ON A FACTORIZATION OF GRADED HOPF ALGEBRAS USING LYNDON WORDS

M. GRAÑA AND I. HECKENBERGER

Abstract. We find a generalization of the restricted PBW basis for pointed Hopf algebras over abelian groups constructed by Kharchenko. We obtain a factorization of the Hilbert series for a wide class of graded Hopf algebras. These factors are parametrized by Lyndon words, and they are the Hilbert series of certain graded Hopf algebras.

Key words: graded Hopf algebra, Nichols algebra, Lyndon word MSC2000: 16W30; 16W50

1. Introduction

Hopf algebras [Swe69] are far from being classified. Up to now there are two main directions of study: semisimple and pointed Hopf algebras. This paper is mainly a contribution to the latter, although all considerations are performed in a more general context. Specifically, we work with Hopf algebras H generated by a Hopf subalgebra H_{0} and a vector space V satisfying properties (4.2) and (4.3) below. This includes in particular pointed Hopf algebras generated by group-like and skew-primitive elements. In Theorem 4.12 we prove a factorization result about the Hilbert series of gr H. Moreover, with Theorem 4.19 we show that to each factor one can associate in a natural way a graded Hopf algebra which projects onto a Nichols algebra. These are the main results of the present paper.

Kharchenko [Kha99] proved that when H_{0} is the group algebra of an abelian group and it acts on V by characters, H admits a restricted PBW basis. The PBW generators in this basis are labelled by Lyndon words on an alphabet given by a set of skew-primitive elements. Examples, where H_{0} is the group algebra of a nonabelian group [MS00] [Gra00b] [AG03], indicate that in general one can not expect Kharchenko’s result to hold in its strong form. Nevertheless, by extending his ideas we were able to construct a basis of gr H using ordered products of subquotients of it. In the particular case of Kharchenko’s setting one recovers the PBW basis. This is possible because graded braided Hopf algebras generated by one primitive element are easy to classify. The difficulty in the more general setting arises from the fact that the structure of graded braided Hopf algebras generated by an irreducible Yetter–Drinfel’d module over H_{0} is not known.

M. G. was partially supported by CONICET and Fundación Antorchas.

M. G. thanks for the warm hospitality of Mathematisches Institut der Universität Leipzig, where the work on this paper begun.

A generalization of Kharchenko’s PBW theorem in a different direction was done by Ufer [Ufe04]. Instead of character Hopf algebras (i.e., Hopf algebras of diagonal type) he considers braided Hopf algebras with “triangular” braidings. On the one hand Ufer is able to give a restricted PBW basis. On the other hand some information about the relations of the Hopf algebra is lost. Although we believe that it is possible to obtain a generalization of Ufer’s approach to our context, we stick to a simpler setting for two reasons. First, valuable additional information can be obtained in our starting context. Second, the proofs in the triangular case would be even more technical, obscuring the essential arguments.

The proof of the main results of the present paper was possible due to taking advantage both from the lexicographic and the inverse lexicographic order on the set of monotonic super-words built from an alphabet of Lyndon words. This leads in a natural way to the construction of subquotients of a graded Hopf algebra. In this way Kharchenko’s PBW theorem becomes more transparent. Note that Ufer’s more technical proof stems from the fact that in his setting the inverse lexicographic order on the set of monotonic super-words can not be used.

Kharchenko’s PBW theorem turned out to be essential in the construction of the Weyl groupoid [Hec06] corresponding to a Nichols algebra of diagonal type. This groupoid played the crucial role in the classification of such Nichols algebras [Hec04]. In turn, the knowledge of these Nichols algebras is important for example for the lifting method of Andruskiewitsch and Schneider [AS98] to classify pointed Hopf algebras. We consider the factorization theorem in this paper to be an important step towards the generalization of the Weyl groupoid to a wider class of Nichols algebras.

In this paper k is an arbitrary field, and all algebras have k as their base field. The symbol ⊗ refers to tensor product over k. We will write m, Δ, ε and S for the product, coproduct, counit, and the antipode of a Hopf algebra.

The authors thank the referee for his helpful comments, especially those regarding Proposition 2.3.

2. Preliminaries

In this section we prove two general results about (braided) bialgebras, for which we did not find references in the literature.

Proposition 2.1. Let A = (A, m, Δ) be a bialgebra, B ⊆ A a subalgebra and I ⊆ B ∩ ker(ε) an ideal of B. Suppose furthermore that

(2.2)  Δ(B) ⊆ B ⊗ B + I ⊗ A, Δ(I) ⊆ B ⊗ I + I ⊗ A.

Then the bialgebra structure on A induces a bialgebra structure on B/I.

Proof. Notice first that B/I is an algebra. Let us take Δ̄ : B → A/I ⊗ A/I as Δ̄ = (π ⊗ π)Δi, for π : A → A/I the projection and i : B → A the inclusion. By using the first formula in (2.2), one gets that Δ̄(B) ⊆ B/I ⊗ B/I. By using the second formula in (2.2), one gets that Δ̄(I) = 0. Then Δ̄ induces a map Δ̃ : B/I → B/I ⊗ B/I. Also, ε induces a map ε̃: B/I → k. It is clear that Δ̃ is coassociative and ε̃ is a counit for Δ̃, whence B/I is a coalgebra. It is immediate that ε̃ is an algebra map. We must prove then that Δ̃ is an algebra map. For a, b ∈ B we compute

Δ̃(π(a)π(b)) = Δ̃(π(ab)) = (π ⊗ π)Δ(ab) = (π ⊗ π)(a_{(1)}b_{(1)} ⊗ a_{(2)}b_{(2)})

= π(a_{(1)}b_{(1)}) ⊗ π(a_{(2)}b_{(2)}) = π(a_{(1)})π(b_{(1)}) ⊗ π(a_{(2)}b_{(2)})

= π(a_{(1)})π(b_{(1)}) ⊗ π(a_{(2)})π(b_{(2)}).

In the first equality above we used that π|_{B} : B → B/I is an algebra map, and the fifth one is obtained from a_{(1)}b_{(1)} ⊗ a_{(2)}b_{(2)} ∈ B ⊗ A. The last equality holds by π(a_{(1)})π(b_{(1)}) ⊗ a_{(2)}b_{(2)} ∈ B/I ⊗ B. □

For the notion of braided Hopf algebras one may consult for example [Tak00].

Proposition 2.3. Let π : R → T be a surjective map of braided ℕ_{0}- graded Hopf algebras (either in the sense of Takeuchi or in a Yetter– Drinfel’d category) which is an isomorphism in degree 0, and assume that R, T are finite dimensional in each degree. Further, suppose that ker π is a categorical braided subspace of R, that is c(ker π ⊗ R) ⊂ R ⊗ ker π, c(R ⊗ ker π) ⊂ ker π ⊗ R. Then the quotient η(R, t)/η(T, t) of Hilbert series is again a series with nonnegative integer coefficients.

Proof. Let R_{n} and T_{n}, where n ∈ ℕ_{0}, denote the homogeneous subspaces of R and T, respectively, of degree n. Since π is a graded map and an isomorphism in degree 0, the ideal ker π is homogeneous and is contained in ⊕_{n=1}^{∞} R_{n}. Moreover, π is an algebra map, and hence (ker π)^{2} ⊂ ker π. Thus (ker π)^{n+1} ⊂ ker π^{n} for all n ∈ ℕ_{0}, where (ker π)^{0} := R. Set

R′(i) :=(ker π)^{i}/(ker π)^{i+1},
R′_{n}(i) :=((ker π)^{i} ∩ R_{n})/((ker π)^{i+1} ∩ R_{n}),  where i, n ∈ ℕ_{0}.

Since (ker π)^{n} ⊂ ⊕_{i=n}^{∞} R_{i}, one obtains that R and

R′ := ⊕_{i=0}^{∞} R′(i), where R′(0) = (ker π)^{0}/(ker π)^{1} = R/(ker π) ≃ T, (2.4)

are isomorphic as graded vector spaces. Here the grading of R′ is induced by the one of R, that is

R′_{n} = ⊕_{i=0}^{n} R′_{n}(i).

The algebra structure of R induces two algebra gradings on R′:

R′_{m}R′_{n} ⊂ R′_{m+n}, R′(m)R′(n) ⊂ R′(m + n) for all m, n ∈ ℕ_{0}.

