ON NICHOLS ALGEBRAS OVER PGL(2, q) AND PSL(2, q)

SEBASTIÁN FREYRE, MATÍAS GRAÑA, LEANDRO VENDRAMIN

Abstract. We compute necessary conditions on Yetter-Drinfeld modules over the groups PGL(2, q) = PGL(2, 𝔽_{q}) and PSL(2, q) = PSL(2, 𝔽_{q}) to generate finite dimensional Nichols algebras. This is a first step towards a classification of pointed Hopf algebras with group of group-likes isomorphic to one of these groups.

As a by-product of the techniques developed in this work, we prove that any finite-dimensional pointed Hopf algebra over the Mathieu groups M_{20} or M_{21} = PSL(3, 4) is the group algebra.

1. Introduction

Quantum groups, introduced by Drinfeld and Jimbo in the 1980’s, were constructed as a mean to solve the Quantum Yang-Baxter Equation. They are deformations of enveloping algebras of semisimple complex Lie algebras. But they are also pointed Hopf algebras, and this marked the beginning of an algebraic program: classifying pointed Hopf algebras. A new chapter in this program was introduced by Lusztig: when the deformation parameter q is a root of unit, we can consider the so called Frobenius-Lusztig kernels. These are now finite dimensional pointed Hopf algebras, which carry the combinatorial information of a Cartan matrix and a Weyl group.

Soon, the question became how to construct new finite dimensional pointed Hopf algebras, and how Cartan matrices and Weyl groups appear in them. To keep this Introduction short, let us just mention that the Lifting Procedure [AS02] was the answer to the first question and the introduction of a Weyl groupoid [Hec06a] was the answer to the second one.

When the group of grouplikes is abelian, the whole procedure is well understood nowadays. However, when the group is not abelian, we are still far away from the classification. Not only so, but even the construction of finite dimensional pointed Hopf algebras over non abelian groups is an elusive matter [Gr].

Let us recall that, according to the Lifting Procedure, to classify finite dimensional pointed Hopf algebras with a specific group of grouplikes, the key step is to compute the Nichols algebras generated by Yetter-Drinfeld modules over the group which are finite dimensional. With the abelian tools at hand, it is possible to rule out, for a given group (or a family of groups), a large class of Yetter-Drinfeld modules which can be shown to produce infinite dimensional Nichols algebras. Therefore, the next step to classify f.d. pointed Hopf algebras over those groups is to study the Nichols algebras produced by the remaining modules.

One of the key tools to produce Yetter-Drinfeld modules over groups is that of racks and 2-cocycles. They allow to work with the braided vector spaces without having to resort to groups. However, when trying to classify pointed Hopf algebras parametrized by groups, sometimes the rack 2-cocycles are “too general”. When the group is fixed, a conjugacy class can not hold any 2-cocycle, but only some of them. Many times, one can prove that all Nichols algebras produced by a conjugacy class within a group are infinite-dimensional, while one is not able to prove that, as a rack, the same thing happens with any 2-cocycle. Therefore, we develop in this article the concept of class of type B, as follows (C_{G}(g) is the centralizer of g ∈ G and 𝔅(C, ρ) stands for the Nichols algebra produced by the Yetter-Drinfeld module M(C, ρ) -see below-):

This work was partially supported by CONICET and ANPCyT (Argentina).

Definition 1.1. Let G be a group, g ∈ G and C = {xgx^{−1} : x ∈ G} its conjugacy class. We say that C is a class of type B if for any representation ρ ∈ { \atop C_{G}(g)} the Nichols algebra 𝔅(C, ρ) is infinite dimensional. We also say that g is of type B if its conjugacy class is. We say that the group G is of type B if all its conjugacy classes are.

Remark 1.2. By the lifting procedure, if G is of type B then any finite dimensional pointed Hopf algebra with group of grouplikes (isomorphic to) G is (isomorphic to) the group algebra of G.

Remark 1.3. It is proved in [AFGV08] that if g ∈ G is of type B and f : G ↪ H is a monomorphism of groups then f(g) is of type B in H. This is the reason why this concept turns out to be a powerful tool.

In [FGV07] Nichols algebras over GL(2, q) and SL(2, q) were studied. In that paper it is proved that SL(2, 2^{n}) is of type B for n > 1. In this paper we deal with the groups PGL(2, q) and PSL(2, q) for q a power of an odd prime number (recall that if q is even then PSL(2, q) = PGL(2, q) = SL(2, q)). For definitions and elementary properties of these groups, see for example [AB95].

One of the main results of this work is Theorem 1.4.

Theorem 1.4. The Mathieu groups M_{20} and M_{21} = PSL(3, 4) are of type B.

