The coradical of the dual of a lifting of a quantum plane

N. Andruskiewitsch ^{∗} Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba CIEM – CONICET (5000) Ciudad Universitaria, Córdoba, Argentina

email: andrus@mate.uncor.edu

and M. Beattie ^{†} Department of Mathematics and Computer Science Mount Allison University Sackville, NB, Canada E4L 1E6

email: mbeattie@mta.ca

Abstract

In this short note we compute the coradical of the dual A^{∗} of a lifting A of a quantum linear space in the case that A^{∗} has only trivial grouplike elements.

1 Introduction

The classification of finite dimensional Hopf algebras is known to be a difficult problem, for which only a few general techniques are available. One technique for studying Hopf algebras whose coradical is a sub Hopf algebra is the “lifting method” of Andruskiewitsch and Schneider. The simplest application of the lifting method is the description of all pointed Hopf algebras with finite abelian group of grouplikes Γ and with space V of (1, g) primitives, V ∈_{K[Γ]}^{K[Γ]} YD, a quantum linear space.

∗This work was partially supported by CONICET, Agencia Córdoba Ciencia, ANPCyT and Secyt (UNC).

†Research supported by NSERC.

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

A lifting of a quantum linear space is a pointed Hopf algebra A whose associated graded Hopf algebra is isomorphic to the Radford biproduct B(V)#K[Γ] where B(V) is the Nichols algebra of the quantum linear space V, and is a Hopf algebra in _{K[Γ]}^{K[Γ]}YD. Liftings of quantum linear spaces were constructed and completely described in [2], and also independently, using repeated Ore extensions, in [5].

Liftings of quantum linear spaces, although fairly easy to describe, have enough interesting structure to prove useful in various situations. They provided counterexamples to Kaplansky’s Tenth Conjecture [2], [5], [9]. Suppose K is an algebraically closed field of characteristic 0. It is shown in [2] or [6] that every pointed nonsemisimple Hopf algebra over K of dimension p^{3}, p prime, is a lifting of a quantum linear space. In [10] it is shown that every pointed Hopf algebra of index p^{2}, p a prime, is the lifting of a quantum line or a quantum linear space; this result generalizes [1, Theorem 1.7] and [8, Theorem 1.2]. In [7], it is shown that over K, every pointed nonsemisimple Hopf algebra of dimension 16 is a lifting of a quantum linear space.

However, although the structure of liftings of quantum linear spaces is well understood, the coradicals of the duals (and thus the irreducible representations) of these Hopf algebras have yet to be completely described. In [12], and more recently in [3], for V a quantum linear space of dimension 1 or 2, the coradicals of the duals of some liftings of B(V)#K[Γ] are described. In particular, if V has dimension 1, then the coradical of the dual A^{∗} of a lifting A of B(V)#K[Γ] is completely described. If V has dimension 2, then the coradical of the dual A^{∗} is described in some cases where A^{∗} has nontrivial grouplikes.

In this short note, we study some cases where V has dimension 2 and A^{∗} has only the trivial grouplike element and we explicitly construct the coradical of the dual of a Hopf algebra in the family of pointed Hopf algebras of dimension 81 = 3^{4} which provided one of the early counterexamples to Kaplansky’s Tenth Conjecture.

2 Preliminaries

Throughout, K will denote an algebraically closed field of characteristic 0, Γ a finite abelian group and _{Γ}^{Γ}YD =_{K[Γ]}^{K[Γ]} YD the category of Yetter-Drinfel’d modules over the group algebra K[Γ]. For V ∈_{Γ}^{Γ} YD, g ∈ Γ, χ ∈ Γ̂, we write V_{g}^{χ} for the set of v ∈ V with the action of Γ on v given by h → v = χ(h)v and the coaction by δ(v) = g ⊗ v. Since Γ is an abelian group, it is well-known that V = ⊕_{{g∈Γ \atop χ∈Γ̂}} V_{g}^{χ}.

Definition 2.1 V = ⊕_{i=1}^{t}Kv_{i} ∈_{Γ}^{Γ} YD with 0 ≠ v_{i} ∈ V_{g_{i}}^{χ_{i}} is called a quantum linear space if χ_{i}(g_{j})χ_{j}(g_{i}) = 1 for i ≠ j . As well, χ_{i}(g_{i}) is a primitive r_{i}th root of unity with 1 < r_{i} < ∞.

Recall that for V a quantum linear space as above, B(V) is the Nichols algebra for V, and the Radford biproduct, H = B(V)#K[Γ], is the coradically graded Hopf algebra generated by the (1, g_{i})-primitives v_{i}#1 (usually written just v_{i}) and the grouplike elements h = 1#h, with h ∈ Γ. Multiplication is given by hv_{i} = χ_{i}(h)v_{i}h and v_{i}v_{j} = χ_{j}(g_{i})v_{j}v_{i}. Also if χ_{i}(g_{i}) is a primitive r_{i}th root of unity, v_{i}^{r_{i}} = 0, and then dim B(V) = Π_{i=1}^{t}r_{i}.

Proposition 2.2 (see [2] or [5]) For V a quantum linear space with χ_{i}(g_{i}) a primitive r_{i}th root of unity, all liftings A of B(V)#K[Γ] are Hopf algebras generated by the grouplikes and by (1, g_{i})-primitives x_{i}, 1 ≤ i ≤ t where

hx_{i} = χ_{i}(h)x_{i}h;
x_{i}^{r_{i}} = α_{ii}(g_{i}^{r_{i}} − 1);
x_{i}x_{j} = χ_{j}(g_{i})x_{j}x_{i} + α_{ij}(g_{i}g_{j} − 1).

We may assume α_{ii} ∈ {0, 1} and then we have that

α_{ii} = 0 if g_{i}^{r_{i}} = 1 or χ_{i}^{r_{i}} ≠ ϵ;
α_{ij} = 0 if g_{i}g_{j} = 1 or χ_{i}χ_{j} ≠ ϵ.

Note that α_{ji} = −χ_{j}(g_{i})^{−1}α_{ij} = −χ_{i}(g_{j})α_{ij}. Thus the lifting A is described by a matrix A = (α_{ij}) with 0’s or 1’s on the diagonal and with α_{ji} = −χ_{i}(g_{j})α_{ij} for i ≠ j.