Next we prove that R induces a braided Hopf algebra structure on R′. Let (c_{R}, Δ_{R}, S_{R}) and (c_{T} , Δ_{T} , S_{T}) denote the triples consisting of the braiding, the coproduct, and the antipode of R and T, respectively. Let ρ_{n} : R → R/(ker π)^{n+1}, where n ∈ ℕ_{0}, be the canonical projections. Define Δ_{R′,n} : R → ⊕_{i=0}^{n} ρ_{i}(R) ⊗ ρ_{n−i}(R) by setting

Δ_{R′,n} := ⊕_{i=0}^{n}(ρ_{i} ⊗ ρ_{n−i})Δ_{R}.

Since ker π is a coideal of R, that is Δ_{R}(ker π) ⊂ R ⊗ ker π + ker π ⊗ R, one gets

Δ_{R′,n}((ker π)^{n+1}) = 0,
Δ_{R′,n}((ker π)^{n}) ⊂ ⊕_{i=0}^{n} R′(i) ⊗ R′(n − i).

Thus the family of maps Δ_{R′,n} induces a coproduct Δ_{R′} on R′ (the coassociativity and compatibility with the counit being obvious) via the definition Δ_{R′}|_{R′(n)} := Δ_{R′,n}. We will show that Δ_{R′} is an algebra homomorphism, but to do so we have to consider first the braiding on R′.

Recall that ker π is a coideal, and by assumption a categorical braided subspace of R. Hence by induction on m and n one gets

c_{R}((ker π)^{m} ⊗ (ker π)^{n}) ⊂ (ker π)^{n} ⊗ (ker π)^{m}, m, n ∈ ℕ_{0}.

Having this, one shows, with the technique used in the construction of Δ_{R′}, that c_{R} induces a braiding c_{R′} on R′ with the property

c_{R′}(R′(m) ⊗ R′(n)) ⊂ R′(n) ⊗ R′(m), m, n ∈ ℕ_{0}.

Now we conclude that Δ_{R′} is an algebra map. Indeed, for x ∈ (ker π)^{m}, y ∈ (ker π)^{n} one gets

Δ_{R′}(ρ_{m}(x)ρ_{n}(y)) = Δ_{R′,m+n}(xy) =

= ∑_{i=0}^{m+n}(ρ_{i} ⊗ ρ_{m+n−i})(m ⊗ m)(id ⊗c_{R} ⊗ id)(x_{(1)} ⊗ x_{(2)} ⊗ y_{(1)} ⊗ y_{(2)}) =

= ∑_{i=0}^{m} ∑_{j=0}^{n}(m ⊗ m)(id ⊗c_{R′} ⊗ id)

(ρ_{i}(x_{(1)}) ⊗ ρ_{m−i}(x_{(2)}) ⊗ ρ_{j}(y_{(1)}) ⊗ ρ_{n−j}(y_{(2)})) =

= ∑_{i=0}^{m} ∑_{j=0}^{n}(ρ_{i}(x_{(1)}) ⊗ ρ_{m−i}(x_{(2)})) (ρ_{j}(y_{(1)}) ⊗ ρ_{n−j}(y_{(2)})) =

= Δ_{R′}(ρ_{m}(x))Δ_{R′}(ρ_{n}(y)).

Analogously, S_{R} induces an antipode S_{R′} of R′. This proves that R′ is a graded braided Hopf algebra.

The fundamental theorem for Hopf modules [Mon93, 1.9.4] implies that a Hopf algebra which admits a projection onto a Hopf subalgebra is isomorphic (via the multiplication map) to the tensor product of the right coinvariants and the Hopf subalgebra. The braided version of this statement holds in our setting: m : R′^{co T} ⊗ T → R′ is an isomorphism of graded vector spaces (even as right T -module comodules), where T has its own grading and R′ has the grading induced by the one of R. Indeed, let π_{0} denote the projection of R′ to T = R′(0) which maps R′(m) to 0 for all m ≥ 1. Then the map (id ⊗π_{0})Δ_{R′} is a right coaction of T on R′, and the map x ↦ m(id ⊗S_{R′} ∘ π_{0})Δ_{R′}(x) is a surjective map from R′ to R′^{co T} (because its restriction to R′^{co T} is the identity). Further, R′ ≃ R′^{co T} ⊗ T as graded vector spaces, which implies the statement. □

Remark 2.5. Suppose that π : R → T in the preceding proposition is a surjective map of braided ℕ_{0}-graded Hopf algebras in a Yetter– Drinfel’d category _{H_{0}}^{H_{0}}YD, where H_{0} is a Hopf algebra with bijective antipode, and both R and T are equipped with the canonical braiding (or both with the inverse of the canonical braiding). Then ker π is an object in the Yetter–Drinfel’d category, too, and the inclusion is a map in the category. Hence, ker π is automatically a categorical braided subspace of R.

Remark 2.6. The subspace R′^{co T} is a subalgebra of R′. Indeed, this follows from the right coaction of T on R′ being an algebra map, which, in turn, follows from ker π being categorical:

(id ⊗π_{0})Δ_{R′}(xy) =(id ⊗π_{0})(Δ_{R′}(x)Δ_{R′}(y))
=(id ⊗π_{0})Δ_{R′}(x) · (id ⊗π_{0})Δ_{R′}(y)

for all x, y ∈ R′.

Remark 2.7. Assume that R and T are graded braided Hopf algebras in a Yetter–Drinfel’d category _{H_{0}}^{H_{0}}YD, where H_{0} is a Hopf algebra with bijective antipode and the assumptions of Proposition 2.3 are fulfilled. Then the algebras R′, defined in Equation (2.4), and R are isomorphic as graded vector spaces, but not necessarily as braided vector spaces, at least if R is not a semisimple Yetter–Drinfel’d module over H_{0}.

The assumption that ker π is a categorical subspace of R is necessary for the definition of the braided Hopf algebra structure on R′. We want to thank the referee to pointing out this fact. For illustration we give an example.

Example 2.8. Assume that V = k{x_{0}, x_{1}, x_{2}} is the braided vector space with braiding c, where

c(x_{i} ⊗ x_{j}) = − x_{2i−j mod 3} ⊗ x_{i}, i, j = 0, 1, 2.

Let R be the associated Nichols algebra B(V) [MS00, Example 6.4]. This is a braided Hopf algebra with primitive generators x_{0}, x_{1}, x_{2}, and as an algebra one has

R ≅ k⟨x_{0}, x_{1}, x_{2}⟩/(x_{0}^{2}, x_{1}^{2}, x_{2}^{2},
x_{0}x_{1} + x_{1}x_{2} + x_{2}x_{0}, x_{1}x_{0} + x_{2}x_{1} + x_{0}x_{2}).

Let T = k[x]/(x^{2}) be the braided Hopf algebra with primitive generator x and braiding c, where c(x ⊗ x) = −x ⊗ x. Then there exists a unique algebra map π : R → T such that

π(x_{0}) = x,  π(x_{1}) = 0,  π(x_{2}) = 0.

Moreover, π is a surjective map of braided ℕ_{0}-graded Hopf algebras. Then ker π is the ideal (x_{1}, x_{2}), but it is not categorical because of the relations

c(R ⊗ ker π) ∋ c(x_{1} ⊗ x_{2}) = −x_{0} ⊗ x_{1} ∉ ker π ⊗ R.

In this case c does not induce a braiding on the algebra R′ defined via Equation (2.4), and R′ does not become a braided Hopf algebra. Nevertheless for this particular example the quotient η(R, t)/η(T, t) of Hilbert series is still a series with nonnegative integer coefficients, see for example [Gra00a, Thm. 3.8], [MS00, Cor. 3.3]. The proofs of the latter statements use a (noncategorical) section of the map R → T instead of requiring that ker π is categorical.

Following the suggestions of the referee we add two remarks.

Remark 2.9. Suppose that R and T are graded connected cocommutative Hopf algebras over a field of characteristic zero. By the Cartier-Kostant-Milnor-Moore theorem one knows that R = U(L_{R}) and T = U(L_{T}) are (isomorphic to) the universal enveloping algebras of the (graded) Lie algebras L_{R} and L_{T} of primitive elements in R and T, respectively. The PBW theorem implies that

ker π = L_{0}U(L_{R}) = U(L_{R})L_{0},

where L_{0} = L_{R} ∩ ker π is an ideal of the Lie algebra L_{R}. In this case one has by definition R′(n) ≃ (L_{0}^{n}/L_{0}^{n+1})U(L_{R}/L_{0}), where L_{0}^{n} has to be interpreted as a subspace of U(L_{0}) ⊂ U(L_{R}). Note that ∩_{n∈ℕ_{0}}L_{0}^{n} ⊂ ∩_{n∈ℕ_{0}} ker π = {0}, and hence the freeness of R′ over R′(0) is equivalent to the isomorphism U(L_{R}) ≃ U(L_{0}) ⊗ U(L_{R}/L_{0}).