It is interesting to note that M_{20} is non-simple. Also, notice that this is the first example of a non-simple group which is of type B. Other results of this work are (see notations below):

Theorem 1.5. Let G = PGL(2, q) (see Table 4 in §5 for the conjugacy classes of G). Then, class C_{2} is of type B. Also, the conditions to obtain finite-dimensional Nichols algebras on the representations for classes C_{3}, C_{4}, C_{5} and C_{6} are given in propositions 5.1, 5.2, 5.3 and 5.4.

Theorem 1.6. Let G = PSL(2, q) (see Tables 2 and 3 in §4). Then, classes C_{i} are of type B for i < 6. Also, the conditions to obtain finite-dimensional Nichols algebras on the representation for class C_{6} are given in Proposition 4.2 if q ≡ 1 (mod 4), or in Proposition 4.4 if q ≡ 3 (mod 4).

2. Preliminaries

2.1. Notations. As said, we use the same notation as in [FGV07]. The number p will be an odd prime number and q a power of p, E = 𝔽_{q^{2}} will be the quadratic extension of 𝔽_{q}, \overline{x} = x^{q} will be the Galois conjugate of x ∈ E, k will be an algebraically closed field of characteristic zero, and we will write R_{n} for the set of Table 1. Character table of 𝔻_{n} (n even)

Degree  r^{k}  sr^{k}
χ_{1}  1  1  1
χ_{2}  1  1  −1
χ_{3}  1  (−1)^{k}  (−1)^{k}
χ_{4}  1  (−1)^{k} (−1)^{k+1}
μ_{h} (0 < h < \frac{n}{2})  2  2 cos \frac{2πhk}{n}  0

primitive n-th roots of 1 in k. If q ≡ 1 (mod 4) we write ±i for the square roots of −1 in 𝔽_{q}. Following the notation in [Ada02], we fix an element Δ ∈ 𝔽_{q} \ 𝔽_{q}^{2} and an element δ ∈ E which is a square root of Δ. We consider elements z ∈ E being written as z = x + δy, with x, y ∈ 𝔽_{q}.

If G is a finite group, C is a conjugacy class of g ∈ G, and ρ is an irreducible representation of the centralizer C_{G}(g), we write 𝔅(C, ρ) (or 𝔅(C) if no confusion can arise) for the Nichols algebra generated by the Yetter-Drinfeld module V (g, ρ) (see for example [AG99]).

2.2. Representations of the Dihedral group. We present the Dihedral group 𝔻_{n} of order 2n by generators r and s and relations

r^{n} = s^{2} = 1, srs = r^{−1}.

When n is even, the irreducible representations are given in Table 1 (see [Ser77, Section 5.3]). We will not need the Dihedral groups 𝔻_{n} with n odd.

2.3. Main tools. Note that, since ρ is irreducible and g ∈ Z(C_{G}(g)) (the center of C_{G}(g)), then ρ(g) is a scalar (by Schur lemma). The following lemmas are contained in [AF07a, AF07b, AZ07, FGV07, Hec06b].

Lemma 2.1. Assume that dim 𝔅(C, ρ) < ∞. If g^{−1} ∈ C then ρ(g) = −1.

Proof. See [AZ07]. □

As in the cited papers, a conjugacy class is called real if it contains the inverses of its elements.

Lemma 2.2. Assume that dim 𝔅(C, ρ) < ∞. If there exist n > 1 such that g^{n} ∈ C then ρ(g) = −1 or ρ(g) ∈ R_{3}. Moreover, if g^{n^{2}} ≠ g then ρ(g) = −1.

Proof. For the proof see for example [FGV07]. □

As in the cited papers, a conjugacy class is called quasireal if it contains proper powers of its elements.

The next lemma is useful to treat, for instance, some conjugacy classes of involutions. A particular case of this Lemma appears also in [Fan07].

Lemma 2.3. Let G be a group.

(1) Assume that g_{0}, g_{1}, g_{2} ∈ G are conjugate and commute to each other, that x_{1}x_{2} and x_{2}x_{1} belong to C_{G}(g_{0}), (where x_{i} are such that g_{i} = x_{i}g_{0}x_{i}^{−1} for i = 1, 2), and that g_{1}g_{2} = g_{0}^{m} for an odd integer m. Then g_{0} is of type B.

(2) The conjugacy class of involutions in the alternating group 𝔸_{4} is of type B.

(3) If g_{0}, x ∈ G are such that g_{0} has order 2 and both x and g_{0}x have order 3, then g_{0} is of type B.