Throughout, we use the usual Sweedler notation for Hopf algebra computations. Also, we use the notation M^{c}(r, K) for an r × r matrix coalgebra over K and we call a basis e(i, j), 1 ≤ i, j ≤ r, a matrix coalgebra basis if △(e(i, j)) = ∑_{k=1}^{r} e(i, k) ⊗ e(k, j) and ϵ(e(i, j)) = δ_{i,j}.

3 The Main Theorem

Let A be a lifting of B(V)#K[Γ] with V = ⊕_{i=1}^{t}Kv_{i} ∈_{Γ}^{Γ} YD a quantum linear space and with χ_{i}(g_{i}) a primitive r_{i}th root of unity. Let x_{i} ∈ A be the lifting of v_{i}#1 to A, i.e. in A, x_{i} is (1, g_{i})-primitive and hx_{i} = χ_{i}(h)x_{i}h for all h ∈ Γ. Then A has a vector space basis hz with h ∈ Γ and z ∈ Z = {x_{1}^{m_{1}} … x_{t}^{m_{t}}|0 ≤ m_{i} ≤ r_{i} − 1}.

For γ ∈ Γ̂, let w(γ, x_{1}^{m_{1}} … x_{t}^{m_{t}}) ∈ A^{∗} be the map which takes hx_{1}^{m_{1}} … x_{t}^{m_{t}} to γ(h) and all other basis elements to 0, so that W = {w(γ, z)|γ ∈ Γ̂, z ∈ Z} is a vector space basis for A^{∗}. The map w(γ, 1) is usually just written γ. Multiplication in A^{∗} is given by the following lemma.

Lemma 3.1 [3, Lemma 3.1] Let A be a lifting of B(V)#K[Γ] with V a quantum linear space as above. Let γ, λ ∈ Γ̂, 0 ≤ m_{i} ≤ r_{i} − 1 and 1 ≤ i < j ≤ t.

(i) Then

w(γ, x_{i}^{m_{i}}) ∗ w(λ, x_{j}^{m_{j}}) = (γχ_{i}^{−m_{i}})(g_{j}^{m_{j}})w(γλ, x_{i}^{m_{i}}x_{j}^{m_{j}});
w(λ, x_{j}^{m_{j}}) ∗ w(γ, x_{i}^{m_{i}}) = λ(g_{i}^{m_{i}})w(γλ, x_{i}^{m_{i}}x_{j}^{m_{j}}).

In particular, w(χ_{i}, x_{i})w(χ_{j}, x_{j}) = w(χ_{i}χ_{j}, x_{i}x_{j}) = χ_{i}(g_{j})w(χ_{j}, x_{j})w(χ_{i}, x_{i}) for i < j.

(ii)

γ ∗ w(λ, x_{1}^{m_{1}} … x_{t}^{m_{t}}) = γ(g_{1}^{m_{1}} … g_{t}^{m_{t}})w(λγ, x_{1}^{m_{1}} … x_{t}^{m_{t}});
w(λ, x_{1}^{m_{1}} … x_{t}^{m_{t}}) ∗ γ = w(γλ, x_{1}^{m_{1}} … x_{t}^{m_{t}}).

In particular γ ∗ w(χ_{i}, x_{i}) = γ(g_{i})w(χ_{i}, x_{i}) ∗ γ = γ(g_{i})w(χ_{i}γ, x_{i}).

(iii) For 0 ≤ m_{i} ≤ r_{i} − 1 we have that

w(λ, x_{i}) ∗ w(γ, x_{i}^{m_{i}}) = λ(g_{i}^{m_{i}})(q^{m_{i}} + q^{m_{i}−1} + … 1)w(λγ, x_{i}^{m_{i}+1})

where q = χ_{i}(g_{i})^{−1}, a primitive r_{i}th root of 1, so that w(χ_{i}, x_{i})^{r_{i}} = 0 in A^{∗}.

The multiplication formulas above show that γ ∈ Γ̂ and the elements w(χ_{i}, x_{i}) generate A^{∗} as an algebra with γ ∗ w(χ_{i}, x_{i}) = γ(g_{i})w(χ_{i}, x_{i}) ∗ γ, w(χ_{i}, x_{i}) ∗ w(χ_{j}, x_{j}) = χ_{i}(g_{j})w(χ_{j}, x_{j}) ∗ w(χ_{i}, x_{i}) for i ≠ j, and w(χ_{i}, x_{i})^{r_{i}} = 0.

From now on, let V ∈_{Γ}^{Γ} YD, V = Kv_{1} ⊕ Kv_{2} be a quantum linear space of dimension 2, with v_{i} ∈ V_{g_{i}}^{χ_{i}} and with g_{1} = g, χ_{1} = χ, χ_{2} = χ^{m}. Also, for i = 1, 2,

1. χ_{i}(g_{i}) is a primitive rth root of unity for some odd integer r > 1.

2. χ_{i} has order r in the group Γ̂, so that (m, r) = 1.

3. g_{i}^{r} ≠ 1.

Let q denote the primitive rth root of unity χ(g). Let H = B(V)#KΓ and

let A be the nontrivial lifting of H with matrix A = [ \begin{matrix}1 & ν \\ −q^{−m}ν & 1 \end{matrix} ]. In A,

let x be the lifting of v_{1} and y the lifting of v_{2}. Then A is the pointed Hopf algebra generated by its group of grouplikes Γ and the skew-primitives x and y where for h ∈ Γ,

hx = χ(h)xh, and hy = χ^{m}(h)yh,  (1)
x^{r} = g^{r} − 1, and y^{r} = g_{2}^{r} − 1,  (2)
xy = χ^{m}(g)yx + ν(g_{1}g_{2} − 1).  (3)

If ν ≠ 0, then m = −1, i.e., χ_{2} = χ^{−1}.

As a K-space, A has basis {gz|g ∈ Γ, z ∈ Z} where Z = {y^{i}x^{j}|0 ≤ i, j ≤ r − 1}. Then as a K-space, A^{∗} has basis W = {w(γ, z)|γ ∈ Γ̂, z ∈ Z}.

Definition 3.2 C(1) is defined to be the sub Hopf algebra of A^{∗} of dimension r^{3} with basis {w(χ^{i}, z)|0 ≤ i ≤ r − 1, z ∈ Z}.