Remark 2.10. There are two natural universal graded braided Hopf algebras associated to a Hopf algebra H_{0} with bijective antipode and a Yetter–Drinfel’d module V ∈ _{H_{0}}^{H_{0}}YD. These are the tensor Hopf algebra T V and the cotensor Hopf algebra T^{c}V [Nic78]. On the one hand, B(V) is a quotient of T V containing V, and hence generally R′ is not isomorphic to T V as a graded algebra, because R′ is not generated by V. On the other hand, B(V) is a braided Hopf subalgebra of both R′ and T^{c}V. However R′ is not isomorphic to T^{c}V as a graded coalgebra. Indeed, R′ may contain non-zero primitive elements of degree at least 2, while T^{c}V does not contain such elements. Therefore the method given in Proposition 2.3 gives a construction of graded (braided) Hopf algebras which are neither tensor nor cotensor Hopf algebras.

3. Lyndon words

Let A = {1, 2, …, d} be a totally ordered set by 1 < 2 < ⋯ < d. We think of A as an alphabet and 1, ⋯, d as the letters of A. Let 𝔸 be the set of non-empty words in this alphabet, and let ∅ denote the empty word. For a word u = a_{1}a_{2} ⋯ a_{r} with a_{i} ∈ A, 1 ≤ i ≤ r, we say that r is the length of u and we write r = |u|. We consider on 𝔸 the lexicographic order <. This means that u < v if and only if v = uu′ for some u′ ∈ 𝔸 or u = wiu′ and v = wjv′, where i < j and w, u′, v′ ∈ {∅} ∪ 𝔸.

A word u ∈ 𝔸 is called a Lyndon word if u = u_{1}u_{2} with u_{1}, u_{2} ∈ 𝔸 implies that u < u_{2}. For example: letters are Lyndon words, ij is a Lyndon word for i < j, 12122 is a Lyndon word, and 1212 is not a Lyndon word. We write L = {u ∈ 𝔸 | u is a Lyndon word}.

Proposition 3.1. [Lot83, Prop. 5.1.3] A word u is a Lyndon word if and only if u ∈ A or u = vw with v, w ∈ L and v < w. More precisely, if w is the proper right factor of maximal length of u = vw ∈ L that belongs to L, then also v ∈ L and v < vw < w. □

If u ∈ L, |u| ≥ 2, then the decomposition u = vw in Prop. 3.1 with w of maximal length is called the Shirshov decomposition of u. We write � (u) = (v, w).

Lemma 3.2. Let u, v ∈ L, u < v, |u| ≥ 2, and let � (u) = (u_{1}, u_{2}). Then exactly one of the following possibilities occurs:

(1) � (uv) = (u, v) and u_{2} ≥ v.

(2) � (uv) = (u′_{1}, u′′_{1}u_{2}v) for some words u′_{1}, u′′_{1} such that u_{1} = u′_{1}u′′_{1} and u_{2} < v (here u′′_{1} may be empty).

Proof. This is equivalent to [Lot83, Prop. 5.1.4] □

Lemma 3.3. [Kha99, Lemma 4] Let u, v ∈ L, u = u_{1}u_{2} for u_{1}, u_{2} ∈ 𝔸 and suppose that u_{2} < v. Then uv < u_{1}v. □

We take on L the lexicographic order. Thus, L is a new alphabet containing the original alphabet A, and following Kharchenko [Kha99] we say that the elements of L are super-letters. Words in super-letters are called super-words. The length |w| of a super-word w is the sum of the lengths of its super-letters. A monotonic super-word is a nonincreasing word on the set of super-letters, i.e., a (possibly empty) word v_{1} … v_{n} such that v_{i} ∈ L and v_{1} ≥ v_{2} ≥ ⋯ ≥ v_{n}. Let M denote the set of monotonic super-words. In what follows the notation v_{1} ⋯ v_{n} ∈ M will mean that v_{1}, …, v_{n} ∈ L and v_{1} ≥ ⋯ ≥ v_{n}. Sometimes we also write v_{1}^{m_{1}} ⋯ v_{n}^{m_{n}} ∈ M, in which case we mean that v_{1}, …, v_{n} ∈ L, v_{1} > ⋯ > v_{n}, and m_{1}, …, m_{n} ≥ 1. Monotonic super-words are lexicographically ordered on the alphabet of super-letters. Notice that the empty super-word is the smallest super-word. For a super-letter u, we shall write

\begin{matrix}L_{>u} = \{v ∈ L | v > u\}, L_{≥u} = \{v ∈ L | v ≥ u\}, \\ M_{>u} = \{v_{1} ⋯ v_{r} ∈ M | r ≥ 1, v_{1} ≥ ⋯ ≥ v_{r} > u\}, \\ M_{≥u} = \{v_{1} ⋯ v_{r} ∈ M | r ≥ 1, v_{1} ≥ ⋯ ≥ v_{r} ≥ u\}. \end{matrix} ; (3.4)

Theorem 3.5. (Lyndon, see [Lot83, Thm. 5.1.5 and Prop. 5.1.6]) A word in 𝔸 can be written in a unique way as a monotonic super-word. Moreover, if u = v_{1} … v_{n} ∈ M then v_{n} is the smallest right factor of u (smallest with respect to lexicographic order in 𝔸). □

As an example, the word 1231233122123 is decomposed as a monotonic super-word as

1231233122123 = (1231233)(122123),

and in turn, � (1231233) = (123, 1233).

Lemma 3.6. Let w = w_{1} ⋯ w_{n} be a super-word with w_{i} ∈ L for 1 ≤ i ≤ n. Then the decomposition of w as a monotonic super-word, w = v_{1} ⋯ v_{m}, satisfies the relation v_{1} ⋯ v_{m} ≥ w_{1} ⋯ w_{n} with respect to the lexicographic order on super-words.

Proof. We proceed by induction on the number of super-letters of w. If w is a super-letter then we are done. Otherwise w = w_{1} ⋯ w_{n}, where n ≥ 2 and w_{i} ∈ L ∀i. Again, if w_{1} ≥ w_{2} ≥ … ≥ w_{n} then we are done. On the other hand, if w_{i} < w_{i+1} for some i, then w′_{i} := w_{i}w_{i+1} ∈ L by Proposition 3.1, and one has w = w_{1} ⋯ w_{i−1}w′_{i}w_{i+2} ⋯ w_{n} with w′_{i} > w_{i}. Hence the claim follows from the induction hypothesis. □

Lemma 3.7. [Kha99, Lemma 5] Let w = w_{1} ⋯ w_{m} and v = v_{1} ⋯ v_{n} be monotonic super-words. Then w < v (with respect to the lexicographic order on M) if and only if w_{1} ⋯ w_{m} < v_{1} ⋯ v_{n} with respect to the lexicographic order on 𝔸. □

The following technical lemma will be needed in the proof of Theorem 4.12.

Lemma 3.8. Let w = w_{1} ⋯ w_{m} and v = v_{1} ⋯ v_{n} be nonempty monotonic super-words with w ≥ v and assume that v_{1} = ⋯ = v_{r} > v_{r+1}, where r ≤ n. For all i ≤ m let (w′_{i}, w′′_{i}) ∈ M × M such that either w′_{i} ≥ w_{i} or w′_{i} = ∅, w′′_{i} = w_{i}. Then the pair (w′_{1} ⋯ w′_{m}, w′′_{1} ⋯ w′′_{m}) satisfies one of the following relations.

(1) w′_{1} ⋯ w′_{m} > v_{r+1} ⋯ v_{n},

(2) w′′_{1} ⋯ w′′_{m} > v_{1}^{r},

(3) w = v and (w′_{1} ⋯ w′_{m}, w′′_{1} ⋯ w′′_{m}) = (v_{r+1} ⋯ v_{n}, v_{1}^{r}).