Proof. Part 2 is a consequence of Part 1 by taking g_{0} = (1 2)(3 4), g_{1} = (1 3)(2 4), g_{2} = (1 4)(2 3), x_{1} = (2 4 3) and x_{2} = x_{1}^{−1} (we have g_{1}g_{2} = g_{0} in this way). Also, Part 3 is a consequence of Part 2 and Remark 1.3 since, as proved in [Dic58, Theorem 265] (or use GAP to check it), 𝔸_{4} can be presented by generators g_{0} and x and relations

g_{0}^{2} = x^{3} = (g_{0}x)^{3} = 1.

Therefore, we need to prove Part 1. First notice that, since g_{i}g_{j} = g_{j}g_{i}, there exists w ∈ V \ {0} and λ_{i} ∈ ℂ such that ρ(g_{i})(w) = λ_{i}w for i = 0, 1, 2. For any 0 ≤ i, j ≤ 2, let γ_{ij} = x_{j}^{−1}g_{i}x_{j} ∈ C_{G}(g_{0}). It is easy to see that

γ = (γ_{ij}) = ( \begin{matrix}g_{0} & g_{2} & g_{1} \\ g_{1} & g_{0} & g_{1}^{m}g_{0}^{−1} \\ g_{2} & g_{2}^{m}g_{1}^{−1} & g_{0} \end{matrix} ) .

Then, W = span{x_{1} ⊗ w, x_{2} ⊗ w, x_{3} ⊗ w} is a braided vector subspace of M(C, ρ) of abelian type with Dynkin diagram given by

[figure: λ_{0}; λ_{0}; λ_{0}; λ_{0}^{m}; λ_{0}^{m^{2}−2}; λ_{0}^{m}]

For 𝔅(C, ρ) to be finite dimensional, we should have λ_{0} = −1 (see [Hec05, Table 3]) and m should be an even number, which contradicts our assumption. □

3. Two examples: M_{20} and M_{21}

In [Fan07] the five simple Mathieu groups are studied. In this section we will study finite dimensional Nichols algebras over the Mathieu groups M_{20} and M_{21} (= PSL(3, 4)). We begin with the non-simple Mathieu group M_{20}. For a definition and elementary properties of this group, see [Hup67, Chapter XII]. Since we do computations with GAP, we use the product in the symmetric groups as they do: the product of two permutations σ and τ means the composition of the permutation σ followed by τ.

We know (see for example [ATLAS]) that M_{20} = ⟨α, β⟩ as a subgroup of 𝕊_{20}, where

α = (1, 2, 4, 3)(5, 11, 7, 12)(6, 13)(8, 14)(9, 15, 10, 16)(1, 19, 20, 18),
β = (2, 5, 6)(3, 7, 8)(4, 9, 10)(11, 17, 12)(13, 16, 18)(14, 15, 19).

This is a group of order 960. The conjugacy classes are 1A, 2A, 2B, 3A, 4A, 4B, 4C, 5A, 5B (we are using the ATLAS notation for conjugacy classes, the name of a class begins with the order of its elements).

Proposition 3.1. The Mathieu group M_{20} is of type B.

Proof. By [AZ07, Remark 1.1] we know that the class 1A gives infinite dimensional Nichols algebras for every representation. Using GAP it is easy to check that all conjugacy classes of M_{20} are real. Therefore, by Lemma 2.1, the conjugacy classes with representatives of odd order (i.e. 3A, 5A, 5B) will also give infinite dimensional Nichols algebras.

All the remaining classes are dealt with Lemma 2.3. We just list the elements g_{1}, g_{2} and x_{1} for each of them. In all the cases, we put g_{0} = g_{1}g_{2} and x_{2} = x_{1}^{−1}. We have used GAP to find those elements, with the help of some scripts which may be downloaded from the third author’s web page: http://mate.dm.uba.ar/~lvendram/.

For class 2A, we take

g_{1} = (2, 3)(5, 19)(6, 13)(7, 18)(8, 14)(11, 17)(12, 20)(15, 16),
g_{2} = (2, 15)(3, 16)(5, 18)(6, 8)(7, 19)(11, 12)(13, 14)(17, 20),
x_{1} = (1, 4, 10)(3, 16, 15)(5, 11, 14)(6, 18, 17)(7, 12, 8)(13, 19, 20).

For class 2B,

g_{1} = (1, 15)(2, 10)(3, 4)(5, 13)(6, 7)(8, 18)(9, 16)(14, 19),
g_{2} = (1, 13)(2, 7)(3, 19)(4, 14)(5, 15)(6, 10)(8, 9)(16, 18),
x_{1} = (2, 19, 8)(3, 18, 6)(4, 9, 10)(5, 13, 15)(7, 14, 16)(12, 17, 20).

