NOTTINGHAM LIE ALGEBRAS WITH DIAMONDS

NOTTINGHAM LIE ALGEBRAS WITH DIAMONDS OF FINITE TYPE A. CARANTI AND S. MATTAREI Abstract. We study a class of positively graded Lie algebras with a pattern of homogeneous components similar to that of the graded Lie algebra associated to the Nottingham group with respect to its lower central series.

1. Introduction The conjectures concerning pro-p-groups of finite coclass proposed in [28] have been proved in [27, 35]. There are a number of related questions that arise naturally in the context of (graded) Lie algebras. One can consider for instance graded Lie algebras of maximal class [34, 13, 15, 20, 16], and also groups and algebras that are of finite coclass or more generally narrow in some technical sense [36, 33]. (We remind the reader that the coclass of a residually nilpotent Lie algebra L is defined as sup{dim(L/γi (L)) − i : i = 1, 2, 3, . . .}, where γi (L) is the ith term of the lower central series of L. Lie algebras of coclass one are said to be of maximal class.) A narrowness condition weaker than being of finite coclass is that of being of finite width [21]. This applies to both groups and (graded) Lie algebras. For instance, a positively graded Lie algebra generated by its first homogeneous component has finite width if there is an upper bound on the dimensions of its homogeneous components. The maximum of these dimensions is the width of the algebra. Groups and algebras of finite width can be sorted according to a finer invariant, introduced in [21], called obliquity. (Pro-)p-groups of width two and obliquity zero have been introduced in [7, 6] under the name of thin groups. The corresponding notion for graded Lie algebras has been introduced in [14]. The present paper is part of a classification effort for thin Lie algebras over fields of positive characteristic. A (positively) graded Lie algebra M L= Li i≥1

Date: 27 September 2002. Key words and phrases. Graded Lie algebras of finite width. The authors are grateful to Ministero dell’Università e della Ricerca Scientifica, Italy, for financial support via the project “Graded Lie algebras and pro-p-groups of finite width”. The first author is very grateful to the School of Mathematical Studies, Australian National University, Canberra, for the kind hospitality and financial support while this paper was being written; and to INdAM-GNSAGA for financial support towards travel expenses. The authors are members of INdAM-GNSAGA.

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A. CARANTI AND S. MATTAREI

is said to be thin if dim(L1 ) = 2, L is generated by L1 , and the following covering property holds: for all i ≥ 1, if u ∈ Li , and u 6= 0, then [uL1 ] = Li+1 . The covering property implies that the non-zero homogeneous components of L have dimension 1 or 2. The components of dimension 2 are called diamonds. We may note at this stage that we will state our results for infinite-dimensional Lie algebras. This we do for the sake of simplicity, as statements about the structure of finite-dimensional Lie algebras that are not quotients of infinite-dimensional ones tend to be involved. Our proofs, however, give implicitly what appears to be a reasonably good description of the structure of the finite-dimensional algebras too. This is to say that the upper bounds we find for the dimensions of the algebras fit with the actual dimensions, as provided by explicit constructions, or suggested by our computer calculations. (See the remarks below concerning computations.) In a thin Lie algebra L, the first homogeneous component L1 is always a diamond. If there are no other diamonds, then L is of maximal class. Now it is proved in [14, 3, 12] that if there are other diamonds, the second one can only occur in weight 3, 5, q, or 2q − 1, where q is a power of the characteristic of the underlying field. The case when the second diamond occurs in weight 3 or 5 has been investigated in [14, 18]; see also [30]. (Here and in the following one has to assume the characteristic of the underlying field to be big enough.) The case of weight 2q − 1 is studied in [12, 1]. The graded Lie algebra associated to the Nottingham group over the field with p elements, with respect to its lower central series, has been studied in [8]. This is a thin Lie algebra, with second diamond in weight p, and can be described as (the positive part of) a twisted loop algebra of the smallest Zassenhaus algebra W (1 : 1). (We use a standard notation for simple Lie algebras of Cartan type, as used, for example, in [24] or [5].) More generally, in [8] certain twisted loop algebras of the Zassenhaus algebras W (1 : n) over the field with p elements have been studied. These are thin Lie algebras, with diamonds occurring exactly in the weights congruent to 1 modulo q − 1, where q = pn . The goal of this paper is to give a complete description of a more general class of thin Lie algebras with diamonds (possibly fake ones: see the next section for the technical details) in these weights. These algebras we call Nottingham Lie algebras. We are naturally led to discuss some related features of thin Lie algebras with the second diamond in weight q. The Lie algebras we study here turn out to be the ones constructed in [4, 1, 2] as the twisted loop algebras of the finite-dimensional, simple, modular Lie algebras H(2 : (1, n); ω1 ) or H(2 : (1, n); ω0 ) (the latter being possibly extended by an outer derivation). What we prove here is a uniqueness result (Theorems 2.3 and 2.4) for these loop algebras, that is best stated by saying that they are nearly finitely presented, in a technical sense to be described in the next section. Our investigations have been guided by extensive computations with the Australian National University p-Quotient Program [19]. As a result, our proofs have a computational flavour, and consist mostly of commutator expansions, that involve

NOTTINGHAM LIE ALGEBRAS

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evaluations of binomial coefficients over a finite prime field via Lucas’ Theorem ([29]; see also [22] for some recent developments). Our proofs tend to be heavy in calculations. One of the reasons for that is that the algebras we describe encode some moderately involved modular combinatorial identities. An example of this is given in Section 8. It might also be noted that Theorems 2.3 and 2.4 yield an upper bound for the second cohomology groups (with coefficients in the trivial module) of the finitedimensional, modular simple algebras involved. For instance, Theorem 2.3 yields that the algebras H(2 : (1, n); ω1 ) have trivial second cohomology group. (See the remark following the statement of Theorem 2.3.) We assume this is well-known to experts, although we have been unable to find a reference for it; the second cohomology group of the algebras H(2 : (1, n); ω0 ) is computed in [17]. 2. Statement of results Here and in the rest of the paper, we will always assume M L= Li i≥1

to be an infinite-dimensional graded Lie algebra over a field F of characteristic p > 5. Such an L is said to be thin if dim(L1 ) = 2, L is generated by L1 , and the following covering property holds: (2.1)

for all i ≥ 1, if u ∈ Li , and u 6= 0, then [uL1 ] = Li+1 .

This implies in particular that L is centreless, that is the centre Z(L) = { a ∈ L : [ab] = 0 for all b ∈ L } of L equals { 0 }. For certain technical reasons that we will be explaining shortly, it is useful to introduce a weaker definition. We say that the algebra L is quasithin if dim(L1 ) = 2, L is generated by L1 , and the following weak covering property holds: (2.2)

for all i ≥ 1, if u ∈ Li , and u 6∈ Z(L), then [uL1 ] = Li+1 .

The homogeneous components of a quasithin algebra have dimension 1 or 2. A component Li of dimension 2 is called a diamond. If Li intersect the centre trivially, then Li is called a true diamond. If Li ∩Z(L) 6= 0, then Li is called a fake diamond. Note that in this case the intersection has dimension 1. In fact, if Li ⊆ Z(L), then because L is generated by L1 , all Lj with j > i are zero. This contradicts our blanket assumption that L is infinite-dimensional. Note that if L is quasithin, then its central quotient L/Z(L) will be thin. Let L be a thin algebra over the field F. From now on we assume that the second diamond occurs in weight q, for some power q = pn of p, with n ≥ 1. Our goal in this paper is to show that certain thin Lie algebras of this kind are uniquely determined by appropriate finite-dimensional quotients. This is equivalent to saying that there is only one thin Lie algebra that satisfies an appropriate finite set of relations. The Lie algebra presented by these relations, however, is not the thin

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Lie algebra itself, but a certain central (or second central) extension of it. This is where the weaker concept of quasithinness turns handy. It follows from [11] and [8] that there are two generators x and y of L such that the following properties hold. First of all h y i = CL1 (L2 ) = · · · = CL1 (Lq−2 ).