Proof. Assume first that w_{1} > v_{1}. Then either w′_{1} > v_{1} > v_{r+1} ⋯ v_{n} or w′_{1} = ∅, w′′_{1} = w_{1} > v_{1}. In the first case we have relation (1) and in the second one relation (2) is fulfilled. On the other hand, if w_{1} ≤ v_{1} then because of w ≥ v and w is monotonic, one has m ≥ r and w_{i} = v_{1} for all i ≤ r. Suppose first that there exists i ≤ r such that w′_{i} ≠ ∅ and w′_{j} = ∅ for all j < i. In this case w′_{i} ≥ v_{1} > v_{r+1} ⋯ v_{n} and hence (1) holds. It remains to consider the case w′_{i} = ∅ for all i ≤ r. Then one has w′′_{1} ⋯ w′′_{r} = v_{1}^{r}. Therefore, if w′′_{i} ≠ ∅ for some i > r then again relation (2) holds. Otherwise w′_{i} ≥ w_{i} for all i > r. Then one has either w′_{1} ⋯ w′_{m} = w′_{r+1} ⋯ w′_{m} > v_{r+1} ⋯ v_{n}, in which case (1) holds, or m = n, w = v, and w′_{i} = w_{i} for all i > r. The latter relations imply (3). □

Let H_{0} be a Hopf algebra with bijective antipode and let V = ⊕_{i=1}^{d}V_{i} be a direct sum of Yetter–Drinfel’d modules over H_{0}. Let T V be the tensor algebra of V. For simplicity, we will omit the ⊗ symbol in roducts of elements of T V. Let . and δ denote the left action and the left coaction of H_{0} on T V, respectively. We will use Sweedler notation δ(x) = x_{(−1)} ⊗ x_{(0)} for x ∈ T V. The algebra T V has a braiding c : T V ⊗ T V → T V ⊗ T V. Note that one has

c(x ⊗ y) =x_{(−1)}.y ⊗ x_{(0)},  c^{−1}(x ⊗ y) =y_{(0)} ⊗ S^{−1}(y_{(−1)}).x

for all x, y ∈ T V. In particular, equations

c(V_{i} ⊗ V_{j}) = V_{j} ⊗ V_{i}, i, j ∈ {1, …, d}

hold. We define

[x, y] = xy − mc^{−1}(x ⊗ y), ⟦x, y⟧ = xy − mc(x ⊗ y).

where m is the multiplication in T V.

Definition 3.9. Let a_{1}, …, a_{m} ∈ A and u = a_{1} … a_{m} ∈ 𝔸. We write V^{u} = V_{a_{1}}V_{a_{2}} ⋯ V_{a_{m}} ≃ V_{a_{1}} ⊗ V_{a_{2}} ⊗ … ⊗ V_{a_{m}}. The elements in V^{u} will be called u-vectors. If x_{u} ∈ V^{u} and u = vw, then we write x_{u} = x_{v} ⊗ x_{w} ∈ V^{v} ⊗ V^{w} (which is in general a sum of tensors) using the canonical isomorphism V^{u} ≃ V^{v} ⊗ V^{w}.

We shall inductively define bracket operations [ ] : ⊕_{n≥0}V^{⊗n} → T V and ⟦ ⟧ : ⊕_{n≥0}V^{⊗n} → T V as follows. Let x_{u} be a u-vector.

(1) If u has length 0 or 1, then [x_{u}] = ⟦x_{u}⟧ = x_{u}.

(2) If the word u is a Lyndon word and � (u) = (v, w), then [x_{u}] = [[x_{v}] , [x_{w}]] and ⟦x_{u}⟧ = ⟦⟦x_{v}⟧ , ⟦x_{w}⟧⟧ (see Def. 3.9).

(3) If the word u is decomposed as a monotonic super-word by u = v_{1} ⋯ v_{r}, then [x] = [x_{v_{1}}] · [x_{v_{2}}] ⋯ [x_{v_{r}}] and ⟦x⟧ = ⟦x_{v_{1}}⟧ · ⟦x_{v_{2}}⟧ ⋯ ⟦x_{v_{r}}⟧.

Remark 3.10. Recall that the braided antipode S_{T V} of the braided Hopf algebra T V satisfies S_{T V} (x) = x_{(−1)}S(x_{(0)}), where S is the antipode of T V #H_{0}. Moreover, S_{T V} (xy) = mc(S_{T V} (x) ⊗ S_{T V} (y)) = (x_{(−1)}.S_{T V} (y))S_{T V} (x_{(0)}). With these formulas it is easy to see that for any u ∈ L and any u-vector x one has S_{T V} ([x]) = (−1)^{|u|−1}⟦x⟧. Therefore most of the following considerations can be performed without difficulties with ⟦ ⟧’s instead of [ ]’s. Even if ⟦ ⟧’s seem to be more natural, we will follow the tradition of Kharchenko [Kha99] with [ ]’s.

Definition 3.11. If u is a Lyndon word and x is a u-vector, then [x] will be called a u-[ ]-letter. If u is a monotonic super-word and x is a u-vector then [x] will be called a u-[ ]-word. Let V^{[u]} = [V^{u}] denote the space of u-[ ]-words. A [ ]-letter ([ ]-word) is a u-[ ]-letter (u-[ ]-word) for some super-letter (monotonic super-word) u.

Lemma 3.12. Let u be a monotonic super-word. For x, y ∈ T V, let c^{−1}(x ⊗ y) =: ∑^{x}y ⊗ x^{y}. Then

(1) if x is a u-vector and h ∈ H_{0} then one has [h.x] = h.[x] and [x]_{(−1)} ⊗ [x]_{(0)} = x_{(−1)} ⊗ [x_{(0)}];

(2) if x is a u-vector and y is a v-vector for a monotonic super-word v, then ^{[x]}y ⊗ [x]^{y} = ^{x}y ⊗ [x^{y}] and ^{x}[y] ⊗ x^{[y]} = [^{x}y] ⊗ x^{y};

(3) for all x, y, z ∈ T V,

[x, yz] = [x, y] z + ∑^{x}y [x^{y}, z];

(4) for all x, y, z ∈ T V,

[[x, y], z] − [x, [y, z]] = ∑[x,^{y}z]y^{z} − ∑^{x}y[x^{y}, z].

Proof. (1) follows from the fact that the brading is a map of Yetter– Drinfel’d modules. (2) follows from (1) and the braid relation. (3) and (4) are straightforward calculations using the definition of [ ] and the braid relation for c^{−1}. □

We prove now a variant of [Kha99, Lemma 6].

Lemma 3.14. Let X, Y be a u-[ ]-letter and a v-[ ]-letter respectively, where u, v ∈ L and u < v. Then [X, Y] is a homogeneous linear combination of products of [ ]-letters corresponding to super-letters in L_{≥uv}.

Proof. Let z = uv. If � (z) = (u, v), then [X, Y] is a z-[ ]-letter and we are done. We proceed by induction on |uv|, since we already know the lemma for the case |u| = |v| = 1.

Let m = |uv| and suppose that the lemma holds for u′, v′ ∈ L, u′ < v′, X ∈ V^{[u′]}, Y ∈ V^{[v′]}, where either |u′v′| < m or |u′v′| = m, u′ < u (notice that there are only finitely many Lyndon words of a given length). As noted above, we may suppose that � (uv) ≠ (u, v). Then, if � (u) = (u_{1}, u_{2}), we must have u_{2} < v because of Lemma 3.2. Let X = [x] for x ∈ V^{u}, Y = [y] for y ∈ V^{v}, and let x = x_{u} = x_{u_{1}} ⊗x_{u_{2}}, X_{u_{1}} ⊗ X_{u_{2}} = [x_{u_{1}}] ⊗ [x_{u_{2}}]. Thanks to Lemma 3.12(4),(2), we have

\begin{matrix}[X, Y] = & [[X_{u_{1}}, X_{u_{2}}], Y] \\ = & [X_{u_{1}}, [X_{u_{2}}, Y]] \\ + ∑[X_{u_{1}},^{X_{u_{2}}}Y](X_{u_{2}})^{Y} − ∑^{X_{u_{1}}}X_{u_{2}}[X_{u_{1}}^{X_{u_{2}}}, Y] \\ = & [X_{u_{1}}, [X_{u_{2}}, Y]] \\ + ∑[X_{u_{1}}, [^{x_{u_{2}}}y]] · [x_{u_{2}}^{y}] − ∑[^{x_{u_{1}}}x_{u_{2}}] · [[x_{u_{1}}^{x_{u_{2}}}], Y]. \end{matrix} ; (3.15)

We start by considering the first summand on the right hand side. By the induction hypothesis, [X_{u_{2}}, Y] is a homogeneous linear combination of products of [ ]-letters corresponding to super-letters in L_{≥u_{2}v}, and the degree of these products is |u_{2}v|. By Lemma 3.12(3), and since u_{1} < u_{2} < u_{2}v, the element [X_{u_{1}}, [X_{u_{2}}, Y]] is a combination of products of [ ]-letters corresponding to super-letters in L_{≥u_{2}v} ⊂ L_{>uv} on one hand, and a bracket [h.X_{u_{1}}, X′] on the other hand, where h ∈ H_{0} and X′ is a w-[ ]-letter with w ≥ u_{2}v. By induction hypothesis again, the latter is a linear combination of products of [ ]-letters corresponding to super-letters in L_{≥u_{1}u_{2}v} = L_{≥uv}.