Note that Lemma 3.1 and the fact that △w(γ, y^{j}x^{i}) is a linear combination of terms of the form w(γ, y^{t}x^{s})⊗w(γχ^{−mt−s}, y^{n}x^{l}) for some 0 ≤ t, s, n, l ≤ r−1, guarantees that C(1) is a sub Hopf algebra of A^{∗}. Also we see that γC(1) is the subcoalgebra of A^{∗} with basis W_{γ} = {w(γχ^{k}, y^{j}x^{l})|0 ≤ k, j, l ≤ r − 1}. From Lemma 3.1, we see that as a coalgebra, A^{∗} is the direct sum of its subcoalgebras γC(1) where γ ∈ \overline{γ} ∈ Γ̂/ < χ > .

Also we note that the only elements w in the basis W_{γ} for γC(1) such that a scalar multiple of w ⊗ w is a summand of △v for some v ∈ W_{γ} are those of the form w = w(γχ^{k}, y^{j}x^{l}), where y^{j}x^{l} ∈ C_{A}(Γ) ∩ Z, i.e., y^{j}x^{l} ∈ Z and commutes with all elements of Γ.

Now let e(1, 1) be a matrix coalgebra basis element for a matrix coalgebra in γC(1). Since e(1, 1) ⊗ e(1, 1) is a summand of △e(1, 1), then e(1, 1) is a linear combination of basis elements w(γχ^{k}, z), where z ∈ C_{A}(Γ) ∩ Z. Since terms of the form γχ^{k} ⊗ γχ^{l}, k ≠ l, do not occur in △(w) for any w ∈ W_{γ}, then exactly one element of Γ̂ is a summand of e(1, 1), say γχ^{k}. Then

e(1, 1) = γχ^{k} + ∑_{1≠z∈C_{A}(Γ)∩Z} a_{z}w(γχ^{k_{z}}, z),

where the a_{z} ∈ K. Also since △a_{z}w(γχ^{k_{z}}, z) contains the summand γχ^{k_{z}} ⊗ a_{z}w(γχ^{k_{z}}, z) and does not contain the summand γχ^{k} ⊗ a_{z}w(γχ^{k_{z}}, z) for k ≠ k_{z}, we must have k_{z} = k for all z.

Theorem 3.3 For A with lifting matrix A = [ \begin{matrix}1 & ν \\ −q^{−m}ν & 1 \end{matrix} ] as above, let

γ ∈ Γ̂ with γ(g_{i}^{r} − 1) ≠ 0 for i = 1, 2. Then γC(1) ≅ ⊕_{j=1}^{r}M^{c}(r, K), the sum of r matrix coalgebras of dimension r^{2}, except possibly when ν belongs to a finite set of nonzero scalars.

Proof. First we show that the matrix coalgebras in γC(1) have dimension at least r^{2}. This is essentially the argument from the proof of Theorem 3.6 in [3] and depends on the fact that γ(g^{r} − 1) ≠ 0.

Suppose e(1, 1) as above is a matrix coalgebra basis element. Since x^{r} = g^{r} − 1 and γχ^{k}(g^{r} − 1) = γ(g^{r} − 1) ≠ 0, then △(γχ^{k}) contains a summand ξw(γχ^{k}, x^{j}) ⊗ w(γχ^{k}χ^{−j}, x^{r−j}) for some ξ ∈ K^{∗}, all 1 ≤ j ≤ r − 1. Thus ξ′w(γχ^{k}χ^{−j}, x^{r−j}) is a summand of e(l, 1) for some l, some ξ′ ∈ K^{∗}. But then γχ^{k−j} is a summand of e(l, l). Thus the matrix coalgebra contains at least r matrix coalgebra basis elements e(l, l) and so has dimension at least r^{2}.

Now we find a family of right A-modules arising from matrix coalgebras in γC(1).

Let E = ⊕_{i=1}^{r}Ke_{i} and, with the understanding that subscripts on the e_{i} are taken modulo r, we define a right A-action on E by

e_{i} ↼ h = (γχ^{i−1})(h)e_{i};  (4)
e_{i} ↼ y = e_{i−m} for 1 ≤ i ≤ r − 1 and e_{r} ↼ y = γ(g_{2}^{r} − 1)e_{r−m}; (5)
e_{i} ↼ x = η_{i−1}e_{i−1} for 1 ≤ i ≤ r, η_{i} ∈ K^{∗}.  (6)

Clearly e_{i} ↼ y^{r} = e_{i} ↼ (g_{2}^{r} − 1), and also we have that

e_{i} ↼ (hx − χ(h)xh) = ((γχ^{i−1})(h) − χ(h)(γχ^{i−2})(h))e_{i} ↼ x = 0;
e_{i} ↼ (hy − χ^{m}(h)yh) = ((γχ^{i−1})(h) − χ^{m}(h)(γχ^{i−m−1})(h))e_{i} ↼ y = 0.

Furthermore, we require that

∏_{i=1}^{r} η_{i} = γ(g^{r} − 1) ≠ 0,  (7)

and this condition ensures that e_{i} ↼ x^{r} = e_{i} ↼ (g^{r} − 1). Thus the action respects relations (1) and (2). As well, in order for the action to respect (3), we must have

η_{r}γ(g_{2}^{r} − 1) − q^{m}η_{r−m} = νγ(g_{1}g_{2} − 1),  (8)
η_{i−1} − q^{m}η_{i−1−m} = νγχ^{i−1}(g_{1}g_{2} − 1) for 2 ≤ i ≤ r − 1, (9)
η_{r−1} − q^{m}γ(g_{2}^{r} − 1)η_{r−m−1} = νγχ^{r−1}(g_{1}g_{2} − 1).  (10)

If ν = 0, so that the right hand sides of equations (8),(9) and (10) are zero, then clearly this system of equations is consistent, each η_{i}, 2 ≤ i ≤ r, is a nonzero multiple of η_{1} and (7) becomes η_{1}^{r} = ζ for ζ ∈ K^{∗}, an equation with r distinct solutions in K (since K is algebraically closed).

