Contemporary Mathematics

Pointed indecomposable Hopf algebras over Coxeter groups

Alexander Milinski and Hans-Jürgen Schneider

1. Introduction

Every pointed Hopf algebra can be decomposed as a crossed product of its link indecomposable component containing 1 (a Hopf subalgebra) and a group algebra [M2]. This means that the structure theory of pointed Hopf algebras can be reduced to the link-indecomposable case. However, the structure of such link-indecomposable Hopf algebras A, in particular in the case when A is finitedimensional and the group of its group-like elements G(A) is non-abelian, is not known. S. Westreich has shown that if A is a pointed, link-indecomposable, and quasitriangular Hopf algebra over a field of characteristic zero, then the group G(A) of its group-like elements is abelian [W]. Moreover it seems that all finitedimensional, link-indecomposable Hopf algebras which have appeared in the literature so far do have an abelian group of group-likes.

In this paper we present a general construction which associates to any Coxeter group W a pointed link-indecomposable Hopf algebra with group-like elements W (over an arbitrary base field 𝕜). If W is one of the symmetric or dihedral groups D_{3}(= S_{3}), D_{4}, S_{4} or S_{5}, the Hopf algebra is finite-dimensional (see Section 6). We do not know whether our construction gives a finite-dimensional Hopf algebra whenever the Coxeter group is finite.

In Section 4 we show that link-indecomposability is an infinitesimal property in the following sense. Let A be a pointed Hopf algebra with coradical 𝕜G, G = G(A) the group-like elements of A. As in [AS1] we consider the associated graded Hopf algebra gr(A), where the grading is defined by the coradical filtration. The coinvariant elements R of gr(A) with respect to the projection onto the degree zero part gr(A)(0) = 𝕜G form a Hopf algebra in the braided category of Yetter-Drinfeld modules over 𝕜G (see [M1] for the notions of braided categories and Yetter-Drinfeld modules). The primitive elements V = P (R) of R are a Yetter-Drinfeld submodule. Then A is link-indecomposable if and only if the elements g ∈ G with non-zero homogeneous component V_{g} generate the group G (Lemma 4.1). In such case the Yetter-Drinfeld module V is called link-indecomposable. Now the subalgebra of R generated by V is a Yetter-Drinfeld Hopf algebra which only depends on V.

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

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

Key words and phrases. Pointed Hopf algebras, link-indecomposable, Yetter-Drinfeld modules, Coxeter groups.

Following [AS2] we call it the Nichols algebra 𝔅(V), since Nichols was the first who considered Hopf algebras of this type in [N].

By a theorem of Radford [R] the smash-product of any Yetter-Drinfeld Hopf algebra with the group-algebra gives a usual Hopf algebra. Thus the main case to study are Nichols algebras of link-indecomposable Yetter-Drinfeld modules V over the group-algebra 𝕜G of a group G. By Proposition 4.2, there exists a finitedimensional link-indecomposable Hopf algebra A with G(A) = G if and only if there is a link-indecomposable Yetter-Drinfeld module V over G such that 𝔅(V) is finite-dimensional and G is finite.

The algebra 𝔅(V) is defined as a certain universal quotient of the tensor algebra T (V). It is one of the main difficulties that the relations are not known explicitly. And if they are known, it can be very hard to decide whether they define a finitedimensional quotient algebra.

In section 2 we review the general theory of Nichols algebras [N], [Wo], also called quantum symmetric algebras [Ro], or bitensor algebras [Sb], in a way suitable for this paper. In Section 5 we construct a class of Yetter-Drinfeld modules over Coxeter groups and study the associated Nichols algebra 𝔅(V) (Theorems 5.7, 5.8). The relations of 𝔅(V) reflect the relations between the generators of the Coxeter group. In section 3 we note an abstract splitting result for braided Hopf algebras which shows in many cases that 𝔅(V) is free over 𝔅(V′) for a subspace V′ ⊂ V which is a Yetter-Drinfeld module over a subgroup, for example a parabolic subgroup of the Coxeter group. As a consequence, the Hilbert series of 𝔅(V′) divides the Hilbert series of 𝔅(V).

In the last section we consider some examples of the general construction. We show the finiteness results already mentioned, and we note that the quadratic algebras E_{n} introduced in [FK] to describe the cohomolgy of the flag variety are in fact braided Hopf algebras over the symmetric groups. Hence the conjectured Poincaré-duality about the dimensions of the homogeneous components of E_{n} in [FK] follows from a general fact about braided Hopf algebras in [N]. As a corollary of the freeness result in section 3, we conclude that E_{n+1} is free over E_{n}, in particular that the Hilbert series of E_{n} divides the Hilbert series of E_{n+1}.

After submission of this paper, the second author learned from C. Procesi that freeness of E_{n+1} over E_{n} was also shown in the preprint [FP] by different methods using explicitly the defining relations. The second author wishes to thank Procesi for stimulating discussions on the subject of this paper.

Both authors would like to thank D. Fischman and S. Westreich whose sketch of a finite-dimensional, link-indecomposable, pointed Hopf algebra H with group S_{3} of group-like elements several years ago motivated the work presented in this paper.

2. Nichols algebras

The ground-field will be denoted by 𝕜, and ⊗ = ⊗_{𝕜}. Let G be a group. Recall that a Yetter-Drinfeld module V over 𝕜G is a G-graded vector space V = ⨁_{g∈G} V_{g}, which is a G-module such that g · V_{h} ⊂ V_{ghg^{−1}} for all g, h ∈ G. The category _{G}^{G}YD of 𝕜G-Yetter-Drinfeld modules is braided: For V, W ∈ _{G}^{G}YD the braiding c : V ⊗ W → W ⊗ V is defined by c(v ⊗ w) = (g · w) ⊗ v, v ∈ V_{g}, w ∈ W. The G-gradation and G-action on V ⊗ W are defined in the usual way by (V ⊗ W)_{g} = ⨁_{ab=g} V_{a} ⊗ W_{b}, and g · (v ⊗ w) = g · v ⊗ g · w, g ∈ G, v ∈ V, w ∈ W.

Let V be a Yetter-Drinfeld module over G, and let T (V) = ⨁_{n≥0} T (V)(n) denote the tensor algebra of the vector space V, that is T (V)(n) = V^{⊗n} and the multiplication μ is defined as tensor product. We consider T (V) as an algebra in _{G}^{G}YD in the obvious way: g · (v_{1} ⊗ ⋯ ⊗ v_{n}) := (g · v_{1}) ⊗ ⋯ ⊗ (g · v_{n}), g ∈ G, v_{1}, …, v_{n} ∈ V; if v_{i} ∈ V_{g_{i}}, 1 ≤ i ≤ n, then v_{1} ⊗ ⋯ ⊗ v_{n} ∈ V_{g}, where g = g_{1} … g_{n}. There is a unique coalgebra structure on T (V), such that v ∈ P (T (V)) for all v ∈ V and T (V) is a bialgebra in _{G}^{G}YD, that is (μ ⊗ μ)(1 ⊗ c ⊗ 1)(Δ ⊗ Δ) = Δμ, where c is the braiding of the Yetter-Drinfeld module T (V) as defined above. For example we have Δ(vw) = 1 ⊗ vw + v ⊗ w + (g · w) ⊗ v + vw ⊗ 1 for v ∈ V_{g}, w ∈ V. We note that the canonical filtration of T (V) is a coalgebra filtration. Therefore the coradical of the coalgebra T (V) equals 𝕜, and T (V) and all its coalgebra quotients are pointed irreducible coalgebras [M1, 5.3.4, 5.3.5]. In particular this implies that the identity of T (V) is convolution invertible by [M1, 5.2.10]. Hence T (V) and all its bialgebra quotients in _{G}^{G}YD are Hopf algebras in _{G}^{G}YD. Note that T (V) is a G × ℕ-graded vector space as each T (V)(n) has a basis consisting of G-homogenous elements.

We consider the set S̃ of all ideals and coideals I of T (V) which are generated as ideals by ℕ-homogeneous elements of degree ≥ 2. Let S be the subset of all I ∈ S̃ which are Yetter-Drinfeld submodules of T (V), that is stable under the action of G and generated by G×ℕ-homogeneous elements. Both sets are closed under addition and contain the zero ideal. Hence the sums I(V) = ∑_{I∈S} I and Ĩ(V) = ∑_{J∈S̃} J are the largest elements in S and S̃.

For all I ∈ S, T (V)/I is a graded Hopf algebra in _{G}^{G}YD with components 𝕜 and V in degree 0 and 1. In particular, V ⊂ P (T (V)/I).

Lemma 2.1. (1) Let I ∈ S. Then I = I(V) if and only if V = P (T (V)/I).

(2) I(V) = Ĩ(V).

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

Now assume that V = P (T (V)/I). Then the surjective map T (V)/I → T (V)/I(V) of irreducible coalgebras is injective by [M1, 5.3.3] since by assumption it is injective on the primitive elements. Hence I = I(V).

(2) As in (1) we see that T (V)/I(V) → T (V)/Ĩ(V) is bijective since V = P (T (V)/I(V)) by (1).

As in [AS1], we call 𝔅(V) := T (V)/I(V) the Nichols algebra of V ∈ _{G}^{G}YD. More generally, if U is a braided subspace of V ∈ _{G}^{G}YD, that is a subspace such that c(U ⊗ U) ⊂ U ⊗ U, where c is the braiding of V, we define 𝔅(U) := T (U)/Ĩ(U).

Remark 2.2. 1) By Lemma 2.1, 𝔅(V) as an algebra and coalgebra only depends on the braiding of V. The relations in each degree of I(V) can be described by the action of the braid group (see for example [Sb, 2.9, 2.12]). However this does not mean that the relations are known explicitly.