We continue with the second and third summands in (3.15). Concerning the u_{2}-[ ]-letters appearing there, notice that u_{2} > uv. Indeed, u_{2} > u_{1}u_{2} and, since u_{1}u_{2} is not the beginning of u_{2}, then we still have u_{2} > u_{1}u_{2}v = uv. On the other factors, which are brackets between u_{1}- [ ]-letters and v-[ ]-letters, we can apply the induction hypothesis since |u_{1}v| < |uv| and u_{1} < u_{1}u_{2} < u_{2} < v. These factors are then linear combinations of products of [ ]-letters corresponding to super-letters in L_{≥u_{1}v}, and u_{1}v > uv by Lemma 3.3. □

Lemma 3.16. Let u ∈ L. Any product of [ ]-letters corresponding to super-letters in L_{≥u} is a linear combination of (monotonic) [ ]-words corresponding to super-words in M_{≥u}.

Proof. Let t = v_{1} · v_{2} ⋯ v_{n} be a super-word, where v_{i} ∈ L_{≥u} ∀i. Let x = x_{t} = x_{v_{1}} ⋯ x_{v_{n}} be a t-vector, where here t is considered as a word on A. We call x_{[t]} := [x_{v_{1}}] ⋯ [x_{v_{n}}] a t-[ ]-vector (notice that the term “t- [ ]-word” is reserved for when t is monotonic). We proceed by induction. We take on the set of super-words the lexicographic order ≺ given by the order < on L. Suppose that the result is true for w-[ ]-vectors when |w| < |t| or when |w| = |t| and w ≻ t. If v_{1} ≥ v_{2} ≥ ⋯ ≥ v_{n}, there is nothing to prove. Otherwise, let i be such that 1 ≤ i < n and v_{i} < v_{i+1}. Let t′ = v_{1} … v_{i+1}v_{i} … v_{n} and t′′ = v_{1} … (v_{i}v_{i+1}) … v_{n},

ON A FACTORIZATION OF GRADED HOPF ALGEBRAS

where the factor (v_{i}v_{i+1}) stands for the Lyndon word v_{i}v_{i+1}. Let

x_{[t′]} := [x_{v_{1}}] ⊗ … ⊗ c^{−1}([x_{v_{i}}] ⊗ [x_{v_{i+1}}]) ⊗ … ⊗ [x_{v_{n}}],
x_{[t′′]} := [x_{v_{1}}] ⊗ … ⊗ [[x_{v_{i}}], [x_{v_{i+1}}]] ⊗ … ⊗ [x_{v_{n}}].

Notice that x_{[t′]} is a t′-[ ]-vector where t′ is a super-word in the same super-letters as t, and t′ ≻ t. Also, by Lemma 3.14, and since v_{i}v_{i+1} > v_{i}, x_{[t′′]} is a linear combination of w-[ ]-vectors, where w runs on superwords ≻ t having only super-letters in L_{≥u}. The last thing to notice is that x_{[t]} = x_{[t′]} + x_{[t′′]}. Therefore, the induction hypothesis implies the claim. □

Definition 3.17. Let x ∈ T V \{0}, x = ∑_{n} x_{n}, where x_{n} ∈ V^{⊗n}. The greatest n such that x_{n} ≠ 0 will be called the degree of x. Let m be the degree of x and let x_{m} = ∑_{u∈𝔸_{m}} x_{u}, where 𝔸_{m} is the set of words of length m and x_{u} is a u-vector for all u. Let v be the least u (with respect to the order in 𝔸_{m}) such that x_{u} ≠ 0. Then x_{v} will be called the leading vector of x.

Lemma 3.18. Let x ∈ T V be a nonzero u-vector for a monotonic super-word u. Then x is the leading vector of [x].

Proof. We prove first the lemma for Lyndon words u by induction on |u|. If |u| is 0 or 1, the result is clear. If � (u) = (v, w), then [x] = [[x_{v}], [x_{w}]] = (m − mc^{−1})([x_{v}] ⊗ [x_{w}]). By the induction hypothesis, [x_{v}][x_{w}] is a sum of products of v′-vectors and w′-vectors, where v′ runs on words ≥ v and w′ runs on words ≥ w. For such v′, w′ we have u = vw ≤ v′w′, whence [x_{v}][x_{w}] is a sum of u′-vectors with u′ ≥ u. Furthermore, the equality holds if and only if v′ = v and w′ = w. Thus, by using induction hypothesis, the leading vector of [x_{v}][x_{w}] is x_{v}x_{w}. For the term mc^{−1}([x_{v}] ⊗ [x_{w}]) we reason similarly: as u is a Lyndon word, u = vw < w < wv, from where mc^{−1}([x_{v}] ⊗ [x_{w}]) is a sum of u′-vectors where u′ runs on words ≥ wv > u. Therefore, such u′-vectors do not contribute to the leading vector of [x_{v}x_{w}].

If u is not a super-letter, then by Theorem 3.5 we have u = v_{1} ⋯ v_{n} ∈ M. By definition [x] = [x_{v_{1}}] ⋯ [x_{v_{n}}], which, by the previous step, is a sum of v′_{1} ⋯ v′_{n}-vectors with v′_{i} ≥ v_{i} ∀i, and its leading vector is x_{v_{1}} ⋯ x_{v_{n}} = x. □

Recall that M is the set of monotonic super-words.

Corollary 3.19. One has T V = ⨁_{u∈M} V^{[u]}.

Proof. Since letters in A are also super-letters, the spaces V^{[u]} generate T V thanks to Lemma 3.16. The linear independence of these spaces follows immediately from Lemma 3.18. □

Corollary 3.20. Let u ∈ 𝔸 and x be a u-vector. Then x − [x] is a linear combination of w-[ ]-words with w > u.

Proof. This assertion can be proven with the help of bases of ⊕_{|v|=|u|}V^{v} and ⊕_{|v|=|u|}V^{[v]} which are obtained from each other using triangular matrices. Alternatively, Lemmas 3.7 and 3.18 imply that x − [x] is a linear combination of w-vectors with w > u, and proceed by induction.

□

Corollary 3.20 allows us to give a description of products of [ ]-letters, which is different from the one in Lemma 3.16.

Corollary 3.21. Let n ∈ ℕ, v_{1}, …, v_{n} ∈ L and let X_{i} be a v_{i}-[ ]-letter for all i ∈ {1, …, n}. Then X_{1} ⋯ X_{n} is a linear combination of w- [ ]-words, where w runs over monotonic super-words ≥ v′_{1} ⋯ v′_{n′}, and v′_{1} ⋯ v′_{n′} is the decomposition of v_{1} ⋯ v_{n} as a monotonic super-word.

□

Definition 3.22. Let K be a totally ordered set and let there be vector spaces W_{k} for each k ∈ K. We define ⊗_{k∈K}^{>}W_{k} to be the direct sum of vector spaces W_{k_{1}} ⊗ … ⊗ W_{k_{r}} where k_{1} > k_{2} > ⋯ > k_{r}.

For a vector space W, we write T^{+}W = ⊕_{n≥1}W^{⊗n}. Notice that for each u ∈ L, T^{+}V^{[u]} is a non-unital subalgebra of T^{+}V in the category _{H_{0}}^{H_{0}}YD. Further, the ℤ-grading of T V induces a ℤ-grading on its subalgebras T V^{[u]}. In particular, the Hilbert series η(T V^{[u]}, t) of T V^{[u]} is a series in the variable t^{|u|}.

Theorem 3.23. One has

T^{+}V ≃ ⊗_{u∈L}^{>}T^{+}V^{[u]}.

More precisely, the map μ : ⊗_{u∈L}^{>}T^{+}V^{[u]} → T^{+}V, which is the multiplication map in each summand, is an isomorphism in the category _{H_{0}}^{H_{0}}YD. In particular, the Hilbert series of T V is η(T V, t) = ∏_{u∈L} η(T V^{[u]}, t).

Proof. This is a reformulation of Corollary 3.19. □

4. Hopf algebras generated by a Hopf subalgebra and a vector space

Our aim is now to apply the results of the previous section to arbitrary Hopf algebras. In many (though not all) cases we will be able to provide some new structure results.

Let H be a Hopf algebra with bijective antipode and a filtration

0 = F_{−1} ⊂ F_{0} ⊂ F_{1} ⊂ F_{2} ⊂ ⋯ ⊆ H.