If ν ≠ 0, so that m = −1, from equations (8) - (10), we obtain

η_{r} = (q^{−1}η_{1} + νγ(g_{1}g_{2} − 1))/γ(g_{2}^{r} − 1)
η_{r−1} = q^{−1}(q^{−1}η_{1} + νγ(g_{1}g_{2} − 1)) + νγχ^{−1}(g_{1}g_{2} − 1)
= q^{−2}η_{1} + νγ(q^{−1} + χ^{−1})(g_{1}g_{2} − 1),

and, in general, for 1 ≤ t ≤ r − 1,

η_{r−t} = q^{−(t+1)}η_{1} + νγ(q^{−t} + q^{−t+1}χ^{−1} + … + χ^{−t})(g_{1}g_{2} − 1).

Then equation (7) is again an equation of degree r in η = η_{1} and thus, since K is algebraically closed, has a solution.

Thus E is a right A-module. Also, from the action of Γ on A, we see that any simple submodule of E must be isomorphic to a right A-module ⊕_{j}Ke(j, k) where e(i, l) are a basis for a matrix coalgebra in γC(1). But we have already shown that such a matrix coalgebra has dimension at least r^{2} and so E is a simple right A-module.

Now we show that any two distinct solutions to equation (7) yield nonisomorphic modules. Let η and η′ be two different solutions, and E, E′ the corresponding modules.

Since Ke_{i} is the eigenspace for γχ^{i−1}(g) = q^{i−1}γ(g) under the action of g, any isomorphism ϕ from E to E′ must map e_{i} to α_{i}e′_{i} for some α_{i} ∈ K^{∗}. But then for j < r, ϕ(e_{j} ↼ y) = ϕ(e_{j−m}) = α_{j−m}e′_{j−m} and ϕ(e_{j}) ↼ y = α_{j}e′_{j} ↼ y = α_{j}e′_{j−m}. Also ϕ(e_{r} ↼ y) = ϕ(γ(g_{2}^{r} − 1)e_{r−m}) = α_{r−m}γ(g_{2}^{r} − 1)e′_{r−m} while ϕ(e_{r}) ↼ y = α_{r}e′_{r} ↼ y = α_{r}γ(g_{2}^{r} − 1)e′_{r−m}, so for all i, j, α_{j} = α_{i} = α. Also ϕ(e_{2} ↼ x) = η_{1}αe′_{1} and ϕ(e_{2}) ↼ x = αη′_{1}e′_{1}, which is impossible unless η_{1} = η′_{1}.

Finally we prove that for ν different from a finite set of nonzero scalars, equation (7) has r distinct solutions, and so γC(1) is the direct sum of r matrix coalgebras, each of dimension r^{2}.

Now suppose ν ≠ 0. Let s_{t} = γ(q^{−t} + q^{−t+1}χ^{−1} + … + χ^{−t})(g_{1}g_{2} − 1), for 0 ≤ t ≤ r − 1; note that s_{r−1} = 0.

Then writing (7) as a polynomial equation in the variable X, we have

∏_{i=o}^{r−1}(q^{i}X + νs_{r−i−1}) − γ(g^{r} − 1)γ(g_{2}^{r} − 1) = 0.

Then, with X = νY, (7) can be written as

g(Y) = Y^{r} + a_{r−1}Y^{r−1} + … + a_{1}Y + a_{0} = 0,

where a_{0} = −γ(g^{r} − 1)γ(g_{2}^{r} − 1)/ν^{r} and the scalar ν does not appear in any other coefficient a_{i}. If g(Y) has a repeated root z then z is also a zero of g′(Y) = 0. Note that g′(Y) is a polynomial whose coefficients do not depend upon ν. Let Z be the finite set of roots of g′(Y). If z ∈ Z is also a zero of g(Y) then ν^{r} = −γ(g^{r} −1)γ(g_{2}^{r} −1)/(z^{r} +a_{r−1}z^{r−1} +…+a_{1}z). If we choose ν so that this equation does not hold for any z ∈ Z, then (7) has r distinct zeros.

Proposition 3.4 For V as above, suppose A is a lifting of B(V)#K[Γ]

with lifting matrix A = [ \begin{matrix}1 & 1 \\ −q & 0 \end{matrix} ]. Let γ ∈ Γ̂ with γ(g^{r} − 1) ≠ 0. Then

γC(1) ≅ ⊕_{j=1}^{r}M^{c}(r, K), the sum of r matrix coalgebras of dimension r^{2}.

Proof. Note that here m = −1. Also since y^{r} = 0, we may assume that if ν is nonzero then ν = 1. The proof that any matrix coalgebra in γC(1) has dimension at least r^{2} still holds and an A-module E may still be constructed using conditions (4) to (10) with γ(g_{2}^{r} − 1) replaced by 0. However, here the equations (8) to (10) determine η_{1}, …, η_{r−1} and equation (7) is a linear equation in η_{r}, so that we obtain one simple A-module of dimension r in this way replacing γ in (8) - (10) by each of the r elements of γ < χ >. Suppose two of these modules are isomorphic, say E = ⊕_{i=1}^{r}Ke_{i}, constructed using γ and F = ⊕_{i=1}^{r}Kf_{i}, constructed using γχ^{s} with s ≠ 0. Under the right action of g, Ke_{i} and Kf_{i−s} are the eigenspaces for γχ^{i−1}(g). Thus any isomorphism from E to F must map e_{i} to α_{i}f_{i−s} for some nonzero scalar α_{i}. But e_{1} ↼ y^{r−1} = e_{r} while f_{1−s} ↼ y^{r−1} = 0 .

To describe the coradical of C(1), note that the sub Hopf algebra C(1) of A^{∗} can be described in the following way. Let J be the Hopf ideal in A generated by g_{i}^{r} − 1 for i = 1, 2 and let A′ = A/J be the quotient Hopf algebra. In A′, x^{′r} = 0 = y^{′r} where x′, y′ are the images of x and y in A′. Then the dual of A′ is isomorphic to a sub Hopf algebra of A^{∗} and C(1) is a sub Hopf algebra of A^{′∗}. The structure of the sub Hopf algebra C(1) of A^{′∗} is different depending on whether ν is zero or nonzero.