2) Let R = ⨁_{n≥0} R(n) be a graded Hopf algebra in _{G}^{G}YD with R(0) = 𝕜1 and R(1) = P (R) = V. Then by Lemma 2.1, 𝔅(V) is the subalgebra 𝕜[V] of R generated by V, since 𝕜[V] is of the form T (V)/I with primitive elements V.

3) Let U be a braided subspace of V ∈ _{G}^{G}YD. Then the canonical map 𝔅(U) → 𝔅(V) is injective since it is injective on the primitive elements by Lemma 2.1.

4) Let R = ⨁_{n≥0} R(n) be a finite-dimensional graded Hopf algebra in _{G}^{G}YD with R(0) = 𝕜1 which is generated as an algebra by R(1), and let x_{1}, …, x_{θ} be a basis of R(1). Then there is a non-zero monomial Λ = x_{i_{1}} ⋯ x_{i_{N}}, 1 ≤ i_{1}, …, i_{N} ≤ θ, of maximal length N. Hence x_{i}Λ = 0 = Λx_{i} for all 1 ≤ i ≤ θ, and Λ is a left and right integral of R. Since R is a Frobenius algebra [FMS], the space of left (or right) integrals is one-dimensional. Hence R = ⨁_{n=0}^{N} R(n), and R(N) = 𝕜Λ. Since R is a graded coalgebra, the linear function λ : R → 𝕜 with λ(Λ) = 1 and λ(R(n)) = 0 for all 0 ≤ n ≤ N − 1 is a two-sided integral of R^{∗}. Therefore for all 0 ≤ i ≤ N,

R(i) ⊗ R(N − i) → 𝕜, x ⊗ y ↦ λ(xy),

is a non-degenerate pairing (cf. [N, 1.5]). In particular, dimR(i) = dimR(N − i).

The crucial property V = P (B(V)) can be characterized as follows. For a graded coalgebra R = ⨁_{n≥0} R(n) with projections π_{n} : R → R(n), n ≥ 0, we denote by

Δ_{i,j} : R(i + j) ⟶^{Δ} R ⊗ R ⟶^{π_{i}⊗π_{j}} R(i) ⊗ R(j), i, j ≥ 0,

the (i, j)-th component of the comultiplication Δ. More generally,

Δ_{i_{1},...,i_{k}} : R(i_{1} + ⋯ + i_{k}) → R(i_{1}) ⊗ ⋯ ⊗ R(i_{k}), i_{1}, …, i_{k} ≥ 0,

denotes the (i_{1}, …, i_{k})-th component of the (k − 1)-fold iteration of Δ.

Lemma 2.3. Let R = ⨁_{n≥0} R(n) be a graded coalgebra with group-like element 1 and R(0) = 𝕜1. Then the following are equivalent:

(1) P(R) = R(1).

(2) For all n ≥ 2, Δ_{1,...,1} : R(n) → R(1)^{⊗n} is injective.

(3) For all i, j ≥ 0, Δ_{i,j} : R(i + j) → R(i) ⊗ R(j) is injective.

(4) For all n ≥ 2, Δ_{n−1,1} : R(n) → R(n − 1) ⊗ R(1) is injective.

Proof. Let x ∈ R(n), n ≥ 2, and Δ(x) = ∑_{i=0}^{n} y_{i} where y_{i} = Δ_{i,n−i}(x) ∈ R(i) ⊗ R(n − i) for all 0 ≤ i ≤ n. For all m ≥ 1, let Φ_{m} be the composition Φ_{m} : R ⟶^{Δ^{m−1}} R^{⊗m} ⟶^{π_{1}^{⊗m}} R(1)^{⊗m}. Hence Δ_{1,...,1} is the restriction of Φ_{m} to R(m). Then for all 1 ≤ i ≤ n − 1,

Φ_{n}(x) = (Φ_{i} ⊗ Φ_{n−i})(Δ(x)) = (Φ_{i} ⊗ Φ_{n−i})(y_{i}).

This formula shows that (1) ⇒ (2) by induction on n (as in [AS2, Lemma 5.5]), and also (2) ⇒ (3).

(3) ⇒ (4) is trivial, and (4) ⇒ (1) is clear since by (4), R(n) ∩ P (R) = 0 for all n ≥ 2.

All the previous results hold more generally for Yetter-Drinfeld modules over Hopf algebras with bijective antipode instead of group algebras. We now recall an important tool introduced by Nichols [N, 3.3] to deal with 𝔅(V) over group algebras without knowing the relations explicitly. Let V ∈ _{G}^{G}YD be of finite dimension θ. We choose a basis x_{i} ∈ V_{g_{i}} with g_{i} ∈ G, 1 ≤ i ≤ θ, of G-homogeneous elements. Let I ∈ S and R = T (V)/I. Then R is a graded Hopf algebra in _{G}^{G}YD with R(0) = 𝕜1 and R(1) = V. For all 1 ≤ i ≤ θ let σ_{i} : R → R be the algebra automorphism given by the action of g_{i}.

Proposition 2.4. Let R, x_{i}, σ_{i}, 1 ≤ i ≤ θ as above.

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

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

Proof. 1) For all 1 ≤ i ≤ θ, let D_{i}(1) = 0, and define for all x ∈ R(n), n ≥ 1, elements D_{i}(x) ∈ R(n − 1) by

Δ_{n−1,1}(x) = ∑_{i=1}^{θ} D_{i}(x) ⊗ x_{i}.

Then for all n, m ≥ 1, x ∈ R(n), y ∈ R(m),

Δ(xy) = Δ(x)Δ(y) = (x ⊗ 1 + D_{i}(x) ⊗ x_{i} + …)(y ⊗ 1 + D_{i}(y) ⊗ x_{i} + …)
= xy ⊗ 1 + xD_{i}(y) ⊗ x_{i} + D_{i}(x)(g_{i} · y) ⊗ x_{i} + …

by the definition of the braided multiplication in R ⊗ R. Hence D_{i}(xy) = xD_{i}(y) + D_{i}(x)σ_{i}(y). Since the elements x_{j} are primitive, D_{i}(x_{j}) = δ_{ij} for all i, j, and D_{i} is uniquely determined by this equation since R is generated by x_{1}, …, x_{θ}.

2) By Lemma 2.1, I = I(V) if and only if P (R) = R(1). Hence the claim follows from Lemma 2.3, since ⋂_{i=1}^{θ} ker(D_{i}) = 𝕜1 if and only if Δ_{n−1,1} is injective for all n ≥ 2.

For later use we note

Lemma 2.5. Let V, x_{i}, D_{i}, 1 ≤ i ≤ θ, be as in Proposition 2.4.

1) For all 1 ≤ i ≤ θ, let 𝔅(V)^{(i)} be the subalgebra of 𝔅(V) generated by all x_{j}, 1 ≤ i ≤ θ, j ≠ i. Assume a, b ∈ 𝔅(V)^{(i)} with a + bx_{i} = 0. Then a = 0 and b = 0.

2) Let 1 ≤ i_{1}, …, i_{n} ≤ θ, n ≥ 1, and i_{k} ≠ i_{l} for all k ≠ l. Then x_{i_{1}}x_{i_{2}} … x_{i_{n}} ≠ 0 in 𝔅(V).

Proof. To prove 1) note that D_{i}(𝔅(V)^{(i)}) = 0. Hence b = D_{i}(a + bx_{i}) = 0.

2) follows from 1) by induction on n.

3. Non-categorical splitting of Yetter-Drinfeld modules

In this section we use a braided version of the fundamental theorem on Hopf modules to prove freeness of certain extensions of Yetter-Drinfeld Hopf algebras.

Let L be a Hopf algebra with bijective antipode, H a Hopf algebra in _{L}^{L}YD (see [M1, 10.6.10] for the general definition) and V a right H-module and a right H-comodule with structure maps μ_{V} : V ⊗ H → V and ρ_{V} : V → V ⊗ H. Then V is a right H-Hopf module, if the braided Hopf module axiom holds, that is if ρ_{V} is a right H-linear map where the H-module structure of V ⊗ H is (μ_{V} ⊗ μ)(id ⊗ c ⊗ id)(id ⊗ id ⊗ Δ). Here, μ and Δ denote the multiplication and comultiplication of H. Note that we do not assume that V is a Yetter-Drinfeld module.

Lemma 3.1. Let L be a Hopf algebra with bijective antipode, H a Hopf algebra in _{L}^{L}YD, and V a right H-Hopf module. Define V^{coH} = {v ∈ V | ρ_{V} (v) = v ⊗ 1}. Then

V^{coH} ⊗ H → V, v ⊗ h ↦ vh,

is bijective.

Proof. This is shown as for usual Hopf algebras (see [M1, 1.9.4]).

Theorem 3.2. Let L′ and L be Hopf algebras with bijective antipode, and α : L′ → L a Hopf algebra map. Let R be a bialgebra in _{L}^{L}YD, R′ a Hopf algebra in _{L′}^{L′}YD, and i : R′ → R, ϕ : R → R′ algebra and coalgebra maps such that ϕi = id. Assume that i is an L′-linear map, and that i, ϕ are L-colinear, where R is a left L′-module and R′ a left L-comodule by restriction via α. Define K = R^{coϕ} = {r ∈ R | (id ⊗ ϕ)(Δ(r)) = r ⊗ 1}. Then the multiplication map

K ⊗ R′ → R, x ⊗ y ↦ xi(y),

is bijective, and K is a subalgebra of R with Δ(K) ⊂ R ⊗ K.

