{i})^{j}(x_{i}^{j}) = (j)_{q_{i}}^{!} , which is nonzero if j < N_{i} (we use the Leibniz rule to make the computations, see 3.5 below). We prove now that the set {x_{1}^{j_{1}} ⋯ x_{n}^{j_{n}} | 0 ≤ j_{i} < N_{i}} is linearly independent. We do it by induction: suppose that the subset with j_{m} = 0 for m ≥ m_{0} is l.i. and consider a linear combination

∑_{j_{1},... ,j_{m_{0}}} α_{j_{1},... ,j_{m_{0}}}x_{1}^{j_{1}} ⋯ x_{m_{0}}^{j_{m_{0}}} = 0.

Applying (∂_{m_{0}})^{N_{m_{0}}−1} we get

∑_{j_{1},... ,j_{m_{0}−1}} α_{j_{1},... ,j_{m_{0}−1},N_{m_{0}}−1}(N_{m_{0}} − 1)_{q_{m_{0}}}^{!} x_{1}^{j_{1}} ⋯ x_{m_{0}−1}^{j_{m_{0}−1}} = 0,

12 MATÍAS GRAÑA

from where the coefficients α_{j_{1},... ,j_{m_{0}−1},N_{m_{0}}−1} vanish. We can apply then (∂_{m_{0}})^{N_{m_{0}}−2}, and get that the coefficients α_{j_{1},... ,j_{m_{0}−1},N_{m_{0}}−2} vanish. Proceeding in this way, we see that all the coefficients α_{j_{1},... ,j_{m_{0}}} = 0, and the inductive thesis follows.

Corollary 2.18. If V = ⊕_{j}M(g_{j}, ρ_{j}) ∈ _{kΓ}^{kΓ}YD (Γ a finite group) and d_{j} = dim M(g_{j}, ρ_{j}), q_{j} = ρ_{j}(g_{j}), N_{j} = N(q_{j}), then ∏_{j} N_{j}^{d_{j}} ≤ dim(𝔅(V)).

3. Main result

Definition 3.1. We say that a finite dimensional braided Hopf algebra (R, c) is rigid if R is a Hopf algebra in a rigid braided category. This is equivalent to (R, c) being an RBP and the structure morphisms of R commuting with the braiding. In particular replacing P by R and by R^{∗} we have

c_{P,R}(id ⊗m_{R}) = (m_{R} ⊗ id)c_{P,(R⊗R)} : P ⊗ R ⊗ R → R ⊗ P,
(c_{R,P})^{−1}(id ⊗m_{R}) = (m_{R} ⊗ id)(c_{(R⊗R),P})^{−1} : P ⊗ R ⊗ R → R ⊗ P,

and analogous equalities for Δ, u, ε, S.

For rigid braided Hopf algebras, most of the “classic” results on finite dimensional Hopf algebras can be stated mutatis mutandis in a braided sense, and we refer to [T2] for this. We shall use the existence of a non-zero integral (in R and R^{∗}) and the non-degenerate bilinear form given by it.

Definition 3.2. Let (V, c) be an RBP such that 𝔅(V) is finite dimensional. We have seen that 𝔅(V^{∗}) ≃ (𝔅(V))^{∗} and that (𝔅(V), c_{𝔅(V)}) is an RBP. It is straightforward to see by the definitions of the structure morphism in 𝔅(V) that it is a rigid braided Hopf algebra. Then 𝔅(V) has a onedimensional space of integrals, which is easily seen to be homogeneous. We denote by top(V) the degree of the integral of 𝔅(V). We further denote by Λ_{𝔅(V)} a nonzero integral of 𝔅(V).

The non-degenerate bilinear form gives the Poincaré duality found by Nichols (see e.g. [AG, Prop. 3.2.2]), namely, that dim 𝔅^{i}(V) = dim 𝔅^{top(V)−i}(V); in particular, dim 𝔅^{top(V)}(V) = 1 and 𝔅^{top(V)+i}(V) = 0 ∀i > 0.

The following lemma, which we have mostly proved, is an addendum to this duality

Lemma 3.3. Let (V, c) be an RBP.