For class 4A,

g_{1} = (1, 2, 10, 15)(3, 4, 16, 9)(5, 14, 7, 8)(6, 19, 13, 18)(11, 12)(17, 20),
g_{2} = (1, 5, 4, 19)(2, 14, 16, 13)(3, 6, 15, 8)(7, 9, 18, 10)(11, 17)(12, 20),
x_{1} = (2, 6, 5)(3, 8, 7)(4, 10, 9)(11, 12, 17)(13, 18, 16)(14, 19, 15).

For class 4B,

g_{1} = (1, 3, 4, 15)(2, 10, 16, 9)(5, 8, 19, 6)(7, 14, 18, 13)(11, 20)(12, 17),
g_{2} = (1, 7, 9, 19)(2, 6, 3, 14)(4, 18, 10, 5)(8, 15, 13, 16)(11, 12)(17, 20),
x_{1} = (2, 6, 5)(3, 8, 7)(4, 10, 9)(11, 12, 17)(13, 18, 16)(14, 19, 15).

For class 4C,

g_{1} = (1, 16, 9, 15)(2, 10, 3, 4)(5, 6, 18, 14)(7, 13, 19, 8)(11, 17)(12, 20),
g_{2} = (1, 18, 10, 19)(2, 13, 15, 6)(3, 8, 16, 14)(4, 7, 9, 5)(11, 20)(12, 17),
x_{1} = (2, 6, 5)(3, 8, 7)(4, 10, 9)(11, 12, 17)(13, 18, 16)(14, 19, 15).
□

We do the same now with M_{21}:

Proposition 3.2. The Mathieu group M_{21} = PSL(3, 4) is of type B.

Proof. As in all groups, the trivial class 1A is of type B. Classes 3A, 3B, 5A and 5B are real, as can be checked with GAP using the function RealClasses. Then these classes are of type B by Lemma 2.1. Classes 7A and 7B are quasireal, as can be checked with GAP with the function PowerMaps. Then these classes are of type B by Lemma 2.2.

We are left with classes 2A, 4A, 4B, 4C. By ATLAS, we know that the semidirect product 2^{4} : 𝔸_{5} = M_{20} is a maximal subgroup of PSL(3, 4) (see [CCN^{+}85, pp. 23]). Now the result follows from 3.1, by looking at how classes of even order in M_{20} sit into M_{21}, by using the function PossibleClassFusions: since M_{20} is a subgroup of M_{21} there exists a monomorphism f : M_{20} ↪ M_{21} and therefore there exist a fusion of conjugacy classes. Moreover, with GAP we calculate all possible fusion of conjugacy classes and by inspection of all of these fusions it is easy to arrive to the desired result (see the file fgv.log for the results).

□

4. Nichols algebras over PSL(2, q)

In this section we deal with the groups PSL(2, q) = SL(2, 𝔽_{q})/{±I}, which have order \frac{(q−1)q(q+1)}{2}. Recall that if q ≠ 2, 3 then PSL(2, q) is simple (see eg. [AB95, Theorem 2.6.8]).

Table 2. Conjugacy classes of PSL(2, 𝔽_{q}) for q ≡ 1(4)

Representative  Number Size
C_{1}  c_{1} = ( \begin{matrix}1 \\ & 1 \end{matrix} )  1  1
C_{2}  c_{2} = \binom{1 1}{1}  1  \frac{q^{2}−1}{2}
C_{3}  c_{3} = \binom{1 Δ}{1}  1  \frac{q^{2}−1}{2}
C_{4} c_{4}(z) = ( \begin{matrix}x & Δy \\ y & x \end{matrix} ) (N(z) = x^{2} − Δy^{2} = 1, y ≠ 0)  \frac{q−1}{4}  q(q − 1)
C_{5}  c_{5} = ( \begin{matrix}i \\ & −i \end{matrix} )  1  \frac{q(q+1)}{2}
C_{6}  c_{6}(x) = ( \begin{matrix}x \\ & x^{−1} \end{matrix} ) (x ∉ \{±1, ±i\})  \frac{q−5}{4}  q(q + 1)

4.1. The case q ≡ 1(4). There are \frac{q+5}{2} conjugacy classes, which are listed in Table 2 (see [Ada02]).

Proposition 4.1. Classes C_{i} are of type B for i = 1, 2, 3, 4, 5.

Proof. The trivial class is of type B for any group. For class C_{2} (resp. C_{3}) we apply Lemma 2.1, since c_{2} and c_{2}^{−1} (resp. c_{3} and c_{3}^{−1}) are conjugate and have odd order p.