If ν = 0, then C(1) is a Radford biproduct. Let \overline{Γ} be the quotient of Γ by the subgroup generated by g_{1}^{r}, g_{2}^{r} and it was noted in [3] that ˆ \overline{Γ} ≅ G(A^{∗}) = {γ ∈ Γ̂|γ(g_{1}^{r}) = 1 = γ(g_{2}^{r})}. The quotient Hopf algebra A′ is isomorphic to the bicrossed product B(V)#K[\overline{Γ}], so that A^{′∗} is isomorphic to the bicrossed product B(W)#K[G(A^{∗})] where W = Kw_{1} ⊕ Kw_{2} ∈Γ̂ Γ̂ YD with w_{i} ∈ W_{χ_{i}}^{g_{i}}. Since χ ∈ G(A^{∗}), then B(W)#K[< χ >] is a sub Hopf algebra of A^{′∗} and this is C(1). Thus C(1) is pointed with coradical K[< χ >]. More details can be found in [3].

If ν ≠ 0, then C(1) is a Frobenius Lusztig kernel of dimension r^{3}. A′ is isomorphic to the nontrivial lifting of B(V)#K[\overline{Γ}] with lifting matrix A =

[ \begin{matrix}0 & 1 \\ −q & 0 \end{matrix} ]. Again, < χ >⊆ ˆ \overline{Γ} = \{γ ∈ Γ̂|γ(g_{1}^{r}) = 1 = γ(g_{2}^{r})\} but here

G(C(1)) = {ϵ} since for i ≠ 0, χ^{i}(g_{1}g_{2}) = q^{2i} ≠ 1 since (r, 2) = 1.

The representation theory for the Frobenius Lusztig kernels is known; however we make the following remark since the construction in the proof of Theorem 3.3 still is valid here.

Remark 3.5 Suppose that A has lifting matrix A = [ \begin{matrix}1 & ν \\ −q^{−m}ν & 1 \end{matrix} ] but

γ is such that γ(g_{i}^{r} − 1) = 0, for example, we could have γ = χ. Then the argument that the simple modules must have dimension greater than or equal to r no longer holds. However, we still may define a right A-module E as in the proof of Theorem 3.3, where equations (4)-(10) hold with γ(g_{i}^{r} − 1) replaced by 0. Suppose γ ∈< χ > and ν ≠ 0. The various choices of γ will yield simple submodules of E of dimensions 1, …, r. For example, if γ = ϵ, then by (8), η_{1} = 0 and then by (10), η_{r−1} = νχ^{r−1}(g_{1}g_{2} −1) = ν(q^{−2} −1), then by (9), η_{r−2} = q^{−1}η_{r−1} +νχ^{r−2}(g^{2} −1) = ν(q^{−3} − q^{−1} + q^{−4} − 1), and then η_{r−3} = ν(q^{−4} − q^{−2} + q^{−5} − q^{−1} + q^{−6} − 1), etc. Thus η_{2}, …, η_{r−1} are nonzero and the submodule Ke_{2} ⊕ … ⊕ Ke_{r} of E is a simple right (r − 1)-dimensional A-module. If η_{2} = 0, and η_{3}, …, η_{r−1} are nonzero, then Ke_{3} ⊕ … ⊕ Ke_{r} is a simple right (r − 2)-dimensional A-module, etc. An example of this construction where C(1) has dimension 27 and has coradical M^{c}(K, 3) ⊕ M^{c}(K, 2) ⊕ Kϵ appears in the next section.

4 Examples of dimension 81

In the remainder of this note, we let Γ = C_{9} =< c >, Γ̂ =< c^{∗} >, V = Kv_{1} ⊕ Kv_{2} with v_{1} ∈ V_{c}^{c^{∗3}}, v_{2} ∈ V_{c}^{c^{∗6}}. Let λ denote c^{∗3} and let q = λ(c), a primitive cube root of unity. Let ν be a nonzero scalar. Let A be the lifting of B(V)#K[Γ] with lifting matrix A = [ \begin{matrix}1 & ν \\ −qν & α_{2,2} \end{matrix} ] and we explicitly de-

termine the matrix coalgebras in each of the subcoalgebras c^{∗i}C(1), i = 1, 2. Noice that here G(A^{∗}) = {ϵ}. D. Stefan has remarked that A cannot satisfy the Chevalley property. Otherwise the coradical of A^{∗} would be a semisimple Hopf algebra of dimension 3^{i}, which, then, by a theorem of Masuoka, would have a nontrivial grouplike element. The irreducible representations of the sub Hopf algebra C(1) are known [11], but the structure of the coradical of C(1) also follows easily from our computations, and so we describe it as well.

By Theorem 3.3, or Proposition 3.4 the coalgebra c^{∗i}C(1), 1 ≤ i ≤ 2, is the sum of three 9-dimensional matrix coalgebras except possibly when ν lies in a finite set of nonzero scalars. We find these coalgebras explicitly in terms of the basis W of A^{∗}, W = {w(γ, z)|γ ∈ Γ̂, z ∈ Z}, Z = {y^{i}x^{j}|0 ≤ i, j ≤ 2}.

4.1 Lifting matrix A with two nonzero diagonal elements

In this section, we let A be the lifting of B(V)#K[Γ] with lifting matrix

A = [ \begin{matrix}1 & ν \\ −qν & 1 \end{matrix} ].

Let γ ∈ Γ̂. From the discussion in the previous section, some matrix coalgebra basis element say e(1, 1) in γC(1) has the form e(1, 1) = γ + aw(γ, yx) + bw(γ, y^{2}x^{2}) and noting that γ(c^{3} − 1) = γλ^{i}(c^{3} − 1) since λ(c) = q, we compute