1. 𝔅^{n}(V^{∗}) ≃ (𝔅^{n}(V))^{∗} via the form [,] of 2.13.

2. If 𝔅(V) is finite dimensional and n ≤ top(V), then [Λ_{𝔅(V)}, 𝔅^{top(V)−n}(V^{∗})] = 𝔅^{n}(V).

Proof. 1. It is part of the content of 2.6, once we see the definition of [,].

A FREENESS THEOREM FOR NICHOLS ALGEBRAS 13

2. If D ∈ 𝔅^{top(V)−n}(V^{∗}) is such that [Λ_{𝔅(V)}, D] = 0, we can take D′ ∈ 𝔅^{n}(V^{∗}) such that Λ_{𝔅(V^{∗})} = DD′, and then [Λ_{𝔅(V)}, Λ_{𝔅(V^{∗})}] = [[Λ_{𝔅(V)}, D], D′] = 0, which contradicts 1. Then we have a monomorphism 𝔅^{top(V)−n}(V^{∗}) ↪ 𝔅^{n}(V). Since by 1 and the Poincaré duality both spaces have the same dimension, we are done.

Definition 3.4. Let (V, c) be an RBP and consider the map ι : 𝔅(V^{∗}) → End 𝔅(V)^{op} of 2.9. Given x ∈ 𝔅(V), we denote also by x the map in End 𝔅(V)^{op} given by right multiplication by x. We have hence 𝔅(V) and 𝔅(V^{∗}) acting on 𝔅(V). We define the quantum differential operators algebra AV as the subalgebra of End 𝔅(V)^{op} generated by 𝔅(V) and 𝔅(V^{∗}).

When V ∈ _{kΓ}^{kΓ}YD, it can be given a nice basis of AV.

Lemma 3.5. Let V ∈ _{kΓ}^{kΓ}YD be finite dimensional.

1. Let {x_{1}, …, x_{n}} be a basis of V such that δ(x_{i}) = g_{i} ⊗ x_{i}. Let {∂_{1}, …, ∂_{n}} be the basis of V^{∗} dual to {x_{1}, …, x_{n}}. Then AV can be presented with generators {∂_{1}, …, ∂_{n}, x_{1}, …, x_{n}} and relations

• ker S^{m} : T V → 𝔅(V), m ≥ 0, i.e. the relations in 𝔅(V),

• ker S^{m} : T V^{∗} → 𝔅(V^{∗}), m ≥ 0, i.e. the relations in 𝔅(V^{∗}), and

• x_{i}∂_{j} = δ_{ij} + ∂_{j}g_{j}(x_{i}) for 1 ≤ i, j ≤ n. Furthermore, for m ≥ 0 take {z_{1}^{m}, …, z_{d_{m}}^{m}} a basis of 𝔅^{m}(V), and let {w_{1}^{m}, …, w_{d_{m}}^{m}} be the dual basis of 𝔅^{m}(V^{∗}) (given by the duality in 3.3). Then the set

{w_{k}^{i}z_{l}^{j} | i, j ≥ 0, 1 ≤ k ≤ d_{i}, 1 ≤ l ≤ d_{j}}

is a basis of AV.

2. Suppose W ⊂ V is a kΓ-subcomodule which is Γ′-stable, where Γ′ ⊂ Γ is the smallest subgroup such that δ(W) ⊆ kΓ′ ⊗ W. Then c(W ⊗ W) = W ⊗ W, the pair (W, c_{W} = c_{V} |_{W ⊗W}) is an RBP and it makes sense to consider 𝔅(W). Suppose furthermore that V = W ⊕ W′, where W′ is a kΓ-subcomodule and a kΓ′-submodule. Let W^{∗} ↪ V^{∗} be the map given by the identification W^{∗} = (W′)^{⊥}. Then there is a monomorphism of algebras

AW ↪ AV,

induced by W ↪ V and W^{∗} ↪ V^{∗}.

Before giving the proof, we note that if Γ is finite, 2.15 tells that the conditions of the second part are fulfilled taking, for i, j, l fixed, W = span{z_{jl}^{i}} and W′ = W_{jl}^{′i}.