The filtration is a Hopf algebra filtration if (1) It is a filtration: H = ⋃_{n∈ℕ} F_{n},

(2) it is an algebra filtration: F_{n}F_{m} ⊆ F_{n+m},

(3) it is a coalgebra filtration: Δ(F_{n}) ⊆ ∑_{i+j=n} F_{i} ⊗ F_{j}, and

(4) it behaves well with the antipode: S(F_{n}) ⊆ F_{n}. Notice that in this case F_{0} is a Hopf subalgebra of H. We will consider Hopf algebras with a Hopf algebra filtration satisfying a stronger version of condition (2):

Assumption 4.1. F_{n}F_{m} = F_{n+m}.

A Hopf algebra with a Hopf algebra filtration satisfying Assumption 4.1 can be presented in the following way.

Suppose that the Hopf algebra H is generated (as an algebra) by a Hopf subalgebra H_{0} and a vector space V, such that

ΔV ⊆ V ⊗ H_{0} + H_{0} ⊗ V + H_{0} ⊗ H_{0}, and (4.2) S(V) ⊆ KV K. (4.3)

We define F_{0} = H_{0}, F_{1} = H_{0} + H_{0}V H_{0}, and F_{n} = (F_{1})^{n}. Then F_{∗} is a Hopf algebra filtration which satisfies Assumption 4.1.

As an important example, assume that H is generated by grouplikes and skew-primitive elements and take H_{0} to be the subalgebra generated by the grouplikes and V to be the subspace generated by skew-primitives.

For n ≥ 0 let H_{n} = F_{n}/F_{n−1}. Let then

gr_{F} H = ⊕_{n≥0} F_{n}/F_{n−1} = ⊕_{n≥0} H_{n}

be the associated graded Hopf algebra. Then we can consider the projection π : gr_{F} H → H_{0} and the inclusion ι : H_{0} → gr_{F} H, and this allows to write gr_{F} H = R#H_{0} as the smash product of H_{0} and the right coinvariants R = {x ∈ gr_{F} H | (id ⊗π)Δ(x) = x ⊗ 1}. This is now a standard procedure: the last part is due to Radford [Rad85] and Majid [Maj94], while the first part is a modification of the one due to Andruskiewitsch and Schneider [AS98].

Also, R is a braided Hopf algebra in _{H_{0}}^{H_{0}}YD, and R = ⊕_{n≥0}R_{n}, R_{0} is the base field and R_{1} is a Yetter–Drinfel’d module which generates R because of Assumption 4.1. We let V := R_{1} and we make the following assumption:

Assumption 4.4. V = ⊕_{i=1}^{d}V_{i} is a direct sum of Yetter–Drinfel’d modules over H_{0}.

Remark 4.5. If V is irreducible, the methods in this paper do not yield any information on R. Otherwise, V has a nontrivial maximal flag V = V_{1} ⊃ V_{2} ⊃ ⋯ ⊃ V_{d} of Yetter–Drinfel’d submodules over H_{0} (which is not necessarily a full flag of vector subspaces). Consider on the vector space R_{n} the ℤ-filtration R_{n} = R_{n,n} ⊃ R_{n,n+1} ⊃ ⋯, where R_{n,m} = ∑_{i_{1}+···+i_{n}≥m} V_{i_{1}}V_{i_{2}} ⋯ V_{i_{n}}. Then

R′ := ⊕_{n=0}^{∞} R′_{n}, where R′_{n} = ⊕_{m≥n} R_{n,m}/R_{n,m+1},

is a graded braided Hopf algebra in the category _{H_{0}}^{H_{0}}YD, and all of the following considerations may be applied to R′ instead of R.

Consider the projection T V → R. We want to study the images under this projection of the components T^{+}V^{[u]} appearing in Theorem 3.23. We begin by considering the comultiplication in T V #H_{0}.

We write the left H_{0}-coaction on r ∈ R by δ(r) = r_{(−1)} ⊗ r_{(0)}. Recall that the coproduct in the smash product R#H_{0} is given by Δ(r#h) = (r^{(1)}#r^{(2)}_{(−1)}h_{(1)}) ⊗ (r^{(2)}_{(0)}#h_{(2)}), where Δ_{R}(r) =: r^{(1)} ⊗ r^{(2)} is the coproduct of the braided Hopf algebra R. This notation applies in particular for R = T V.

Proposition 4.6. Let X ∈ T V ⊆ T V #H_{0} be a u-[ ]-letter. Then the coproduct Δ of T V #H_{0} satisfies

Δ(X) = X ⊗ 1 + X_{(−1)} ⊗ X_{(0)} + ∑_{i}(X′_{i}h_{i}) ⊗ X′′_{i},

where X′_{i}, X′′_{i} ∈ T^{+}V, h_{i} ∈ H_{0}, and each X′_{i} is a w_{i}-[ ]-word with w_{i} ∈ M_{>u}.

Proof. We proceed by induction on |u|. If u ∈ A, we get X ∈ V and then Δ(X) = X ⊗ 1 + X_{(−1)} ⊗ X_{(0)}, whence we are done. Assume now that |u| ≥ 2, � (u) = (v, w) and x is a u-vector. Then x = x_{v}x_{w}, and we write X = [x], Y = [x_{v}], Z = [x_{w}]. By standard computations,

\begin{aligned}Δ(X) & = Δ([Y, Z]) = Δ(Y Z − Z_{(0)}(S^{−1}(Z_{(−1)}).Y)) \\ & = (Y^{(1)}Y^{(2)}_{(−1)} ⊗ Y^{(2)}_{(0)}) · (Z^{(1)}Z^{(2)}_{(−1)} ⊗ Z^{(2)}_{(0)}) \\ & − (Z_{(0)}^{(1)}Z_{(0)}^{(2)}_{(−1)} ⊗ Z_{(0)}^{(2)}_{(0)}) \\ & · ((S^{−1}(Z_{(−1)}).Y^{(1)})S^{−1}(Z_{(−2)})Y^{(2)}_{(−1)}Z_{(−4)} \\ & ⊗ S^{−1}(Z_{(−3)}).Y^{(2)}_{(0)}). \end{aligned} ; (4.7)

Note that Y is a v-[ ]-letter and Z is a w-[ ]-letter. According to the induction hypothesis and Lemma 3.12(1), for any h ∈ H_{0}, h.Y^{(1)} can be taken to be either ε(h)1, h.Y or a v′-[ ]-word with v′ ∈ M_{>v}. Similarly, h.Z^{(1)} can be taken to be either ε(h)1, h.Z or a w′-[ ]-word with w′ ∈ M_{>w}, and Z_{(0)}^{(1)} to be either 1, Z_{(0)} or a w′-[ ]-word with w′ ∈ M_{>w}.

We begin by considering the summand

(Y^{(1)}Y^{(2)}_{(−1)} ⊗ Y^{(2)}_{(0)}) · (Z^{(1)}Z^{(2)}_{(−1)} ⊗ Z^{(2)}_{(0)})
= Y^{(1)}Y^{(2)}_{(−1)}Z^{(1)}Z^{(2)}_{(−1)} ⊗ Y^{(2)}_{(0)}Z^{(2)}_{(0)}

= Y^{(1)}(Y^{(2)}_{(−2)}.Z^{(1)})Y^{(2)}_{(−1)}Z^{(2)}_{(−1)} ⊗ Y^{(2)}_{(0)}Z^{(2)}_{(0)}.

We consider the summands in which Y^{(1)} is a v′-[ ]-word with v′ ∈ M_{>v}. Notice that since these v′ are shorter than v, they belong to M_{>vw}. Therefore, since Z^{(1)} is either 1 or a t-[ ]-word with t ∈ M_{>vw}, by Lemma 3.16 these summands satisfy the claim of the Proposition. The summands in which Y^{(1)} = 1 and Z^{(1)} ≠ 1 also satisfy the claim, because w > u. We are thus left with

(4.8)  Y_{(−1)}Z_{(−1)} ⊗ Y_{(0)}Z_{(0)} + Y Z^{(1)}Z^{(2)}_{(−1)} ⊗ Z^{(2)}_{(0)}.

