GRADED LIE ALGEBRAS OF MAXIMAL CLASS

A. Caranti, S. Mattarei and M. F. Newman

19 February 1996

Abstract. We study graded Lie algebras of maximal class over a field F of positive characteristic p. A. Shalev has constructed infinitely many pairwise non-isomorphic insoluble algebras of this kind, thus showing that these algebras are more complicated than might be suggested by considering only associated Lie algebras of p-groups of maximal class. Here we construct |F|^{ℵ_{0}} pairwise non-isomorphic such algebras, and max{|F|, ℵ_{0}} soluble ones. Both numbers are shown to be best possible. We also exhibit classes of examples with a non-periodic structure. As in the case of groups, two-step centralizers play an important role.

1. Introduction

In a recent paper Shalev [Sh2] has shown that graded Lie algebras of maximal class are more complicated than might be suggested by considering only associated Lie algebras of p-groups with maximal class. A nilpotent graded Lie algebra L has maximal class if it has class c and dimension c + 1. Thus,

L = L_{1} ⊕ L_{2} ⊕ ⋯ ⊕ L_{c}

with dim L_{1} = 2 and, for 2 ≤ i ≤ c, dim L_{i} = 1 and [L_{i−1}L_{1}] = L_{i}. Shalev shows that over fields with positive characteristic there are such algebras with arbitrarily large soluble length. In this paper we exhibit many more examples and prove some structure theorems which show that for every field F with positive characteristic there are max(|F|, ℵ_{0}) isomorphism types of nilpotent graded Lie algebras of maximal class over F.

As usual we include under the heading ‘maximal class’ algebras L graded by the positive integers: that is

L = L_{1} ⊕ L_{2} ⊕ ⋯ ⊕ L_{i} ⊕ …

with dim L_{1} = 2 and, for 2 ≤ i, dim L_{i} = 1 and [L_{i−1}L_{1}] = L_{i}. These algebras can be viewed as (projective) limits of nilpotent graded Lie algebras of maximal

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 17B70 17B65 17B05 17B30 17B40 20D15 20F40.

Key words and phrases. Graded Lie algebras of maximal class, p-groups of maximal class, pro-p-groups of finite coclass, nilpotent Lie algebras.

First two authors are members of CNR–GNSAGA, Italy, and acknowledge support of MURST, Italy. Third author acknowledges support from CNR-GNSAGA, Italy, and University of Trento, Italy.

class. They are just infinite-dimensional in the sense that, correctly interpreted, all their proper quotients are finite-dimensional. Shalev ([Sh2], Theorem 1) shows that there are insoluble algebras of this kind. We show that over every field F with positive characteristic there are |F|^{ℵ_{0}} isomorphism types of graded Lie algebras of maximal class (Corollary 9.4).

As with p-groups with maximal class (and indeed finite co-class), the two-step centralizers C(L^{i}/L^{i+2}), where L^{j} is a Lie power, play an important role [Bla]. In fact more so because, in a sense described in Section 3, graded Lie algebras of maximal class are determined by their two-step centralizers (Theorem 3.2). We associate with every graded Lie algebra of maximal class L over a field F a (coefficient) sequence of elements of the projective line F_{∞} over F. The coefficient sequence of the graded Lie algebra of maximal class, M, which comes from pro-p-groups with maximal class is the constant sequence (0, …). All terms of a coefficient sequence except 0 are isolated (Lemma 3.3). There are further strong restrictions on the coefficient sequences (Section 5).

The algebras constructed by Shalev are (just) infinite-dimensional algebras based on some finite-dimensional algebras of Albert and Frank [AF]. Their coefficient sequences are periodic with period length p^{n} − 1 where p is the characteristic. We call a graded Lie algebra of maximal class whose coefficient sequence is ultimately periodic centralizer periodic.

The basic construction, which we call inflation, is described in Section 6 (we are indebted to Aner Shalev and Georgia Benkart for drawing our attention to the work of Block [Blo] in which this construction is also used – and hence, to its use in the fundamental paper of Zassenhaus [Z]). The inflations of the algebra M are soluble and have period length a power of the characteristic. Inflation can be reversed by a process we call deflation (Section 7). Every just infinite soluble graded Lie algebra of maximal class deflates to M (Theorem 8.3). On the other hand deflation is not necessarily reversed by inflation. Still we get that a graded Lie algebra of maximal class is soluble if and only if it is centralizer periodic with period length a power of the characteristic (Theorem 8.2). It follows that there are max(|F|, ℵ_{0}) soluble graded Lie algebras of maximal class over F. And there are the same number of insoluble centralizer periodic ones (Proposition 6.3).

In the work on groups of finite coclass there is the notion of a covered group, one in which all maximal subgroups are two-step centralizers (see [M2] and [Ma]). Some of our results are similar to some in [M2]. There are no groups of maximal class which are covered; in fact they have at most 2 distinct two-step centralizers There are covered graded Lie algebras of maximal class for every finite (Proposition 6.2) and every countable field with positive characteristic (Theorem 9.2).

These we obtain by repeating the inflation process countably many times, via an inverse limit (Section 9). The same procedure allows us to construct graded Lie algebra of maximal classs that are not centralizer periodic. This answers a question of Shalev [Sh3, Problem 28], and is further evidence that graded Lie algebras of finite coclass in positive characteristic are distinctly more complicated than the analogous groups. In characteristic zero, the situation appears to be much better [SZ].

Our results are inspired by, though independent of, computations (Section 10).

2. Preliminaries

An algebra of maximal class is a graded Lie algebra

L = ⨁_{i=1}^{∞} L_{i}

over a field F, where dim(L_{1}) = 2, dim(L_{i}) ≤ 1, for i ≥ 2, and [L_{i}L_{1}] = L_{i+1}, for all i ≥ 1, or equivalently, L_{1} generates L as an algebra. All subalgebras and ideals will be taken to be graded.

It is sometimes useful to consider the Cartesian product

L = ∏_{i=1}^{∞} L_{i}

instead of the direct product (or sum). When this is done, the algebra acquires a topology by taking the Lie powers

L^{i} = ∏_{j≥i} L_{j},

that is, the terms of the lower central series, for j ≥ 2, as a fundamental system of neighbourhoods of 0. With respect to this topology, L is complete. All subalgebras and ideals will be taken to be graded and closed in this case. We will use the topological setting only in Section 9, where it allows us to consider inverse limits. On the other hand, we use the discrete setting in Section 4, for uniformity with the paper of Shalev [Sh2] we’re building upon there.

Algebras of maximal class are either finite dimensional, or just-(infinite dimensional), that is, all their proper factors are finite-dimensional. In fact, if M ≠ 0 is an ideal of L, take 0 ≠ m ∈ M ∩ L_{i}, for some i. Then M ≥ ⨁_{j≥i} L_{j}, so that M has codimension at most i.

We denote by F_{q} the finite field with q elements, for q a prime-power. When F is a field, we write F_{∞} = F ∪ {∞}.

A notation like

a_{1}, a_{2}^{m}, (a_{3}, a_{4}^{n})^{∞},

where a_{i} are arbitrary elements, and m, n are non-negative integers, denotes the sequence

a_{1}, \underbrace{a_{2}, …, a_{2}}_{m}, a_{3}, \underbrace{a_{4}, …, a_{4}}_{n}, a_{3}, \underbrace{a_{4}, …, a_{4}}_{n}, a_{3}, \underbrace{a_{4}, …, a_{4}}_{n}, …

We say such a sequence is periodic, a period being in this case a_{3}, \underbrace{a_{4}, …, a_{4}}_{n}, and a

pre-period being a_{1}, \underbrace{a_{2}, …, a_{2}}_{m}.

The Jacobi identity in Lie algebras

[u[yx]] = [uyx] − [uxy],

and its consequence

[u[y \underbrace{x ⋯ x}_{n}]] = ∑_{i=0}^{n}(−1)^{i}\binom{n}{i}[u \underbrace{x ⋯ x}_{i} y \underbrace{x ⋯ x}_{n−i}],

will be used without specific mention.

We will also use several times Lucas’ Theorem [L]. Let a and b be two nonnegative integers, and p a prime. Write a and b in p-adic form,

\begin{aligned}a & = a_{n}p^{n} + a_{n−1}p^{n−1} + ⋯ + a_{1}p + a_{0}, \\ b & = b_{n}p^{n} + b_{n−1}p^{n−1} + ⋯ + b_{1}p + b_{0}, \end{aligned}

that is, with 0 ≤ a_{i}, b_{i} < p, for all i. Then we can compute the binomial coefficient modulo p as

\binom{a}{b} ≡ \binom{a_{n}}{b_{n}}\binom{a_{n−1}}{b_{n−1}} … \binom{a_{1}}{b_{1}}\binom{a_{0}}{b_{0}} (mod p).

This is easily proved by comparing coefficients in the polynomial equality over F_{p}:

(1 + t)^{a} = (1 + t^{p^{n}})^{a_{n}} … (1 + t^{p})^{a_{1}}(1 + t)^{a_{0}}.

As a simple example, if q is a power of p we have

\binom{2q − 1}{q} ≡ \binom{q}{q}\binom{q − 1}{0} ≡ 1 (mod p).

Our standard reference for groups and Lie algebras are [Hu], resp. [Jac].

3. Two-step Centralizers

Let L = ⨁_{i≥1} L_{i} be an algebra of maximal class over the field F of positive characteristic p. In this context, the two-step centralizers

C_{L} (L^{i} /L^{i+2})

are equivalent to the centralizers in L_{1} of the homogeneous components L_{i} for i ≥ 2, as

C_{L} (L^{i} /L^{i+2}) = C_{L_{1}}(L_{i}) + L^{2}.

Therefore we define the two-step centralizers to be the subspaces of L_{1}

C_{i} = C_{L_{1}}(L_{i}),

for i ≥ 2. Now C_{i} is a subspace of L_{1} of dimension 1, except when L is finite dimensional, of class at most i, so that C_{i} = L_{1} in this case. The sequence of two-step centralizers is the sequence (C_{i})_{i≥2}. If one of the terms of this sequence equals L_{1}, then so do all the following ones. In this case we may omit all the terms of the sequence following the first L_{1}; if we do this, the sequence is infinite, if L is infinite dimensional, and has length c − 1 if L is finite dimensional, of class c. The abelian algebra of maximal class L = L_{1}, of dimension 2, has an empty sequence of two-step centralizers.