Proof. 1. The relations in 𝔅(V) hold in AV because it acts by the regular representation. The relations in 𝔅(V^{∗}) hold in AV because of 2.9. Of course, AV, CV and 𝔅(V) lie in _{kΓ}^{kΓ}YD, whence we may consider g_{i} acting on Aut(𝔅(V)). It is immediate to see from the definition that the derivations ∂_{i} verify the Leibniz rule

∂_{i}(ab) = ∂_{i}(a)g_{i}(b) + a∂_{i}(b),

which implies at once the last set of relations taking b = x_{j}.

Thus, if B is the algebra defined with generators {∂_{1}, …, ∂_{n}, x_{1}, …, x_{n}} and the stated relations, we have a surjection B ↠ AV. Moreover, the last set of relations implies that any monomial in B can be written in the form ∑_{r} a_{r}b_{r}, where the a_{r}’s are generated by the ∂_{i}’s and the b_{r}’s are generated by the x_{i}’s. Now, the first and second set of relations imply that we can choose a_{r} and b_{r} to be respectively in the sets {w_{k}^{i}} and {z_{l}^{j}}. The last thing to be proved is the linear independence of the set {w_{k}^{i}z_{l}^{j}}. Since AV is generated by {x_{i}} and {∂_{i}}, which act on 𝔅(V) in a graded way, then AV is a graded algebra with degrees given by deg(x_{i}) = 1 and deg(∂_{i}) = −1, and then it is enough to prove the linear independence of the subsets with the same degree. Let then

T = ∑_{\begin{matrix}−i+j=n \\ k,l \end{matrix}} α_{ijkl}w_{k}^{i}z_{l}^{j} = 0,

and suppose some of the α_{ijkl} ≠ 0. We take m = min{i | ∃j, k, l s.t. α_{ijkl} ≠ 0} and let

T̃ = ∑_{\begin{matrix}j=n+m, i=m \\ k,l \end{matrix}} α_{ijkl}w_{k}^{i}z_{l}^{j}.

Since T and T − T̃ act trivially on 𝔅^{m}(V) then T̃ also does. Thus, for any k′ we have

0 = T̃(z_{k′}^{m}) = [z_{k′}^{m}, ∑_{k} w_{k}^{m}] ∑_{l} α_{m,n+m,k,l}z_{l}^{n+m}

= ∑_{l} α_{m,n+m,k′,l}z_{l}^{n+m},

but then the linear independence of the set {z_{l}^{n+m}}_{l} implies that α_{m,n+m,k,l} = 0 ∀k, l, which contradicts the definition of m.

A FREENESS THEOREM FOR NICHOLS ALGEBRAS 15

2. The pair (W, c_{W}) is an RBP thanks to 2.14. Let {x_{1}, …, x_{m}} be a basis of W such that δ(x_{i}) = g_{i} ⊗ x_{i}. We consider {∂_{1}, …, ∂_{m}} the dual basis of W^{∗} and extend it by ∂_{i}|_{W′} = 0 (i.e., W^{∗} ↪ V^{∗} as W^{∗} = (W′)^{⊥}). The condition on W′ to be a kΓ-subcomodule implies that in AV we have x_{i}∂_{j} = δ_{ij} + ∂_{j}g_{j}(x_{i}) for 1 ≤ i, j ≤ m. The condition on W′ to be a kΓ′-submodule implies that c_{V^{∗}}(W^{∗} ⊗ W^{∗}) = W^{∗} ⊗ W^{∗}, and it is easy to see that the restriction c_{V^{∗}}|_{W^{∗}⊗W^{∗}} coincides with c_{W^{∗}}, the braiding given by considering W^{∗} as an object of _{kΓ′}^{kΓ′}YD. Thus,

ker(S^{n} : (W^{∗})^{⊗n} → (W^{∗})^{⊗n}) = ker(S^{n} : (V^{∗})^{⊗n} → (V^{∗})^{⊗n}) ∩ (W^{∗})^{⊗n}.