(2.3)

Here CL1 (Li ) = { a ∈ L1 : [ab] = 0 for all b ∈ Li } is the centralizer of Li in L1 . Let v2 = [yxq−2 ] ∈ Lq−1 . Here we use two pieces of notation: Notation 2.1. (1) Repeated commutators are left-normed, that is, recursively [abc] = [[ab]c]. (2) [yxn ] = [y |x .{z . . x}]. n

Then v2 6= 0, and [v2 xx] = [v2 yy] = 0,

[v2 yx] = −2 [v2 xy].

For the purposes of this paper, it is convenient to use a different set of generators, namely z = x + y and y. Because of (2.3), the value of v2 is not affected by this change, and we have [v2 zz] = [v2 zy] + [v2 yz],

[v2 yy] = 0,

[v2 zy] = −[v2 zz].

It follows that [v2 yz] = 2[v2 zz]. We say Lq is a diamond of type −1. More generally, suppose that in a quasithin algebra L there is a diamond Li , for some i > 1, with the following properties. The previous component Li−1 is either one-dimensional, or a fake diamond. Given a non-central element v ∈ Li−1 , then the following hold (possibly modulo Z(L)) (2.4)

[vzz] = [vzy] + [vyz],

[vyy] = 0,

[vzy] = µ[vzz],

for some µ ∈ F. (Here we have [vyz] = (1−µ)[vzz]. Note that our assumption that L be infinite-dimensional implies [vzz] 6= 0 here.) Then Li is said to be a diamond of (finite) type µ. (Note that we do not attach a type to the first diamond L1 .) Computations suggest that among thin algebras with second diamond in weight q there are some where [vzy] + [vyz] = 0,

[vyy] = 0,

[vzz] = 0.

We refer to these diamonds as being of infinite type. In this paper we concentrate on those algebras where all diamonds are of finite type. Note then when µ = 1 the diamond Li is a fake, as [vy] is central in this case; and the same happens when µ = 0, as [v, z − y] is central then. For all other values of µ it is easy to see that the diamond Li is a true one. In Section 5 we begin with proving Lemma 2.2. Let L be an infinite-dimensional quasithin algebra with second diamond in weight q and true. Then L has no true diamonds in any weight k, for q < k < 2q − 1.


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We may note that there are infinite-dimensional thin (soluble) algebras with just two (true) diamonds, the first one in weight 1, and the second one in weight q. The main result of this paper is Theorem 2.4 below, or rather the uniqueness part of it (see the comments after the statement). For the sake of clarity, we state separately the neater case when the type µ3 of the third diamond does not lie in the prime field. Theorem 2.3 (Big field case). Let L be a thin algebra with second diamond in weight q = pn , where p is the characteristic of the underlying field, and n > 0, and fix generators z and y for L as above. Suppose L has a diamond of finite type µ3 in weight 2q − 1, and that µ3 does not lie in the prime field. Consider the set R of homogeneous relations in z and y of weight at most 2q satisfied by L. Then the algebra M = Mq,µ3 given by the presentation M = h z, y : R i is isomorphic to L. M has diamonds of finite types in all weights congruent to 1 modulo q − 1. The types of these diamonds form an arithmetic progression with difference µ3 − µ2 = µ3 + 1, so that the type of the k-th diamond L1+(k−1)(q−1) , for k > 1, is µk = −1 + (k − 2)(µ3 + 1). As we said in the Introduction, in this paper we prove the uniqueness part of Theorems 2.3 and 2.4: see the remarks after Theorem 2.4. The existence of an algebra M as described in Theorem 2.3 follows from [4], where M is constructed as a twisted loop algebra of an algebra H(2 : (1, n); ω1 ). The periodic structure of such a loop algebra is reflected in the periodicity of the sequence of the µi . We have mentioned in the Introduction that Theorem 2.3 can be used to prove that the second cohomology group of H(2 : (1, n); ω1 ) is trivial. (We always assume our cohomology groups to have coefficients in the trivial module.) In fact, let M 0 be the corresponding twisted loop algebra of the universal covering algebra of H(2 : (1, n); ω1 ). If the second cohomology of H(2 : (1, n); ω1 ) is nonzero, then M 0 has infinite-dimensional centre Z(M 0 ). This contradicts the fact that M ∼ = M 0 /Z(M 0 ) is finitely presented (see for instance [31], [32, 2.2.3] and [9]). Therefore Theorem 2.3, together with the results of [4], yields that the second cohomology group of H(2 : (1, n); ω1 ) is trivial, as claimed in the Introduction. Note, here and in Theorem 2.4, that it suffices to take as relations only homogeneous linear combinations of left-normed commutators in z and y. Here and in Theorem 2.4, the relations R can be taken as an appropriate subset of the relations described in Section 4. We now turn to the case when µ3 lies in the prime field. Actually, Theorem 2.4 as stated deals with the general case, and specializes to Theorem 2.3 when µ3 does not lie in the prime field.

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Since this case is rather technically involved, we have discussed the special cases that occur in it later in this section. Also, we have deferred a thorough discussion of the complete structural details for Section 4. Theorem 2.4 (General/prime field case). Let L be a thin algebra with second diamond in weight q = pn , where p is the characteristic of the underlying field, and n > 0, and fix generators z and y for L as above. Suppose L has a diamond of finite type µ3 in weight 2q − 1. Suppose µ3 6= −2, −1, 0, 1. Consider the set R of homogeneous relations in z and y of weight at most 2q satisfied by L. Consider the algebra M = Mq,µ3 given by the presentation M = h z, y : R i . Then M has a central ideal I such that M/I is quasithin, with diamonds of finite types in all weights congruent to 1 modulo q −1. The types of these diamonds form an arithmetic progression with difference µ3 − µ2 = µ3 + 1, so that the type of the k-th diamond L1+(k−1)(q−1) , for k > 1, is µk = −1 + (k − 2)(µ3 + 1). The thin algebra L is uniquely determined as M/Z(M ). The ideal I has a basis as follows. If the k-th diamond has type 1, for some k, then I contains a basis element in the homogeneous component immediately following the (k + 1)-th diamond. If the k-th diamond has type 2, for some k, then I contains n − 1 basis elements in distinct weights between the k-th diamond, and the (k + 1)-th one. We discuss what happens for the remaining values µ3 = −2, −1, 0, 1 below. In [1, 2] the positive parts of certain twisted loop algebras N = Nq,µ3 (for parameters q and µ3 as in Theorem 2.4, with µ3 lying in the prime field) of suitable central extensions of the finite-dimensional, simple, modular Hamiltonian Lie algebras H(2 : (1, n); ω0 ) (see [23], [37], [24]) are constructed. (In two cases, one has to take the extension of such an algebra by an outer derivation.) One can choose two elements z, y ∈ N such that N is generated by z, y as a Lie algebra, and z and y satisfy the relations R of Theorem 2.4. Therefore N is a homomorphic image of M . Note that the n + 1 central basis elements which we obtain every period (where q = pn ) correspond to a basis for the second cohomology group of H(2 : (1, n); ω0 ) with values in the trivial module. This constitutes the existence part of Theorem 2.4. In Sections 5–7 we prove the uniqueness part of Theorem 2.4, by providing upper bounds for the dimensions of the homogeneous components of M , and enough relations to show that the diamond structure of M is as stated. The exact location of the elements of the ideal I lying after a diamond of type 2 is discussed in Sections 4 and 7. The construction of M given in [1, 2, 4] shows that M has infinite-dimensional centre when µ3 lies in the prime field. It follows that L = M/Z(M ) itself is not