Now consider the set C = { C_{i} : i ≥ 2 }\{L_{1}} in an algebra L. If C has n elements, we say that L has n (distinct) two-step centralizers. We also number the distinct two-step centralizers in order of occurrence, so C_{2} is the first two-step centralizer; if C_{2} = C_{3} = ⋯ = C_{k−1} ≠ C_{k}, we say that C_{k} is the second (distinct) two-step centralizer, and so on. It is to be noted that we will apply ordinal numbers to two-step centralizer in this sense only.

Let M = F[t] be the (abelian) Lie algebra derived from the ring of polynomials. Consider the derivation D of M, given by multiplication by t, that is, fD = tf for f ∈ M. Let M be the Lie algebra Fx+M, where x induces the derivation D on M. Write y = 1 ∈ M. It is easily seen that M_{i} = Ft^{i−1}, and that C_{M_{1}} (M_{i}) = Fy, for i ≥ 2. Note that M has a presentation with generators x and y, and relators [y \underbrace{x ⋯ x}_{i} y], for i ≥ 1. We recall that a Lie algebra L is said to be metabelian

when [LL] is abelian. When F is a prime field F_{p}, this is the graded Lie algebra associated to the unique infinite pro-p-group of maximal class. We first have

3.1 Lemma. If all the two-step centralizers distinct from L_{1} coincide in the algebra L of maximal class, then L is a factor of M.

Proof. Let C_{2} = Fy and let x ∈ L_{1} \Fy. Then L is generated by {x, y} and satisfies the defining relations of M, and is thus a quotient of it.

It follows that every metabelian algebra is a quotient of M, and M is the unique infinite-dimensional metabelian algebra of maximal class. In fact, let L be a metabelian algebra of maximal class, and Fy = C_{2}. Choose x ∈ L_{1} \ Fy, so that L_{2} = F · [yx], and [yxy] = 0. We have, by induction on i,

0 = [[y \underbrace{x ⋯ x}_{i}][yx]] = [y \underbrace{x ⋯ x}_{i} yx] − [y \underbrace{x ⋯ x}_{i+1} y] = −[y \underbrace{x ⋯ x}_{i+1} y],

for all i ≥ 1, so that Fy is the unique two-step centralizer. We call M the infinitedimensional metabelian algebra of maximal class.

It follows immediately from the standard theory of p-groups of maximal class ([Hu, III.14]) that the family of algebras of maximal class associated with the prop-groups of maximal class consists of the infinite-dimensional metabelian algebra of maximal class over F_{p}, of its quotients, and, for p > 3, of one algebra for each even dimension 6, 8, …, p + 1. The latter algebras are associated with the exceptional p-groups of maximal class, those that have two distinct two-step centralizers.

One can actually show that the sequence of two-step centralizers of an algebra L of maximal class determines the isomorphism type of L in all cases. To formulate this precisely, we have to be able to compare the two-step centralizer sequences of two algebras, and say when they are the same. We do this by attaching a canonical form to the sequence of two-step centralizers.

Choose a non-zero element y in the first two-step centralizer of a non-abelian algebra L of maximal class. If there are no other distinct two-step centralizers, we have already seen that L is determined. Suppose then that L has at least two distinct two-step centralizers, and choose a non-zero element x in the second two-step centralizer. If there is a third distinct two-step centralizer, then one can assume, possibly replacing x, say, with a scalar multiple of it, that the third distinct two-step centralizer contains the element y−x. Now all two-step centralizers contain a uniquely determined element of the form y − αx, for some α ∈ F_{∞}, where we understand y−∞x = x. This procedure amounts to introducing an affine coordinate on a projective line.

Define the coefficient sequence of L to be the (normalised) sequence (α_{i})_{i≥2} in F_{∞}, where C_{L_{1}}(L_{i}) = F · (y − α_{i}x). A simple calculation shows that the coefficient sequence depends only on L. We say that two algebras of maximal class have the same sequences of two-step centralizers when their coefficient sequences, determined as above, are the same. We have the

3.2 Theorem. An algebra of maximal class is determined up to isomorphism by its sequence of two-step centralizers.

Proof. We show that the coefficient sequence of the algebra L determines uniquely a presentation of L, or equivalently its adjoint representation on L^{2}. If all two-step centralizers of L coincide, we have just seen that L is a factor algebra of the unique infinite-dimensional metabelian one. So assume L has at least two distinct two-step centralizers, and write the two-step centralizers in the canonical form described above.

We first have that L is generated by x and y. We have L_{2} = Fu_{2} ≠ 0, where u_{2} = [yx]. We define recursively, if L_{i} ≠ 0,

u_{i} = \{ \begin{matrix}[u_{i−1}x] & if [u_{i−1}x] ≠ 0, \\ [u_{i−1}y] & if [u_{i−1}x] = 0, \end{matrix}

so that L_{i} = Fu_{i} for all i ≥ 2. This determines completely a presentation of L in terms of x, y and the u_{i}. In fact, if Fx ≠ C_{i−1} = F · (y − αx), for some α ∈ F, we have

[u_{i−1}y] = [u_{i−1}, αx + y − αx] = α[u_{i−1}x] = αu_{i},

so that the action of x and y on the u_{i} is completely determined in any case.

We will say that an algebra of maximal class is periodic if its sequence of twostep centralizers is (ultimately) periodic. For instance, the infinite-dimensional metabelian algebra of maximal class is periodic. And a finite-dimensional algebra L of maximal class is also periodic, as the elements of its sequence of two-step centralizers ultimately equal L_{1}.

It follows from the above theorem that over a field F there are at most |F|^{ℵ_{0}} algebras of maximal class, and max{|F|, ℵ_{0}} periodic ones. We will see that these bounds are actually attained over every field F of positive characteristic (6.2 and 9.4(1)).

We now need the following simple 3.3 Lemma. If C_{2} = C_{i−1} ≠ C_{i}, and the class of L is at least i+1, then C_{i+1} = C_{2}.

Proof. Let C_{2} = Fy, and C_{i} = Fz, for some y, z ∈ L_{1}. We have L_{2} = F · [zy], so that [zyy] = 0. If 0 ≠ u ∈ L_{i−1}, we have from our assumptions on the two-step centralizers

0 = [u[zyy]] = [uzyy] − 2[uyzy] + [uyyz] = [uzyy].

Now [uzy] ≠ 0, so that L_{i+1} = F · [uzy], and our claim follows.

There is another method which is very useful for describing algebras of maximal class, especially those with with two distinct two-step centralizers. In the sequence of two-step centralizers of an algebra of maximal class, the first two-step centralizer occurs in contiguous subsequences, separated by isolated occurrences of a different two-step centralizer. We call each of these subsequences a constituent of the algebra. More precisely, a constituent is a subsequence

(C_{i}, C_{i+1}, …, C_{j})

of the sequence of two-step centralizers such that C_{2} = C_{i} = C_{i+1} = ⋯ = C_{j}, but C_{j+1} ∉ { C_{2}, L_{1} } and either i = 2, or C_{i−1} ≠ C_{2}. We make more precise statements on the pattern of constituents in Section 5. Note that if L has exactly two distinct two-step centralizers, the sequence of its constituents lengths determines the sequence of two-step centralizers, and thus the isomorphism type of L. In the case of two distinct two-step centralizers, the constituent sequence turns out indeed to be the most practical way of describing the algebra.

We mention at this point that it will be sometimes convenient to add formally a first element C_{1} to the sequence of two-step centralizers of an algebra of maximal class. This will be a one-dimensional subspace of L_{1}, which is not necessarily a two-step centralizer. If C_{1} = C_{2}, the length of the first constituent has thus to be increased by one.

4. The algebras of Albert-Frank-Shalev

Albert and Frank [AF] have constructed certain simple algebras over a prime field F_{p}. They depend on four parameters a, b, n and p, where 0 ≤ a < b < n, so that n ≥ 2. The algebra of Albert and Frank S = AF (a, b, n, p) corresponding to these parameters has dimension p^{n} − 1, and is described below. Shalev [Sh2] shows that the algebras S can be given a cyclic grading

S = ⨁_{i=1}^{p^{n}−1} S_{i},

where dim(S_{i}) = 1 for all i, and that there is a (non-singular) derivation D of S such that S_{i}D = S_{i+1} cyclically. Let S_{i} = F_{p}u_{i}. Consider the algebra obtained by extending S by D, and tensoring with the polynomial ring F_{p}[t]

(S + F_{p}D) ⊗ F_{p}[t],

and let L = AF S(a, b, n, p) be the subalgebra spanned by u_{1} ⊗ t, D ⊗ t. It is easy to see that L is an algebra of maximal class, with

L_{1} = F_{p} · (u_{1} ⊗ t) + F_{p} · (D ⊗ t) L_{i} = F_{p} · (u_{i} ⊗ t^{i}) for i > 1,

where subscripts are taken cyclically.

In this section we shall show that the algebras of Albert-Frank-Shalev have two distinct two-step centralizers, and we will describe their constituents, as defined in the previous section. We claim that the algebra AF S(a, b, n, p) has a cyclic sequence of constituent lengths

\{ \begin{matrix}(2p^{b} − 2, (p^{b} − 1)^{p^{n−b}−2})^{∞} & if a = 0, \\ \begin{matrix}(2p^{a} − 1, & (p^{a} − 1)^{p^{b−a}−2}, 2p^{a} − 2, \\ ((p^{a} − 1)^{p^{b−a}−2}, 2p^{a} − 1)^{p^{n−b}−2}, (p^{a} − 1)^{p^{b−a}−2})^{∞} \end{matrix} & if a > 0. \end{matrix}

Here, as explained in the preliminaries, an exponent s denotes repetition s times of the basic pattern. If s = 0, the pattern is omitted altogether. Note that we make use of the possibility of regarding the first constituent in two different ways.

For instance the algebras AF S(0, 1, n, 3) have constituent length sequence

(4, 2)^{∞} for n = 2,
(4, 2^{7})^{∞} for n = 3,
(4, 2^{25})^{∞} for n = 4.

The algebras AF S(1, 2, n, 3) have constituent length sequence

(4, 2, 4, 2, 5, 2)^{∞} for n = 3,
(4, 2, 4, (2, 5)^{7}, 2)^{∞} for n = 4,
(4, 2, 4, (2, 5)^{25}, 2)^{∞} for n = 5.