For i = 4, we notice that c_{4}(z)^{−1} = c_{4}(z^{−1}) = ( \begin{matrix}x & −Δy \\ −y & x \end{matrix} ), which is conjugate

to c_{4}(z). Indeed, c_{5}c_{4}(z)c_{5}^{−1} = c_{4}(z^{−1}). Then the result follows from Lemma 2.1, since the order of c_{4}(z) is a divisor of \frac{q+1}{2}, which is odd.

Table 3. Conjugacy classes of PSL(2, q) for q ≡ 3(4)

Representative  Number Size
C_{1}  c_{1} = ( \begin{matrix}1 \\ & 1 \end{matrix} )  1  1
C_{2}  c_{2} = \binom{1 1}{1}  1  \frac{q^{2}−1}{2}
C_{3}  c_{3} = \binom{1 Δ}{1}  1  \frac{q^{2}−1}{2}
C_{4} c_{4}(z) = ( \begin{matrix}x & Δy \\ y & x \end{matrix} ) (N(z) = x^{2} − Δy^{2} = 1, xy ≠ 0)  \frac{q−3}{4}  q(q − 1)
C_{5}  c_{5} = ( \begin{matrix}& −1 \\ 1 \end{matrix} )  1  \frac{q(q−1)}{2}
C_{6}  c_{6}(x) = ( \begin{matrix}x \\ & x^{−1} \end{matrix} ) (x ∉ \{±1\})  \frac{q−3}{4}  q(q + 1)

For i = 5, we notice that 𝔸_{4} is a subgroup of PSL(2, q) (see [Dic58, Ch. XII] or [Hup67, Satz II.8.18 (b)]). Since the only involutions in PSL(2, q) lie in class C_{5}, Lemma 2.3 part 2 and Remark 1.3 say that C_{5} is of type B. □

The centralizer of c_{6}(x) is isomorphic to 𝔽_{q}^{∗}/\{±1\} ≃ ℤ/\frac{q−1}{2}.

Proposition 4.2. If dim 𝔅(C_{6}, ρ) < ∞ then c_{6}(x) has even order and ρ(c_{6}(x)) = −1.

Proof. Again, C_{6} is a real class, since conjugating c_{6}(x) by ( \begin{matrix}& −1 \\ 1 \end{matrix} ) gives c_{6}(x^{−1}) = c_{6}(x)^{−1}. The result then follows from Lemma 2.1. □

4.2. The case q ≡ 3(4). There are \frac{q+5}{2} conjugacy classes, which are listed in Table 3 (see [Ada02]).

Proposition 4.3. Classes C_{i} are of type B for i = 1, 2, 3, 5, 6.

Proof. It is entirely analogous to that of Proposition 4.1. The only difference is that now classes C_{6} have elements of odd order, instead of classes C_{4}. □

The centralizer of c_{4}(x) is isomorphic to ℤ/\frac{q+1}{2}. Again, by using Lemma 2.1 we have

Proposition 4.4. If dim 𝔅(C_{4}, ρ) < ∞, then c_{4}(z) has even order and ρ(c_{4}(z)) = −1.

5. Nichols algebras over PGL(2, q)

We deal in this section with PGL(2, q) = GL(2, q)/{tI : t ∈ 𝔽_{q}^{×}}, which has order (q −1)q(q +1). Since PGL(2, 3) ≃ 𝕊_{4} and PGL(2, 5) ≃ 𝕊_{5} (which are treated in [AZ07]) we may assume that q > 5. There are q + 2 conjugacy classes, which are listed in Table 4 (see [Ada02]).

Table 4. Conjugacy classes of PGL(2, q)

Representative  Number Size
C_{1}  c_{1} = ( \begin{matrix}1 \\ & 1 \end{matrix} )  1  1
C_{2}  c_{2} = \binom{1 1}{1}  1  q^{2} − 1
C_{3} c_{3}(x) = ( \begin{matrix}x \\ & 1 \end{matrix} ) (x ≠ ±1)  \frac{q−3}{2}  q(q + 1)
C_{4}  c_{4} = ( \begin{matrix}−1 \\ & 1 \end{matrix} )  1  \frac{q(q+1)}{2}
C_{5} c_{5}(z) = ( \begin{matrix}x & Δy \\ y & x \end{matrix} ) (xy ≠ 0)  \frac{q−1}{2}  q(q − 1)
C_{6}  c_{6} = ( \begin{matrix}& Δ \\ 1 \end{matrix} )  1  \frac{q(q−1)}{2}

By considering the inclusion PSL(2, q) ↪ PGL(2, q), we get from the previous section that the following classes are of type B:

C_{1}, C_{2}
C_{3} if q ≡ 3 (mod 4) and x ∈ 𝔽_{q}^{2}
C_{4} if q ≡ 1 (mod 4) (because it is class C_{5} in PSL(2, q))
C_{5} if q ≡ 1 (mod 4) and N(z) = x^{2} − Δy^{2} ∈ 𝔽_{q}^{2}
C_{6} if q ≡ 3 (mod 4) (because it is class C_{5} in PSL(2, q))

We consider now the other cases. They are not of type B, but one can restrict the representations anyway.