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finitely presented – this is the situation we had in mind in the Introduction when we dubbed L as nearly finitely presented. The case µ3 = −1. The case when µ3 = −1 in Theorem 2.4 is dealt with for p > 2 in [8, 10, 9] – the true diamonds occur in the same places, and have constant type −1. Depending on the characteristic of the underlying field, there are various fake diamonds here. When q = p in this case, the resulting algebra L is the graded Lie algebra associated to the Nottingham group with respect to its lower central series, that is, the positive part of a twisted loop algebra of the smallest Zassenhaus algebra. The case µ3 = −2. The case when µ3 = −2 in Theorem 2.4 differs from the general one only in one respect. The algebra M/Z(M ) is not thin, as it has a nonzero centre. Here L is still uniquely determined, though, as the quotient M/Z2 (M ), where Z2 (M ) = { a ∈ M : [abc] = 0 for all b, c ∈ L } is the second centre of M . This is explained in detail in Section 4. This is one of the two cases in [1, 2] in which an outer derivation enters in the construction. (The other case occurs for µ3 = −3.) It is this outer derivation that is responsible for the occurrence of a second centre. The case µ3 = 1. When µ3 = 1, the relations up to weight 2q do not suffice to force the conclusions of Theorem 2.4. We need to enforce a further relation, of weight 3q − 1, that prescribes the type µ4 of the fourth diamond. Only the two choices µ4 ∈ { 0, 3 } lead to an infinite-dimensional algebra. When µ4 = 3, the statements in Theorem 2.4 are still valid once the extra relation is added. When µ4 = 0, there are other algebras, as shown by David Young [38]. The case µ3 = 0. When µ3 = 0, we obtain that the fourth diamond occurs in weight 3q − 2, and has type µ4 = 1. But here too the relations up to weight 2q do not suffice to force the conclusions of Theorem 2.4. We need to enforce a further relation, of weight 4q − 2, that prescribes the type µ5 of the fifth diamond. The only choice of µ5 that leads to an infinite-dimensional algebra is µ5 = 2. The statements in Theorem 2.4 are still valid once the extra relation is added. 3. Methods As mentioned in the Introduction, our methods are elementary and direct, consisting mainly of commutator expansions. For this we use repeatedly the generalized Jacobi identity n n X X n i n i n−i j n [u[yz ]] = (−1) [uz yz ] ≈ (−1) [uz n−j yz j ]. i j i=0 j=0 Here a ≈ b stands for a = ±b. We also use Lucas’ theorem to evaluate binomial coefficients modulo a prime p, usually in the following form: if q is a power of p, a, b, c, d are non-negative integers, and b, d < q, then aq + b a b ≡ · (mod p). cq + d c d

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4. The adjoint representation In this section we provide a compact form of the multiplication table of the algebras M of Section 2, by specifying the values of the adjoint representation of M on its generators z and y. This means in practice describing the (homogeneous) relations among leftnormed commutators in z and y. By taking these relations up to the appropriate weights, as described in Section 2, one obtains the set of defining relations R referred to in Theorems 2.3 and 2.4. The resulting set of (homogeneous) relations is definitely not minimal. The algebra M is generated by two elements z and y, of weight one. The first relation states that y centralizes L2 = [L1 L1 ] = h [yz] i: (4.1)

[yzy] = 0.

Our main assumption that L is thin with the second diamond occurs in weight q = pn , and the result of [11], lead to the set of relations [yz l y] = 0, for 1 ≤ l < q − 2.

(4.2)

Now the discussion of [8] forces the diamond in weight q to be of type −1, as discussed in Section 2. That is, if we write v2 = [yz q−2 ], we have the relations (4.3)

[v2 zz] = [v2 zy] + [v2 yz], [v2 zy] = −[v2 zz],

[v2 yy] = 0,

[v2 yz] = 2 [v2 zz].

In Section 5 we prove Lemma 2.2 of Section 2. This leads to the further relations (4.4)

[v2 z l y] = 0, for 2 ≤ l < q − 1.

Actually, the results of Section 5 show that it is enough to take the single relation [v2 z q−3 y] = 0. Write v3 = [v2 z q−1 ] = [yz 2q−3 ]. We can now prescribe the third diamond to be of arbitrary finite type µ3 : (4.5)

[v3 zz] = [v3 zy] + [v3 yz], [v3 zy] = µ3 [v3 zz],

[v3 yy] = 0,

[v3 yz] = (1 − µ3 ) [v3 zz].

When we are in the general case of Theorem 2.4, that is, µ3 6= −2, −1, 0, 1, the relations we have listed so far determine M . This yields the following recursive description of M . Define (4.6)

vi = [vi−1 z q−1 ] = [yz (i−1)(q−1)−1 ],

for i > 2, and write recursively µi = µi−1 + (µi−1 − µi−2 ). Therefore the µi form an arithmetic progression of difference µ3 − µ2 = µ3 + 1, and µi = −1 + (i − 2)(µ3 + 1).


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If µi−1 6= 1, we have [vi zz] = [vi zy] + [vi yz], (4.7)

[vi zy] = µi [vi zz],

[vi yy] = 0,

[vi yz] = (1 − µi ) [vi zz]

[vi zzy] = 0. Therefore L(i−1)(q−1)+1 is a diamond of type µi . If µi−1 = 1 we have µi [vi yz] − (1 − µi ) [vi zy] = 0 (4.8)

[vi yy] = 0,

[vi zzy] = [vi zyy] = [vi yzy] = 0, [[vi zz] − [vi zy] − [vi yz], z] = 0, [[vi zy] − µi [vi zz], z] = 0,

[[vi yz] − (1 − µi ) [vi zz], z] = 0.

(Note that of the last three relations only one is needed.) In this case the element [vi zy] − µi [vi zz] is central in M and lies in I, as in the statement of Theorem 2.4. Note what happens when µi−2 = 2 and µi−1 = 1. Here the difference in the arithmetic progression of the µ is −1, so that µ3 = −2, and µi = 0. The relations in this case yield [vi , z −y, z, z] = 0, so that the element [vi , z −y, z] is central. The construction of [1, 2] shows that [vi , z − y, z] is non-trivial. It follows that [vi , z − y] is non-central, but belongs to the second centre of M . This is the situation when µ3 = −2, referred to in Section 2. As explained in Section 2, computations suggest that when µ3 ∈ { 0, 1 } the relations above up to weight 2q are not enough to make M quasithin. When µ3 = 1 we need to prescribe the fourth diamond, that is, the relations up to weight 3q − 1. We will see in Section 6 that there are two choices here. The choice µ4 = 0 appears lo lead to several different possibilities, that we are not exploring here. The other choice that gives an infinite-dimensional algebra is µ4 = 3. In this case, too, the types of the diamonds form an arithmetic progression. Also, when µ3 = 0, we will see that the fourth diamond has the correct type µ4 = 1. To make M quasithin, however, we need to prescribe the type of the fifth diamond, that is, the relations up to weight 4q − 2. We will see in Section 6 that the only choice that gives an infinite-dimensional algebra is µ5 = 2, so that again we obtain that the types form an arithmetic progression. Finally for µi 6= 2 we have for all 2 < l < q − 1 (4.9)

[vi z l y] = 0.

When µi = 2 the relation (4.9) does not hold when l = q − r, where r = ph , with 0 < h < n, and we recall the notation q = pn . In this case we have (4.10)

[vi z q−r yz] = [vi z q−r yy] = 0.

In other words, for these values of l the element [vi z l y] does not vanish, but it is central. These are the n − 1 elements of the ideal I that follow a diamond of type 2, referred to in Theorem 2.4.