Proof. We consider R as a right R′-module by restriction via i, and as a right R′-comodule with structure map Δ̄ : R ⟶^{Δ} R ⊗ R ⟶^{id ⊗ϕ} R ⊗ R′. Here we used that i is an algebra map and ϕ is a coalgebra map. Then one checks that R is a right R′-Hopf module using that i is a coalgebra map and L′-linear, ϕ is an algebra map and L-colinear, and ϕi = id. Hence the multiplication map K ⊗ R′ → R is bijective by 3.1. Moreover it follows from the definition of K that Δ(K) ⊂ R ⊗ K. To see that K is a subalgebra of R, one checks that (id ⊗ i)Δ̄ : R → R ⊗ R is an algebra map, where the algebra structure on R ⊗ R is given by the braiding of R, using that i, ϕ are L-colinear algebra maps.

Theorem 3.2 is true in a more general version which only uses the braidings c : R ⊗ R → R ⊗ R and c′ : R′ ⊗ R′ → R′ ⊗ R′ of R and R′. It suffices to assume that i, ϕ are algebra and coalgebra maps with ϕi = id, which satisfy the following identities: (i ⊗ id)c′(ϕ ⊗ id) = (id ⊗ ϕ)c(id ⊗ i), and c(iϕ ⊗ id) = (id ⊗ iϕ)c.

If X = ⨁_{n=0}^{∞} X_{n} is a graded vector space with finite-dimensional components X_{n}, n ≥ 0, the Hilbert series of X is the formal power series P_{X}(t) = ∑_{i=0}^{∞} dim(X_{n})t^{n}.

Corollary 3.3. Let L be a Hopf algebra with bijective antipode, V ∈ _{L}^{L}YD, and L′ ⊂ L a Hopf subalgebra with bijective antipode, and V′ ⊂ V a subspace such that L′V′ ⊂ V′, δ(V′) ⊂ L′ ⊗V′, where δ : V → L⊗V is the coaction of V. Then V is a left L′-module by restriction and V′ is a left L-comodule via δ. Let φ : V → V′ be an L′-linear and L-colinear map with φ | V′ = id.

Then V′ is a Yetter-Drinfeld module over L′, 𝔅(V′) is a graded subalgebra of 𝔅(V), and 𝔅(V) is a free right 𝔅(V′)-module. If V is finite-dimensional, then P_{𝔅(V′)} divides P_{𝔅(V)}.

Proof. The tensor algebras T (V) and T (V′) are Hopf algebras in _{L}^{L}YD and _{L′}^{L′}YD as in the first section where the elements of V and V′ are primitive. The natural map ϕ : T (V) → T (V′) given by φ : V → V′ is an algebra and a coalgebra map with ϕi = id, where i : T (V′) → T (V) denotes the inclusion map. Moreover, ϕ is L-colinear and L′-linear, where T (V′) is an L-comodule and T (V) is an L′-module via the inclusion map α : L′ → L. Therefore ϕ induces an L-colinear algebra and coalgebra map 𝔅(V) → 𝔅(V′) which will again be denoted by ϕ. Since 𝔅(V) and 𝔅(V′) are graded algebras and coalgebras and ϕ is a graded algebra map (of degree 0), K = 𝔅(V)^{coϕ} is a graded subalgebra of 𝔅(V). Hence we conclude from Theorem 3.2 with R = 𝔅(V) and R′ = 𝔅(V′) that K ⊗ 𝔅(V′) ≅ 𝔅(V). In particular, 𝔅(V) is free over 𝔅(V′), and if V is finite-dimensional, P_{𝔅(V′)}P_{K} = P_{𝔅(V)}.

When 𝔅(V′) is finite-dimensional, Corollary 3.3 also follows from [G1, Theorem 3.11] with a completely different proof. But the argument in Theorem 3.2 is quite general and does not depend on Nichols algebras.

An example of the situation of Corollary 3.3 will be given in Section 5. The next remark shows that the existence of the splitting map φ in Corollary 3.3 is a rather mild assumption.

Remark 3.4. Let G be a group, G′ ⊂ G a finite subgroup such that the characteristic of 𝕜 does not divide the order of G′. Let V ∈ _{G}^{G}YD and V′ ⊂ V a subspace such that G′V′ ⊂ V′, and δ(V′) ⊂ 𝕜G′ ⊗ V′, where δ : V → 𝕜G ⊗ V is the coaction of V. Then there is always a 𝕜G′-linear and 𝕜G-colinear map φ : V → V′ with φ | V′ = id. Hence we are in the situation of Corollary 3.3 with L′ = 𝕜G′ and L = 𝕜G.

Proof. Since 𝕜G is cosemisimple, there is a 𝕜G-colinear map f : V → V′ with f | V′ = id. Define φ : V → V′ by φ(v) = \frac{1}{ord(G′)} ∑_{g∈G′} gf(g^{−1}v) for all v ∈ V. Then φ is a G′-linear map with φ | V′ = id, and φ is 𝕜G-colinear, since for all h ∈ G and v ∈ V_{h}, by the Yetter-Drinfeld condition g^{−1}v ∈ V_{g^{−1}hg} and gf(g^{−1}v) ∈ V′_{h} for all g ∈ G′.

4. Link-indecompsable pointed Hopf algebras Let A be a pointed Hopf algebra with group-like elements G = G(A). For

g, h ∈ G, P_{g,h}(A) = {x ∈ A | Δ(x) = g ⊗ x + x ⊗ 1} will denote the (g, h)-primitive elements of A. P_{g,h}(A) is called non-trivial if 𝕜(g − h) ⫋ P_{g,h}(A).

We recall the definition of the quiver Γ_{A} of A in [M2]. The vertices of Γ_{A} are the elements of G; for g, h ∈ G, there exists an arrow from h to g if P_{g,h}(A) is non-trivial.

Note that P_{g,h}(A) is non-trivial if and only if P_{gh^{−1},1}(A) or P_{h^{−1}g,1}(A) is nontrivial. Then g, h ∈ G are connected (or g ∼ h) if they are in the same connected component of Γ_{A} as an undirected graph. The Hopf algebra A is called linkindecomposable if any two elements in G are connected.

We call Supp(A) = {g ∈ G | P_{g,1}(A) is non-trivial} the support of A. If V = ⨁_{g∈G} V_{g} is a left 𝕜G-comodule with homogeneous components V_{g}, then we define Supp(V) = {g ∈ G | V_{g} ≠ 0}.

Let A_{0} = 𝕜G ⊂ A_{1} ⊂ … be the coradical filtration of A and gr(A) = A_{0} ⊕ A_{1}/A_{0} ⊕ A_{2}/A_{1} ⊕ … the associated graded Hopf algebra. Then

gr(A) ≅ R#𝕜G,

where R = (gr(A)^{coA_{0}} is the diagram of A ; R is a graded Hopf algebra in _{G}^{G}YD and P (R) is in _{G}^{G}YD (see [AS1]).

Lemma 4.1. Let A be a pointed Hopf algebra with G = G(A) and diagram R. Define N = {g ∈ G | g ∼ 1}, and V = P (R). Then

1) N is a normal subgroup of G and the connected components of Γ_{A} are the cosets of N.

2) N is generated by Supp(A).

3) Supp(A) = Supp(V).

Proof. 1) is noted in [M2, 3.2]. By definition there exists an arrow between g, h ∈ G if and only if gh^{−1} or hg^{−1} ∈ Supp(A). Suppose g ∼ 1. Then there are g_{1}, …, g_{n} ∈ G, n ≥ 1, g_{1} = g, g_{n} = 1 with g_{i}g_{i+1}^{−1} ∈ Supp(A) or (Supp(A))^{−1} for all i, and g = g_{1}g_{2}^{−1}g_{2}g_{3}^{−1} … g_{n−1}g_{n}^{−1} is generated by Supp(A). Conversely, if u_{1}, …, u_{n−1} ∈ Supp(A) or (Supp(A))^{−1}, n ≥ 1, and g = u_{1} … u_{n−1}, define g_{1} = g, and g_{i+1} by u_{i} = g_{i}g_{i+1}^{−1}, 1 ≤ i ≤ n − 1. Then g is connected to g_{n} = 1. This shows 2). For 3), note that P (R)#𝕜G ≅ A_{1}/A_{0} by [AS1, Lemma 2.4]. For all g, h ∈ G choose a decomposition of vector spaces P_{g,h}(A) = 𝕜(g − h) ⊕ P_{g,h}(A)′. Then by the Theorem of Taft and Wilson [M1, 5.4.1], ⨁_{g,h∈G} P_{g,h}(A)′ ≅ A_{1}/A_{0}. Thus for all g ∈ G we obtain an isomorphism P (R)_{g}#1 ≅ P_{g,1}(A)/𝕜(1 − g).

As suggested by the previous Lemma, we say a Yetter-Drinfeld module V over G is link-indecomposable if G is generated by Supp(V).

Proposition 4.2. Let G be a finite group. Then the following are equivalent:

(1) There exists a finite-dimensional pointed and link-indecomposable Hopf algebra A such that G ≅ G(A).

(2) There exists a link-indecomposable Yetter-Drinfeld module V over G such that 𝔅(V) is finite-dimensional.

Proof. Assume (1) and let R be the diagram of A. Then V = P (R) is linkindecomposable by Lemma 4.1. By 2.2 2), 𝕜[V] ≅ 𝔅(V) is finite-dimensional.

Assume (2). Define R = 𝔅(V), and let A = R#𝕜G be the Radford biproduct. It is well-known and not difficult to see that P_{g,1}(A) = P (R)_{g}#1 + 𝕜(g − 1) (if x ∈ P_{g,1}(A), write (ε_{R} ⊗ id)(x) = α(g − 1), α ∈ 𝕜, and check that x − α(g − 1) ∈ P (A)_{g}^{co𝕜G}). Hence Supp(A) = Supp(P (R)) = Supp(V) generates G, and A is linkindecomposable by 4.1.