The simple Lie algebras AF (a, b, n, p) of Albert and Frank [AF] are defined as follows. Let a, b, n, p as above. Let \overline{S} be a vector space of dimension p^{n} − 1 over F_{p^{n}}, having a basis {e_{x} | x ∈ F_{p^{n}}^{∗}} indexed by the non-zero elements of F_{p^{n}} (we also put e_{0} = 0). We make \overline{S} into a Lie algebra over F_{p^{n}} by defining a Lie product on the basis elements and extending linearly, via the formula

[e_{x}, e_{y}] = f(x, y) e_{x+y},

where f : F_{p^{n}} × F_{p^{n}} → F_{p^{n}} is the bilinear form defined by

f(x, y) = x^{p^{a}}y^{p^{b}} − x^{p^{b}}y^{p^{a}}.

It is clear from the definition that AF (a, b, n, p) depends on n and p, and on the cosets a + nZ, b + nZ. Therefore it makes sense to define AF (a, b, n, p) for all unordered pairs of integers {a, b}, with a ≢ b (mod n). So for instance we can regard the algebra AF S(0, b, n, p) as AF S(b, n, n, p).

We will make a sharper statement later.

It is known that \overline{S} is a simple Lie algebra over F_{p^{n}}, but Shalev shows in [Sh2] that \overline{S} is actually defined over the prime field F_{p}. Indeed, another basis of \overline{S} is given by {f_{0}, …, f_{p^{n}−2}}, where

f_{i} = ∑_{x∈F_{p^{n}}^{∗}} x^{i}e_{x},

for 0 ≤ i < p^{n} − 1, and, according to Proposition 2.4 of [Sh2] (but see also below), the multiplication constants of \overline{S} with respect to this basis belong to the prime field. Hence

S = ∑_{i=0}^{p^{n}−2} F_{p}f_{i}

is an F_{p}-Lie subalgebra of \overline{S}, which we shall denote by AF (a, b, n, p).

As remarked in [Sh2], it is convenient to put

u_{j} = f_{j−p^{a}−p^{b}},

for all j (viewed modulo p^{n} − 1), because S_{j} = F_{p}u_{j} form a cyclic grading of S. Also, S has a derivation D, such that u_{j}D = u_{j+1} for all j.

In terms of the u_{j}, Proposition 2.4 of [Sh2] asserts that

(Rel)  \{ \begin{matrix}[u_{p^{a}+p^{b}}, u_{p^{a}}] = u_{2p^{a}+p^{b}}, \\ [u_{p^{a}+p^{b}}, u_{p^{b}}] = −u_{p^{a}+2p^{b}}, \\ [u_{p^{a}+p^{b}}, u_{j}] = 0 & otherwise. \end{matrix}

Note that these relations, which describe ad(u_{p^{a}+p^{b}}) on S, together with the relations u_{j}D = u_{j+1} for all j, form a finite presentation (in terms of the generators u_{0}, …, u_{p^{n}−2}, and D) for the split extension of S by D, as a Lie algebra over F_{p} (because L is generated as a Lie algebra by u_{p^{a}+p^{b}} and D). This yields a proof of the fact, mentioned above, that \overline{S} is defined over the prime field.

A remarkable fact is that, once n and p are fixed, the isomorphism type of AF (a, b, n, p) depends only on the distance between the two subsets a + nZ and b + nZ of Z, that is to say, on

min{b − a, n + a − b}.

More precisely, for any a, b we shall set up an isomorphism

δ : AF (a, b, n, p) → AF (a − 1, b − 1, n, p).

In fact, let \overline{S} = AF (a, b, n, p) be as above, and let \overline{S}′ = AF (a − 1, b − 1, n, p) have a basis {e′_{x} | x ∈ F_{p^{n}}^{∗}}, with Lie multiplication given by

[e′_{x}, e′_{y}] = f′(x, y) e′_{x+y},

where

f′(x, y) = x^{p^{a−1}}y^{p^{b−1}} − x^{p^{b−1}}y^{p^{a−1}}.

Thus f(x, y) = f′(x^{p}, y^{p}), and one easily checks that the map

\overline{δ} : e_{x} ↦ e′_{x^{p}}

defines by linear extension a Lie algebra isomorphism of \overline{S} onto \overline{S}′. Furthermore, if f′_{i} and u′_{i} denote the elements of \overline{S}′ defined in the same way as f_{i} and u_{i} for \overline{S}, then \overline{δ}(f_{ip}) = f′_{i}, and \overline{δ}(u_{ip}) = u′_{i}. Hence \overline{δ} restricts to the desired isomorphism δ, which sends u_{ip} to u′_{i}. Furthermore, if D′ denotes the derivation of S′ = AF (a − 1, b − 1, n, p) such that u′_{j}D′ = u′_{j+1} for all j, then the derivation of S which corresponds to D′ under the isomorphism δ is D^{p}. This will be of relevance in Section 7, when we shall describe the process of deflation for algebras of maximal class. It is perhaps worth mentioning, however, that our formulas for the constituent lengths sequence will show that the algebras of maximal class AF S(a, b, n, p) and AF S(a − 1, b − 1, n, p) are not isomorphic, unless a = 0, b = 1 and n = 2.

Now we proceed to the determination of the two-step centralizers, and hence the constituent lengths, of L = AF S(a, b, n, p). By the very definition of L, this reduces to finding the action of ad(u_{1}) on the basis {u_{j}} of the simple Lie algebra S = AF (a, b, n, p). In fact, if we define the coefficients α_{j} ∈ F_{p} via

[u_{j}, u_{1}] = α_{j}u_{j+1}

(where subscripts are taken cyclically modulo p^{n} − 1), then

C_{S_{1}+F_{p}D}(S_{i}) = F_{p} · (u_{1} − α_{i}D),

and, in turn, for i ≥ 2,

C_{L_{1}}(L_{i}) = F_{p} · (u_{1} ⊗ t − α_{i}D ⊗ t).

We also put

[u_{j}, u_{p^{a}}] = β_{j}u_{j+p^{a}},

and, recalling that in characteristic p the p-th power of a derivation is itself a derivation, we find

[u_{p^{a}+p^{b}}, u_{j}] = [u_{p^{a}}, u_{j}]D^{p^{b}} − [u_{p^{a}}, u_{j+p^{b}}].

From formulas (Rel) we deduce that

β_{j+p^{b}} − β_{j} = \{ \begin{matrix}1 & if j = p^{a}, \\ −1 & if j = p^{b}, \\ 0 & otherwise. \end{matrix}

Starting from the known value β_{p^{a}} = 0, we may find all coefficients β_{j} recursively, for example by the following argument. If the coefficients β_{j} are arranged cyclically, with β_{j} immediately followed by β_{j+p^{b}}, then this cyclic sequence will split into a sequence of 0’s, starting with β_{2p^{b}} and ending with β_{p^{a}}, followed by a sequence of 1’s. We shall compute the value of β_{p^{b}+p^{a}+j} for 0 ≤ j < p^{n} − 1 (note that we know β_{p^{b}+p^{a}} = 1). We shall write

j = ip^{b} + r,

with 0 ≤ i < p^{n−b}, and 0 ≤ r < p^{b} (note that the pair of values i = p^{n−b} − 1, r = p^{b} − 1, is never attained, since j < p^{n} − 1). We observe that the smallest non-negative multiple of p^{b} which is congruent to j modulo p^{n} − 1 is

ip^{b} + r + r(p^{n} − 1) = p^{b}(i + rp^{n−b}).

In particular, for j = 2p^{b} − (p^{b} + p^{a}) = p^{b} − p^{a}, such a multiple of p^{b} will be p^{b}(p^{n} − p^{n−b+a}), and for j = p^{a} − (p^{b} + p^{a}) + (p^{n} − 1), it will be p^{b}(p^{n} − 2). It follows from our previous discussion, that β_{p^{b}+p^{a}+j} = 0 if and only if

p^{n} − p^{n−b+a} ≤ i + rp^{n−b} ≤ p^{n} − 2,

which is equivalent to

p^{b} − p^{a} ≤ r ≤ p^{b} − \frac{i + 2}{p^{n−b}}.

Now the inequality on the right is actually superfluous, as j < p^{n} − 1. We conclude that

(B)  β_{p^{b}+p^{a}+j} = \{ \begin{matrix}0 & if r ≥ p^{b} − p^{a}, \\ 1 & otherwise. \end{matrix}

Now the case a = 0 is easily settled by noting that α_{j} = β_{j} for all j then. Explicitly, we have α_{1} = 0, and α_{kp^{b}} = 0 for 2 ≤ k < p^{n−b}, while the remaining α_{j} are 1. Hence AF S(0, b, n, p) has only two distinct two-step centralizers, namely C_{2} = F_{p} · (u_{1} − D), and C_{2p^{b}} = F_{p}u_{1}. The constituent lengths can be found by counting the number of 1’s between two consecutive 0’s, in the (cyclic) sequence (α_{j}). Thus we see that AF S(0, b, n, p) has constituent lengths sequence

(2p^{b} − 2, (p^{b} − 1)^{p^{n−b}−2})^{∞}.

We shall now turn to the case a > 0. A relation between the coefficients (α_{j}) and (β_{j}) can be found by computing the commutator [u_{p^{a}+1}, u_{j}] in two different ways, namely

[u_{p^{a}+1}, u_{j}] = [u_{j+p^{a}}, u_{1}] − [u_{j}, u_{1}]D^{p^{a}},

and

[u_{p^{a}+1}, u_{j}] = [u_{j+1}, u_{p^{a}}] − [u_{j}, u_{p^{a}}]D.