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5. The Second Chain In this section we prove Lemma 2.2. The term chain, here and in Section 7, refers to the sequence of one-dimensional homogeneous components between two diamonds. We assume M L= Li i≥1

to be an infinite-dimensional thin Lie algebra with second diamond in weight q, a power of the characteristic p > 5 of the underlying field. In the notation of Section 2, let v2 = [yz q−2 ] 6= 0, so that [v2 yy] = 0, [v2 yz] = 2[v2 zz], and [v2 zy] = −[v2 zz]. We show here that there cannot be a true diamond in weight less than 2q − 1. More precisely, we will show that (5.1)

[v2 zzz l y] = 0 or [v2 zzz l y] ∈ Z(L)

for 0 ≤ l < q − 3. We will use (2.3) in the form [yz j y] = 0 for 0 ≤ j < q − 2. We proceed by induction on l ≥ 0. Let first l be odd, and compute 0 = [v2 z[yz l y]] = [v2 z[yz l ]y] − [v2 zy[yz l ]] = [v2 zyz l y] − (−1)l [v2 zyz l y] = (−1 + (−1)l ) · [v2 zzz l y]. This provides the induction step for l odd. Let thus l be even. We have 0 = [v2 [yz l+1 y]] = [v2 [yz l+1 ]y] − [v2 y[yz l+1 ]] = [v2 yz l+1 y] − (l + 1)[v2 zyz l y] − (−1)l+1 [v2 yz l+1 y] = (2 + l + 1 + 2) · [v2 zzz l y], as l is even. This yields the induction step, except when l + 5 ≡ 0 (mod p). In this case, write l + 5 = βr, where r > 1 is a power of p, and β 6≡ 0 (mod p). Suppose first l + 5 < q − 2, so that r < q. Note that we have [yz i y] = 0 for 0 ≤ i < q + l, provided i 6= q − 2, q − 1. Now l + r + 1 < q + l, and l + r + 1 ≡ −4 (mod p). Therefore [yz l+r+1 y] = 0. We now introduce a critical piece of Notation 5.1. If u = [yz n ] ∈ L, then we write u−m = [yz n−m ], for m ≤ n. Clearly [u−m z m ] = u.


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In this notation we have 0 = [v2−r [yz l+r+1 y]] l+r+1 l+r+1 −r r l+1 =− [v2 z yz y] + [v2−r z r+1 yz l y] r r+1 = (−2β + 4β) · [v2 zzz l y]. This yields the induction step in this case. The only remaining case is thus when l + 5 ≥ q − 2, so that l = q − 5. Let w = [v2 z q−3 ], and assume by way of contradiction [wy] = [v2 zzz q−5 y] to be non-central. From 0 = [w−1 [yzy]] we get [wyy] = 0. Therefore [wyz] 6= 0, and by the (weak) covering property, [wyz] spans L2q−2 . Now 0 = [v2 v2 ] = [v2 [yz q−2 ]] = [v2 yz q−2 ] − (−2)[v2 zyz q−2 ] + (−2)[v2 z q−3 yz] − [v2 z q−2 y] = −2[wyz] − [wzy], or [wzy] + 2[wyz] = 0. Since [wyz] spans L2q−2 , we have [wzz] = λ[wyz], for some λ. Now 0 = [w−1 [yzzy]] = [w−1 [yzz]y] = [w−1 yzzy] − 2[w−1 zyzy] + [w−1 zyyy] = −2[wyzy], so that [wyzy] = 0. Similarly 0 = [w[yzzy]] = [w[yzz]y] − [wy[yzz]] = [wyzzy] − 2[wzyzy] + [wzzyy] − [wyzzy] = (1 − 2(−2) − 1)[wyzzy] = 4[wyzzy] yields [wyzzy] = 0. Finally 0 = [[v2 z][v2 z]] = [v2 z[yz q−1 ]] = [v2 zyz q−1 ] + [wyzzz] + [wzyzz] = −(λ + 1)[wyzzz]. If λ 6= −1, this shows L2q = 0, a contradiction to L being infinite-dimensional. Let thus λ = −1. In this case we will prove L3q−1 = 0. We may note here that the description we give in the following of the (finite-dimensional) algebra L corresponds to the structure suggested by our computer calculations. This is one of the cases, referred to in the Introduction, where we just stop short of providing exact information about finite-dimensional algebras; to do that, one would need to construct such a finite-dimensional algebra here.

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We begin with showing [wyzz l y] = 0 for 0 ≤ l < q − 3. We have first 0 = [w[yz l+1 y]] = [w[yz l+1 ]y] − [wy[yz l+1 ]] = [wyz l+1 y] − (l + 1)[wzyz l y] − (−1)l+1 [wyz l+1 y] = (1 + 2(l + 1) − (−1)l+1 )[wyzz l y]. This proves [wyzz l y] = 0 unless l is odd (and l ≡ −1 (mod p)), or l ≡ −2 (mod p) (and l is even). The case of l odd is dealt with via 0 = [wz[yz l y]] = [wz[yz l ]y] − [wzy[yz l ]] = [wzyz l y] − (−1)l [wzyz l y] = 2[wzyz l y]. To deal with l ≡ −2 (mod p), we write l + 2 = βr, where β 6≡ 0 (mod p) and r is a power of p greater than 1. If l 6= q − r − 2 then l + r + 1 < q − 1, and we conclude by expanding 0 = [w−r [yz l+r+1 y]] = [w−r [yz l+r+1 ]y] l+r+1 l+r+1 l =− [wyzz y] + [wzyz l y] r r+1 βr + r − 1 βr + r − 1 = − −2 · [wyzz l y] r r+1 = β[wyzz l y]. In the remaining case where l = q − r − 2 we reach the same conclusion by expanding 0 = [[yz q+(q−r)/2−3 ][yz q+(q−r)/2−3 ]] q + (q − r)/2 − 3 q + (q − r)/2 − 3 q−r−2 ≈− [wyzz ]+ [wzyz q−r−2 ] q − (q − r)/2 − 2 q − (q − r)/2 − 1 q−r−2 + [wzzz y] = [wzzz q−r−2 y]. Here the two binomial coefficients vanish according to Lucas’ Theorem, because q + (q − r)/2 − 3 ≡ p − 3 (mod p), while q − (q − r)/2 − 2 ≡ p − 2 (mod p) and q − (q − r)/2 − 1 ≡ p − 1 (mod p). See also the argument after (7.5). Now define u = [wyzz q−3 ], an element of weight 3q − 5.


13

We have easily [uyy] = 0 from the expansion of 0 = [u−1 [yzy]]. The calculation 0 = [w, [yz q ] + [yz q−1 y]] = [w[yz q ]] + [w[yz q−1 ]y] − [wy[yz q−1 ]] = [wyz q ] − [wz q y] + [wyz q−1 y] + [wzyz q−2 y] − [wyz q−2 yz] − [wyz q−1 y] = [uzz] + (1 + 1 − 2 − 1)[uzy] − [uyz] yields [uzz] = [uyz] + [uzy]. The expansion 0 = [[yz (3q−5)/2 ][yz (3q−5)/2 ]] (3q − 5)/2 (3q − 5)/2 q−1 ≈− [wyzz ] + [wzyz q−1 ] q q−1 (3q − 5)/2 − [wz q−1 yz] + [wz q−1 zy] 1 5 = −[uzz] − [uyz] − [uzy] 2 thus yields 7 [uzy] = − [uyz] 4

and

3 [uzz] = − [uyz]. 4

Now 0 = [u[yzy]] = −[uzyy] + 2[uyzy] − [uyyz] =

15 [uyzy]. 4

We get [uyzy] = 0. Using this, one gets also [uyzzy] = 0 from the expansion 0 = [[wyz], −2 [yz q ] + [yz q−2 yz]] = −2[wyz[yz q ]] + [wyz[yz q−2 y]z] − [wyzz[yz q−2 ]y] = 2 [wyzz q y] q−4 q − 2 − (−1) [wyzzz q−4 yzzy] q−4 q−3 q − 2 − (−1) [wyzzz q−4 zyzy] q−3 = 2 [uzzzy] + 3 [uyzzy] + 2 [uzyzy] 3 7 +3+2· − · [uyzzy] = 2· − 4 4 3 7 = − +3− · [uyzzy] 2 2 = −2 [uyzzy].

14


We finally obtain 0 = [[yz (3q−3)/2 ][yz (3q−3)/2 ]] (3q − 3)/2 (3q − 3)/2 q+2 ≈− [wyz ] + [wzyz q+1 ] q+2 q+1 (3q − 3)/2 (3q − 3)/2 q−1 − [wz yzzz] + [wz q−1 zyzz] 3 2 15 3 35 15 = − +2 · [uzzzz] − [uyzzz] − [uzyzz] 8 2 16 8 9 3 35 15 7 = · − − + − · − · [uyzzz] 8 4 16 8 4 1 = [uyzzz]. 4 This proves L3q−1 = 0, as claimed. 6. Going around a diamond Here we assume by induction to have proved all relations of Section 4, for i < k, and prove (4.7) (resp. (4.8) for µk−1 = 1) for i = k. First of all, we have (6.1)

[vk yy] = 0,

[vk yzy] = [vk zyy] = 0.