△(γ + aw(γ, yx) + bw(γ, y^{2}x^{2}))  (11)
= [γ ⊗ γ + νγ(c^{2} − 1)w(γ, x) ⊗ w(γλ^{2}, y)
+ γ(c^{3} − 1)w(γ, x) ⊗ w(γλ^{2}, x^{2})
+ γ(c^{3} − 1)w(γ, y) ⊗ w(γλ, y^{2}) + γ(c^{3} − 1)w(γ, x^{2}) ⊗ w(γλ, x)
+ ν^{2}γ(q − c^{2})γ(c^{2} − 1)w(γ, x^{2}) ⊗ w(γλ, y^{2})
+ νγ(q − c^{2})γ(c^{3} − 1)w(γ, x^{2}) ⊗ w(γλ, yx^{2})
+ qγ(c^{3} − 1)^{2}w(γ, yx) ⊗ w(γ, y^{2}x^{2}) + γ(c^{3} − 1)w(γ, y^{2}) ⊗ w(γλ^{2}, y)
+ q^{2}γ(c^{3} − 1)^{2}w(γ, yx^{2}) ⊗ w(γλ^{2}, y^{2}x)
+ νγ(c^{3} − 1)γ(q − c^{2})w(γ, y^{2}x) ⊗ w(γλ, y^{2})
+ q^{2}γ(c^{3} − 1)^{2}w(γ, y^{2}x) ⊗ w(γλ, yx^{2})
+ qγ(c^{3} − 1)^{2}w(γ, y^{2}x^{2}) ⊗ w(γ, yx)
+ qνγ(c^{2}q − 1)γ(c^{3} − 1)^{2}w(γ, y^{2}x^{2}) ⊗ w(γ, y^{2}x^{2})]
+ a[γ ⊗ w(γ, yx) + w(γ, yx) ⊗ γ + q^{2}w(γ, x) ⊗ w(γλ^{2}, y)

+ qνγ(1 − c^{2}q)w(γ, x) ⊗ w(γλ^{2}, y^{2}x) + w(γ, y) ⊗ w(γλ, x)
+ qνγ(c^{2} − 1)w(γ, x^{2}) ⊗ w(γλ, y^{2}) + qγ(c^{3} − 1)w(γ, x^{2}) ⊗ w(γλ, yx^{2})
+ νγ(c^{2}q^{2} − 1)w(γ, yx) ⊗ w(γ, yx) + γ(c^{3} − 1)w(γ, y^{2}) ⊗ w(γλ^{2}, y^{2}x)
+ qνγ(1 − c^{2}q)w(γ, yx^{2}) ⊗ w(γλ^{2}, y) + γ(c^{3} − 1)w(γ, yx^{2}) ⊗ w(γλ^{2}, x^{2})
+ ν^{2}γ(1 − c^{2}q)γ(c^{2} − q)w(γ, yx^{2}) ⊗ w(γλ^{2}, y^{2}x)
+ qγ(c^{3} − 1)w(γ, y^{2}x) ⊗ w(γλ, y^{2})
+ q^{2}γ(c^{3} − 1)^{2}w(γ, y^{2}x^{2}) ⊗ w(γ, y^{2}x^{2})]
+ b[γ ⊗ w(γ, y^{2}x^{2}) + w(γ, y^{2}x^{2}) ⊗ γ + qw(γ, x) ⊗ w(γλ^{2}, y^{2}x)
+ w(γ, y) ⊗ w(γλ, yx^{2}) + q^{2}w(γ, x^{2}) ⊗ w(γλ, y^{2})
+ q^{2}w(γ, yx) ⊗ w(γ, yx)
+ qνγ(1 − c^{2})w(γ, yx) ⊗ w(γ, y^{2}x^{2}) + w(γ, y^{2}) ⊗ w(γλ^{2}, x^{2})
+ qw(γ, yx^{2}) ⊗ w(γλ^{2}, y) + νγ(c^{2} − q)w(γ, yx^{2}) ⊗ w(γλ^{2}, y^{2}x)
+ w(γ, y^{2}x) ⊗ w(γλ, x) + νγ(c^{2}q − 1)w(γ, y^{2}x) ⊗ w(γλ, yx^{2})
+ qνγ(1 − c^{2})w(γ, y^{2}x^{2}) ⊗ w(γ, yx)
+ qν^{2}γ(1 − c^{2})γ(c^{2}q − 1)w(γ, y^{2}x^{2}) ⊗ w(γ, y^{2}x^{2})].

From the first two paragraphs of the proof of Theorem 3.3, we recall that γ is a summand of the matrix coalgebra basis element e(j, j) only when j = 1. But then since for i = 1, 2, we have γ ⊗ aw(γ, y^{i}z^{i}) a summand of △aw(γ, y^{i}z^{i}), then aw(γ, y^{i}z^{i}) cannot be a summand of e(j, 1), j ≠ 1. Then no scalar multiple of w(γ, y^{i}x^{i})⊗w(γ, y^{i}x^{i}) can occur in e(1, j)⊗e(j, 1) unless j = 1. Thus:

a^{2} = aνγ(c^{2}q^{2} − 1) + bq^{2};  (12)
b^{2} = (qνγ(c^{2}q − 1) + aq^{2})γ(c^{3} − 1)^{2} + bqν^{2}γ(1 − c^{2})γ(c^{2}q − 1), (13)
ab = qγ(c^{3} − 1)^{2} + bqνγ(1 − c^{2}).  (14)

Then from equation (12) and equation (14) we have

b = (a^{2} − aνγ(c^{2}q^{2} − 1))q  (15)
b(a − qνγ(1 − c^{2})) = qγ(c^{3} − 1)^{2}.  (16)

so that

a(a − νγ(c^{2}q^{2} − 1))(a − qνγ(1 − c^{2})) = γ(c^{3} − 1)^{2}.  (17)

Equation (17) is consistent with equation (13) so the system has solutions a and b. If γ(c^{3} − 1) ≠ 0 then both a and b are nonzero. (Note that equation (17) is just equation (7) with r = 3, a = η_{1}, χ = λ.)

4.1.1 Case: γ = c^{∗} or c^{∗2}.

Now let γ = c^{∗i} with i = 1 or 2, let a be a solution to equation (17) and let b be defined by equation (15). Let e(1, 1) = γ + aw(γ, yx) + bw(γ, y^{2}x^{2}) as above.

Then define

e(2, 1) = e(1, 1) ↼ y = γ(c^{3} − 1)w(γλ, y^{2}) + aw(γλ, x) + bw(γλ, yx^{2}) (18)

and

e(3, 1) = e(2, 1) ↼ y = e(1, 1) ↼ y^{2}  (19)
= γ(c^{3} − 1)w(γλ^{2}, y) + aγ(c^{3} − 1)w(γλ^{2}, y^{2}x) + bw(γλ^{2}, x^{2}).