This gives us the relation

α_{j+p^{a}} − α_{j} = β_{j+1} − β_{j},

for all j. After a change of indices, and taking into account our previous computation of the coefficients β_{j}, we obtain

α_{p^{b}+2p^{a}−1+j} − α_{p^{b}+p^{a}−1+j} = β_{p^{b}+p^{a}+j} − β_{p^{b}+p^{a}−1+j} = \{ \begin{matrix}1 & if r = 0, \\ −1 & if r = p^{b} − p^{a}, \\ 0 & otherwise. \end{matrix}

It is not difficult to deduce that

α_{p^{b}+2p^{a}−1+j} = \{ \begin{matrix}α_{p^{b}+2p^{a}−1} & if r = 0, p^{a}, …, (p^{b−a} − 2)p^{a}, \\ α_{p^{b}+2p^{a}−1} − 1 & otherwise. \end{matrix}

Since we know that

0 = α_{p^{a}+p^{b}} = α_{p^{a}+p^{b}+(p^{n}−1)} = α_{p^{b}+2p^{a}−1+(p^{n}−p^{a})},

we conclude that

α_{p^{b}+2p^{a}−1+j} = \{ \begin{matrix}1 & if r = 0, p^{a}, …, (p^{b−a} − 2)p^{a}, \\ 0 & otherwise. \end{matrix}

This proves that AF S(a, b, n, p) has exactly two distinct two-step centralizers. Note that the above formula specializes to (B) when a = 0. However, in contrast to the case a = 0, when a > 0 we have α_{2} = 0 (since α_{2} can be written as α_{p^{b}+2p^{a}−1+j}, for some 0 ≤ j < p^{n} − 1 with r = p^{b} − 2p^{a} + 2), and thus the first two-step centralizer is C_{2} = F_{p}u_{1}, and the second distinct two-step centralizer is C_{2p^{b}} = F_{p} · (u_{1} − D). By counting the number of 0’s between consecutive 1’s in the sequence (α_{j}), one sees that the constituent length sequence for AF S(a, b, n, p) when a > 0, is

\begin{matrix}2p^{a} − 2, (p^{a} − 1)^{p^{b−a}−2}, 2p^{a} − 2, \\ (((p^{a} − 1)^{p^{b−a}−2}, 2p^{a} − 1)^{p^{n−b}−1}, (p^{a} − 1)^{p^{b−a}−2}, 2p^{a} − 2)^{∞}. \end{matrix}

Here we take advantage of the possibility of taking formally an extra centralizer C_{1} to write the constituent length sequence in the more convenient purely periodic form stated at the beginning of this section.

5. Constituents

Let L = ⨁_{i≥1} L_{i} be an algebra of maximal class over the field F of positive characteristic p. In this section we describe the possible lengths of constituents. Let C_{2} = Fy be the first two-step centralizer in L. If there are no more two-step centralizers distinct from C_{2}, we have seen that L is uniquely determined. Thus suppose there are at least two distinct two-step centralizers. Assume that C_{k} = Fx is the first occurrence of the second distinct two-step centralizer. Therefore

Fy = C_{2} = C_{3} = ⋯ = C_{k−1} ≠ C_{k} = Fx,

and L has class at least k + 1. We begin by showing 5.1 Lemma. k is even. If the class of L is at least k + 2, then 2p divides k.

The first part of the lemma reflects a well-known property of exceptional groups of maximal class ([Hu], III.14.9 and III.14.12), and relates to [M2, Lemma 3.1].

Proof. We have L_{i} = F · [y \underbrace{x ⋯ x}_{i−1}] for i = 2, …, k, and L_{k+1} = F · [y \underbrace{x ⋯ x}_{k−1} y]. If

k = 2h + 1 is odd, then

0 = [[y \underbrace{x ⋯ x}_{h}][y \underbrace{x ⋯ x}_{h}]] = (−1)^{h}[y \underbrace{x ⋯ x}_{k−1} y].

Therefore k is even. Thus we have

0 = [[y \underbrace{x ⋯ x}_{k/2}][y \underbrace{x ⋯ x}_{k/2}]]

= ±\binom{k/2}{k/2 − 1}[y \underbrace{x ⋯ x}_{k−1} yx]

= ± \frac{k}{2} [y \underbrace{x ⋯ x}_{k−1} yx].

Since C_{k} ≠ C_{2}, we have by Lemma 3.3 that C_{2} centralizes L_{k+1} = F · [y \underbrace{x ⋯ x}_{k−1} y].

Since by assumption L_{k+1} is not central, we have in fact C_{2} = C_{k+1}, and [y \underbrace{x ⋯ x}_{k−1} yx] ≠ 0. From the above calculations we obtain that p divides k/2.

We note that it is because of the fact that the characteristic has to divide k that we have

5.2 Corollary. An algebra of maximal class over a field of characteristic zero is centre-by-metabelian.

Because of Lemma 5.1, we can write the length of the first constituent in the form 2qn − 2, where q = p^{h}, with h > 0, and p does not divide n.

5.3 Lemma. If p is odd, L has first constituent of length 2qn − 2, and L has class at least 2qn + q + 1, then the second constituent has length q − 1.

Proof. First we have, for 0 ≤ j < q − 1,

[y \underbrace{x ⋯ x}_{2qn−1} y \underbrace{x ⋯ x}_{j} y] = 0.

To prove this, work by induction on j. The case j = 0 is given by Lemma 3.3. For j > 0 write j + 1 = βp^{t}, with β ≢ 0 (mod p). Note that as j < q − 1, we have p^{t} < q, so that j + p^{t} < 2q − 1, and [y \underbrace{x ⋯ x}_{j+p^{t}} y] = 0. Therefore

\begin{aligned}0 & = [[y \underbrace{x ⋯ x}_{2qn−p^{t}−1}][y \underbrace{x ⋯ x}_{j+p^{t}} y]] \\ & = (−1)^{p^{t}} \binom{j + p^{t}}{p^{t}}[y \underbrace{x ⋯ x}_{2qn−1} y \underbrace{x ⋯ x}_{j} y]. \end{aligned}

14 A. CARANTI, S. MATTAREI AND M. F. NEWMAN

This implies [y \underbrace{x ⋯ x}_{2qn−1} y \underbrace{x ⋯ x}_{j} y] = 0, as

\binom{j + p^{t}}{p^{t}} = \binom{βp^{t} + p^{t} − 1}{p^{t}} ≡ β ≢ 0 (mod p),

by Lucas’ Theorem.

Finally we show [y \underbrace{x ⋯ x}_{2qn−1} y \underbrace{x ⋯ x}_{q}] = 0. Suppose [y \underbrace{x ⋯ x}_{2qn−1} y \underbrace{x ⋯ x}_{q−1}] is not central. Since the class of L is thus at least 2qn + q + 1, we will obtain that

Fx = C_{2qn} ≠ Fy = C_{2qn+1} = ⋯ = C_{2qn+q−1} ≠ C_{2qn+q} = Fx,

so that the second constituent has length q − 1 as claimed. Because qn + (q − 1)/2 is an integer, and

\begin{aligned}0 & = [[y \underbrace{x ⋯ x}_{qn+(q−1)/2}][y \underbrace{x ⋯ x}_{qn+(q−1)/2}]] \\ & = ±\binom{qn + (q − 1)/2}{q}[y \underbrace{x ⋯ x}_{2qn−1} y \underbrace{x ⋯ x}_{q}], \end{aligned}

Lucas’ Theorem gives

\binom{qn + (q − 1)/2}{q} ≡ n ≢ 0 (mod p),

so that our claim is proved. 5.4 Lemma. Suppose n > 1.

(1) If p is odd, then L has class at most 2qn + q.

(2) If p = 2, then L has class at most 2qn + 2q − 1.

Part (1) of this lemma is related to part (a) of Corollary 4.4 of [M2]. Proof. As n > 1, we have [y \underbrace{x ⋯ x}_{2q−1} y] = 0, so that

\begin{aligned}0 & = [[y \underbrace{x ⋯ x}_{2qn−q−1}][y \underbrace{x ⋯ x}_{2q−1} y]] \\ & = (−1)^{q}\binom{2q − 1}{q}[y \underbrace{x ⋯ x}_{2qn−1} y \underbrace{x ⋯ x}_{q−1} y]. \end{aligned}

Since \binom{2q − 1}{q} = \binom{q + q − 1}{q} ≡ 1 (mod p), we obtain [y \underbrace{x ⋯ x}_{2qn−1} y \underbrace{x ⋯ x}_{q−1} y] = 0.

This concludes the proof when p is odd, in view of Lemma 5.3.

Suppose now p = 2. The only value j < 2q − 1 such that q divides j + 1 is j = q − 1. Since we have

[y \underbrace{x ⋯ x}_{2qn−1} y \underbrace{x ⋯ x}_{q−1} y] = 0

here, the proof of Lemma 5.3 shows that

[y \underbrace{x ⋯ x}_{2qn−1} y \underbrace{x ⋯ x}_{j} y] = 0

for all j < 2q − 1, Now

\begin{aligned}0 & = [[y \underbrace{x ⋯ x}_{q(n+1)−1}][y \underbrace{x ⋯ x}_{q(n+1)−1}]] \\ & = (−1)^{qn−q}\binom{q(n + 1) − 1}{qn − q}[y \underbrace{x ⋯ x}_{2qn−1} y \underbrace{x ⋯ x}_{2q−1}], \end{aligned}

so that L_{2qn+2q} = 0, as

\binom{q(n + 1) − 1}{qn − q} = \binom{qn + q − 1}{q(n − 1)} ≡ \binom{n}{n − 1} ≡ n ≢ 0 (mod 2).

We may summarize the above in the following

5.5 Theorem. Let L be an algebra of maximal class over a field of characteristic p. If the class of L is large enough, then the first constituent has length of the form 2p^{h} − 2, for some h ≥ 1. If p is odd, then the second constituent has length p^{h} − 1.

This theorem, and Lemma 5.3 above, are related to Theorem 4.1 of [M2].

Because of this result, we will confine ourselves from now on to algebras of maximal class whose first constituent has length 2q − 2, where q = p^{h} is a power of the characteristic p. We are therefore covering in particular the infinite-dimensional case. The number q = p^{h} will play an important role in the description of the structure of L: we call it the parameter of the algebra L.

Note that the Albert-Frank-Shalev algebra AF S(h, h+1, n, 2), for 1 ≤ h < n−1, has parameter q = 2^{h} and second constituent of length 2q−2, so that the restriction on the length of the second constituent of Theorem 5.5 really does not apply to p = 2.

We can now state

5.6 Proposition. Let L be an algebra of maximal class with parameter q = p^{h}. Then the lengths of the constituents of L can only assume the values

2q − 1,  and  2q − p^{s} − 1, for 0 ≤ s ≤ h.

We will show in the next section that all these numbers actually occur as constituent lengths in some algebra of maximal class.

Proof. Let L be an algebra of maximal class with parameter q = p^{h}. Let the first two-step centralizer be C_{2} = Fy and the second one be C_{2q} = Fx.

First note that q − 1 is a lower bound for the length of a constituent. This is basically a calculation we have already employed. In fact, proceed by induction on the occurrence of constituents, the base being provided by the statement about the length of the first constituent. Consider the constituent starting at C_{i+m+1}, where m ≥ q − 1 is the length of the previous constituent (C_{i}, C_{i+1}, …, C_{i+m−1}). Let 0 ≠ u ∈ L_{i−1}, or u = x if i = 2. We have thus L_{i+m} = F · [uy \underbrace{x ⋯ x}_{m}]. We claim

that

[uy \underbrace{x ⋯ x}_{m} y \underbrace{x ⋯ x}_{j} y] = 0 for 1 ≤ j < q − 1,

so that by the definition of constituents [uy \underbrace{x ⋯ x}_{m} y \underbrace{x ⋯ x}_{j+1}] ≠ 0 for 1 ≤ j < q.