The first one follows from the expansion of 0 = [vk−1 [yzy]], and the others from 0 = [vk−1 [yzzy]] = −2[vk yzy] + [vk zyy] and 0 = [vk−2 [yzzzy]] = 3[vk yzy] − [vk zyy]. We will see that [vk zzy] = 0 also holds true, but it is a bit harder to get. 6.1. The case µk−1 6= 1. Consider first the case µk−1 6= 1. 6.1.1. The first relation. We want to prove [vk zz] = [vk zy] + [vk yz]. We compute 0 = [vk−1 , [yz q ] + [yz q−1 y]] = [vk−1 [yz q ]] + [vk−1 [yz q−1 ]y] − [vk−1 y[yz q−1 ]] = [vk−1 yzz q−3 zz] − [vk−1 zzz q−3 zy] + [vk−1 yzz q−3 zy] + [vk−1 zyz q−3 zy] − [vk−1 yzz q−3 yz] − [vk−1 yzz q−3 zy] = (1 − µk−1 ) · ([vk zz] − [vk zy] − [vk yz]) Since we are assuming for the time being µk−1 6= 1, we obtain (6.2)

[vk zz] = [vk zy] + [vk yz],

and from it and (6.1) [vk zzy] = 0.


15

6.1.2. The second relation. We expand the following identity 0 = [vk−2 , −µ3 [yz 2q−1 ] + [yz 2q−2 y]].

(6.3) We first have

[vk−2 , [yz 2q−1 ]] = [vk−2 yz 2q−1 ] − (−1)[vk−2 zyz 2q−2 ] 2q − 1 2q − 1 q−1 q−1 + [vk−2 z yzz ] − [vk−2 z q−1 zyz q−1 ] q−1 q 2q − 1 + [vk−2 z 2q−2 yz] − [vk−2 z 2q−2 zy] 2q − 2 Now we work out [vk−2 , [yz 2q−2 y]] = [vk−2 [yz 2q−2 ]y] − [vk−2 y[yz 2q−2 ]] = [vk−2 yz 2q−2 y] + 2[vk−2 zyz 2q−3 y] 2q − 2 2q − 2 q−1 q−2 + [vk−2 z yzz y] − [vk−2 z q−1 zyz q−2 y] q−1 q 2q − 2 2q − 2 q−2 q−1 + [vk−2 yz yzz ] − [vk−2 yz q−2 zyz q−1 ] q−2 q−1 2q − 2 + [vk−2 yz q−2 z q−1 yz] − [vk−2 yz 2q−2 y]. 2q − 3 Summing up, (6.3) yields (1 − µk−1 )(1 − µk−2 − 2µ3 ) · [vk zz]+ +(µ3 − 2 + 2µk−2 ) · [vk yz]+

(6.4)

+(µ3 − µk−1 + 2µk−2 ) · [vk zy] = 0. Now the 2 × 3 matrix of the relations (6.2) and (6.4) with respect to [vk zz], [vk yz], [vk zy] is −1 1 1 . (1 − µk−1 )(1 − µk−2 − 2µ3 ) µ3 − 2 + 2µk−2 µ3 − µk−1 + 2µk−2 Looking at the second and third column, we see that this has rank two when µk−1 6= 2. Thus L(k−1)(q−1)+2 is at most 1-dimensional, and it is easy to see that the relations are compatible with the intended [vk zy] = µk [vk zz], where µk = µk−1 + µ3 + 1. 6.1.3. The third relation. To deal with the case µk−1 = 2, we consider a third relation, coming from the balanced expansion 0 = [[yz λ ][yz λ ]], where λ=

(k − 1)(q − 1) (k − 1) ≡− 2 2

(mod p).

16


This expands to (6.5)

0 = σ[vk zz] − λ[vk yz] + [vk zy] = σ[vk zz] +

(k − 1) [vk yz] + [vk zy]. 2

We need not determine σ, but see Section 9 for a remark on how to compute it. The matrix of the relations (6.2) and (6.5) is "

−1 σ

# 1 1 . (k − 1) 1 2

The last two columns are independent, and we can proceed as above, unless k ≡ 3 (mod p), and thus µk−1 = −1 + (k − 3)(µ3 + 1) = −1. In particular, this covers the remaining case µk−1 = 2, as p > 3 here. 6.2. The case µk−1 = 1. In the case µk−1 = 1 the component following the k-th diamond is two-dimensional, generated by [vk zz] and [vk zy]. We intend to show that the relations (4.8) hold here, for i = k. Relation (6.4) holds here. Since we are assuming by induction 1 − µk−2 = µk−1 − µk−2 = µ3 + 1, or µ3 = −µk−2 , this can be rewritten as (−µk−2 + 2)[vk yz] + (−µk−2 + 1)[vk zy] = 0. We now work out two further relations in the next weight.

6.2.1. A relation in the next weight. We begin with the following expansion, where we omit the terms which are known to vanish from (6.1), 0 = [vk−1 z, [yz q−1 y] + [yz q ]] = [vk−1 z[yz q−1 ]y] − [vk−1 zy[yz q−1 ]] + [vk−1 z[yz q ]] (6.6)

= [vk−1 zyz q−1 y] − [vk−1 zyz q−3 yzz] − [vk−1 zyz q−3 zyz] − [vk−1 zyz q−3 zzy] + [vk−1 zyz q ] − [vk−1 zz q y] = −[[vk zz], y] + µk−1 · ([[vk zz] − [vk yz] − [vk zy], z]) .

6.2.2. A further relation in the next weight. We expand the following identity 0 = [vk−2 z, [yz 2q−2 y] − µ3 [yz 2q−1 ]], making no assumptions on the values of the µi .


17

We first have [vk−2 z, [yz 2q−2 y]] = [vk−2 z[yz 2q−2 ]y] − [vk−2 zy[yz 2q−2 ]] = [vk−2 zyz 2q−2 y] 2q − 2 2q − 2 q−1 q−1 − [vk−2 z yzz y] + [vk−2 z q−1 zyz q−1 y] q−2 q−1 2q − 2 2q − 2 q−3 q − [vk−2 zyz yzz ] + [vk−2 zyz q−3 zyz q ] q−3 q−2 2q − 2 2q − 2 2q−4 − [vk−2 zyz yzz] + [vk−2 zyz 2q−4 zyz] 2q − 4 2q − 3 2q−4 − [vk−2 zyz zzy]. Now we work out −µ3 [vk−2 z,[yz 2q−1 ]] = −µ3 [vk−2 zyz 2q−1 ] 2q − 1 2q − 1 q−2 q + µ3 [vk−2 zz yzz ] − µ3 [vk−2 zz q−2 zyz q ] q−2 q−1 2q − 1 2q − 1 2q−3 + µ3 [vk−2 zz yzz] − µ3 [vk−2 z 2q−2 zyz] 2q − 3 2q − 2 + µ3 [vk−2 z 2q−2 zzy]. We obtain (µk−2 − (1 − µk−1 ) − µk−2 + µ3 ) · [vk zzy]+ + (2µk−2 (1 − µk−1 ) + µk−2 µk−1 − µ3 µk−2 − µ3 (1 − µk−1 ) − µ3 µk−1 ) · [vk zzz]+ + (−3µk−2 + µ3 ) · [vk yzz]+ + (−2µk−2 + µ3 ) · [vk zyz] = 0, that is (µk−1 − 1 + µ3 ) · [vk zzy]+ (6.7)

+ (2µk−2 − µk−2 µk−1 − µ3 µk−2 − µ3 ) · [vk zzz]+ + (−3µk−2 + µ3 ) · [vk yzz]+ + (−2µk−2 + µ3 ) · [vk zyz] = 0,

We use (6.7) for µk−1 = 1, and µ3 = −µk−2 to get (6.8)

−µk−2 · [vk zzy] + (µ2k−2 + 2µk−2 ) · [vk zzz] −4µk−2 · [vk yzz] − 3µk−2 · [vk zyz] = 0

6.2.3. The general case for µk−1 = 1. The relations we have obtained are  −[vk zzy] + [vk zzz] − [vk yzz] − [vk zyz] = 0    (−µ k−2 + 2) · [vk yz] + (−µk−2 + 1) · [vk zy] = 0 (6.9)  −µk−2 · [vk zzy] + (µ2k−2 + 2µk−2 ) · [vk zzz]    −4µk−2 · [vk yzz] − 3µk−2 · [vk zyz] = 0.