Since clearly

ker(S^{n} : W^{⊗n} → W^{⊗n}) = ker(S^{n} : V^{⊗n} → V^{⊗n}) ∩ W^{⊗n},

by the first part this gives a well defined morphism of algebras AW → AV. We must prove that this map is injective, but this is also a consequence of the first part, for if we take {z_{i}} a basis of 𝔅(W) and {w_{j}} a basis of 𝔅(W^{∗}) then the set {w_{j}z_{i}} is a basis of AW, and it is linearly independent in AV (completing the bases to bases of 𝔅(V) and 𝔅(V^{∗})). The assertion follows.

Example 3.6. If V = kx and c(x ⊗ x) = qx ⊗ x, let ∂ ∈ V^{∗} such that ∂(x) = 1. Then AV has basis {∂^{i}x^{j} | 0 ≤ i, j < N(q)}, and it can be seen by induction that the relation x∂ = q∂x + 1 implies that for i, j ≥ 0 we have

x^{i}∂^{j} = ∑_{l≥0} ( \begin{matrix}i \\ l \end{matrix} )_{q} ( \begin{matrix}j \\ l \end{matrix} )_{q} (l)_{q}^{!} q^{(j−l)(i−l)}∂^{j−l}x^{i−l}.

Remark 3.7. Let V ∈ _{kΓ}^{kΓ}YD. We note that the relations in AV imply that if ∂ ∈ V^{∗} is such that ∂|_{V^{h}} = 0 ∀h ≠ g and z ∈ 𝔅(V), then z∂ = [z, ∂] + ∂g(z) in AV. Thus, if w ∈ 𝔅(V^{∗}), we have

zw = [z, w] + terms beginning with elements of positive degree in 𝔅(V^{∗}).

Now, for a finite dimensional graded space A = ⊕_{i=0}^{n}A_{i} we denote the Hilbert polynomial P_{A}(t) = ∑_{i=0}^{n} dim(A_{i})t^{i}. We are in position to prove the main theorem.

Theorem 3.8. Let V ∈ _{kΓ}^{kΓ}YD be finite dimensional.

1. Suppose there exists W as in 3.5 part 2 such that 𝔅(W) is finite dimensional. Take the basis {x_{1}, …, x_{m}} of W and the dual basis {∂_{1}, …, ∂_{m}} of W^{∗} ⊂ V^{∗} (extend them to V by ∂_{i}|_{W′} = 0 ∀i). Consider then the ∂_{i}’s as operators on 𝔅(V) and let K = ∩_{i=1}^{m} ker(∂_{i}). Then 𝔅(V) ≃ K ⊗ 𝔅(W) as right 𝔅(W)-modules and as left K-modules. In particular,

P_{𝔅(V)}(t) = P_{K}(t)P_{𝔅(W)}(t),

and thus dim 𝔅(W) divides dim 𝔅(V).

2. Suppose that 𝔅(V) is finite dimensional and V = V_{1} ⊕ ⋯ ⊕ V_{θ} is a decomposition of (V, c) (that is, c(V_{i} ⊗ V_{j}) = V_{j} ⊗ V_{i}). For each 1 ≤ i ≤ θ, let W_{i} ⊆ V_{i} be a (possibly zero) subspace of V_{i} as in 3.5 part 2, i.e., for each i, W_{i} is a kΓ′_{i}-comodule for Γ′ ⊆ Γ a subgroup, it is kΓ′_{i}-stable and there exists W′_{i} ⊆ V a kΓ-subcomodule and a kΓ′_{i}-submodule such that V = W_{i} ⊕ W′_{i}. Thus, if P_{i}(t) is the Hilbert polynomial of 𝔅(W_{i}), then the product ∏_{i=1}^{θ} P_{i}(t) divides P_{𝔅(V)}(t) and the quotient lies in ℤ[t]. In particular, ∏_{i=1}^{θ} dim 𝔅(W_{i}) divides dim 𝔅(V).

Proof. 1. Since 𝔅(W) is finite dimensional, we can fix Λ_{𝔅(W)} and Λ_{𝔅(W^{∗})} integrals such that

[Λ_{𝔅(W)}, Λ_{𝔅(W^{∗})}] = 1.