Proceeding by induction on j, write j + 1 = βp^{t}, with β ≢ 0 (mod p). Since j + 1 < q, we have p^{t} < q, so that j + p^{t} < 2q − 1. Also, 1 ≤ p^{t} ≤ p^{h−1} < q − 1 ≤ m. Thus we have [y \underbrace{x ⋯ x}_{j+p^{t}} y] = 0, and

\begin{aligned}0 & = [[uy \underbrace{x ⋯ x}_{m−p^{t}}][y \underbrace{x ⋯ x}_{j+p^{t}} y]] \\ & = (−1)^{p^{t}} \binom{j + p^{t}}{p^{t}}[uy \underbrace{x ⋯ x}_{m} y \underbrace{x ⋯ x}_{j} y], \end{aligned}

where the binomial coefficient is congruent to β ≢ 0 (mod p). It may be worth noting that the situation is more delicate here than it was in the similar argument in the proof of Lemma 5.3, as we might have that both [uy \underbrace{x ⋯ x}_{m} y] and [uy \underbrace{x ⋯ x}_{m+1}]

are non-zero here.

Next note that 2q − 1 is an upper bound for the length of a constituent. As above, consider the constituent starting at C_{i+m+1}, where m ≥ q − 1 is the length of the previous constituent (C_{i}, C_{i+1}, …, C_{i+m−1}). Let 0 ≠ u ∈ L_{i−1}, and write v = [uy \underbrace{x ⋯ x}_{m}]. We have thus L_{i+m} = Fv. We want to show that the following

constituent has length at most 2q − 1.

We have [y \underbrace{x ⋯ x}_{2q}] = 0. Thus if [vx] = 0, we have

\begin{aligned}0 & = [v[y \underbrace{x ⋯ x}_{2q}]] \\ & = [vy \underbrace{x ⋯ x}_{2q}], \end{aligned}

so that one of the two-step centralizers C_{i+m+1}, …, C_{i+m+2q} must be Fx. So suppose [vz] = 0 for some z ∈ L_{1} \ \{Fy, Fx\}. We have [y \underbrace{z ⋯ z}_{2q−1} x] = 0, so that

\begin{aligned}0 & = [v[y \underbrace{z ⋯ z}_{2q−1} x]] \\ & = [v[y \underbrace{z ⋯ z}_{2q−1}]x] − [vx[y \underbrace{z ⋯ z}_{2q−1}]]. \end{aligned}

Suppose, by way of contradiction, that C_{i+m+1} = ⋯ = C_{i+m+2q} = Fy. Thus

[vx[y \underbrace{z ⋯ z}_{2q−1}]] = ∑_{j=0}^{2q−1}(−1)^{j}\binom{2q − 1}{j}[vx \underbrace{z ⋯ z}_{j} y \underbrace{z ⋯ z}_{2q−1−j}] = 0.

It follows that

0 = [v[y \underbrace{z ⋯ z}_{2q−1}]x] = [vy \underbrace{z ⋯ z}_{2q−1} x],

so that either one of the centralizers C_{i+m+1}, …, C_{i+m+2q−1} is Fz, or C_{i+m+2q} = Fx, a contradiction.

Now suppose by induction (C_{i}, C_{i+1}, …, C_{i+m−1}) is a constituent of length m = 2q − p^{s} − 1, where we admit formally s = −∞, and p^{−∞} = 0, to take care of the case m = 2q − 1. We want to show that the length of the next constituent can only assume one of the stated values. Proceeding by way of contradiction, we may thus assume that the next constituent has length n, with q − 1 < n < 2q − 2.

Write n + 1 = βp^{t}, with β ≢ 0 (mod p), so that we have p^{t} < q. Suppose n + p^{t} = (β + 1)p^{t} − 1 ≥ 2q − 1 = 2p^{h} − 1, so that β ≥ 2p^{h−t} − 1. Since βp^{t} = n + 1 < 2q − 1, we have also β < 2p^{h−t}, so that the only possibility is β = 2p^{h−t} − 1, and n = 2q − p^{t} − 1, one of the allowed values.

Therefore we may assume n + p^{t} < 2q − 1, so that [y \underbrace{x ⋯ x}_{n+p^{t}} y] = 0. We have also

2q − p^{s} − p^{t} − 1 ≥ 0. Take as above 0 ≠ u ∈ L_{i−1}. We get

\begin{aligned}0 & = [[uy \underbrace{x ⋯ x}_{2q−p^{s}−p^{t}−1}][y \underbrace{x ⋯ x}_{n+p^{t}} y]] \\ & = (−1)^{p^{t}} \binom{n + p^{t}}{p^{t}}[uy \underbrace{x ⋯ x}_{2q−p^{s}−1} y \underbrace{x ⋯ x}_{n} y], \end{aligned}

a contradiction.

When p is odd, we can actually make a slightly stronger statement, which extends Lemma 5.3.

5.7 Lemma. If p is odd, and the class of L is large enough, then a constituent of length m > q − 1 is followed by a constituent of length q − 1.

Proof. If m > q −1, we have m ≥ 2q −q/p−1 > q, as q > 2, so that, in the notation already employed,

\begin{aligned}0 & = [[uy \underbrace{x ⋯ x}_{m−q}][y \underbrace{x ⋯ x}_{2q}]] \\ & = (−1)^{q}\binom{2q}{q}[uy \underbrace{x ⋯ x}_{m} y \underbrace{x ⋯ x}_{q}], \end{aligned}

where the binomial coefficient is congruent to 2 ≢ 0 (mod p).

6. Inflation

In this section we describe a construction procedure for algebras of maximal class. The arguments we employ here are self-contained, but the construction itself goes back to Zassenhaus [Z] and Block [Blo].

Let F be a field of positive characteristic p. Let ε be an element with minimal polynomial X^{p} over F, and consider the extension F[ε]. Now let M be a graded Lie algebra over F, and consider the tensor product

M^{↑} = M ⊗_{F} F[ε],

which can be seen as a Lie algebra over F or F[ε]. Given a derivation D ∈ Der_{F}(M) of degree 1, this extends naturally to an element of Der_{F[ε]}(M^{↑}), which we also call D. We now construct a derivation E ∈ Der_{F}(M^{↑}), with the property that E^{p} = D. It is well known that in characteristic p the p-th power of a derivation is again a derivation.

Consider a graded basis B of M over F. Then the elements

b ⊗ \frac{ε^{i}}{i!}

form a basis for M^{↑} over F, as b ranges in B, and p > i ≥ 0. If w is the original grading on M, we see that

w′(b ⊗ \frac{ε^{i}}{i!} ) = pw(b) − i

defines a grading of M^{↑}.

Let ∂ be the “standard” derivation of F[ε] defined by ε∂ = 1, so that ε^{i}∂ = iε^{i−1} for i > 0, and 1∂ = 0. Consider the element of Der_{F}(M^{↑})

E = D ⊗ \frac{ε^{p−1}}{(p − 1)!} + 1 ⊗ ∂,

in the notation of [Blo, Theorem 7.1], that is, E acts on the basis b ⊗ ε^{i}/i! by

E : \{ \begin{matrix}b ⊗ \frac{ε^{i}}{i!} ↦ b ⊗ \frac{ε^{i−1}}{(i − 1)!} & if i > 0, and \\ b ⊗ 1 ↦ bD ⊗ \frac{ε^{p−1}}{(p − 1)!} . \end{matrix}

It is easy to verify that E has degree 1, and that E^{p} = D. We call E the p-th root of D, and write E = D^{1/p}.

What is important here is that the inflation of an algebra of maximal class is an algebra of maximal class. Let L be an algebra of maximal class over the field F, and let M be any maximal ideal of L. Choose an element s ∈ L_{1} \ M, and consider the derivation D = ad(s) ∈ Der_{F}(M). We define the inflation of L at M to be the algebra ^{M}L which is the extension of M^{↑} by an element s′ inducing on M^{↑} the derivation E = D^{1/p}. As above, M^{↑} is a graded algebra.

Take an element 0 ≠ t ∈ M ∩ L_{1}, and define

t′ = t ⊗ \frac{ε^{p−1}}{(p − 1)!} ∈ M^{↑}.

As above, E is a derivation of M^{↑} of degree 1. To show that ^{M}L is of maximal class, it suffices thus to show that s′ and t′ generate ^{M}L, or equivalently, [^{M}L_{k},^{M}L_{1}] = ML_{k+1} for all k ≥ 1. In fact we have, for b in a graded basis of M,

[b ⊗ \frac{ε^{i}}{i!} , s′] = b ⊗ \frac{ε^{i−1}}{(i − 1)!},

and thus [^{M}L_{k}, s′] = ^{M}L_{k+1}, if i > 0, or equivalently if

k = w′(b ⊗ \frac{ε^{i}}{i!} ) = pw(b) − i ≢ 0 (mod p).

If i = 0, we have

[b ⊗ 1, s′] = [bs] ⊗ \frac{ε^{p−1}}{(p − 1)!}.

If this is zero, then [bs] = 0 in L, so that either Fb is the centre of L, and Fb ⊗ 1 is the center of ^{M}L, or C_{w(b)} = Fs. In the latter case [bt] ≠ 0, so that

[b ⊗ 1, t′] = [b ⊗ 1, t ⊗ \frac{ε^{p−1}}{(p − 1)!}] = [bt] ⊗ \frac{ε^{p−1}}{(p − 1)!}

is non-zero. So in all cases we have [^{M}L_{k},^{M}L_{1}] = ^{M}L_{k+1} for all k ≥ 1.

Observe that if L has class c then ^{M}L has class cp.