We consider now the other summand of Δ(X). By similar reasons, we are left with

\begin{matrix}− Y_{(−1)}Z_{(−2)} ⊗ Z_{(0)}(S^{−1}(Z_{(−1)}).Y_{(0)}) \\ − Z_{(0)}^{(1)}Z_{(0)}^{(2)}_{(−1)}(S^{−1}(Z_{(−1)}).Y )S^{−1}(Z_{(−2)})Z_{(−4)} \\ ⊗ Z_{(0)}^{(2)}_{(0)}(S^{−1}(Z_{(−3)}).1) \\ = −Y_{(−1)}Z_{(−2)} ⊗ Z_{(0)}(S^{−1}(Z_{(−1)}).Y_{(0)}) \\ − Z_{(0)}^{(1)}Z_{(0)}^{(2)}_{(−1)}(S^{−1}(Z_{(−1)}).Y ) ⊗ Z_{(0)}^{(2)}_{(0)} \\ = −Y_{(−1)}Z_{(−2)} ⊗ Z_{(0)}(S^{−1}(Z_{(−1)}).Y_{(0)}) \\ − Z^{(1)}_{(0)}((Z^{(2)}_{(−2)}S^{−1}(Z^{(2)}_{(−3)})S^{−1}(Z^{(1)}_{(−1)})).Y )Z^{(2)}_{(−1)} ⊗ Z^{(2)}_{(0)} \\ = −Y_{(−1)}Z_{(−1)} ⊗ mc^{−1}(Y_{(0)} ⊗ Z_{(0)}) \\ − mc^{−1}(Y ⊗ Z^{(1)})Z^{(2)}_{(−1)} ⊗ Z^{(2)}_{(0)}. \end{matrix} (4.9)

Adding (4.8) and (4.9) we get

Y_{(−1)}Z_{(−1)} ⊗ [Y_{(0)}, Z_{(0)}] + [Y, Z^{(1)}]Z^{(2)}_{(−1)} ⊗ Z^{(2)}_{(0)}.

Notice that Y_{(−1)}Z_{(−1)} ⊗ [Y_{(0)}, Z_{(0)}] = X_{(−1)} ⊗ X_{(0)}. Also, when in the second summand we put Z^{(1)} = Z, we get X ⊗ 1. Finally, when Z^{(1)} is a w′-[ ]-word with w′ ∈ M_{>w}, by using Lemmas 3.12(3), 3.14 and 3.16 we obtain terms which satisfy the claim. □

Corollary 4.10. Let u ∈ L and v ∈ M_{≥u}. Let X ∈ T V ⊆ T V #H_{0} be a v-[ ]-word. Then the coproduct Δ of T V #H_{0} satisfies

Δ(X) = X ⊗ 1 + X_{(−1)} ⊗ X_{(0)} + ∑_{i}(X′_{i}h_{i}) ⊗ X′′_{i},

where X′_{i}, X′′_{i} ∈ T^{+}V, h_{i} ∈ H_{0}, and each X′_{i} is a w_{i}-[ ]-word with w_{i} ∈ M_{≥u}.

Proof. This follows at once from the Proposition and Lemma 3.16. □

We study the structure of R now. For that, we will use the following notation.

Definition 4.11. For u ∈ L, let V^{[≥u]} be the subalgebra of T V generated by (∑_{v∈L,v≥u} V^{[v]}). Let I^{[≥u]} be the ideal of V^{[≥u]} generated by (∑_{v∈L,v>u} V^{[v]}). We define also V_{[u]} = π(V^{[u]}), V_{[≥u]} = π(V^{[≥u]}), V_{[≥u]}^{+} = π(V^{[≥u]}) ∩ ker ε, and I_{[≥u]} = π(I^{[≥u]}), where π : T V → R is the canonical projection.

The grading on R induces a grading on all of the algebras and ideals defined above. Take u ∈ L. Notice that the graded algebras V^{[≥u]}/I^{[≥u]} and V_{[≥u]}/I_{[≥u]} have only elements in degrees m|u| where m ∈ ℕ_{0}. Moreover, Lemma 3.16 and Corollary 3.21 imply that V^{[≥u]} is the subspace of T V generated by w-[ ]-words with w ∈ M_{≥u} and I^{[≥u]} is the subspace of V^{[≥u]} generated by w-[ ]-words with w = w_{1} ⋯ w_{n} ∈ M_{≥u} with w_{1} > u.

Notice that the leading vector of any X ∈ V^{[≥u]} \ I^{[≥u]} of degree m|u| is a u^{m}-vector. Thus, we can choose for any u ∈ L a graded linear map ι_{u} : V_{[≥u]}^{+}/I_{[≥u]} → T^{+}V^{[u]} = ⊕_{m∈ℕ}V^{[u^{m}]} ⊂ V^{[≥u]} such that π_{u} ∘ π|_{T^{+}V^{[u]}} ∘ ι_{u} = id, where π : T V → R and π_{u} : V_{[≥u]} → V_{[≥u]}/I_{[≥u]} are the canonical maps.

Recall Def. 3.22 for the definition of ⊗^{>}.

Theorem 4.12. The map ϕ : ⊗_{u∈L}^{>}V_{[≥u]}^{+}/I_{[≥u]} → R^{+} defined by

(4.14)  ϕ(X_{u_{1}} ⊗ ⋯ ⊗ X_{u_{n}}) = π(ι_{u_{1}}(X_{u_{1}}) ⋯ ι_{u_{n}}(X_{u_{n}})),

where u_{1}, …, u_{n} ∈ L, u_{1} > ⋯ > u_{n}, and X_{u_{i}} ∈ V_{[≥u_{i}]}^{+}/I_{[≥u_{i}]} for all i, is an isomorphism of graded vector spaces.

To prove the theorem we will use the following lemma.

Lemma 4.15. Let u = u_{1}^{n_{1}} ⋯ u_{r}^{n_{r}} ∈ M, and let X_{i} ∈ V^{[u_{i}^{n_{i}}]} for all i. Take Y_{i} = π_{u_{i}}(π(X_{i})) ∈ V_{[≥u_{i}]}^{+}/I_{[≥u_{i}]}. Then

X_{1} ⋯ X_{r} − ι_{u_{1}}(Y_{1}) ⋯ ι_{u_{r}}(Y_{r}) ∈ ker π + ∑_{\begin{matrix}w>u \\ w∈M \end{matrix}} V^{[w]}.

Proof. Since π_{u_{i}}π(X_{i}−ι_{u_{i}}(Y_{i})) = 0, we have X_{i}−ι_{u_{i}}(Y_{i}) ∈ ker π+I^{[≥u_{i}]}. We then consider

Z = X_{1} ⋯ X_{r} − ι_{u_{1}}(Y_{1}) ⋯ ι_{u_{r}}(Y_{r})

= ∑_{j=1}^{r} X_{1} ⋯ X_{j−1}(X_{j} − ι_{u_{j}}(Y_{j}))ι_{u_{j+1}}(Y_{j+1}) ⋯ ι_{u_{r}}(Y_{r})

∈ ∑_{j=1}^{r} V^{[u_{1}^{n_{1}}]} ⋯ V^{[u_{j−1}^{n_{j−1}}]}I^{[≥u_{j}]}V^{[u_{j+1}^{n_{j+1}}]} ⋯ V^{[u_{r}^{nr}]} + ker π.

As mentioned above the theorem, I^{[≥u_{j}]} consists of sums of w-[ ]-words where w runs on monotonic super-words > u_{j}^{n_{j}}. Thus, by Corollary 3.21 and Lemma 3.6, Z is a sum of w-[ ]-words, where w runs on super-words > u. □

Proof of the Theorem. Since the set of words of a given length is finite, the surjectivity of ϕ follows easily from the previous lemma.

We now prove injectivity of ϕ. To do so define ϕ′ : ⊗_{u∈L}^{>}V_{[≥u]}^{+}/I_{[≥u]} → T^{+}V by

ϕ′(X_{u_{1}} ⊗ ⋯ ⊗ X_{u_{n}}) = ι_{u_{1}}(X_{u_{1}}) ⋯ ι_{u_{n}}(X_{u_{n}}),

where u_{1}, …, u_{n} ∈ L, u_{1} > ⋯ > u_{n}, and X_{u_{i}} ∈ V_{[≥u_{i}]}^{+}/I_{[≥u_{i}]} for all i. Assume then that there exists a smallest integer m such that ϕ is not injective in degree m. For all u ∈ L let B_{u} = {b_{u,i} | i ∈ I_{u}} be a homogeneous basis of V_{[≥u]}^{+}/I_{[≥u]}, where I_{u} is an appropriate index set, and let X_{u,i} := ι_{u}(b_{u,i}) for all u ∈ L, i ∈ I_{u}.