5. Link-indecomposable pointed Hopf algebras over Coxeter groups

Let W be a group and T ⊂ W a subset such that for all g ∈ W, t ∈ T, gtg^{−1} ∈ T. Let χ : W × T → 𝕜 \ {0} be a function such that for all g, h ∈ W and t ∈ T,

(5.1)  χ(1, t) = 1,
(5.2)  χ(gh, t) = χ(g, hth^{−1})χ(h, t).

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

(5.3)  gx_{t} = χ(g, t)x_{gtg^{−1}},
(5.4)  δ(x_{t}) = t ⊗ x_{t}

for all g ∈ W, t ∈ T.

Conversely, if the function χ defines a Yetter-Drinfeld module on the vector space V by (5.3), (5.4), then χ satisfies (5.1), (5.2).

Note that the braiding c of V (W, T, χ) is determined by

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

hence by the values of χ on T × T.

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

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

Definition 5.1. Let (W, S) be a Coxeter system, and T = {gsg^{−1} | g ∈ W, s ∈ S}. Define χ : W × T → 𝕜 \ {0} by

χ(g, t) = (−1)^{l(g)} for all g ∈ W, t ∈ T. (5.5)

Let V = V (W, T, χ) ∈ _{W}^{W} YD. The associated Hopf algebras will be denoted by R(W, S) = 𝔅(V) and A(W, S) := 𝔅(V)#𝕜W.

By Proposition 4.2, A(W, S) is a pointed link-indecomposable Hopf algebra with group-like elements W.

Remark 5.2. We assume W, T, χ as in the beginning of this section, and define V = V (W, T, χ).

1) By definition, T is a union of conjugacy classes of W. Let t_{1} ∈ T and T_{1} the conjugacy class containing t_{1}. Let s_{i}, 1 ≤ i ≤ θ, be representatives of the right residue classes of the centralizer C_{W} (t_{1}), W = ⋃_{i=1}^{θ} s_{i}C_{W} (t_{1}), and define t_{i} := s_{i}t_{1}s_{i}^{−1}, 1 ≤ i ≤ θ. Then W/C_{W} (t_{1}) → T_{1}, s̄_{i} ↦ t_{i}, 1 ≤ i ≤ θ, is bijective. Let χ̂ : C_{W} (t_{1}) → 𝕜 \ {0} be the group homomorphism defined by χ̂(g) = χ(g, t_{1}) for all g ∈ C_{W} (t_{1}). We now change the basis to

x_{i} := χ(s_{i}, t_{1})x_{t_{i}}, 1 ≤ i ≤ θ. (5.6)

For all g ∈ W, 1 ≤ i ≤ θ, define 1 ≤ j ≤ θ by t_{j} = gt_{i}g^{−1}. Then s_{j}^{−1}gs_{i} ∈ C_{W} (t_{1}), and

(5.7)  χ(g, t_{i}) = χ̂(s_{j}^{−1}gs_{i})χ(s_{j}, t_{1})χ(s_{i}, t_{1})^{−1},
(5.8)  gx_{i} = η(g, t_{i})x_{j} where η(g, t_{i}) = χ̂(s_{j}^{−1}gs_{i}).

Thus η : W × T_{1} → 𝕜 \ {0} satisfies (5.1), (5.2), and ⨁_{1≤i≤θ} 𝕜x_{t_{i}} = V (W, T_{1}, η) with respect to the new basis y_{t_{i}} = x_{i}, 1 ≤ i ≤ θ.

Proof. From t_{j} = gt_{i}g^{−1} and the definition of t_{i}, t_{j} it follows that s_{j}^{−1}gs_{i} ∈ C_{W} (t_{1}). Hence by (5.2), χ(s_{j}(s_{j}^{−1}gs_{i}), t_{1}) = χ(s_{j}, t_{1})χ(s_{j}^{−1}gs_{i}, t_{1}). On the other hand by (5.2), χ(gs_{i}, t_{1}) = χ(g, t_{i})χ(s_{i}, t_{1}), and (5.7) and (5.8) follow.

2) Conversely, any character χ̂ : C_{W} (t_{1}) → 𝕜\{0} defines a Yetter-Drinfeld module ⨁_{t∈T_{1}} 𝕜x_{t} over W by (5.8) and (5.6), and by (5.7) a function χ : W × T_{1} → 𝕜 \ {0} satisfying (5.1) and (5.2) where χ(s_{i}, t_{1}), 1 ≤ i ≤ θ, can be arbitrary non-zero scalars. This construction is a special case of the well-known description of the simple Yetter-Drinfeld modules over a finite group (cf. [CR], [AG]).

3) Assume that 𝔅(V) is finite-dimensional. Then for all t ∈ T there is a natural number n ≥ 2 such that 1 + χ(t, t) + ⋯ + (χ(t, t))^{n−1} = 0, and if t^{2} = 1 and char(𝕜) = 0, then χ(t, t) = −1. Note that in 1), χ(t_{i}, t_{i}) = χ̂(t_{i}) = η(t_{i}, t_{i}) for all 1 ≤ i ≤ θ.

Proof. Let D_{t}, t ∈ T be the skew-derivations corresponding to the basis x_{t}, t ∈ T. Let t ∈ T and q = χ(t, t). It follows easily by induction on n that D_{t}(x_{t}^{n}) = (1 + q + ⋯ + q^{n−1})x_{t}^{n−1} for all n ≥ 1. Since x_{t}^{n} ∈ 𝔅(V)(n), x_{t}^{n} must be zero for large n, hence 1 + q + ⋯ + q^{n−1} = 0, if x_{t}^{n} = 0 and x_{t}^{n−1} ≠ 0. In particular, if t^{2} = 1, then (χ(t, t))^{2} = 1 by (5.1), (5.2), and χ(t, t) = −1 if char(𝕜) = 0.

We describe the possible functions χ up to base change as in Remark 5.2, 1) for the symmetric and the dihedral groups assuming χ(t, t) = −1 for all t ∈ T as in Remark 5.2, 3).

Example 5.3. Let W = S_{n} be the symmetric group of permutations of n ≥ 2 elements, and S = {(12), (23), …, (n − 1, n)}. Then (W, S) is a Coxeter system, and T is the set of all transpositions in S_{n}. Up to base change as described in 5.2 1), there are exactly two functions χ (which coincide if char(𝕜) = 2) satisfying (5.1), (5.2) and χ(t, t) = −1 for all t ∈ T. Either χ is defined as in (5.5) or for all g ∈ S_{n}, 1 ≤ i < j ≤ n,

χ(g, (ij)) = \begin{cases}1 & , if g(i) < g(j), \\ −1 & , if g(i) > g(j). \end{cases} (5.9)

Proof. Let t_{1} = (12). The centralizer of t_{1} is {g ∈ S_{n} | {g(1), g(2)} = {1, 2}} = ⟨(34), (45), …, (n − 1, n)⟩ ∪ ⟨(34), (45), …, (n − 1, n)⟩(12). Hence the characters χ̂ : C_{S_{n}}(t_{1}) → 𝕜\{0} are χ_{ε_{1},ε_{2}}, with ε_{1}, ε_{2} ∈ {1, −1}, and χ_{ε_{1},ε_{2}}((ij)) = ε_{1} if 3 ≤ i < j ≤ n, and χ_{ε_{1},ε_{2}}((12)) = ε_{2}. Moreover ε_{2} = −1 since we assumed η(t_{1}, t_{1}) = −1. The characters χ_{−1,−1} and χ_{1,−1} define the functions χ in (5.5) and (5.9).

Example 5.4. Let W = D_{m} = ⟨t, t′⟩, m ≥ 1, be the dihedral group of order 2m, where t, t′ ∈ T are generators of order 2 and tt′ is of order m. Let S = {t, t′}. Then (W, S) is a Coxeter sytem. Define t_{i} = (tt′)^{i−1}t for all i ∈ ℤ. Then T = {t_{i} | 1 ≤ i ≤ m}, and for all i, j, t_{i}t_{j}t_{i} = t_{2i−j}, and t_{i} = t_{j} if and only if i ≡ j mod m. If m is odd, then T is one conjugacy class. For m = 2l even, T is the union of the two conjugacy classes T_{0} = {t_{2k} | 1 ≤ k ≤ l} and T_{1} = {t_{2k−1} | 1 ≤ k ≤ l}, and z = (tt′)^{l} is a central element of order 2.

Let χ be a function satisfying (5.1), (5.2) and χ(t, t) = −1 for all t ∈ T.

a) If m is odd, then the centralizer of t_{1} is C_{W} (t_{1}) = {1, t_{1}}. We choose representatives s_{i}, 1 ≤ i ≤ m, of the residue classes of C_{W} (t_{1}) by s_{i} = t_{k} with 2k − 1 ≡ i mod m, hence s_{i}t_{1}s_{i}^{−1} = t_{i}. In (5.8) we then have η(t_{k}, t_{i}) = −1 for all k, i, that is we are in case (5.5).