The above construction may appear to depend on the choice of s, but all choices lead to isomorphic algebras. In fact, suppose without loss of generality we choose s + t instead of s. The underlying ideal being in both cases M, it is enough to show that the elements s′ + t′ and (s + t)′ induce the same derivation on M^{↑}. Here the element (s + t)′ induces on M^{↑} the derivation ad(s + t)^{1/p}. Now if i > 0 one has

[b ⊗ \frac{ε^{i}}{i!} , s′ + t′] = b ⊗ \frac{ε^{i−1}}{(i − 1)!} + [bt] ⊗ \frac{ε^{i+p−1}}{i!} = b ⊗ \frac{ε^{i−1}}{(i − 1)!} = [b ⊗ \frac{ε^{i}}{i!} , (s + t)′],

as i + p − 1 ≥ p. We also have

[b ⊗ 1, s′ + t′] = [b ⊗ 1, s′] + [b ⊗ 1, t′]

= [b, s] ⊗ \frac{ε^{p−1}}{(p − 1)!} + [b, t] ⊗ \frac{ε^{p−1}}{(p − 1)!}

= [b, s + t] ⊗ \frac{ε^{p−1}}{(p − 1)!}

= [b ⊗ 1, (s + t)′].

Next we describe how two-step centralizers behave under inflation. It is handy, when going from an algebra L to its inflation ^{M}L, to consider the linear map L_{1} → ^{M}L_{1} defined by αs + βt ↦ αs′ + βt′, for α, β ∈ F, in the above notation. This allows an easy description of ^{M}L via its sequence of two-step centralizers. To do this, we first take advantage of the possibility of prepending a first element C_{1} to the sequence of two-step centralizers of L. We choose here formally C_{1} = C_{L_{1}}(L_{1}) = M ∩ L_{1} = Ft, in the above notation. It is now easy to see 6.1 Lemma.

(1) If i ≡ 0 (mod p), then

C_{^{M}L_{1}}(^{M}L_{i}) = C_{L_{1}}(L_{i/p})′.

(2) If i ≢ 0 (mod p), and i ≥ 2, then

C_{^{M}L_{1}}(^{M}L_{i}) = C_{L_{1}}(L_{2})′ = Ft′.

Note that the element t determines M = Ft + ⨁_{i≥2} L_{i}, and conversely M determines Ft = M ∩ L_{1}. We may thus speak equivalently of inflating L at M or at Ft or at t. We also write ^{M}L = ^{t}L.

Also note that if we identify the elements of L_{1} with the elements of ^{M}L_{1} = ^{t}L_{1} via the prime operator, Ft is the first two-step centralizer in ^{M}L. If Ft was already the first two-step centralizer in L, the number and order of occurrence of distinct two-step centralizers in L and ^{M}L is the same. If Ft was a centralizer other than the first one, the number of distinct two-step centralizers in L and ^{M}L is the same, with Ft moving to the first place in the order of occurrence. Finally, if Ft was not a two-step centralizer in L, then ^{M}L has one two-step centralizer more than L, with the first two-step centralizer in L occurring as the second distinct one in ^{M}L, and so on.

This identification makes it possible to consider inflating for instance the abelian algebra L of maximal class with respect to a finite sequence t_{n}, t_{n−1}, …, t_{1} of nonzero elements in L_{1}. In particular, if the t_{i} are pairwise linearly independent, so that F has at least n − 1 elements, one sees with Lemma 6.1 that in the resulting algebra N of maximal class the i-th two-step centralizer is Ft_{i}, and that it occurs first as C_{N}(N_{2p^{i−1}})

Now in analogy with the case of groups (see [Ma]), we say that an algebra L of maximal class is covered if the set { C_{i} : i ≥ 2 } consists of all one-dimensional subspaces of L_{1}. If F is a finite field with n − 1 elements, we can construct covered algebras, starting with the abelian algebra L of maximal class, and inflating all n subspaces of dimension 1 of L_{1} to two-step centralizers. The covered algebra constructed this way has class p^{n+1}. It is however not hard to see that the smallest covered factor of this algebra has class 2p^{n}.

We have obtained

6.2 Proposition. Over a field of positive characteristic p and order n there are covered algebras of maximal class of class 2p^{n}.

We return to this theme in Section 9.

Now consider inflating the abelian algebra of maximal class L with respect to four distinct subspaces of L of dimension 1, for fields with at least 3 elements. We obtain an algebra of class p^{4} with four distinct two-step centralizers. We have noted that we may put these four two-step centralizers in canonical form as

Fy, Fx, F · (y − x), F · (y − αx)

where x and y form a basis of L = L_{1}, and the value α ∈ F \ {0, 1} is uniquely determined by the isomorphism type of the algebra. Therefore two algebras of this kind are isomorphic if and only if they have the same value of α. We thus see that if the field F of characteristic p is infinite, there are infinitely many pairwise non-isomorphic algebras of maximal class of class p^{4} which have a factor of class 2p^{3} with four distinct two-step centralizers.

We also have

6.3 Proposition. Over a field F of positive characteristic there are max{|F|, ℵ_{0}} periodic, insoluble algebras of maximal class.

Proof. The algebras of Albert-Frank-Shalev deal with the case of F finite. If F is infinite, it suffices to start with an algebra of Albert-Frank-Shalev, and inflate it twice with respect to suitable elements, to obtain an insoluble algebra with four distinct two-step centralizers. By the above, we get |F| pairwise non-isomorphic algebras.

There is another ‘projective’ description of the transformation of two-step centralizers when going from L to ^{M}L which is sometimes useful, and deals with the coefficient sequence. Let this sequence be (α_{i})_{i≥2} for L. We recall that we have set the first two distinct two-step centralizers to be Fy and Fx, and the third distinct one, if it exists, to be F · (y − x). The two-step centralizer C_{i} can then be written uniquely as F(y − α_{i} · x), for α_{i} ∈ F_{∞}, where we understand y − ∞ · x = x. Here too we have chosen C_{1} = Ft as above. Now if (α′_{i})_{i≥2} is the coefficient sequence for ML, we have as above α′_{i} = 0 if i ≢ 0 (mod p), and i ≥ 2, whereas

α′_{pi} = 1 − \frac{β}{α_{i}} ,

if t is a nonzero multiple of y − βx, where β ≠ 0, ∞. If β = 0, then α′_{pi} = α_{i}, whereas for β = ∞ we have

α′_{pi} = \frac{1}{α_{i}} .

It is also easy to see what happens to the length of constituents under inflation. The inflation of an algebra L with parameter q with respect to the first two-

step centralizer has parameter pq, and a constituent of length m in L becomes a constituent of length pm + p − 1 = p(m + 1) − 1 in the inflation.

In the inflation with respect to an element which is not already a two-step centralizer, the parameter becomes p, and all constituents except the first one have length p − 1.

In the inflation with respect to a two-step centralizer which is not the first one, the parameter is also p, and the length of a constituent other than the first one is either p − 1 or 2p − 1.

In the particular case of an algebra L with two distinct two-step centralizers Fy, Fx, inflation replaces a constituent of length m with a subsequence of constituent lengths 2p − 1, (p − 1)^{m−1}, and the first constituent by the subsequence 2p − 2, (p − 1)^{2q−3}, where q is the parameter of L.

Hence 6.4 Lemma. In an inflated algebra over a field of positive characteristic p, all constituents except the first one have length m ≡ −1 (mod p).

We are now able to prove the following converse of Proposition 5.6.

6.5 Theorem. Let p be a prime number, h ≥ 1, and 0 ≤ s ≤ h. Then there exists an algebra of maximal class with two distinct two-step centralizers over the field F_{p} with parameter p^{h}, and a constituent other than the first one of length 2p^{h} − p^{s} − 1.

Proof. Given the ambiguity on the length of the first constituent, we will realize 2q − 1 and 2q − 2 as lengths of constituents which are not the first ones.

In fact, proceed by induction on h. For h = 1, the algebra AF S(1, 2, 3, p) has constituents of all possible lengths p − 1, 2p − 2, 2p − 1. Assume we have algebras L^{−∞}, L^{0}, …, L^{h} with two distinct two-step centralizers such that L^{s} has parameter p^{h}, and at least one constituent of length 2p^{h}−p^{s}−1, where we understand p^{−∞} = 0. By inflating L^{s} with respect to the first two-step centralizer, we obtain an algebra with parameter p^{h+1}, and a constituent of length

p(2p^{h} − p^{s} − 1 + 1) − 1 = 2p^{h+1} − p^{s+1} − 1.

This leaves out only constituents of length 2p^{h+1} − 2. But such a constituent appears in the algebra AF S(h + 1, h + 2, h + 3, p). Note that our inflation method can provide algebras with at most three different constituent lengths, as in the Albert-Frank-Shalev algebras we use as starting points.

7. Deflation

An algebra L of maximal class can be recovered from any of its inflations N = ML. In fact, choose any element z ∈ N_{1} which does not lie in the first two-step centralizer. Let ζ denote an element that induces on ⨁_{i≥1} N_{ip} the derivation ad(z)^{p}. Then the algebra

N^{↓} = Fζ + ⨁_{i≥1} N_{ip},

is clearly isomorphic to L. Actually, the definition of N^{↓} makes sense for every algebra N of maximal class: we call N^{↓} the deflation of N.

We claim that deflation does not depend on the choice of the element z. In fact, let Fy be the first two-step centralizer in N. To prove our claim, we will show that

ad(z + y)^{p} = ad(z)^{p} + ad([y \underbrace{z ⋯ z}_{p−1}]),

as derivations of ⨁_{i≥1} N_{ip}. In fact, let 0 ≠ u ∈ N_{ip}. Because the minimum length of a constituent is p − 1, we have that

[u \underbrace{z ⋯ z}_{i}] ≠ 0,  and  [u \underbrace{z ⋯ z}_{i−1} y] = 0

GRADED LIE ALGEBRAS OF MAXIMAL CLASS 23

for all 1 ≤ i ≤ p, except possibly for exactly one value i_{0}, for which we have [u \underbrace{z ⋯ z}_{i_{0}−1} y] ≠ 0, whereas we do not make any assumption about [u \underbrace{z ⋯ z}_{i_{0}}]. If there is no such i_{0} we thus have

u ad([y \underbrace{z ⋯ z}_{p−1}]) = [u[y \underbrace{z ⋯ z}_{p−1}]]

= ∑_{i=0}^{p−1}(−1)^{i}\binom{p − 1}{i}[u \underbrace{z ⋯ z}_{i} y \underbrace{z ⋯ z}_{p−1−i}]

= 0,

so that

u ad(z + y)^{p} = [u, \underbrace{z + y, …, z + y}_{p}]

= [u \underbrace{z ⋯ z}_{p}]

= u ad(z)^{p}

= u(ad(z)^{p} + ad([y \underbrace{z ⋯ z}_{p−1}])).