Suppose that there exists a nonempty finite subset M′ of M with |w| = m for w ∈ M′, and for each w = w_{1}^{n_{1}} ⋯ w_{r}^{n_{r}} ∈ M′ there exist nonzero elements b_{w} ∈ V_{[≥w_{1}]}^{+}/I_{[≥w_{1}]} ⊗ ⋯ ⊗ V_{[≥w_{r}]}^{+}/I_{[≥w_{r}]} such that ϕ′(∑_{w∈M′} b_{w}) ∈ ker π. Let u = u_{1}^{m_{1}} ⋯ u_{s}^{m_{s}}, where u_{1} > ⋯ > u_{s}, be the minimal element of M′, and write b_{u} := ∑_{i_{1},...,i_{s}} λ_{i_{1},...,i_{s}}b_{u_{1},i_{1}} ⊗⋯⊗b_{u_{s},i_{s}} with λ_{i_{1},...,i_{s}} ∈ k. We consider T V as a subalgebra of T V #H_{0}, and then we have

Δ(ϕ′( ∑_{w∈M′} b_{w})) =: ∑_{i} Z′_{i} ⊗ Z′′_{i}
∈ (ker π#H_{0}) ⊗ T V + (T V #H_{0}) ⊗ ker π.

Therefore,

(4.16)  ∑_{i} S^{−1}(Z′′_{i}_{(−1)})Z′_{i} ⊗ Z′′_{i}_{(0)} ∈ ker π ⊗ T V + T V ⊗ ker π.

We apply Prop. 4.6 to each [ ]-letter in ϕ′(∑_{w∈M′} b_{w}), and we use Lemma 3.8 to obtain a description of the tensor factors of (4.16). Afterwards, we apply Corollary 3.21 and Lemma 3.6 to rearrange the tensor factors as sums of [ ]-words. This gives

∑_{i}S^{−1}(Z′′_{i}_{(−1)})Z′_{i} ⊗ Z′′_{i}_{(0)}
∈ ∑_{i_{1},...,i_{s}} λ_{i_{1},...,i_{s}}S^{−1}(X_{u_{1},i_{1}}_{(−1)})X_{u_{1},i_{1}}_{(−2)}X_{u_{2},i_{2}} ⋯ X_{u_{s},i_{s}} ⊗ X_{u_{1},i_{1}}_{(0)}

+ ∑_{\begin{matrix}w′,w′′∈M \\ w′>u_{2}^{m_{2}}···u_{s}^{ms} or w′′>u_{1}^{m_{1}} \end{matrix}} V^{[w′]} ⊗ V^{[w′′]}.

By repeatedly using Lemma 4.15 and since ϕ′(∑_{w∈M′} b_{w}) ∈ ker π, we get

∑_{i}S^{−1}(Z′′_{i}_{(−1)})Z′_{i} ⊗ Z′′_{i}_{(0)}
∈ ∑_{i_{1},...,i_{s}} λ_{i_{1},...,i_{s}}ϕ′(b_{u_{2},i_{2}}) ⋯ ϕ′(b_{u_{s},i_{s}}) ⊗ ϕ′(b_{u_{1},i_{1}})

+ ∑_{\begin{matrix}w′,w′′∈M, |w′|,|w′′|<|u|, \\ w′>u_{2}^{m_{2}}···u_{s}^{ms} or w′′>u_{1}^{m_{1}} \end{matrix}}(Im ϕ′ ∩ V^{[w′]}) ⊗ (Im ϕ′ ∩ V^{[w′′]})

+ ker π ⊗ T V + T V ⊗ ker π.

Therefore, (4.16) shows that

∑_{i_{1},...,i_{s}}λ_{i_{1},...,i_{s}}ϕ′(b_{u_{2},i_{2}} ⋯ b_{u_{s},i_{s}}) ⊗ ϕ′(b_{u_{1},i_{1}})
∈ ∑_{\begin{matrix}w′,w′′∈M, |w′|,|w′′|<|u|, \\ w′>u_{2}^{m_{2}}···u_{s}^{ms} or w′′>u_{1}^{m_{1}} \end{matrix}}(Im ϕ′ ∩ V^{[w′]}) ⊗ (Im ϕ′ ∩ V^{[w′′]})

+ ker π ⊗ T V + T V ⊗ ker π.

(4.17)

By the assumption on m, π ∘ ϕ′ = ϕ is injective in degrees < m and hence the sums

(ker π ∩ V^{⊗n}) + ( ⨁_{\begin{matrix}w′∈M, |w′|=n, \\ w′>u_{2}^{m_{2}}···u_{s}^{ms} \end{matrix}} Im ϕ′ ∩ V^{[w′]}) + (Im ϕ′ ∩ V^{[u_{2}^{m_{2}}···u_{s}^{ms}]})

(ker π ∩ V^{⊗n}) + ( ⨁_{\begin{matrix}w′′∈M, |w′′|=n, \\ w′′>u_{1}^{m_{1}} \end{matrix}} Im ϕ′ ∩ V^{[w′′]}) + (Im ϕ′ ∩ V^{[u_{1}^{m_{1}}]})

are direct in T V whenever 1 ≤ n < m. Thus (4.17) implies that λ_{i_{1},...,i_{s}} = 0 for all i_{1}, …, i_{s}, which contradicts to the choice of b_{u}. □

Corollary 4.18. The Hilbert series of R factors as

η(R, t) = ∏_{u∈L} η(V_{[≥u]}/I_{[≥u]}, t)

□

The importance of Corollary 4.18 becomes clearer with the following theorem.

Theorem 4.19. For each u ∈ L, the algebra V_{[≥u]}/I_{[≥u]}#H_{0} is a |u|ℤgraded Hopf algebra, where the grading is induced by that of R#H_{0}. Equivalently, V_{[≥u]}/I_{[≥u]} is a |u|ℤ-graded braided Hopf algebra in _{H_{0}}^{H_{0}}YD. Moreover, V_{[≥u]}/I_{[≥u]} is generated by V_{[u]}/(I_{[≥u]} ∩ V_{[u]}) and it projects onto the Nichols algebra B(V_{[u]}/(I_{[≥u]} ∩ V_{[u]})). The quotient

η(V_{[≥u]}/I_{[≥u]}, t) / η(B(V_{[u]}/(I_{[≥u]} ∩ V_{[u]})), t^{|u|})

is a power series with nonnegative integer coefficients.

Proof. Since V_{[≥u]}/I_{[≥u]} is graded and its degree 0 part is k, in order to show the first statement it is sufficient to prove that V_{[≥u]}/I_{[≥u]}#H_{0} is a bialgebra (see [Tak71]). This follows from Proposition 2.1, by taking A = R#H_{0}, B = V_{[≥u]}, and I = I_{[≥u]}. It remains to show that Equations (2.2) hold in this case. Indeed, it suffices to prove this for generators of B and I, and [ ]-letters satisfy (2.2) thanks to Proposition 4.6.

By the definition of V_{[≥u]} and I_{[≥u]}, V_{[≥u]}/I_{[≥u]} is generated as an algebra by the space V_{[u]}/(I_{[≥u]} ∩ V_{[u]}). Further, V_{[≥u]}/I_{[≥u]} can be considered as the quotient of T (V_{[u]}/(I_{[≥u]}∩V_{[u]})) by a graded Hopf ideal consisting of elements of degree ≥ 2. Since B(V_{[u]}/(I_{[≥u]} ∩ V_{[u]})) is the quotient of T (V_{[u]}/(I_{[≥u]}∩V_{[u]})) by the maximal Hopf ideal consisting of elements of degree ≥ 2, there exists a natural projection V_{[≥u]}/I_{[≥u]} → B(V_{[u]}/(I_{[≥u]} ∩ V_{[u]})). The last statement follows from Proposition 2.3 (see also Remark 2.5). □

Remark 4.20. In Theorem 4.19 it is necessary to put t^{|u|} as the variable of the Hilbert series of the Nichols algebra, since in B(V_{[u]}/(I_{[≥u]} ∩V_{[u]})) the elements of V_{[u]}/(I_{[≥u]} ∩ V_{[u]}) are considered to be in degree 1.

Open Problems 4.21.

(1) Assume that R is a Nichols algebra. Are the graded Hopf algebras V_{[≥u]}/I_{[≥u]} appearing in Theorem 4.19 again Nichols algebras? This is true in the case where char k = 0, H_{0} is the group algebra of an abelian group, and R is finite dimensional, by Kharchenko’s PBW theorem. More generally, if R has a finite number of PBW generators, the statement follows by using the Weyl groupoid.

(2) Generalize Theorems 4.12 and 4.19 to a more general setting which covers also Ufer’s PBW basis.

(3) Is it possible to generalize results of this paper to arbitrary (say finite dimensional) non-semisimple Hopf algebras?

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Matías Graña, Depto de Matemática - FCEyN, Universidad de Buenos Aires, Pab I - Ciudad Universitaria (1428) Buenos Aires - Argentina

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

István Heckenberger, Mathematisches Institut, Universität Leipzig, Augustusplatz 10-11, 04109 Leipzig, Germany.

E-mail address: heckenberger@math.uni-leipzig.de