The centralizer of c_{3}(x) is given by the classes in PGL(2, q) of matrices ( \begin{matrix}σ \\ & 1 \end{matrix} ) and therefore it is isomorphic to 𝔽_{q}^{×}.

Proposition 5.1. Let q ≡ 1 (mod 4). If dim 𝔅(C_{3}, ρ) < ∞, then x has even order and ρ(c_{3}(x)) = −1.

Proof. Notice that C_{3} is a real class, and use Lemma 2.1. □

The centralizer of c_{4} is given by the classes in PGL(2, q) of matrices ( \begin{matrix}1 \\ & σ \end{matrix}) and ( \begin{matrix}& 1 \\ σ \end{matrix}) for σ ∈ 𝔽_{q}^{×}. It is easy to see that this group is isomorphic to 𝔻_{q−1}. In the presentation by r, s given at §2.2, c_{4} corresponds to r^{\frac{q−1}{2}} .

Proposition 5.2. Let q ≡ 3 (mod 4) and let ρ be a representation of C_{PGL(2,q)}(c_{4}) with character χ. If dim 𝔅(C_{4}) < ∞, then χ ∈ {χ_{3}, χ_{4}, μ_{h} (h odd)} (see Table 1).

Proof. Since c_{4} is an involution, we must have ρ(c_{4}) = −1. Then, the result follows by inspection of the Table. □

The centralizer of c_{5}(z) is isomorphic to E^{∗}/𝔽_{q}^{∗} ≃ ℤ/q + 1 (see [Ada02]).

Proposition 5.3. Let q ≡ 3 (mod 4). If dim 𝔅(C_{5}, ρ) < ∞, then c_{5}(z) has even order and ρ(g) = −1.

Proof. In a similar way as the proof of Prop. 4.1, we see that classes C_{5} are real by conjugating c_{5}(z) with c_{4}. Now, we apply Lemma 2.1. □

The centralizer of c_{6} is given by the classes in PGL(2, q) of matrices ( \begin{matrix}x & ±Δy \\ y & x \end{matrix} ). It is easy to see that this group is isomorphic to 𝔻_{q+1}. In the presentation by r, s given at §2.2, c_{6} corresponds to r^{\frac{q+1}{2}} .

Proposition 5.4. Let q ≡ 1 (mod 4) and let χ be the character of the representation ρ. If dim 𝔅(C_{6}) < ∞, then χ ∈ {χ_{3}, χ_{4}, μ_{h} (h odd)} (see Table 1).

Proof. Analogous to that of Prop. 5.2. □

6. The class of order 4 in SL(2, q)

In this section we use the tools in the present paper to improve a result in [FGV07]. There, we considered in SL(2, q) the classes C_{7} of c_{7}(x) = ( \begin{matrix}x & 0 \\ 0 & x^{−1} \end{matrix} ), and C_{8} of c_{8}(z) = ( \begin{matrix}x & Δy \\ y & x \end{matrix} ), where z = x + δy ∈ E \ 𝔽_{q}. Propositions 3.4 (resp. 3.5) in [FGV07] prove that in order for 𝔅(C_{7}) (resp. 𝔅(C_{8})) to be finite dimensional, the order of x (resp. z) must be even. On the other hand, when q = 3 there is only one class C_{8}, and it is proved in [AFGV08] that it is of type B. We prove here that SL(2, 3) is a subgroup of SL(2, q). This implies that when q ≡ 1 (mod 4) and x has order 4, the class of c_{7}(x) ∈ SL(2, q) is of type B, and when q ≡ 3 (mod 4) and z has order 4, the class of c_{8}(z) ∈ SL(2, q) is of type B.

One possible presentation of SL(2, 3) is given by generators x, y and relations x^{3} = y^{4} = (xy^{3})^{3} = 1, xy^{2} = y^{2}x. In fact, one can take

x = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) , y = (\begin{matrix}1 & 1 \\ 1 & 2 \end{matrix}) .