In the other case

\begin{aligned}u ad([y \underbrace{z ⋯ z}_{p−1}]) & = (−1)^{i_{0}−1}\binom{p − 1}{i_{0} − 1}[u \underbrace{z ⋯ z}_{i_{0}−1} y \underbrace{z ⋯ z}_{p−i_{0}}] \\ & = [u \underbrace{z ⋯ z}_{i_{0}−1} y \underbrace{z ⋯ z}_{p−i_{0}}], \end{aligned}

so that

u ad(z + y)^{p} = [u, \underbrace{z + y, …, z + y}_{p}]

= [u \underbrace{z ⋯ z}_{p}] + [u \underbrace{z ⋯ z}_{i_{0}−1} y \underbrace{z ⋯ z}_{p−i_{0}}]

= u(ad(z)^{p} + ad([y \underbrace{z ⋯ z}_{p−1}])).

It also follows that N^{↓} is of maximal class. We have N_{1}^{↓} = Fz′ + Fy′, and N_{i}^{↓} = N_{ip}, for i ≥ 2. Now if all the two-step centralizers

C_{ip}, C_{ip+1}, …, C_{(i+1)p−1},

are different from Fz, then for 0 ≠ u ∈ N_{ip} we have

0 ≠ [uz′] = [u \underbrace{z ⋯ z}_{p}] ∈ N_{(i+1)p} = N_{i+1}^{↓}.

If C_{ip+i_{0}−1} = Fz ≠ Fy, for some 1 ≤ i_{0} ≤ p, then

0 ≠ [u \underbrace{z ⋯ z}_{i_{0}−1} y \underbrace{z ⋯ z}_{p−i_{0}}] = [u[y \underbrace{z ⋯ z}_{p−1}]] = [uy′] ∈ N_{(i+1)p} = N_{i+1}^{↓}.

Note that in this case exactly one of the above two-step centralizers is distinct from Fy, as constituents have length at least p − 1.

It is not difficult to compare two-step centralizers in L and its deflation L^{↓}. As in the case of inflation, define a linear map

\begin{matrix}′ : & L_{1} ⟶ L_{1}^{↓} \\ αz + βy ⟼ αζ + β[y \underbrace{z ⋯ z}_{p−1}], \end{matrix}

for α, β ∈ F. Let C_{i} = C_{L_{1}}(L_{i}) and C′_{i} = C_{L_{1}^{↓}}(L_{i}^{↓}).

We have

7.1 Proposition.

(1) If all the two-step centralizers

C_{ip}, C_{ip+1}, …, C_{(i+1)p−1},

equal C_{2} = Fy, for i ≥ 2, then

C_{L_{1}^{↓}}(L_{i}^{↓}) = Fy′.

(2) If there is a 1 ≤ i_{0} ≤ p such that C_{ip+i_{0}−1} = F · (z − αy), for some α, then

C_{L_{1}^{↓}}(L_{i}^{↓}) = F · (z′ − αy′).

Proof. Note that, as above, exactly one of the above two-step centralizers is thus distinct from Fy.

The first possibility is clear. As to the second one we have, for 0 ≠ u ∈ C_{ip},

[u, z′ − αy′] = [uz′] − α[uy′]

= [u \underbrace{z ⋯ z}_{p}] − α[u \underbrace{z ⋯ z}_{i_{0}−1} y \underbrace{z ⋯ z}_{p−i_{0}}]

= [u \underbrace{z ⋯ z}_{i_{0}−1}, z − αy, \underbrace{z ⋯ z}_{p−i_{0}}]

= 0.

Note the following immediate consequences 7.2 Corollary.

(1) In going from L to L^{↓}, no centralizer can disappear except possibly the first one.

(2) Suppose that in going from L to L^{↓} the first centralizer disappears. Then L has parameter p, and all constituents except the first one have length p − 1.

(3) Suppose that in going from L to L^{↓} the first centralizer becomes the i-th one, for some i > 1. Then L has parameter p, and all constituents, distinct from the first one, leading to the first occurrences of the second, third, …, i-th centralizer in L have length p − 1.

In general, it is not true that inflation is the inverse of deflation. It can happen, as we now show, that ^{t}(L^{↓}) ≅ L̸ for all 0 ≠ t ∈ L_{1}. In fact, by our observations above, an inflated algebra with parameter q and two distinct two-step centralizers has no constituents, except the first one, of length 2q − 2.

Now it follows readily from the theory we have developed for the Albert-Frank-Shalev algebras that

7.3 Proposition. For 0 ≤ a < b < n, and p a prime, we have

AF S(a, b, n, p)^{↓} = AF S(a − 1, b − 1, n, p),

where we understand AF S(0, b, n, p) = AF S(b, n, n, p) in the left-hand term.

Therefore all algebras of Albert-Frank-Shalev can be obtained via deflation from another algebra of this kind. However they have more than one constituent of length 2q −2 while, as we have shown above, all algebras obtained by inflation have only one constituent of length 2q − 2.

On the other hand we have

7.4 Proposition. If an algebra of maximal class L with parameter q has at most one constituent of length 2q − 2, then it can be recovered from its deflation N = L^{↓} with a suitable inflation.

Proof. In fact, all constituents except the first have thus lengths congruent to p − 1 modulo p. Hence values i for which C_{i} ≠ C_{2} are spaced apart by multiples of p. The first such value is 2q, where q is the parameter of the algebra. Since this is itself a multiple of p, we obtain that C_{i} = C_{2} = Fy in L, for i ≢ 0 (mod p). It follows that L ≅ ^{y′}(L^{↓}), as the two algebras have the same sequence of two-step centralizers.

We can use deflation to show

7.5 Lemma. Let L be an infinite-dimensional algebra of maximal class over a field of positive characteristic p. Suppose the i-th distinct two-step centralizer appears for the first time as C_{L_{1}}(L_{j}). Then j ≥ 2p^{i−1}, and the bound is attained.

We consider infinite-dimensional algebras to avoid the exceptional behaviour of the second distinct two-step centralizer in low-dimensional algebras, as in those coming from the exceptional groups of maximal class ([Hu, III.14.24]).

Proof. We proceed by induction on i. The case i = 1 is clear and the case i = 2 is settled in Lemma 5.1. Suppose L is an infinite-dimensional algebra of maximal class in which the i-th distinct two-step centralizer T, for some i > 2, occurs as early as possible as T = C_{L_{1}}(L_{k}), and suppose by way of contradiction that k < 2p^{i−1}. In the deflated algebra L^{↓}, it is readily seen that the earliest occurrence of T as a two-step centralizer is as T = C_{L_{1}^{↓}}(L_{⌈k/p⌉}^{↓}), with ⌈k/p⌉ ≤ 2p^{i−2}. If T is the i-th distinct two-step centralizer in L^{↓}, we get a contradiction, as ⌈k/p⌉ < k.

So suppose T is the (i − 1)-th distinct two-step centralizer in L^{↓}. By the description of centralizers in L^{↓}, this can only happen if all occurrences of the first centralizer in L prior to T vanish during deflation, and in turn this can only happen if all constituents in L leading to T, except the first one, have length exactly p − 1, so that L has parameter p. It follows that k is a multiple of p. By induction, we have ⌈k/p⌉ = 2p^{i−2}, so that 2p^{i−1} − p < k < 2p^{i−1}, and k is not a multiple of p, a contradiction.

8. Soluble algebras

In this section we will be concerned with the soluble algebras of maximal class. A finite-dimensional algebra of maximal class is nilpotent, and thus soluble. We will therefore take all algebras of maximal class in this section to be infinite-dimensional.

Let L = ⨁_{i=1}^{∞} L_{i} be an algebra of maximal class over the field F of positive characteristic p. Note first that if L is soluble, then it is abelian-by-(finite dimensional), as the last non-trivial term of the derived series will be an abelian ideal of finite codimension, since L is just-(infinite dimensional).

We begin by proving

8.1 Proposition. Let L be an infinite-dimensional, soluble algebra of maximal class. Let M = ⨁_{i>k} L_{i} be the maximal abelian ideal of L, or M = C_{L} (⨁_{i≥2} L_{i}) if L is metabelian. Then M is a free k-dimensional module over a polynomial ring F[t], and L/M embeds into End_{F[t]}(M).

Proof. We employ an argument modeled on one of [Sh1].

If L is metabelian, then L is the unique infinite-dimensional metabelian algebra of maximal class, as constructed in Section 3. So suppose M = ⨁_{i>k} L_{i}, with k ≥ 2.

Let 0 ≠ τ ∈ L_{k}. Since M is the maximal abelian ideal, τ does not centralize M. Let z be any element of L, and m ∈ M. Since [τz] ∈ M, and M is abelian, we have

0 = [m[τz]] = [mτz] − [mzτ].

We claim [L_{k+1}τ] ≠ 0. Assume, by way of contradiction, [L_{k+1}τ] = 0. Let 0 ≠ m_{0} ∈ L_{k+1}, and 0 ≠ m ∈ L_{k+s+1}, for some s. Then there are z_{1}, …, z_{s} ∈ L_{1} such that m = [m_{0}, z_{1}, …, z_{s}]. Thus

0 = [m_{0}, τ, z_{1}, …, z_{s}]
= [m_{0}, z_{1}, τ, …, z_{s}]
= …
= [m_{0}, z_{1}, …, z_{s}, τ]
= [mτ],

so that τ centralizes M, a contradiction. It follows that [L_{k+1}τ] = L_{2k+1}, and more generally, by the same argument, that [L_{s}τ] = L_{s+k}, for all s > k. Completing the proof is now a straightforward matter. The Lie algebra morphism

\begin{matrix}φ : & L ⟶ End_{F}(M) \\ z ↦ (u ↦ [uz]) \end{matrix}

has kernel M. The element t = φ(τ) centralizes φ(L) in End_{F}(M), so that φ(L) ⊆ End_{F[t]}(M). Now if f is a polynomial with coefficients in F, it is easy to see that uf(t) = 0 for some 0 ≠ u ∈ M implies f = 0. Therefore F[t] is a polynomial ring, and M is a finite-dimensional, torsion-free F[t]-module.

Clearly if C_{s} = Fz, for some s > k, then

[L_{s+k}z] = [L_{s}τz] = [L_{s}zτ] = 0,

that is, C_{s+k} = C_{s}. It follows that the sequence of two-step centralizers of L is periodic, with period length k, starting with C_{k+1}. Therefore there are at most max{|F|, ℵ_{0}} soluble algebras of maximal class.

We can actually make a stronger statement on the periodicity of soluble algebras of maximal class.