18


By looking at columns 1, 2, 3, and then 1, 2, 4, of the underlying matrix of coefficients, we see that if µ2k−2 + µk−2 6= 0, the underlying matrix has rank three. Elementary linear algebra then yields   [vk zzy] = 0 [vk zyz] = µk [vk zzz]  [v yzz] = (1 − µ )[v zzz], k k k where µk = µk−1 + µk−1 − µk−2 = 2 − µk−2 , as required. 6.2.4. The subcase µk−1 = 1 and µk−2 = −1. Let us consider first the case when µk−2 = −1. Since the µ are in arithmetic progression, this will first occur for k = 4, when µ3 = 1. In fact, computational evidence shows that in this case the relations up to weight 2q are not enough to make the algebra quasithin. We need to prescribe the fourth diamond as well here. So let us set [v4 zy] = µ4 [v4 zz], so that the second relation in (6.9) gives [v4 yz] = − 32 µ4 [v4 zz]. As mentioned in Section 2, computations appear to show that the case µ4 = 0 yields to a great variety of algebras, that we are not considering here. If µ4 6= 0, (6.1) yields [vk zzy] = 0. We obtain µ4 0 = [vk zzy] = 1 − [vk zzz]. 3 This is a contradiction, unless µ4 = 3 = µ3 + µ3 + 1, the expected value. When µk−2 = −1 and µk−1 = 1 for values k > 4, we can now recover the missing relation from the more complicated expansion of (6.10)

0 = [vk−3 z, [yz 3q−3 y] − µ3 [yz 3q−2 ]].

This is dealt with in Section 9, and yields 3[vk zzz] + 9[vk yzz] + 5[vk zyz] = 0. This yields an underlying matrix of rank three, so we are done here. 6.2.5. The subcase µk−1 = 1 and µk−2 = 0. We deal here with the complications arising when µk−2 = 0 and µk−1 = 1. Since the µ are in arithmetic progression, we have here µ3 = 0, so that µ4 = 1, and the problem will first occur for k = 5. Here, too, computations show that relations up to weight 2q are not enough to determine the structure of algebra as in Section 4. One has to prescribe the fifth diamond to get that. The relations we have obtained so far are ( −[vk zzy] + [vk zzz] − [vk yzz] − [vk zyz] = 0, (6.11) 2[vk yzz] + [vk zyz] = 0. If we insist on L4(q−1)+1 being a diamond of finite type, we will have a relation of the form (6.12)

[v5 zy] = µ5 [v5 zz]


19

for some µ5 . This yields [vk yz] = (1 − µ5 )[vk zz], and then µ5 − 2 [vk zzy] = [vk zzz]. 2 But we have also 0 = [vk [yzy]] = 2[vk yzy] − [vk yyz] − [vk zyy] = (2(1 − µ5 ) − µ5 )[vk zzy] = (2 − 3µ5 )[vk zzy] and 0 = [vk−2 [yzzzy]] = 3[vk yzy] − [vk zyy] = (3(1 − µ5 ) − µ5 )[vk zzy] = (3 − 4µ5 )[vk zzy], so that [vk zzy] = 0. If µ5 6= 2, we obtain [vk zzz] = 0, a contradiction. Now when µk−2 = 0, µk−1 = 1, µk = 2, for some k > 5, we expand (6.13)

0 = [vk−4 , [yz 4q−3 y] − µ5 [yz 4q−2 ]].

This is also dealt with in Section 9, and yields 2[vk zzz] + 16[vk yzz] + 7[vk zyz] = 0. This is the third relation that added to (6.11) gives (4.8) in this case. 7. The chain Here we assume all identities of Section 4 to hold for i ≤ k, except (4.9) and (4.10) for i = k, and show the latter two to hold, proceeding by induction on l. We will mainly rely on [yz l y] = 0, for l ≤ q − 3. We proceed briskly, as several arguments here are similar to those of Section 5. 7.1. The case µk = 0. Suppose first µk = 0. We compute, for l ≤ q − 2, (7.1)

0 = [vk [yz l−1 y]] = [vk [yz l−1 ]y] − [vk y[yz l−1 ]] = (1 − µk ) − µk (l − 1) − (−1)l−1 (1 − µk ) · [vk z l y].

Since µk = 0, we obtain (1 + (−1)l )[vk zzz l y] = 0. This yields (4.9) if l is even. Provided l 6= q − 2, we have thus [yz l y] = 0. In this case we compute

(7.2)

0 = [vk−1 [yz l y]] = [vk−1 [yz l ]y] l = −(1 − µk ) · l + µk · [vk z l y] 2 µk (l + 1) − 2 · [vk z l y]. = l· 2

20


If l 6≡ 0 (mod p), this yields (4.9) as µk = 0 here. When l = q − 2, note that [yz 2q−5 y] = 0, and expand 0 = [vk−1 z[yz 2q−5 y]] = [vk−1 z[yz 2q−5 ]y] − [vk−1 zy[yz 2q−5 ]] 2q−5 q−2 2q − 5 = [vk−1 zyz y] + (−1) [vk−1 zz q−2 yzz q−4 y] q−2 2q − 5 − (−1)q−3 [vk−1 zyz q−3 yzz q−4 z] − (−1)2q−5 [vk−1 zyz 2q−5 y] q−3 2µk−1 [vk z q−2 y], as the two binomial coefficients vanish, by Lucas’ theorem. We obtain [vk z q−2 y] = 0, since by our induction assumptions we cannot have both µk = 0 and µk−1 = 0. Now consider the case when l ≡ 0 (mod p), and l is odd. Write l = βr, where β 6≡ 0 (mod p), so that r is a power of p. Note that we cannot have l = q − r here, as l is odd. Therefore l ≤ q − 2r < q − r − 2, and [yz l+r−1 y] = 0, so that we can compute

(7.3)

0 = [vk−r [yz l+r−1 y]] l+r−1 l+r−1 l−2 ≈ [vk yzz y] − [vk zyz l−2 y] r r+1 βr + r − 1 βr + r − 1 = (1 − µk ) − µk · [vk z l y] r r+1 = (β(1 − µk ) − (−β)µk ) · [vk z l y] = β[vk z l y].

This yields (4.9) here. We have completed the proof in the case µk = 0. 7.2. The case µk 6= 0. We now turn to µk 6= 0. We first compute 0 = [vk z[yz l−2 y]] (7.4)

= [vk z[yz l−2 ]y] − [vk zy[yz l−2 ]] = µk · 1 − (−1)l−2 · [vk z l y]

This yields (4.9) when l is odd. So let l be even. 7.2.1. The case l 6≡ 0 (mod p). In this case we obtain from (7.2), as l is even, [vk z l y] = 0 unless µk (l + 1) = 2. Consider first the case when l = q − 3 and thus µk = −1. A technical difficulty has to be expected here, because in the case of the graded Lie algebra associated to the Nottingham group (in which all µi are −1), dealt with in [8], the corresponding element [vk z q−3 y] does not vanish (but can be shown to be central). We deal with


21

this by evaluating −1 0 = [vk−1 [yz l+q−1 y]] −1 = [vk−1 [yz l+q−1 ]y]

l+q+1 −1 = −(l − + [vk−1 zzyz l+q−3 y] 2 l+q−1 l+q−1 −1 −1 q−1 l−2 − [vk−1 zz yzz y] − [vk−1 zz q−1 zyz l−2 y] q q+1 −1 1)[vk−1 zyzz l+q−3 y]