Let {z_{1}, …, z_{n}} be a basis of homogeneous elements of 𝔅(W), and let {z̃_{1}, …, z̃_{n}} be the dual basis with respect to the Poincaré duality, i.e. [z̃_{i}z_{j}, Λ_{𝔅(W^{∗})}] = δ_{ij}. Let p_{i} ∈ AW be defined by p_{i} = z_{i}Λ_{𝔅(W^{∗})}z̃_{i}. Now, by 3.7 we have that if z ∈ 𝔅(W) and w ∈ 𝔅(W^{∗}) then

Λ_{𝔅(W^{∗})}zw = Λ_{𝔅(W^{∗})}[z, w].

Thus, by definition we have p_{i}p_{j} = δ_{ij}p_{i}, i.e., the p_{i}’s are mutually orthogonal idempotents. Furthermore, let z ∈ 𝔅(W), z = ∑_{j} α_{j}z̃_{j}. Then

(∑_{i} p_{i})(z) = ∑_{i,j} α_{j}[z̃_{j}z_{i}, Λ_{𝔅(W^{∗})}]z̃_{i} = ∑_{j} α_{j}z̃_{j} = z,

whence ∑_{i} p_{i} = 1. Because of the immersion AW ↪ AV, the equality ∑_{i} p_{i} = 1 also holds in AV. Now, since the image of Λ_{𝔅(W^{∗})} lies in K, we have that

𝔅(V) = ∑_{i} 𝔅(V)z_{i}Λ_{𝔅(W^{∗})}z̃_{i} ⊆ ∑_{i} Kz̃_{i}.

This tells that the canonical map μ : K ⊗ 𝔅(W) → 𝔅(V) given by multiplication is surjective. We shall prove that the sum is direct and that μ restricted to K ⊗ z̃_{i} is injective ∀i, which

A FREENESS THEOREM FOR NICHOLS ALGEBRAS 17

completes the proof. To see this, if 0 = ∑_{i} k_{i}z̃_{i} with k_{i} ∈ K, then for each j we have

0 = p_{j}(∑_{i} k_{i}z̃_{i})

= ∑_{i} k_{i}z̃_{i}z_{j}Λ_{𝔅(W^{∗})}z̃_{j}

= ∑_{i} k_{i} ([z̃_{i}z_{j}, Λ_{𝔅(W^{∗})}] + terms beginning with elements of positive degree in 𝔅(W^{∗})) z̃_{j}

= k_{j}z̃_{j}.

This proves that the sum is direct. Now, if k ∈ K and, for some i, kz̃_{i} = 0, then let d be the degree of z̃_{i} and let w_{i} be an element in 𝔅^{d}(W^{∗}), such that [z̃_{i}, w_{i}] = 1. We have

0 = kz̃_{i}w_{i} = k[z̃_{i}, w_{i}] + k ( terms beginning with elements of positive degree in 𝔅(W^{∗})) = k,

which implies the injectivity of μ.

2. Consider in 𝔅(V) the multi-degree given by the decomposition V = ⊕_{i}V_{i}. The condition on the V_{i}’s implies that the map S^{n} is (multi-)graded for each n, whence we can take the Hilbert polynomial of 𝔅(V) in ℤ[t_{1}, …, t_{θ}], i.e.

P_{𝔅(V)}(t_{1}, …, t_{θ}) = ∑_{i_{1},... ,i_{θ}} dim 𝔅^{i_{1},... ,i_{θ}}(V)t_{1}^{i_{1}} ⋯ t_{θ}^{i_{θ}}