Proof. By (5.8) with g = t_{k}, η(t_{k}, t_{i}) = χ(s_{j}^{−1}t_{k}s_{i}, t_{1}) = −1, since χ(t_{1}, t_{1}) = −1 by assumption, and s_{j}^{−1}t_{k}s_{i} = t_{1} (since s_{j}^{−1}t_{k}s_{i} ≠ 1 as a product of three reflections).

b) If m = 2l is even, we pick t_{0} in the conjugacy class T_{0}. Then C_{W} (t_{0}) = {1, t_{0}, z, t_{0}z}. We choose representatives s_{i}, 1 ≤ i ≤ l, of the residue classes of C_{W} (t_{0}) by s_{i} = t_{i}, 1 ≤ i ≤ l. Then s_{i}t_{0}s_{i}^{−1} = t_{2i},1 ≤ i ≤ l, and for clarity we denote the elements of T_{0} by τ_{i} = t_{2i}, 1 ≤ i ≤ l. Let g = t_{k}, 1 ≤ k ≤ m, and 1 ≤ i ≤ l. Define 1 ≤ j ≤ l by τ_{j} = gτ_{i}g^{−1}, that is t_{2j} = t_{2(k−i)}, and j ≡ k − i mod l. In (5.6) we write x_{2i} instead of x_{i} and extend this definition by x_{2i} = χ(t_{ϕ(i)}, t_{0})x_{2i}, where ϕ(i) is the integer with 1 ≤ ϕ(i) ≤ l and ϕ(i) ≡ i mod l. We then have t_{k}x_{2i} = η(t_{k}, t_{2i})x_{2(k−i)}. Define ε_{0} ∈ {1, −1} by χ(z, t_{0}) = −ε_{0}. Then for all 1 ≤ k ≤ m, 1 ≤ i ≤ l,

(5.10) η(t_{k}, t_{2i}) = \begin{cases}−1 & , if 1 ≤ k − i ≤ l, \\ ε_{0} & otherwise. \end{cases}

Proof. By (5.8),

η(t_{k}, τ_{i}) = χ̂_{0}((tt′)^{j−k+i}t′), since s_{j}^{−1}t_{k}s_{i} = t_{j}t_{k}t_{i} = (tt′)^{j−k+i}t′.

The homomorphism

χ̂_{0} : {1, t_{0}, z, t_{0}z} → 𝕜 \ {0}

is given by χ̂_{0}(t_{0}) = χ(t_{0}, t_{0}) = −1 and χ̂_{0}(z) = χ(z, t_{0}) = −ε_{0}. If 1 ≤ k − i ≤ l, then j = k−i, and (tt′)^{j−k+i}t′ = t′ = t_{0}, and η(t_{k}, τ_{i}) = χ̂_{0}(t_{0}) = −1. If k−i = j±l, then (tt′)^{j−k+i}t′ = zt_{0}, and η(t_{k}, τ_{i}) = χ̂_{0}(zt_{0}) = ε_{0}.

c) If m = 2l, we pick t_{1} in the conjugacy class T_{1}. Then C_{W} (t_{1}) = {1, t_{1}, z, t_{1}z}. We choose representatives s_{i}, 1 ≤ i ≤ l, by s_{i} = t_{i}. Then s_{i}t_{1}s_{i}^{−1} = t_{2i−1}, and for clarity we set τ_{i} = t_{2i−1}, 1 ≤ i ≤ l. Let g = t_{k}, 1 ≤ k ≤ m, and 1 ≤ i ≤ l. Define 1 ≤ j ≤ l by τ_{j} = gτ_{i}g^{−1}, that is t_{2j−1} = t_{2(k−i+1)−1}, or j ≡ k − i + 1 mod l. In (5.6) we write x_{2i−1} instead of x_{i}. As in b) we define x_{2i−1} = χ(t_{ϕ(i)}, t_{1})x_{t_{2i−1}} for all integers i. We then have t_{k}x_{2i−1} = η(t_{k}, t_{2i−1})x_{2(k−i+1)−1}. Define ε_{1} ∈ {1, −1} by χ(z, t_{1}) = −ε_{1}. Then for all 1 ≤ k ≤ m, 1 ≤ i ≤ l,

(5.11) η(t_{k}, t_{2i−1}) = \begin{cases}−1 & , if 1 ≤ 1 + k − i ≤ l, \\ ε_{1} & otherwise. \end{cases}

Proof. By (5.8) with g = t_{k}, η(t_{k}, τ_{i}) = χ̂_{1}((tt′)^{j−k+i}t′) as before. The homomorphism

χ̂_{1} : {1, t_{1}, z, t_{1}z} → 𝕜 \ {0}

is given by χ̂_{1}(t_{1}) = −1, χ̂_{1}(z) = −ε_{1}. If 1 ≤ k − i + 1 ≤ l, then j = k − i + 1, and (tt′)^{j−k+i}t′ = t = t_{1}, hence η(t_{k}, τ_{i}) = χ̂_{1}(t_{1}) = −1. If k − i + 1 = j ± l, then (tt′)^{j−k+i}t′ = z(tt′)t′ = zt_{1}, hence η(t_{k}, τ_{i}) = ε_{1}.

d) Conversely, if m = 2l is even and ε_{0}, ε_{1} are arbitrary elements in {1, −1}, then (5.10), (5.11) define a function η satisfying (5.1), (5.2) and η(t, t) = −1 for all t ∈ T.

Note that the case ε_{0} = −1 = ε_{1} is the example (5.5) for W = D_{m}.

Lemma 5.5. Assume W, T and χ as in the beginning of this section and let V = V (W, T, χ). Then the antipode S : 𝔅(V) → 𝔅(V) can be explicitly computed as follows. Let s_{1}, …, s_{q} ∈ T and define t_{1}, …, t_{q} ∈ T by t_{i} = (s_{1} ⋯ s_{i−1})s_{i}(s_{1} ⋯ s_{i−1})^{−1}, 1 ≤ i ≤ q. Then

S(x_{s_{1}}x_{s_{2}} ⋯ x_{s_{q}}) = ε(s_{1}, s_{2}, …, s_{q})x_{t_{q}} ⋯ x_{t_{2}}x_{t_{1}},

where ε(s_{1}, …, s_{q}) = (−1)^{q} ∏_{i=2}^{q} χ(s_{1}s_{2} ⋯ s_{i−1}, s_{i}).

Proof. For all x, y ∈ 𝔅(V), S(xy) = S(x_{−1} · y)S(x_{0}). Hence
S(x_{s_{1}} ⋯ x_{s_{q−1}}x_{s_{q}}) = S((s_{1} ⋯ s_{q−1}) · x_{s_{q}})S(x_{s_{1}} ⋯ x_{s_{q−1}})
= −χ(s_{1} ⋯ s_{q−1}, s_{q})x_{t_{q}}S(x_{s_{1}} ⋯ x_{s_{q−1}}),

and the claim follows by induction on q.

We now describe some of the relations of 𝔅(V). In general we do not know whether these are defining relations. We begin with the dihedral group of order 4. In the terminology of [AS2], the Hopf algebras are then quantum linear spaces or of type A_{2}. For completeness we give a proof in the context of this paper.

Lemma 5.6. Let W = D_{2} be generated by t = t_{1}, t′ = t_{2} as in Example 5.4 with T = {t_{1}, t_{2}}. Let ε_{1}, ε_{0} ∈ {1, −1} and define χ : W × T → 𝕜 \ {0} by (5.10) and (5.11). Let V = V (D_{2}, T, η), and x = x_{t}, y = x_{t′}. Thus the action is given by t · x = −x, t′ · y = −y and t′ · x = ε_{1}x, t · y = ε_{0}y.

1) Assume ε_{1} = ε_{0} and let ε = ε_{1} = ε_{0}. Then

(5.12)  x^{2} = 0, y^{2} = 0,
(5.13)  xy = εyx,

are defining relations of 𝔅(V), and 1, x, y, xy is a basis of 𝔅(V).

2) Assume ε_{1} = −ε_{0}, and char(𝕜) ≠ 2. Then

(5.14)  x^{2} = 0, y^{2} = 0,
(5.15)  xyxy + yxyx = 0,

are defining relations of 𝔅(V), and 1, x, y, xy, yx, xyx, yxy, xyxy is a basis of 𝔅(V).

Proof. Let D_{1}, D_{2} be the skew-derivations in Proposition 2.4 with respect to the basis x, y of V. By Proposition 2.4 it suffices to show that the identities hold after applying D_{i} for all i.

The relations (5.12) and (5.14) follow from η(t, t) = η(t′, t′) = −1, since D_{1}(x^{2}) = x + η(t, t)x = 0, D_{2}(x^{2}) = 0, and by the same argument D_{i}(y) = 0, i = 1, 2.

We compute D_{1}(xy) = η(t_{1}, t_{2})y = ε_{0}y, D_{2}(xy) = x, D_{1}(yx) = y and D_{2}(yx) = η(t_{2}, t_{1})x = ε_{1}x. Then D_{1}((xy)^{2}) = −yxy, D_{2}((xy)^{2}) = xyx, and D_{1}((yx)^{2}) = yxy, D_{2}((yx)^{2}) = −xyx, since x^{2} = 0, y^{2} = 0. This implies (5.13) and (5.15).

It is clear from the relations that the above elements generate the vector space 𝔅(V). Hence linear independency of these elements (of the same degree in the variables x, y) implies that the relations are defining relations. This is clear in case 1) since D_{1}(xy) ≠ 0, hence xy ≠ 0 in 𝔅(V), and follows easily in case 2) by using the values of the derivations D_{i} which we have just computed. For example, suppose αxy + βyx = 0 in case 2), where α, β ∈ 𝕜. Then 0 = D_{1}(αxy + βyx) = (αε_{0} + β)y, and 0 = D_{2}(αxy + βyx) = (α + βε_{1})x, hence α = β = 0 since char(𝕜 ≠ 2.

Theorem 5.7. Let W be a group, T ⊂ W a subset which is stable under conjugation with elements in W, χ : W × T → 𝕜 \ {0} a function satisfying (5.1), (5.2) and V = V (W, T, χ) ∈ _{W}^{W} YD. Let t, t′ ∈ T be elements of order 2 and assume that tt′ is of order m ≥ 1.