Then, if a group G has elements A, B, C of orders 3, 3 and 4 respectively such that ABC = [B, C^{2}] = 1, then G has a subgroup isomorphic to SL(2, 3). Indeed, take the map x ↦ B, y ↦ C^{3}. It is easy to see that this defines a map SL(2, 3) → G. It is injective because otherwise the orders of A, B or C would be smaller than stated, as can be seen by considering the normal subgroups of SL(2, 3) (the nontrivial ones being isomorphic to C_{2} and the quaternion group Q of order 8). Also, any C ∈ SL(2, q) has order 4 if and only if tr C = 0. Indeed, it is easy to see that if α = tr C, then C^{4} = (α^{3} − 2α)C + (1 − α^{2}), which implies the claim. Therefore, there is only one class in SL(2, q) of order 4. This means that the embedding we shall find SL(2, 3) ↪ SL(2, q) sends the class of ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) ∈ SL(2, 3) of type B to it.

We prove now the existence of the embedding SL(2, 3) ↪ SL(2, q). Since SL(2, p) embeds in SL(2, q), it is enough to prove that SL(2, 3) embeds in SL(2, p). Let x ∈ 𝔽_{p}^{×} be of order p − 1 and let G_{l} be the class of c_{7}(x^{l}). Let z ∈ E generate the group {a + δb ∈ E | a^{2} − Δb^{2} = 1} and let H_{m} be the class of c_{8}(z^{m}). The character table of SL(2, p) restricted to the classes G_{l} and H_{m} is given in Table 5, where we write c[\frac{a}{b}] = e^{\frac{2πia}{b}} + e^{− \frac{2πia}{b}}. There, 1 ≤ i ≤ \frac{p−3}{2} for ζ_{i} and 1 ≤ i ≤ \frac{p−1}{2} for θ_{i}.

Table 5. Character table on classes G_{l} and H_{m}.

1 ψ  ζ_{i}  ξ_{1}  ξ_{2}  θ_{i}  η_{1}  η_{2}
deg 1 p p + 1 \frac{1}{2}(p + 1) \frac{1}{2}(p + 1) p − 1 \frac{1}{2}(p − 1) \frac{1}{2}(p − 1)
G_{l} 1 1 c[\frac{il}{p−1}] (−1)^{l}  (−1)^{l}  0  0  0
H_{m} 1 −1  0  0  0  −c[\frac{im}{p+1}] (−1)^{m+1} (−1)^{m+1}

When p ≡ 1 (mod 3), classes G_{l} have order 3 for l = \frac{p−1}{3} and l = \frac{2(p−1)}{3}. On the other hand, when p ≡ 2 (mod 3), classes H_{m} have order 3 for m = \frac{p+1}{3} and m = \frac{2(p+1)}{3}. We use then classes G_{l} and H_{m} for A, B or C depending on the class of p (mod 12).

We apply then the well-known formula (see e.g. [Go68, Theorem 2.12])

S(C_{i}, C_{j}, C_{k}) = \frac{|C_{i}||C_{j}||C_{k}|}{|G|} ∑_{χ} \frac{χ(C_{i})χ(C_{j})χ(C_{k})}{χ(1)}

which counts solutions of the equation abc = 1 with a, b, c respectively in classes C_{i}, C_{j} and C_{k}. Therefore, it is enough to see that for any p we have S(C_{i}, C_{j}, C_{k}) > 0, where classes C_{i}, C_{j} and C_{k} have orders 3, 3 and 4. We have then the following possibilities:

• If p ≡ 5 (mod 12), the element of order 4 belongs to the class G with l = \frac{p−1}{4} and the element of order 3 belongs to the class H with m = \frac{p+1}{3}.

ON NICHOLS ALGEBRAS OVER PGL(2, q) AND PSL(2, q) 10

Then,

S(H, H, G) = \frac{p^{3}(p − 1)^{2}(p + 1)}{p(p − 1)(p + 1)} (1 + \frac{1}{p}) = (p − 1)p(p + 1) > 0.

• If p ≡ 7 (mod 12), the element of order 4 belongs to the class H with m = \frac{p+1}{4} and the element of order 3 belongs to the class G with l = \frac{p−1}{3}. Then,

S(G, G, H) = \frac{p^{3}(p + 1)^{2}(p − 1)}{p(p − 1)(p + 1)} (1 − \frac{1}{p}) = (p − 1)p(p + 1) > 0.