(𝔅^{i_{1},... ,i_{θ}}(V) the homogeneous component of degree (i_{1}, …, i_{θ})). If K_{i} ⊂ 𝔅(V) is the intersection of the kernels of ∂ for all ∂ ∈ W_{i}^{∗}, then we can take also the Hilbert polynomial of K_{i} in ℤ[t_{1}, …, t_{θ}] since the elements in W_{i}^{∗} act in an homogeneous way. By the first part, we have that P_{𝔅(V)} = P_{K_{i}}P_{𝔅(W_{i})}, where P_{𝔅(W_{i})} ∈ ℤ[t_{i}]. Thus, for each i the polynomial P_{𝔅(W_{i})} divides P_{𝔅(V)}, and since these polynomials are coprime, their product also divides P_{𝔅(V)}. Moreover, since the product is a monic polynomial (because dim 𝔅^{top(W_{i})}(W_{i}) = 1) the quotient lies in ℤ[t_{1}, …, t_{θ}]. The result follows taking t_{1} = ⋯ = t_{θ} = t.

Remark 3.9. In the situation of the first part of the theorem, the Leibniz rule tells immediately that K is a subalgebra of 𝔅(V). It is in general not a coalgebra, but with the help of (2.8) it is easy to see that it is a left coideal. The fact that the morphism K ⊗ 𝔅(W) → 𝔅(V) is the multiplication of 𝔅(V) shows that they are isomorphic as (K, 𝔅(W))-bimodules.

Remark 3.10. Let V = ⊕_{i}M(g_{i}, ρ_{i}) a module in _{kΓ}^{kΓ}YD, Γ finite. Let z = z_{jl}^{i} ∈ M(g_{i}, ρ_{i}) be as in 2.15. Then c(z ⊗ z) = qz ⊗ z, where q = ρ_{i}(g_{i}). By the first part of Theorem 3.8 (taking W′ = W_{jl}^{′i}), we

18 MATÍAS GRAÑA

have that N(q) divides dim 𝔅(V). Furthermore, a basis of 𝔅(V) is given by {y_{a}z^{b} | 0 ≤ b < N(q)}, where {y_{a}} is a basis of ker ∂_{jl}^{i}.

Taking one element z_{j_{i}l_{i}}^{i} in each M(g_{i}, ρ_{i}) and q_{i} = ρ_{i}(g_{i}), N_{i} = N(q_{i}), by the second part of Theorem 3.8 (W_{i} being the subspace generated by z_{j_{i}l_{i}}^{i} and W′_{i} = W_{j_{i}l_{i}}^{′i}) we have that ∏_{i} N_{i} divides dim 𝔅(V).

Remark 3.11. We notice the difference between this result, 2.2 and 2.17. Let Γ be finite and V ∈ _{kΓ}^{kΓ}YD be finite dimensional, V = ⊕_{i}M(g_{i}, ρ_{i}) = ⊕_{i}V_{i}. Let d_{i} = dim(V_{i}), q_{i} = ρ_{i}(g_{i}), N_{i} = N(q_{i}), let D_{i} = dim(𝔅(V_{i})). Then by 2.2 we have dim(𝔅(V)) ≥ ∏_{i} D_{i} and by 2.17 we have D_{i} ≥ N_{i}^{d_{i}}. Finally, 3.8 tells that if dim(𝔅(V)) is finite, then it can be divided by ∏_{i} N_{i}.

We warn that it is not in general true that dim(𝔅(V)) is divisible by ∏_{i} N_{i}^{d_{i}}. For instance taking V with basis {x_{0}, x_{1}, x_{2}} and braiding c(x_{i} ⊗ x_{j}) = −x_{−i−j} ⊗ x_{i} (we take the subindices in ℤ/3), then d = 3 and N = 2, but dim(𝔅(V)) = 12 (it is computed in [MS], see also [AG]).

This example shows also why 3.8 is not a consequence of a braided version of the Nichols–Zoeller theorem. Taking W = span(x_{0}), we have c(V ⊗W) ⊄ (W ⊗V), and in particular c(𝔅(V)⊗𝔅(W)) ⊄ (𝔅(W) ⊗ 𝔅(V)), from where 𝔅(W) is not a categorical braided Hopf subalgebra of 𝔅(V) in the sense of [T3], which is one of the hypotheses of Nichols–Zoeller theorem for braided Hopf algebras.