Then

e(1, 1) ↼ y^{3} = e(3, 1) ↼ y
= γ(c^{3} − 1)γ + aγ(c^{3} − 1)w(γ, yx) + bγ(c^{3} − 1)w(γ, y^{2}x^{2})
= γ(c^{3} − 1)e(1, 1) = e(1, 1) ↼ (c^{3} − 1).

Thus e(i, 1) ↼ y^{3} = e(i, 1) ↼ (c^{3} − 1) for 1 ≤ i ≤ 3.

Now from equation (11), we see that △e(1, 1) = ∑_{i=1}^{3} e(1, j) ⊗ e(j, 1) where

e(1, 2) = αw(γ, x^{2}) + w(γ, y) + \frac{b}{a}w(γ, y^{2}x),
e(1, 3) = βw(γ, x) + w(γ, y^{2}) + aβw(γ, yx^{2})

with β = \frac{γ(c^{3}−1)}{b}, and α = \frac{γ(c^{3}−1)}{a}. Now define

e(2, 2) = γλ + \frac{b}{a}w(γλ, yx) + αγ(c^{3} − 1)w(γλ, y^{2}x^{2}),
e(2, 3) = w(γλ, y) + aβw(γλ, x^{2}) + βγ(c^{3} − 1)w(γλ, y^{2}x),
e(3, 2) = \frac{b}{a}w(γλ^{2}, x) + γ(c^{3} − 1)w(γλ^{2}, y^{2}) + αγ(c^{3} − 1)w(γλ^{2}, yx^{2}),
e(3, 3) = γλ^{2} + βγ(c^{3} − 1)w(γλ^{2}, yx) + aβγ(c^{3} − 1)w(γλ^{2}, y^{2}x^{2}).

It is tedious but straightforward to check that the e(i, j) are a matrix coalgebra basis. The 3-dimensional spaces V_{j} = ⊕_{i=1}^{3}Ke(i, j) are irreducible right A-modules. In general, the right action of g ∈ Γ or y on V_{j} is given by

e(i, j) ↼ g = (λ^{i−1}γ)(g)e(i, j);

e(1, j) ↼ y = e(2, j), e(2, j) ↼ y = e(3, j), e(3, j) ↼ y = γ(c^{3} − 1)e(1, j); We compute the action of x on e(1, 1), e(2, 1) and e(3, 1). From equation (11) we see that

e(1, 1) ↼ x = (νγ(c^{2} − 1)w(γλ^{2}, y) + γ(c^{3} − 1)w(γλ^{2}, x^{2}))
+ a(q^{2}w(γλ^{2}, y) + qνγ(1 − c^{2}q)w(γλ^{2}, y^{2}x)) + bqw(γλ^{2}, y^{2}x)
= (νγ(c^{2} − 1) + aq^{2})w(γλ^{2}, y)
+ (aqνγ(1 − c^{2}q) + bq)w(γλ^{2}, y^{2}x) + γ(c^{3} − 1)w(γλ^{2}, x^{2}).

We see that e(1, 1) ↼ x = βe(3, 1) if and only if

νγ(c^{2} − 1) + aq^{2} = βγ(c^{3} − 1) = \frac{γ(c^{3} − 1)^{2}}{b}  (20)
and aqνγ(1 − c^{2}q) + bq = βaγ(c^{3} − 1) = \frac{aγ(c^{3} − 1)^{2}}{b} .  (21)

Now, (20) holds iff νγ(c^{2} − 1)b + abq^{2} = γ(c^{3} − 1)^{2} iff νγ(c^{2} − 1)b + γ(c^{3} − 1)^{2} + bνγ(1 − c^{2}) = γ(c^{3} − 1)^{2} by equation (14),

and this last equation clearly holds. Also, equation (21) holds if and only if

abqνγ(1 − c^{2}q) + b^{2}q = aγ(c^{3} − 1)^{2}

and using (13), we see that this equation reduces to equation (14). Thus e(1, 1) ↼ x = βe(3, 1). Similar computations now show that

e(1, 1) ↼ x^{2} = βe(3, 1) ↼ x = αe(2, 1), and

e(1, 1) ↼ x^{3} = βe(3, 1) ↼ x^{2} = αe(2, 1) ↼ x = αae(1, 1) = γ(c^{3} − 1)e(1, 1), and that, in general, the right action of x on the e(i, j) is given by

e(1, j) ↼ x = βe(3, j), e(2, j) ↼ x = ae(1, j), e(3, j) ↼ x = \frac{b}{a}e(2, j).

This action preserves the relations (2) and (3) since,

e(i, j) ↼ y^{3} = γ(c^{3} − 1)e(i, j) = e(i, j) ↼ (c^{3} − 1);
e(i, j) ↼ x^{3} = (\frac{γ(c^{3} − 1)}{b}) (a)(\frac{b}{a})e(i, j) = e(i, j) ↼ (c^{3} − 1),

and also denoting xy − q^{2}yx by z and using (12),(13), (14), we have

e(1, j) ↼ z = (\frac{γ(c^{3} − 1)^{2}}{b} − q^{2}a) e(1, j) = νγ(c^{2} − 1)e(1, j),
e(2, j) ↼ z = (a − q^{2} \frac{b}{a}) e(2, j) = ν(γλ)(c^{2} − 1)e(2, j),
e(3, j) ↼ z = (\frac{b}{a} − q^{2} \frac{γ(c^{3} − 1)^{2}}{b} ) e(3, j) = ν(γλ^{2})(c^{2} − 1)e(3, j).

We note that similarly, U_{1} = ⊕_{i=1}^{3}Ke(1, i) is an irreducible left A-module. The left action of Γ is given by h ⇀ e(1, j) = γλ^{j−1}(h)e(1, j). The left action of x is given by

x ⇀ e(1, 1) = ae(1, 2); x ⇀ e(1, 2) = \frac{b}{a}e(1, 3); x ⇀ e(1, 3) = βe(1, 1),

and the left action of y is given by

y ⇀ e(1, 3) = e(1, 2); y ⇀ e(1, 2) = e(1, 1); y ⇀ e(1, 1) = γ(c^{3} − 1)e(1, 3).