• If p ≡ 1 (mod 12) S(G_{\frac{p−1}{3}} , G_{\frac{p−1}{3}} , G_{\frac{p−1}{4}} )

= \frac{p^{2}(p + 1)^{2}}{p − 1} (1 + \frac{1}{p} + ∑_{j=1}^{\frac{p−3}{2}} \frac{1}{p + 1}c[\frac{j}{3}]^{2}c[\frac{j}{4}] + 4\frac{(−1)^{\frac{p−1}{4}}}{p + 1} )

≥ \frac{p^{2}(p + 1)^{2}}{p − 1} (1 + \frac{1}{p} + ∑_{j=1}^{\frac{p−3}{2}} \frac{8}{p + 1} cos(\frac{2πj}{3} )^{2} cos(\frac{2πj}{4} ) − \frac{4}{p + 1})

= \frac{p^{2}(p + 1)^{2}}{p − 1} (1 + \frac{1}{p} + \frac{8}{p + 1} ∑_{l=1}^{[ \frac{p−3}{4} ]} cos(\frac{4πl}{3} )^{2} cos(πl) − \frac{4}{p + 1})

= \frac{p^{2}(p + 1)^{2}}{p − 1} (1 + \frac{1}{p} + \frac{8}{p + 1}( ∑_{\begin{matrix}l=1 \\ l≡0(3) \end{matrix}}^{[ \frac{p−3}{4} ]} cos(πl) + ∑_{\begin{matrix}l=1 \\ l≢0(3) \end{matrix}}^{[ \frac{p−3}{4} ]} \frac{1}{4} cos(πl)) − \frac{4}{p + 1})

= \frac{p^{2}(p + 1)^{2}}{p − 1} (1 + \frac{1}{p} + \frac{2}{p + 1}( ∑_{\begin{matrix}l=1 \\ l≡0(3) \end{matrix}}^{[ \frac{p−3}{4} ]} 3 cos(πl) + ∑_{l=1}^{[ \frac{p−3}{4} ]} cos(πl)) − \frac{4}{p + 1})

≥ \frac{p^{2}(p + 1)^{2}}{p − 1} (1 + \frac{1}{p} − \frac{12}{p + 1}) > 0

• If p ≡ 11 (mod 12)

S(H_{\frac{p+1}{3}} , H_{\frac{p+1}{3}} , H_{\frac{p+1}{4}} )

= \frac{p^{2}(p − 1)^{2}}{p + 1} (1 − \frac{1}{p} − ∑_{j=1}^{\frac{p−1}{2}} \frac{1}{p − 1}c[\frac{j}{3}]^{2}c[\frac{j}{4}] + 4\frac{(−1)^{\frac{p+5}{4}}}{p − 1} )

≥ \frac{p^{2}(p − 1)^{2}}{p + 1} (1 − \frac{1}{p} − ∑_{j=1}^{\frac{p−1}{2}} \frac{8}{p − 1} cos(\frac{2πj}{3} )^{2} cos(\frac{2πj}{4} ) − \frac{4}{p − 1})

= \frac{p^{2}(p − 1)^{2}}{p + 1} (1 − \frac{1}{p} − \frac{8}{p − 1} ∑_{l=1}^{[ \frac{p−1}{4} ]} cos(\frac{4πl}{3} )^{2} cos(πl) − \frac{4}{p − 1})

= \frac{p^{2}(p − 1)^{2}}{p + 1} (1 − \frac{1}{p} − \frac{8}{p − 1}( ∑_{\begin{matrix}l=1 \\ l≡0(3) \end{matrix}}^{[ \frac{p−1}{4} ]} cos(πl) + ∑_{\begin{matrix}l=1 \\ l≢0(3) \end{matrix}}^{[ \frac{p−1}{4} ]} \frac{1}{4} cos(πl)) − \frac{4}{p − 1})

= \frac{p^{2}(p − 1)^{2}}{p + 1} (1 − \frac{1}{p} − \frac{2}{p − 1}( ∑_{\begin{matrix}l=1 \\ l≡0(3) \end{matrix}}^{[ \frac{p−1}{4} ]} 3 cos(πl) + ∑_{l=1}^{[ \frac{p−1}{4} ]} cos(πl)) − \frac{4}{p − 1})

≥ \frac{p^{2}(p − 1)^{2}}{p + 1} (1 − \frac{1}{p} − \frac{4}{p − 1}) > 0

Acknowledgement. We thank Nicolás Andruskiewitsch and Fantino Fantino for fruitful discussions. We have used GAP [GAP] to do some of the computations.

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SF, MG, LV: Departamento de Matemática - FCEyN, Universidad de Buenos Aires, Pab I - Ciudad Universitaria, (1428) Buenos Aires - Argentina

LV: Instituto de Ciencias, Universidad Nacional General Sarmiento, J. M. Gutierrez 1150 - Los Polvorines (1613), Provincia de Buenos Aires - Argentina

   E-mail address: (sfreyre|matiasg|lvendramin)@dm.uba.ar