4. Applications

As a consequence of the main theorem, we have a generalization of [D] and [AS2, Lemma 7.4]:

Corollary 4.1. Let A be a finite dimensional pointed Hopf algebra, let Γ = G(A) be its group of group-like elements, and let [A : Γ] be the index of Γ in A. If [A : Γ] = p is a prime number, then there exists g ∈ Z(Γ), χ ∈ Γ̂ a character and λ ∈ {0, 1} such that

• χ(g) is a primitive p-th root of unity

• λ = 0 if χ^{p} ≠ 1 or g^{p} = 1, and the algebra A is isomorphic to the algebra generated by G(A) plus an element x with relations

x^{p} = λ(g^{p} − 1),
hx = χ(h)xh.

The coalgebra structure is given by Δ(x) = g ⊗ x + x ⊗ 1.

Proof. The lifting procedure (see [AS1]) applied to A, we get R a coradically graded Hopf algebra in _{kΓ}^{kΓ}YD of dimension p. Let R′ be the subalgebra generated by R(1). By the Nichols–Zoeller theorem

A FREENESS THEOREM FOR NICHOLS ALGEBRAS 19

[NZ], R′ = R, and thus R is a Nichols algebra. Take x a non-zero element in R(1). The algebra generated by x in R is isomorphic to k[x]/x^{t}, but by 3.8 and 3.10 we must have t = p, and then R = k[x]/x^{p}. The equality x^{p} = λ(g^{p} − 1) is a straightforward computation of the lifting procedure, as done for instance in [G1].

Corollary 4.2. Let A be a pointed Hopf algebra, let Γ = G(A) be its group of group-like elements, and let [A : Γ] be the index of Γ in A. If [A : Γ] = p^{2} is a square prime, then either

1. the situation is the same as in 4.1, replacing each occurrence of “p” by “p^{2}”,

2. there exist two elements g_{1}, g_{2} ∈ Z(Γ), two characters χ_{1}, χ_{2} ∈ Γ̂ and λ_{1}, λ_{2}, λ_{3} ∈ k such that • χ_{i}(g_{i}) are p-th roots of unity,

• χ_{1}(g_{2})χ_{2}(g_{1}) = 1

• λ_{i} = 0 (i = 1, 2) if χ_{i}^{p} ≠ 1 or g_{i}^{p} = 1,

• λ_{3} = 0 if χ_{1}χ_{2} ≠ 1 or g_{1}g_{2} = 1, and the algebra A is isomorphic to the algebra generated by Γ(A) plus two elements x_{1}, x_{2} with relations

x_{i}^{p} = λ_{i}(g_{i}^{p} − 1), i = 1, 2; x_{1}x_{2} − χ_{2}(g_{1})x_{2}x_{1} = λ_{3}(g_{1}g_{2} − 1);
hx_{i} = χ_{i}(h)x_{i}h.

The coalgebra structure is given by

Δ(x_{i}) = g_{i} ⊗ x_{i} + x_{i} ⊗ 1,

3. p = 2, there exists an element g ∈ Z(Γ) and ρ an irreducible representation of Γ of degree 2 such that ρ(g) = − id; there exists a Γ-invariant quadratic form λ : span{x_{1}, x_{2}} → k such that λ = 0 if g^{2} = 1, and the algebra A is isomorphic to the algebra generated by Γ plus two elements x_{1}, x_{2} with relations

\begin{matrix}hxh^{−1} = ρ(h)(x) \\ x^{2} = λ(x)(g^{2} − 1) \end{matrix} \} for x ∈ span\{x_{1}, x_{2}\}.

The coalgebra structure is given by Δ(x) = g ⊗ x + x ⊗ 1, or

4. there exists an element g ∈ Z(Γ) with [Γ : Γ_{g}] = 2, and A may be presented as an extension

k → B → A → k(Γ/H) → k, (4.3)

where H is the subgroup of Γ generated by O_{g} and B fits into the case 2.