Finally we consider the number of solutions to equation (17),

X(X − νγ(c^{2}q^{2} − 1))(X − qνγ(1 − c^{2})) − γ(c^{3} − 1)^{2} = 0,

or equivalently,

Y^{3} + ωY^{2} + θY = γ(c^{3} − 1)^{2}/ν^{3}

where X = νY, ω = −γ(c^{2}q^{2} − 1) − qγ(1 − c^{2}),and θ = qγ(c^{2}q^{2} − 1)γ(1 − c^{2}). Then the solutions to 3Y^{2} + 2ωY + θ = 0 are Z = \{(−ω ± \sqrt{ω^{2} − 3θ})/3\}, and in order to guarantee that γC(1) is ⊕_{i=1}^{3}M^{c}(3, K), we should choose ν so that γ(c^{3} −1)^{2} ≠ ν^{3}(z^{3} +ωz^{2} +θz) for either z ∈ Z, i.e. 6 possible choices for ν are excluded.

Note that if γ = c^{∗}, then ω^{2}−3θ = −3ζ^{2}q^{2} and if γ = c^{∗2}, then ω^{2}−3θ = −3ζ where ζ = c^{∗}(c) is a primitive 9th root of unity with ζ^{3} = q. Thus (17) always has at least 2 distinct solutions.

4.1.2 Case: γ ∈< λ >.

Now consider the matrix coalgebras in C(1), as described in Remark 3.5. For γ ∈< λ >, γ(c^{3} − 1) = 0 so that equations (13), (14) are different, and we have

a^{2} = aνγ(c^{2}q^{2} − 1) + bq^{2};
b^{2} = bqν^{2}γ(1 − c^{2})γ(c^{2}q − 1);
ab = bνqγ(1 − c^{2}).

If γ = ϵ, then b = 0 and then either a = 0 or a = ν(q^{2} − 1). If γ = λ, then b = 0 and then either a = 0 or a = ν(q − 1). If γ = λ^{2}, then a^{2} = bq^{2} and then either a = b = 0 or a = νq(1 − q) and b = ν^{2}q(1 − q)(q^{2} − 1).

Then we obtain the 1 dimensional coalgebra K ·ϵ, a 9-dimensional coalgebra E with basis e(i, j), and a 4-dimensional F coalgebra with basis f(i, j) where

e(1, 1) = λ^{2} + νq(1 − q)w(λ^{2}, yx) + ν^{2}(1 − q)^{2}w(λ^{2}, y^{2}x^{2}),
e(1, 2) = w(λ^{2}, y) + ν(q^{2} − 1)w(λ^{2}, y^{2}x);
e(1, 3) = w(λ^{2}, y^{2});
e(2, 1) = νq(1 − q)w(ϵ, x) + ν^{2}(1 − q)^{2}w(ϵ, yx^{2});
e(2, 2) = ϵ + ν(q^{2} − 1)w(ϵ, yx),
e(2, 3) = w(ϵ, y);
e(3, 1) = ν^{2}(1 − q)^{2}w(λ, x^{2});
e(3, 2) = ν(q^{2} − 1)w(λ, x);
e(3, 3) = λ,

and

f(1, 1) = λ^{2},
f(1, 2) = ν(q − 1)w(λ^{2}, x);
f(2, 1) = w(λ, y);
f(2, 2) = λ + ν(q − 1)w(λ, yx).

4.2 Lifting matrix A with one nonzero diagonal element

Now let A have lifting matrix [ \begin{matrix}1 & ν \\ −qν & 0 \end{matrix} ], i.e., x^{3} = c^{3} − 1, y^{3} = 0 and we

are in the situation of Proposition 3.4. Suppose γ ∉< λ >. Then equations (12),(13) and (14) become

a^{2} = aνγ(c^{2}q^{2} − 1) + bq^{2};  (22)
b^{2} = bqν^{2}γ(1 − c^{2})γ(c^{2}q − 1),  (23)
ab = bqνγ(1 − c^{2}).  (24)

The three solutions to this set of equations are a = b = 0, b = 0 and a = νγ(c^{2}q − 1) or b = qν^{2}γ(1 − c^{2})γ(c^{2}q − 1) and a = qνγ(1 − c^{2}). Let us choose the solution where both a and b are nonzero and then we can construct a matrix coalgebra exactly as in the previous subsection. As before, let β = \frac{γ(c^{3}−1)}{b}, and α = \frac{γ(c^{3}−1)}{a}. Then Ke(1, 1) ⊕ Ke(2, 1) ⊕ Ke(3, 1) is a simple right A-module and e(i, j) with 1 ≤ i, j ≤ 3 is a basis for a matrix coalgebra in γC(1) where

e(1, 1) = γ + aw(γ, yx) + bw(γ, y^{2}x^{2}),
e(1, 2) = αw(γ, x^{2}) + w(γ, y) + \frac{b}{a}w(γ, y^{2}x),
e(1, 3) = βw(γ, x) + w(γ, y^{2}) + aβw(γ, yx^{2}),
e(2, 1) = aw(γλ, x) + bw(γλ, yx^{2}),
e(2, 2) = γλ + \frac{b}{a}w(γλ, yx),
e(2, 3) = w(γλ, y) + aβw(γλ, x^{2}),
e(3, 1) = bw(γλ^{2}, x^{2}),
e(3, 2) = \frac{b}{a}w(γλ^{2}, x),
e(3, 3) = γλ^{2}.

As in the previous section e(1, j) ↼ y = e(2, j), e(2, j) ↼ y = e(3, j) and e(3, j) ↼ y = 0. Had we not chosen a, b nonzero, then the action of y on e(1, 1) would not have yielded a 3-dimensional K-vector space.

To obtain the other two matrix coalgebras in γC(1), replace γ by γλ or γλ^{2} in equations (22)-(24) to obtain matrix coalgebras generated by elements

e′(1, 1) = γλ + a′w(γλ, yx) + b′w(γλ, y^{2}x^{2}),
e′′(1, 1) = γλ^{2} + a′′w(γλ^{2}, yx) + b′′w(γλ^{2}, y^{2}x^{2}),

where a′ = qνγ(1 − c^{2}q^{2}), b′ = qν^{2}γ(1 − c^{2}q^{2})γ(c^{2} − 1), a′′ = qνγ(1 − c^{2}q), b′′ = qν^{2}γ(1 − c^{2}q)γ(c^{2}q^{2} − 1).

Acknowledgement

Thanks to the referee for his/her careful reading of this paper and helpful comments.

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