Then the subgroup D ⊂ W generated by t, t′ is the dihedral group of order 2m. For all i ∈ ℤ, define t_{i} = (tt′)^{i−1}t as in Example 5.4. Then t_{i} = t_{j} if and only if i ≡ j mod m, and the sequence t_{1}, …, t_{m} is defined inductively by

t_{0} = t′, t_{1} = t and t_{i}t_{i−1}t_{i} = t_{i+1}, 1 ≤ i ≤ m − 1.

Assume χ(t_{i}, t_{i}) = −1 for all i.

If m = 2l + 1 is odd, define x_{i} = χ(t_{k}, t_{1})x_{t_{i}} for all i ∈ ℤ where k ∈ ℤ with i ≡ 2k − 1 mod m. If m = 2l is even, define x_{2i−1} = χ(t_{ϕ(i)}, t_{1})x_{t_{2i−1}} and x_{2i} = χ(t_{ϕ(i)}, t_{1})x_{t_{2i}} for all i ∈ ℤ, where ϕ(i) is the natural number with 1 ≤ ϕ(i) ≤ l and ϕ(i) ≡ i mod l.

Then x_{a} = x_{b} if a, b ∈ ℤ, a ≡ b mod m, and for all i, j ∈ ℤ,

t_{i}x_{j} = η(t_{i}, t_{j})x_{2i−j}

where the function η is defined by (5.5) if m is odd, and by (5.10), (5.11) if m = 2l is even, and where ε_{0}, ε_{1} ∈ {1, −1} are given by χ((tt′)^{l}, t_{i}) = −ε_{i}, 0 ≤ i ≤ 1.

The following relations are satisfied in 𝔅(V):

(5.16)  x_{i}^{2} = 0 for all i.

If m is even, assume ε_{0} = ε_{1}, and define ε = ε_{0} = ε_{1}. Then

(5.17)  x_{1}x_{2} + x_{2}x_{3} + ⋯ + x_{m−1}x_{m} = εx_{m}x_{1},
(5.18)  x_{1} ⋯ x_{m} = εx_{m} ⋯ x_{1},
(5.19)  (x_{m}x_{1})^{l} = ε(x_{1}x_{m})^{l}.

If m = 2l + 1 is odd, then

(5.20)  x_{1}x_{2} + x_{2}x_{3} + ⋯ + x_{m−1}x_{m} + x_{m}x_{1} = 0,
(5.21)  x_{1}x_{2} ⋯ x_{m} = x_{m}x_{m−1} ⋯ x_{1},
(5.22)  (x_{m}x_{1})^{l}x_{m} = (x_{1}x_{m})^{l}x_{1}.

Assume m = 2l and l are even and ε_{0} = ε_{1} = −1.

Let x_{±} = ∑_{i=1}^{l}(±1)^{i−1}x_{2i−1}, and y_{±} = ∑_{i=1}^{l}(±1)^{i−1}x_{2i}.

If x = x_{+}, y = y_{−} or x = x_{−}, y = y_{+}, then

(5.23)  x^{2} = 0, y^{2} = 0,
(5.24)  xyxy + yxyx = 0,

and if char(𝕜) ≠ 2, then the subalgebra generated by x, y has dimension 8.

Proof. Let U ⊂ V be the braided subspace with basis x_{1}, …, x_{m}. Then U is a Yetter-Drinfeld module over D and U = V (D, {t_{1}, …, t_{m}}, η) with respect to the basis y_{t_{i}} = x_{i}, 1 ≤ i ≤ m, as shown in Example 5.4. Since the obvious natural map 𝔅(U) → 𝔅(V) is an algebra map, we can assume that U = V.

For all 1 ≤ i ≤ m, let D_{i} be the skew-derivation defined in Proposition 2.4 with respect to x_{i} ∈ V_{t_{i}}. As in the previous Lemma it suffices to show that the identities hold after applying D_{i} for all i.

The relations (5.16) follow from η(t_{i}, t_{i}) = χ(t_{i}, t_{i}) = −1 as in the proof of (5.12).

To show (5.17) and (5.20), let a = x_{1}x_{2} + x_{2}x_{3} + ⋯ + x_{m−1}x_{m} − εx_{m}x_{1}, where ε = −1 when m is odd. Then

D_{1}(a) = t_{1} · x_{2} − εx_{m} = (η(t_{1}, t_{2}) − ε)x_{0},
D_{2}(a) = x_{1} + t_{2} · x_{3} = (η(t_{2}, t_{3}) + 1)x_{1},
…
D_{m−1}(a) = x_{m−2} + t_{m−1} · x_{m} = (η(t_{m−1}, t_{m}) + 1)x_{m−2},
D_{m}(a) = x_{m−1} − εt_{m} · x_{1} = (−εη(t_{m}, t_{1} + 1)x_{m−1}.

Assume m is even. From (5.10) and (5.11) we see that η(t_{i}, t_{i+1}) = −1 for all 2 ≤ i ≤ m − 1, and η(t_{1}, t_{2}) = ε_{0}, η(t_{m}, t_{1}) = ε_{1}. Thus D_{i}(a) = 0 for all i since ε_{0} = ε_{1}.

If m is odd, then D_{i}(a) = 0 for all i, since −1 = ε = η(t_{k}, t_{i}) for all k, i.

To prove (5.18) and (5.21), we first note that

D_{1}(x_{1}(x_{2} ⋯ x_{m})) = t_{1} · (x_{2} ⋯ x_{m}) = η(t_{1}, t_{2}) … η(t_{1}, t_{m})x_{m}x_{m−1} ⋯ x_{2}.

On the other hand, D_{1}((x_{m} ⋯ x_{2})x_{1}) = x_{m} ⋯ x_{2}.

Assume m = 2l. By (5.10) and (5.11) we compute η(t_{1}, t_{2}) … η(t_{1}, t_{m}) = ε_{0}^{l}ε_{1}^{l−1} = ε. Hence D_{1}(x_{1} ⋯ x_{m}) = D_{1}(εx_{m} ⋯ x_{1}). Similarly we obtain the same equality for D_{m} since η(t_{m}, t_{m−1}) … η(t_{m}, t_{1}) = ε_{0}^{l−1}ε_{1}^{l} = ε.

If m is odd, then η(t_{k}, t_{i}) = −1 for all k, i, and D_{i}(x_{1} ⋯ x_{m}) = D_{i}(x_{m} ⋯ x_{1}) for i = 1 and i = m.

Finally, if 2 ≤ i ≤ m − 1, then

D_{i}((x_{1} ⋯ x_{i−1}x_{i})(x_{i+1} ⋯ x_{m})) =
D_{i}((x_{1} ⋯ x_{i−1})x_{i}) t_{i} · (x_{i+1} ⋯ x_{m}) =
(x_{1} ⋯ x_{i−1}) t_{i} · (x_{i+1} ⋯ x_{m}) = 0,

since t_{i} · (x_{i+1} ⋯ x_{m}) begins with the element x_{i−1}, and x_{i−1}^{2} = 0 by (5.16).

Similarly we have D_{i}(x_{m} ⋯ x_{1}) = (x_{m} ⋯ x_{i+1}) t_{i} · (x_{i−1} ⋯ x_{m}) = 0, since x_{i+1}^{2} = 0.

To show (5.19), consider the sequence (s_{1}, …, s_{2l}) = (t, t′, t, t′, …, t, t′). Then s_{1} = t = t_{1}, s_{1}s_{2}s_{1}^{−1} = tt′t = t_{2}, s_{1}s_{2}s_{3}(s_{1}s_{2})^{−1} = tt′tt′t = t_{3}, …. Thus t_{1}, …, t_{2l} coincide with the t_{i}’s corresponding to s_{1}, …, s_{2l} in Lemma 5.5. Hence by Lemma 5.5,

S((x_{1}x_{m})^{l}) = ε(t, t′, …, t, t′)x_{m}x_{m−1} ⋯ x_{1}. (5.25)

Changing the roles of t, t′ we obtain in the same way

S((x_{m}x_{1})^{l}) = ε(t′, t, …, t′, t)x_{1}x_{2} ⋯ x_{m}, (5.26) since (t′t)^{m−i}t′ = (tt′)^{i}t′ = (tt′)^{i−1}t = t_{i} for all i. By the definition in Lemma 5.5

ε(t, t′, t, t′, …, t, t′) = ∏_{i=2}^{2l} η(s_{1} ⋯ s_{i−1}, s_{i}) = η(t, t′)η(tt′, t) ⋯ η(tt′ ⋯ tt′t, t′).

From (5.10), (5.11),

η((tt′)^{i−1}t, t′) = η(t_{i}, t_{2l}) = ε_{0} for all 1 ≤ i ≤ l,

and

η((t′t)^{i}, t) = η(t_{i+1}t, t) = η(t_{i+1}, t)η(t, t) = 1 for all 1 ≤ i ≤ l − 1.

Hence we have

ε(t, t′, t, t′, …, t, t′) = ε_{0}^{l}, and similarly ε(t′, t, t′, t, …, t′, t) = ε_{1}^{l}.

Therefore we obtain from (5.25), (5.26) and (2) the equality

S((x_{1}x_{m})^{l}) = S(ε(x_{m}x_{1})^{l}),

and the claim follows from the bijectivity of the antipode S.