20 MATÍAS GRAÑA

Proof. Take R′ and R as in the proof of 4.1. Then the dimension of R′ may be p or p^{2}. If it were equal to p, then 3.8 tells that the rank of R′ is 1, and thus by [AS1, Thm. 3.2] R = R′, a contradiction. Then R = R′ = 𝔅(V), and R is a Nichols algebra. If dim V = 1, then V = M(g, χ) and we are in case 1. If dim V = 2, then either V = M(g_{1}, χ_{1}) ⊕ M(g_{2}, χ_{2}) and we are in case 2, V = M(g, ρ) with deg ρ = 2 and we are in case 3 (ρ(g) = − id is an equivalent condition for 𝔅(V) to be p^{2}-dimensional, because of 2.2, the condition on λ to be Γ-invariant follows computing hx^{2}h^{−1}), or V = M(g, ρ) with [Γ : Γ_{g}] = 2, and then by [AG, Lemma 3.1.9] R#kΓ is an extension as 4.3 and we are in case 4. The rank of R is necessarily ≤ 2 because of 2.17.

As a last application of 3.8, we classify Nichols algebras of dimension p^{3} that arise in Yetter–Drinfeld categories over finite groups. Together with the lifting procedure of [AS1], this is a first step in order to classify finite dimensional pointed Hopf algebras of index p^{3}.

Let then V ∈ _{kΓ}^{kΓ}YD be such that dim 𝔅(V) = p^{3}. By 2.17 and 3.8, we have dim V ≤ 3. It can be proved (see [G2]) that if (V, c) does not come from the abelian case then dim 𝔅(V) > p^{3}.

If dim V = 3, 𝔅(V) is a QLS because of 2.2 and 3.8, i.e. V has a basis {x_{1}, x_{2}, x_{3}} such that the set {x_{1}^{n_{1}}x_{2}^{n_{2}}x_{3}^{n_{3}} | 0 ≤ n_{i} < p} is a basis of 𝔅(V).

If dim V = 2 then c has a matrix ( \begin{matrix}q_{11} & q_{12} \\ q_{21} & q_{22} \end{matrix} ) where the q_{ii}’s are either p-roots or p^{2}-roots of unity.

If one of them has order p^{2}, say q_{22}, then N(q_{11}) = p and 𝔅(V) is a QLS. If N(q_{11}) = N(q_{22}) = p then by the results in [G2] we have two possibilities:

1. p = 2 and q_{12}q_{21} ≠ 1. Furthermore, for 𝔅(V) to be 8-dimensional q_{12}q_{21} = −1. The algebra 𝔅(V) is of type A_{2}.

2. p > 2 and q_{12}q_{21}q_{22} = 1. Interchanging q_{11} with q_{22} we see that q_{12}q_{21}q_{11} = 1, whence q_{11} = q_{22}

and 𝔅(V) is of type A_{2}. Thus we reach in both cases to the same situation. The algebra 𝔅(V) has a PBW basis given by {x_{1}^{m_{1}}z^{n}x_{2}^{m_{2}} | 0 ≤ m_{1}, m_{2}, n < p}, z = x_{2}x_{1} − q_{21}x_{1}x_{2}.

The last case is dim V = 1. We have simply {x} a basis of V such that c(x⊗x) = qx⊗x, N(q) = p^{3}. Thus 𝔅(V) = k[x]/(x^{p^{3}}).

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A FREENESS THEOREM FOR NICHOLS ALGEBRAS 21

[AG] N. Andruskiewitsch & M. Graña, Braided Hopf algebras over non abelian finite groups, Boletín de la Acad. Nac. Cs. Córdoba, Argentina 63 (1999), 45–78. Also in www.mate.uncor.edu/andrus/articulos.html.

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[T1] M. Takeuchi, The coquasitriangular Hopf algebra associated with a rigid Yang–Baxter coalgebra, Proc. Coll. Hopf algs. & quantum gps., ed. S. Caenepeel, M. Dekker. Brussels (1998), 251–270.

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[T3] M. Takeuchi, The Nichols–Zoeller theorem for braided Hopf algebras, (1999) preprint.

Matías Graña: Departamento de Matemática, FCEyN, Pab. I,

(1428) Ciudad Universitaria, Buenos Aires, Argentina

  E-mail address: matiasg@dm.uba.ar