= (4(1 − µk−1 ) + 10µk−1 − (1 − µk ) − 4µk ) [vk z l y]. Here the coefficient evaluates to 6µk−1 + 6. This is nonzero, because µk = −1, so that µk−1 6= −1, as we are not in the case of the graded Lie algebra associated to the Nottingham group. We may thus suppose l < q − 3, that is, l + 1 < q − 2, so that [yz l+1 y] = 0. We compute 0 = [vk−2 [yz l+1 y]] = [vk−2 [yz l+1 ]y] l+1 l+1 = (1 − µk ) · − µk · · [vk z l y]. 2 3 The overall coefficient of [vk z l y] here is (l + 1)l(3 − µk (l + 2)) l(3(l + 1) − µk (l + 1)(l + 2)) 1 = = l(l − 1), 6 6 6 since we are assuming µk (l + 1) = 2. This yields (4.9), except when l − 1 ≡ 0 (mod p), and thus µk = 1. To deal with this case, we note that [yz l+q−2 y] = 0, as q < l + q − 2 < 2q − 3. We expand 0 = [vk−1 [yz l+q−2 y]] = [vk−1 [yz l+q−2 ]y] − [vk−1 y[yz l+q−2 ]] = [vk−1 yz l+q−2 y] − (l − 2)[vk−1 zyz l+q−3 y] l+q−2 l+q−2 q−1 l−2 + [vk−1 z zyz y] − [vk−1 z q−1 zyz l−2 y] q−1 q l+q−2 l+q−2 q−3 l−1 + [vk−1 yzz yzz ] − [vk−1 yzz q−3 zyz l−1 ] q−2 q−1 − (−1)l+q−2 [vk−1 yzz q−1 z l−2 y] = ((1 − µk−1 ) − (l − 2)µk−1 − µk + 1) · [vk z l y] = [vk z l y]. Here we have used the fact l+q−2 l−1+q−1 (7.5) = ≡0 q−1 q−1

(mod p);

22


this is because all the non-zero digits of the expansion of q − 1 in base p are p − 1. Therefore in the addition of q − 1 and l − 1 < q in base p there will be a carry. It follows that the binomial coefficient vanishes modulo p: this is a consequence of Lucas’ theorem, and it is a particular case of a result of Kummer [25, 26], stating a+b the the number of times a prime p divides a binomial coefficient equals b the number of carries in the addition of a and b in base p. Similarly, l+q−2 ≡ 0 (mod p), q−2 as l > 1. 7.2.2. The case l ≡ 0 (mod p). Finally, we are left with the case when l ≡ 0 (mod p). The argument of (7.3) will work except when l = q−r here, for 1 < r < q a power of p. This is where the exceptional case (4.10) occurs: because of (7.1) we will indeed have µk = 2 here. Now [vk zyz l−2 yy] = 0 is easy to get, and (7.2) and (7.4) yield [vk yzz l−2 zy] = 0 = 2[vk zyz l−2 zy] + [vk zyz l−2 yz]. This completes the proof. 8. Modular combinatorial identities In this section we prove two modular combinatorial identities. These will be used in the next section to compute the coefficient σ of Subsection 6.1.3, and the results of the expansions of (6.10) and (6.13). First we want to compute the sum X a b

b(q − 1) − r

modulo p, where a is a positive integer, q is a power of the prime p, and 0 ≤ r < q − 1. We follow the convention that all unqualified summation indices run over the integers; however, our sum is a finite sum since a is positive integer, if ac is defined to be 0 for c < 0, as it is customary. Consider the generating polynomial of the binomial coefficients, X a a (1 + x) = xb . b b This can also be viewed as a polynomial with coefficients in the field of p elements, or some extension P of it. If A(x) = i ai xi is a formal power series (here we assume ai = 0 for i < 0) with coefficients in any field, and ω is a primitive nth root of unity in the field (in particular, n must be prime to the characteristic of the field), then X X 1 X jr ω A(ω j x) = aj x j = ank−r xnk−r , n j k j≡−r

(mod n)


because n

1 X ij ω = n j=1

23

( 1 if i ≡ 0 (mod n), 0 otherwise.

We apply this trick to our series, taking as ω a primitive (q − 1)th root of unity in the finite field of q elements Fq , that is to say, a generator of the multiplicative group F× q of the field. We obtain: q−1 X a 1 X ir b(q−1)−r ω (1 + ω i x)a x = q − 1 b(q − 1) − r j=1 b X =− αr (1 + αx)a . α∈F× q

By evaluating for x = 1 we obtain X X a (8.1) =− αr (1 + α)a . b(q − 1) − r × b α∈Fq

This would be easy to evaluate directly when r = 0, and would give X a X = 1− (1 + α)a b(q − 1) α∈Fq b ( q−1 X 2 if a ≡ 0 (mod q − 1), = 1− ω ja = 1 otherwise. j=1 (Note that the above formula would be wrong for a = 0, which we have excluded.) However, the general case of arbitrary r is best dealt with by observing that the second member of formula (8.1) depends only on the smallest positive remainder of a modulo q − 1. Thus, if we put c¯ ≡ c (mod q − 1) with 0 < c¯ < q for any integer c, we obtain that X X a a ¯ = b(q − 1) − r b(q − 1) − r b b a ¯ a ¯ = + . −r q−1−r We also need to compute the related sum X X a a = (−b − r) . c c b(q − 1) − r b c≡−r

(mod q−1)

The standard way of doing that is to apply the operator xD (that is, formal derivation of polynomials followed by multiplication by x) to both members of the equality X X a xb(q−1)−r = − αr (1 + αx)a b(q − 1) − r × b α∈Fq

24


computed above, thus obtaining X X a (−b − r) xb(q−1)−r = −a αr+1 (1 + αx)a−1 . b(q − 1) − r × b α∈Fq

By evaluating for x = 1 we obtain X X a (−b − r) = −a αr+1 (1 + α)a−1 b(q − 1) − r b α∈F× q X a−1 a−1 a−1 =a +a . = a b(q − 1) − r − 1 q − 2 − r 2q − 3 − r b Note that the last binomial coefficient vanishes, unless r = q − 2 and a ≡ 1 (mod q − 1). 9. Some calculations 9.1. A commutator expansion. The purpose of this subsection is that of computing explicitly the commutator of two elements of our Lie algebra, namely [[yz r−1 ][yz s−1 ]]. If we assume that r + s 6≡ 1 (mod q − 1), we will have X r−1 s−1 i s−1 [[yz ][yz ]] = (−1) [yz r−1+i yz s−1−i ] i (9.1) i ≡ c(r, s) · [yz r+s−1 ] modulo the second centre. Note that [yz r−¯r−1 ] = vk(r) , whence [yz r−1 ] = [vk(r) z r¯], r − r¯ where k(r) = + 1 (and r¯ is defined to be the smallest positive remainder of q−1 r modulo q − 1, as in the previous section). It follows that X s−1 r¯ c(r, s) = (−1) (1 − µk(r)+j ) j(q − 1) − r¯ j ! X s−1 − µk(r)+j j(q − 1) − r¯ + 1 j X s − 1 r¯ = (−1) j(q − 1) − r¯ j ! X s − µk(r)+j . j(q − 1) − r ¯ + 1 j The above formula is valid without any assumption on the types of the diamonds (only that they appear at the correct weights).


Now we have X j

25

X s − 1 s − 1 s−1 = = , j(q − 1) − r¯ j(q − 1) − r¯ q − 1 − r¯ j

s−1 since the term vanishes unless r ≡ 0 and s ≡ 1 (mod q − 1), which 2q − 2 − r¯ we have excluded. In order to proceed further we assume that the coefficients µi involved all coincide with the values given by the expected formula, namely that µk(r)+j = −1 + (k(r) + j − 2)(µ3 + 1) ≡ −1 + (−r − 1 + r¯ + j)(µ3 + 1)

(mod p).