To prove (5.22), we proceed as in the previous case. Let (s_{1}, …, s_{2l+1}) be the sequence (t, t′, t, t′, …, t, t′, t). Then t_{1}, …, t_{2l+1} coincide with the t_{i}’s corresponding to s_{1}, …, s_{2l+1} in Lemma 5.5. By Lemma 5.5,

S((x_{1}x_{m})^{l}x_{1}) = ε(t, t′, …, t)x_{m}x_{m−1} ⋯ x_{1},

where ε(t, t′, …, t) = −η(t, t′)η(tt′, t) ⋯ η(tt′ ⋯ tt′, t) = (−1)^{1+l}. By changing t, t′ we obtain

S((x_{m}x_{1})^{l}x_{m}) = (−1)^{1+l}x_{1}x_{2} ⋯ x_{m},

since (t′t)^{i−1}t′ = t_{m−i+1}, 1 ≤ i ≤ 2l + 1. Therefore the claim follows from (5.21).

The last part of the Theorem follows from the fact that the elements x, y span a braided subspace of type A_{2}. Write x = ∑_{i=1}^{l} α^{i−1}x_{2i−1} and y = ∑_{i=1}^{l} β^{i−1}x_{2i}, where α, β ∈ {±1} and αβ = −1. We compute the braiding c with respect to x, y:

c(x ⊗ y) = ∑_{i,j} α^{i−1}β^{j−1}c(x_{2i−1} ⊗ x_{2j}),

where the indices i, j are in ℤ/(l) (since l is even). Since

c(x_{2i−1} ⊗ x_{2j}) = −x_{2(2i−j−1)} ⊗ x_{2i−1}

, we substitute k = 2i − j − 1 and obtain

c(x ⊗ y) = − ∑_{i,k} α^{i−1}β^{k}x_{2k} ⊗ x_{2i−1} = b_{12}(y ⊗ x) ,where b_{12} = −β.

In the same way one shows c(y ⊗ x) = b_{21}(x ⊗ y), c(x ⊗ x) = b_{11}(x ⊗ x), c(y ⊗ y) = b_{22}(y ⊗ y), where b_{21} = −α, b_{11} = b_{22} = −1. Hence the braiding on x, y is the braiding of case 2) in Lemma 5.6. Since the relations only depend on the braiding (see Remark 2.2, 1)), the claim follows from Lemma 5.6.

In the next theorem we assume that for all elements t, t′ such that the product tt′ has even order 2l, we have ε_{0} = ε_{1} in the sense of Theorem 5.7. Note that this condition is satisfied in the example (5.5) where ε_{0} = ε_{1} = −1, and also for the symmetric group in (5.9) where ε_{0} = ε_{1} = 1 if l = 1.

Theorem 5.8. Let (W, S) be a Coxeter system, T the set of all W -conjugates of elements in S, χ : W × T → 𝕜 \ {0} a function satisfying (5.1), (5.2), V = V (W, T, χ), and R = 𝔅(V). Assume for all t, t′ ∈ T such that the product tt′ has even order 2l that χ((tt′)^{l}, t) = χ((tt′)^{l}, t′). For all t ∈ T let R^{(t)} be the subalgebra of R generated by all x_{t′}, t′ ∈ T, t′ ≠ t. Then the generators x_{t}, t ∈ T of R satisfy the relations (5.16) – (5.22), and

1) For all t ∈ T and x ∈ R there are uniquely determined elements a, b ∈ R^{(t)} such that x = a+bx_{t}. If x is a W -homogeneous element in R_{g}, g ∈ W, then a ∈ R_{g} and b ∈ R_{gt}.

2) For all g ∈ W, the W -homogeneous components R_{1} and R_{g} are isomorphic as vector spaces.

3) For all g ∈ W, choose a reduced representation g = s_{1} ⋯ s_{q}, s_{1}, ⋯, s_{q} ∈ S, of g, and define

x_{g} = x_{s_{1}} ⋯ x_{s_{q}}.

Then x_{g} is a non-zero element in R_{g}.

4) For all g = t_{1} ⋯ t_{p}, where t_{1}, ⋯, t_{p} are elements of S,

(5.27) x_{t_{1}} ⋯ x_{t_{p}} = \begin{cases}±x_{g}, & if (t_{1}, ⋯, t_{p}) is a reduced representation of g \\ 0, & otherwise. \end{cases}

Proof. 1) To prove the existence of the representation we have to show for any s_{1}, …, s_{q} ∈ T and x = x_{s_{1}}, ⋯ x_{s_{q}} the existence of W -homogeneous elements a, b ∈ R^{(t)} with x = a + bx_{t}. This follows easily by induction on the length q from (5.17), (5.20), since these relations are W -homogeneous since all their monomials have the same degree t_{i−1}t_{i} = t_{i}t_{i+1}. The uniqueness of the representation is clear from Lemma 2.5 1).

2) Fix an element t ∈ T. By 1) there is well-defined linear isomorphism

ϕ_{t} : R → R, with ϕ_{t}(a + bx_{t}) = b + ax_{t} for all a, b ∈ R^{(t)}.

For any g ∈ W, the restriction of ϕ_{t} defines an isomorphism R_{g} ≅ R_{gt}, and 2) follows, since T generates the group W.

3) Define the sequence (t_{1}, …, t_{q}) of elements in T with respect to (s_{1}, ⋯, s_{q}) as in Lemma 5.5. Since (s_{1}, …, s_{q}) is a reduced representation of g, the elements t_{i} are pairwise distinct by [B, 1.4, Lemma 2]. Hence S(x_{g}) ≠ 0 by Lemma 5.5 and Lemma 2.5 2).

4) We consider M = 𝔅(V)/{1, −1} as a multiplicative monoid, and denote by π : 𝔅(V) → M the quotient map. We claim that there exists a map Φ : W → M satisfying Φ(g) = π(x_{s_{1}} ⋯ x_{s_{q}}) whenever (s_{1}, …, s_{q}) is a reduced representation of g. By [B, 1.6, Proposition 5] it is sufficient to show that a(s, s′) = a(s′, s) for all s, s′ ∈ S, s ≠ s′, where

(5.28) a(s, s′) = \begin{cases}π(x_{s}x_{s′})^{l}, & if ord(ss′) = 2l, \\ π((x_{s}x_{s′})^{l}x_{s}), & if ord(ss′) = 2l + 1. \end{cases}

This follows directly from (5.19), (5.22).

Finally we have to show that x_{t_{1}} ⋯ x_{t_{p}} = 0 if the length of g is < p. We may assume that (t_{2}, ⋯, t_{p}) is reduced. Then by the exchange property in [B, 1.4, Lemma 3] there exists 2 ≤ j ≤ p such that t_{1}t_{2} ⋯ t_{j−1} = t_{2} ⋯ t_{j−1}t_{j}. Hence (x_{t_{1}} ⋯ x_{t_{j−1}})(x_{t_{j}} ⋯ x_{t_{p}}) = ±(x_{t_{2}} ⋯ x_{t_{j}})(x_{t_{j}} ⋯ x_{t_{p}}) = 0 by what we have already shown, since (x_{t_{j}})^{2} = 0 by (5.16).

The following Corollary is an immediate consequence of Theorem 5.8.

Corollary 5.9. Assume the situation of Theorem 5.8. Then the subalgebra of R generated by all x_{s}, s ∈ S has the 𝕜-basis x_{g}, g ∈ W.

If R is finite-dimensional, then W is finite and dim(R) = ord(W)dim(R_{1}). Here, R_{1} is the W -homogeneous component of R of the unit element in W.

Finally we note that the results of Section 3 give some information about the Hopf algebras considered in this Section, in particular about the parabolic subalgebras of the examples in Theorem (5.8).

Proposition 5.10. Let W be a finite group, T ⊂ W a subset closed under conjugation, and χ : W × T → 𝕜 \ {0} a function satisfying (5.1), (5.2). Let W′ ⊂ W be a subgroup, T′ ⊂ T a subset which is stable under conjugation with elements in W′, and let χ′ be the restriction of χ. Define V = V (W, T, χ) and V′ = V (W′, T′, χ′). Then

(1) 𝔅(V) is a free right 𝔅(V′)-module, and the Hilbert series P_{𝔅(V′)} divides P_{𝔅(V)}.

(2) As in Theorem (5.8) assume for all t, t′ ∈ T such that the product tt′ has even order 2l that χ((tt′)^{l}, t) = χ((tt′)^{l}, t′). Let 𝔅̃(V) resp. 𝔅̃(V′) be the tensor algebra T (V) resp. T (V′) modulo all the quadratic relations (5.16) and (5.17), (5.20). Then 𝔅̃(V) is a free right 𝔅̃(V′)-module, and P_{𝔅̃(V′)} divides P_{𝔅̃(V)}.

Proof. Define φ : V → V′ for all t ∈ T by φ(x_{t}) = x_{t}, if t ∈ T′, and φ(x_{t}) = 0, otherwise. Then φ is a 𝕜W -colinear map, and it is W′-linear, since for all g ∈ W′, gtg^{−1} ∈ T′ if and only if t ∈ T′. Hence (1) follows from Corollary 3.3.

To prove (2) let ϕ : T (V) → T (V′) be the map induced by φ. Since the quadratic relations define in fact primitive elements in the tensor algebras, 𝔅̃(V) and 𝔅̃(V′) are braided Hopf algebras. The map ϕ induces a map on the quotients since the quadratic relations are preserved. This is clear for (5.16), and is also true for (5.17), (5.20), since all the t_{i}, 1 ≤ i ≤ m, are in T′, if for one i both t_{i} and t_{i+1} are in T′. The claim now follows from Theorem 3.2 in the same way as Corollary 3.3.

6. Examples

We consider some special cases of Theorem 5.8.