It follows that s µk(r)+j = j(q − 1) − r¯ + 1 j X s = (−1 − r(µ3 + 1)) j(q − 1) − r¯ + 1 j X s +(µ3 + 1) (j + r¯ − 1) . j(q − 1) − r¯ + 1 j

X

The first summation is X

X s s¯ = j(q − 1) − r¯ + 1 j(q − 1) − r − 1 j j s¯ = q−1−r−1 r−1 s−r+1 = (−1) q − 1 − s¯ s¯ where we note, again, that the term vanishes unless r ≡ 1 and 2q − 2 − r − 1 s ≡ 0 (mod q − 1), which we have excluded, and in the last step we have used the congruence a b b a ≡ (−1) (mod p), (−1) q−1−b q−1−a which holds as long as 0 ≤ a, b < q. The other summation is X X s − 1 s s−1 (j + r¯ − 1) =s =s j(q − 1) − r ¯ + 1 j(q − 1) − r ¯ q − 1 − r¯ j j

26


Putting all pieces together, we conclude that s−1 c(r, s) = (−1) (1 + s(µ3 + 1)) q − 1 − r¯ r−1 s − (−1) (1 + r(µ3 + 1)) . q − 1 − s¯ r

(9.2)

Note that the expression c(r, s) is antisymmetric in r and s. Of course, this is to be expected once we know that a thin Lie algebra exists with diamonds at the correct places and diamond types µi as we have prescribed them. However, this antisymmetry and, in particular, the fact that c(r − 1, r − 1) = 0, must be regarded as genuinely new information if all coefficients µi but the last one, say µk , involved in [[yz r−1 ][yz s−1 ]] are known and we want to use the expansion of such a commutator to draw conclusions about µk , as we do in the next subsection. 9.2. A balanced commutator. Now we suppose that r + s ≡ 2 (mod q − 1), but we assume that only the diamonds before the last one are in place. In other words, we regard [vk zz], [vk zz], [vk zz] as independent elements (assuming only [vk yy] = 0). However, we still assume that µk is defined as usual, that is, µk = −1 − (r + s − 1)(µ3 + 1). Then we will have [[yz r−1 ][yz s−1 ]] = c(r, s)[vk zz] + (−1)s (s − 1) ([vk yz] − (1 − µk )[vk zz]) − (−1)s ([vk zy] − µk [vk zz]) = (c(r, s) + (−1)s (1 − s + sµk )) [vk zz] +(−1)s (s − 1)[vk yz] − (−1)s [vk zy], This formula allows one, for example, to compute more explicitly the balanced expansion 0 = [[yz λ ][yz λ ]], that we used in (6.1.3), where λ = (k − 1)(q − 1)/2. Since c(λ + 1, λ + 1) = 0 we obtain that (k − 1) λ λ λ [[yz ][yz ]] = (−1) σ[vk zz] + [vk yz] + [vk zy] , 2 where

k−3 σ = λ − (λ + 1)µk = (k − 2) −1 + (µ3 + 1) 2

= (k − 2)µ1−λ .

9.3. The two “lengthy calculations”. Now we would like to show how our commutator formula simplifies the lengthy direct calculation needed to expand the identities which we used in (6.10) and (6.13). In (6.10) we had µ3 = 1, and the µ followed the regular pattern up to µk−1 = 1, where k > 4, and k ≡ 4 (mod p). There we had to compute 0 = [vk−3 z, [yz 3q−3 y] − µ4 [yz 3q−2 ]] = [vk−3 z, [yz 3q−3 ], y] − [vk−3 zy, [yz 3q−3 ]] − µ4 [vk−3 z, [yz 3q−2 ]] = [vk−3 z, [yz 3q−3 ], y] + 3[vk−3 zz, [yz 3q−3 ]] − 3[vk−3 z, [yz 3q−2 ]]. We compute the various summands. We recall that vk−3 has weight (k − 4)(q − 1).


27

Since our formula (9.2) for the coefficients c(r, s) of (9.1) works under the assumption the µk has the expected value 3, we must take suitable correction terms into account. We have [vk−3 z, [yz 3q−3 ], y] = c((k − 4)(q − 1) + 1, 3q − 2)[vk zzy] 3q − 3 3q − 3 − [vk yzy] + [vk zyy] 3q − 4 3q − 3 = −6[vk zzy], since we already know that [vk yzy] = [vk zyy] = 0, and c((k − 4)(q − 1) + 1, 3q − 2) = q−1 q−1 = −(1 − 2(µ3 + 1)) + (1 − (k − 5)(µ3 + 1)) = −6. q−2 q−2 Next, we have [vk−3 zz, [yz 3q−3 ]] = c((k − 4)(q − 1) + 2, 3q − 2)[vk zzz] 3q − 3 + ([vk yzz] + 2[vk zzz]) 3q − 5 3q − 3 − ([vk zyz] − 3[vk zzz]) 3q − 4 3q − 3 + [vk zzy] 3q − 3 = 6[vk yzz] + 3[vk zyz] + [vk zzy], since

q−1 c((k − 4)(q − 1) + 2, 3q − 2) = (1 − 2(µ3 + 1)) = −3. q−3 Finally, we have [vk−3 z, [yz 3q−2 ]] = c((k − 4)(q − 1) + 1, 3q − 1)[vk zzz] 3q − 2 − ([vk yzz] + 2[vk zzz]) 3q − 4 3q − 2 + ([vk zyz] − 3[vk zzz]) 3q − 3 3q − 2 [vk zzy] − 3q − 2 = −3[vk zzz] − 3[vk yzz] − 2[vk zyz] − [vk zzy],

since q−1 c((k − 4)(q − 1) + 1, 3q − 1) = −(1 − (k − 5)(µ3 + 1)) = −3. q−3 Putting all pieces together, and dividing by 3, we obtain the relation 3[vk zzz] + 9[vk yzz] + 5[vk zyz] = 0.

28


In (6.13) we had µ3 = 0, and the µ followed the regular pattern up to µk−1 = 1, where k > 5, and k ≡ 5 (mod p). There we had to compute 0 = [vk−4 z, [yz 4q−4 y] − µ5 [yz 4q−3 ]] = [vk−4 z, [yz 4q−4 ], y] − [vk−4 zy, [yz 4q−4 ]] − µ5 [vk−4 z, [yz 4q−3 ]] = [vk−4 z, [yz 4q−4 ], y] + 2[vk−4 zz, [yz 4q−4 ]] − 2[vk−4 z, [yz 4q−3 ]]. We compute the various summands. We recall that vk−4 has weight (k − 5)(q − 1). We have, as before, [vk−4 z, [yz 4q−4 ], y] = c((k − 5)(q − 1) + 1, 4q − 3)[vk zzy] = −4[vk zzy]. Similarly we have [vk−4 zz, [yz 4q−4 ]] = c((k − 5)(q − 1) + 2, 4q − 3)[vk zzz] 4q − 4 + ([vk yzz] + [vk zzz]) 4q − 6 4q − 4 − ([vk zyz] − 2[vk zzz]) 4q − 5 4q − 4 + [vk zzy] 4q − 4 = 10[vk yzz] + 4[vk zyz] + [vk zzy]. Finally, we have [vk−4 z, [yz 4q−3 ]] = c((k − 5)(q − 1) + 1, 4q − 2)[vk zzz] 4q − 3 − ([vk yzz] + [vk zzz]) 4q − 5 4q − 3 + ([vk zyz] − 2[vk zzz]) 4q − 4 4q − 3 − [vk zzy] 4q − 3 = 2[vk zzz] − 6[vk yzz] − 3[vk zyz] − [vk zzy]. Putting all pieces together, and dividing by 2, we obtain the relation 2[vk zzz] + 16[vk yzz] + 7[vk zyz] = 0. References 1. M. Avitabile, Some loop algebras of Hamiltonian Lie algebras, Ph.D. thesis, Trento, November 1999. 2. , Some loop algebras of Hamiltonian Lie algebras, to appear, Internat. J. Algebra Comput., 2002. 3. M. Avitabile and G. Jurman, Diamonds in thin Lie algebras, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 4 (2001), no. 3, 597–608. 4. M. Avitabile and S. Mattarei, Some loop algebras of Hamiltonian Lie algebras. II, in preparation, 2002.


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