Example 6.1. Let W = S_{n}, n ≥ 2, and T = {(ij) | 1 ≤ i < j ≤ n} the set of all transpositions. Define χ by (5.5) and let V = V (W, T, χ). Then the following relations hold in 𝔅(V) for all 1 ≤ i < j ≤ n, 1 ≤ k < l ≤ n:

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

Example 6.2. Let W = S_{n}, n ≥ 2, and T = {(ij) | 1 ≤ i < j ≤ n} the set of all transpositions. Define χ by (5.9) and let V = V (W, T, χ). Then the following relations hold in 𝔅(V) for all 1 ≤ i < j ≤ n, 1 ≤ k < l ≤ n:

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

The algebras generated by all x_{(ij)}, 1 ≤ i < j ≤ n, with the quadratic relations in the examples 6.1 and 6.2 are examples of the braided Hopf algebras 𝔅̃(V) in Corollary (5.10). 𝔅̃(V) in example 6.2 is the algebra E_{n} introduced in [FK] to describe the cohomology ring of the flag variety. We believe that indeed the quadratic relations in the examples 6.1 and 6.2 are defining relations for 𝔅(V), that is 𝔅̃(V) = 𝔅(V) in these cases. Since the algebras E_{n} are braided Hopf algebras we immediately obtain the

Corollary 6.3. Let n ≥ 1.

(1) E_{n+1} is a free module over E_{n}, and P_{E_{n}} divides P_{E_{n+1}}.

(2) If E_{n} is finite-dimensional with highest non-zero homogeneous component E_{n}(N) , then for all 0 ≤ i ≤ N, dimE_{n}(i) = dimE_{n}(N − i).

Proof. (1) follows from Corollary 5.10 (2) with W = S_{n+1} and W′ = {g ∈ W | g(n + 1) = n + 1} = S_{n}, and (2) is a special case of Remark 2.2 4).

The ‘Poincaré-duality” in 6.3 was a conjecture in [FK], and 6.3 (1) is also shown in [FP] by different methods.

Example 6.4. Let 𝔅(V_{n}) be the Hopf algebra of Example 6.1. If n = 3 or 4, then (6.1), (6.2) and (6.3) are defining relations of B(V_{n}), and dim𝔅(V_{3}) = 12, dim𝔅(V_{4}) = 24^{2}. In both cases the integral can be described in terms of the longest element in the Coxeter groups S_{3}, S_{4}.

Proof. 1) We first consider R = 𝔅(V_{3}). Let t_{1} = (12), t_{2} = (23), t_{3} = (13) and x_{i} = x_{t_{i}}, 1 ≤ i ≤ 3.. Then S = {t_{1}, t_{2}}, T = {t_{1}, t_{2}, t_{3}}. By Corollary 5.9 R^{(s_{3})} = 𝕜[x_{1}, x_{2}] has dimension ordS_{3} = 6. Thus dimR = 2dimR^{(s_{3})} by Theorem 5.8 1). On the other hand it is not difficult to check (using the diamond lemma) that the algebra with relations (6.1), (6.2) and (6.3) in this case has dimension 12. The longest element is s_{1}s_{2}s_{1}. Therefore x_{1}x_{2}x_{1}x_{3} is a non-zero monomial of maximal length, hence an integral.

2) Now let R = 𝔅(V_{4}). We denote s_{1} = (12), s_{2} = (23), s_{3} = (34), s_{4} = (24), s_{5} = (14), s_{6} = (13), and x_{i} = x_{s_{i}}, 1 ≤ i ≤ 6. Hence S = {s_{1}, s_{2}, s_{3}}, T = {s_{1}, …, s_{6}}. The relations are x_{i}^{2} = 0, for all 1 ≤ i ≤ 6, and

x_{1}x_{3} + x_{3}x_{1} = 0,
x_{2}x_{5} + x_{5}x_{2} = 0,
x_{4}x_{6} + x_{6}x_{4} = 0,
x_{1}x_{2} + x_{2}x_{6} + x_{6}x_{1} = 0, x_{2}x_{1} + x_{1}x_{6} + x_{6}x_{2} = 0,
x_{1}x_{4} + x_{4}x_{5} + x_{5}x_{1} = 0, x_{4}x_{1} + x_{1}x_{5} + x_{5}x_{4} = 0,
x_{2}x_{3} + x_{3}x_{4} + x_{4}x_{2} = 0, x_{3}x_{2} + x_{2}x_{4} + x_{4}x_{3} = 0,
x_{3}x_{5} + x_{5}x_{6} + x_{6}x_{3} = 0, x_{5}x_{3} + x_{3}x_{6} + x_{6}x_{5} = 0.

From these relations one can see that 𝔅(V_{4}) = 𝕜[x_{1}, x_{2}, x_{3}]·𝕜[x_{4}, x_{5}, x_{6}]. By Corollary 5.9, 𝕜[x_{1}, x_{2}, x_{3}] has dimension ordS_{4} = 24.

Let σ = (1243) in S_{4}. Then σs_{1}σ^{−1} = s_{4}, σs_{2}σ^{−1} = s_{5}, σs_{3}σ^{−1} = s_{6}. Hence the action of σ defines an algebra isomorphism σ· : 𝕜[x_{1}, x_{2}, x_{3}] → 𝕜[x_{4}, x_{5}, x_{6}]. Hence dim𝔅(V_{4}) ≤ 24^{2}. Using the method of the proof of Lemma 5.6, it is possible to show that dim𝔅(V_{4}) = 24^{2}. On the other hand it follows from the diamond lemma that the algebra with the above realations also has dimension 24^{2}.

The longest element is w_{0} = s_{1}s_{2}s_{1}s_{3}s_{2}s_{1}. Since σw_{0}σ^{−1} = s_{4}s_{5}s_{4}s_{6}s_{5}s_{4}, x_{1}x_{2}x_{1}x_{3}x_{2}x_{1}x_{4}x_{5}x_{4}x_{6}x_{5}x_{4} is a non-zero monomial of maximal length, hence an integral.

According to [FK, (2.8)] also E_{5} is finite-dimensional. However we do not know whether E_{5} = 𝔅(V_{5}).

Example 6.5. Assume char(𝕜) ≠ 2. We consider the dihedral group D_{4} in Example 5.4. Then T = {t_{1}, …, t_{4}}. Let χ be defined by (5.5), and V = V (D_{4}, T, χ). Then R = 𝔅(V) is generated by the elements x_{i} = x_{t_{i}}, 1 ≤ i ≤ 4 and defining relations are the quadratic relations x_{i}^{2} = 0, 1 ≤ i ≤ 4 and

x_{1}x_{3} + x_{3}x_{1} = 0,
x_{2}x_{4} + x_{4}x_{2} = 0,
x_{1}x_{2} + x_{2}x_{3} + x_{3}x_{4} + x_{4}x_{1} = 0,
x_{1}x_{4} + x_{4}x_{3} + x_{3}x_{2} + x_{2}x_{1} = 0,

and the relations of degree 4

xyxy + yxyx = 0, uvuv + vuvu = 0,

where x = x_{1} + x_{3}, y = x_{2} − x_{4}, u = x_{1} − x_{3}, v = x_{2} + x_{4}.

The dimension of R is 64, and the integral can be described in terms of the longest element of the Coxeter group D_{4}.

Proof. These relations are part of the relations of Theorem 5.7. Hence R is an epimorphic image of the algebra R̃ generated by x, y, u, v with the relations

xyxy + yxyx = 0, uvuv + vuvu = 0,

and x^{2} = 0, y^{2} = 0, u^{2} = 0, v^{2} = 0, xu+ux = 0, xv+vx = 0, yu−uy = 0, yv+vy = 0, which follow from the quadratic relations above. Using Lemma 5.6 it is easy to see that dimR̃ = 64.

On the other hand, dim𝕜[x_{1}, x_{2}] = ordD_{4} = 8, by Corollary 5.9. Since t_{1}t_{2}t_{1}t_{2}t_{1} = t_{3}, and t_{1}t_{2}t_{2}t_{2}t_{1} = t_{4}, the action of t_{1}t_{2} defines an isomorphism 𝕜[x_{1}, x_{2}] → 𝕜[x_{3}, x_{4}]. Again one can show similarly to the proof of Lemma 5.6 that 𝕜[x_{1}, x_{2}] · 𝕜[x_{3}, x_{4}] has dimension 8^{2}. Hence R = 𝕜[x_{1}, x_{2}] · 𝕜[x_{3}, x_{4}], and we have a monomial basis in the elements x_{i}.

The longest element in D_{4} is t_{1}t_{2}t_{1}t_{2}. Therefore x_{1}x_{2}x_{1}x_{2}x_{3}x_{4}x_{3}x_{4} is a nonzero integral.

In an earlier version of this paper we showed dimR = 64 in the last example in a different way. In addition to the quadratic relations, we proved the following relations: If y_{1} = x_{1}x_{2} + x_{4}x_{1}, y_{2} = x_{1}x_{4} + x_{2}x_{1}, then

y_{1}^{2} = 0, y_{2}^{2} = 0, y_{1}y_{2} + y_{2}y_{1} = 0.

It is not difficult to see that these relations define an algebra of dimension 64. Then Graña [G2, 5.2.1] noted that the braiding in the basis x, y, u, v of V reduces the computation of 𝔅(V) to the case of A_{2} in Lemma 5.6.

Remark 6.6. The examples we have constructed show that the symmetric groups S_{3}, S_{4} and S_{5} and the dihedral group D_{4} occur as the groups of group-like elements of finite-dimensional pointed and link-indecomposable Hopf algebras. In the other examples of link-indecomposable Hopf algebras over finite Coxeter groups we do not know whether the algebras 𝔅(V) are finite-dimensional.

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