8.2 Theorem. Let L be a non-metabelian algebra of maximal class over a field F of positive characteristic p. The following are equivalent:

(1) L is soluble.

(2) L is periodic, of period a power of p, namely

C_{L_{1}}(L_{i}) = C_{L_{1}}(L_{i+p^{m}}),

for some m, and for all i > k, for a suitable k. Furthermore, if k and m in (2) are chosen to be minimal, then k = p^{m}, and L^{k+1} is the maximal abelian ideal of L.

Proof. Without loss of generality, we may assume L is uncovered; tensoring with a suitable extension field will ensure this. Let z ∈ L_{1} be an element that is not in any two-step centralizer, y ≠ 0 an element in the first centralizer, and define u_{i} = [y \underbrace{z ⋯ z}_{i−1}], for i ≥ 2, so that L_{i} = Fu_{i}, for i ≥ 2. For i ≥ 2, let [u_{i}y] = α_{i}u_{i+1},

for some α_{i} ∈ F. Clearly L is periodic if and only if the sequence (α_{i})_{i≥2} is periodic.

Suppose L is soluble, and let L^{k+1} be its maximal abelian ideal. Let p^{m} ≥ k. Then for i > k we have

0 = [u_{i}u_{p^{m}+1}] = [u_{i}[y \underbrace{z ⋯ z}_{p^{m}}]]

= [u_{i}y \underbrace{z ⋯ z}_{p^{m}}] − [u_{i} \underbrace{z ⋯ z}_{p^{m}} y]

= (α_{i} − α_{i+p^{m}}) u_{i+p^{m}+1}, so the sequence (α_{i})_{i≥2} is periodic, of period a power of p. Note the consequence of Proposition 8.1: if L^{k+1} is the maximal abelian ideal of L, then

C_{L}(L^{i}) = L^{k+1}, for all i > k.

Take the smallest m such that there is a k for which

α_{i} = α_{i+p^{m}} for all i > k,

and take k to be minimal with this property. Then the above calculation, read backwards, shows that

[u_{i}u_{p^{m}+1}] = 0 for i > k, and [u_{k}u_{p^{m}+1}] ≠ 0.

By means of

[u_{i}u_{j+1}] = [u_{i}[u_{j}z]] = [u_{i}u_{j}z] − [u_{i}zu_{j}]

it is easy to prove by induction on j that

[u_{i}u_{j}] = 0 for all i > k, and j > p^{m},

and that

[u_{k}u_{j}] = −[u_{k}u_{p^{m}+1}] ≠ 0 for all j > p^{m}.

The latter equality yields k ≤ p^{m}, and the former shows that L^{p^{m}+1} is abelian, whence L is soluble.

Furthermore, we deduce that

C_{L}(L^{p^{m}+1}) = L^{k+1},

and hence L^{k+1} is the maximal abelian ideal of L. But then k is a period of L, and thus it is a multiple of the minimum period p^{m}. As k ≤ p^{m}, we obtain k = p^{m}, concluding the proof.

Note that the codimension of the maximal abelian ideal of a soluble algebra decreases under deflation, unless the algebra is metabelian.

In fact, let this codimension be k + 1 in an algebra L. If L is not metabelian, then k > 1. In the deflated algebra L^{↓}, the codimension of the maximal abelian ideal is at most

k′ + 1 = ⌈\frac{k + 1}{p}⌉ .

This is less than k + 1. In fact we have

⌈\frac{k + 1}{p}⌉ ≤ \frac{k + 1}{p} + 1.

Now (k + 1)/p + 1 < k + 1 if and only if (k + 1) + p < (k + 1)p, that is, k(p − 1) > 1, which always holds.

This implies 8.3 Proposition. Soluble algebras of maximal class reduce to a metabelian one under repeated deflation.

Conversely, we can construct soluble algebras of maximal class by repeatedly inflating a metabelian one. In fact, one has the easy

8.4 Lemma. If L is a soluble algebra of maximal class, then so are all of its inflations.

Proof. This can be seen directly from the definition of inflation, or exploiting the characterization of soluble algebras of maximal class in terms of their periodicity.

To provide an example, let us construct a set of cardinality ℵ_{0} of infinitedimensional soluble algebras with two distinct two-step centralizers, for p = 3. Here we make use of the description of inflation we have given in the previous section. We start with the metabelian one. We first inflate it with respect to any element not in the first two-step centralizer, to get an algebra with constituent length sequence 5, 2^{∞}. Now we inflate with respect to the first, resp. to the second, two-step centralizer to get algebras with constituent length sequences 16, 8^{∞}, resp. 4, 2^{3}, (5, 2)^{∞}. Repeating the argument, we get four algebras with constituent length sequences

52, 26^{∞},  4, 2^{15}, (5, 2^{7})^{∞},  16, 8^{3}, (17, 8)^{∞},  4, 2^{3}, (5, 2)^{3}, (5, 2^{4}, 5, 2)^{∞},

and so on. Note that in writing the constituent length sequence this way, although the period is correctly displayed, periodicity may occur earlier than the writing appears to suggest.

In the next section, we will construct some more infinite-dimensional soluble algebras of maximal class with parameter q, and exactly one constituent, distinct from the first one, of length 2q − 2.

9. Limits

In this section we illustrate a way of constructing further infinite-dimensional algebras of maximal class by using inverse limits.

A first straightforward application of this method allows us to construct soluble algebras of maximal class with two distinct centralizers and parameter q = p^{h}, which have exactly two constituents of length 2q − 2. By Lemma 6.4 such algebras cannot be obtained by inflation.

Let N = AF S(h, h + k, k + n, p), for h, k, n ≥ 1. Here we consider h and k fixed, and n variable. Consider the factor algebra M^{(n)} of N with class p^{h+k+n} + p^{2h}; it has constituent length sequence

\begin{matrix}2p^{h} − 2, & (p^{h} − 1)^{p^{k}−2}, 2p^{h} − 2, \\ ((p^{h} − 1)^{p^{k}−2}, 2p^{h} − 1)^{p^{n}−1}. \end{matrix}

Since the constituent length sequence of an algebra of maximal class with two distinct centralizers determines the sequence of two-step centralizers, and thus its isomorphism type, these algebras M^{(n)} form an inverse system. The limit algebra has constituent length sequence

\begin{matrix}2p^{h} − 2, & (p^{h} − 1)^{p^{k}−2}, 2p^{h} − 2, \\ ((p^{h} − 1)^{p^{k}−2}, 2p^{h} − 1)^{∞}, \end{matrix}

and is thus periodic, with period p^{h+k}. By Theorem 8.2, it is soluble.

For instance, for p = 3 we get the following algebras with two distinct two-step centralizers, besides those already constructed in the previous section. For h = 1 and k = 1 we get the algebra with constituent length sequence 4, 2, 4, (2, 5)^{∞}, and period 9. This algebra inflates to the two algebras 16, 8, 14, (8, 17)^{∞} and 4, 2^{3}, 5, 2, 5, 2^{3}, (5, 2, 5, 2^{4})^{∞}, both of period 27. For h = 1 and k = 2, and h = 2 and k = 1, we get two more algebras of period 27, with constituent length sequences

4, 2^{7}, 4, (2^{7}, 5)^{∞},  and  16, 8, 16, (8, 17)^{∞}

and so on. In the same way for all primes p and for each field F of characteristic p we are able to construct

2^{m−1} + ∑_{i=1}^{m−1} 2^{m−i−1}i

pairwise non-isomorphic, infinite-dimensional, soluble algebras of maximal class with two distinct two-step centralizers and period of length p^{m}.

We now describe a slightly more complicated process of taking limits of algebras of maximal class. This involves repeating the inflation process countably many times.

Let L be an algebra of maximal class, and 0 ≠ t ∈ L_{1}. Recall that if we identify L_{1} and ^{t}L_{1} via the prime operator, we can describe the centralizer sequence of ^{t}L in terms of that for L.

This identification allows us, as seen before, to define the inflation of an algebra L with respect to a finite subsequence (t_{1}, …, t_{n}) of a sequence τ = (t_{i})_{i≥1} of non-zero elements of L_{1} as follows

(t_{1},...,t_{n})L = ^{t_{1}}(^{(t_{2},...,t_{n})}L) = ^{t_{1}}(^{t_{2}} … (^{t_{n}}L) …) .

It follows from our previous observations that the sequence of two-step centralizers of L up to class p^{n} is completely determined by the sequence (t_{1}, …, t_{n}) alone. Therefore the isomorphism type of the class p^{n} factor ^{(t_{1},...,t_{n})}L^{(n)} of ^{(t_{1},...,t_{n})}L is also determined by (t_{1}, …, t_{n}), and so ^{(t_{1},...,t_{n})}L^{(n)} is isomorphic to the class p^{n} factor of the algebra

(t_{1},...,t_{n+1})L = ^{t_{1}}(^{(t_{2},...,t_{n+1})}L) .

We obtain that^{(t_{1},...,t_{n})}L^{(n)} is a homomorphic image of the class p^{n+1} factor algebra (t_{1},...,t_{n+1})L^{(n+1)} of the algebra ^{(t_{1},...,t_{n+1})}L. It follows that the algebras ^{(t_{1},...,t_{n})}L^{(n)}, with the described homomorphisms, form an inverse system.

Now consider an infinite sequence τ = (t_{i})_{i≥1} of non-zero elements of L_{1}. We define the algebra ^{τ}L as the limit of the inverse system ^{(t_{1},...,t_{n})}L^{(n)}.

It is easily seen that distinct sequences may lead to isomorphic algebras. To ensure a one-to-one correspondence between sequences τ and isomorphism types of algebras ^{τ}L, we consider a set of normalized sequences, by going projective as before. Fix a basis u, v for L = L_{1}, and consider the set T of sequences τ = (t_{i})_{i≥1} with the following properties

(1) For each i ≥ 1, we have t_{i} = u−α_{i}v for some α_{i} ∈ F_{∞}. Here we understand as always u − ∞v = v.

(2) t_{1} = u.

(3) If not all t_{i} equal u, then the second element in order of occurrence in τ is v.

(4) If not all t_{i} equal u or v, then the third element in order of occurrence in τ is u − v.

The two-step centralizers in ^{τ}L are Ft_{1}, …, Ft_{n}, … in this order of occurrence. Ft_{n}, for n ≥ 1, occurs for the first time as a two-step centralizer as C_{^{τ}L_{1}}(^{τ}L_{2p^{n−1}}). It follows that ^{τ}L uniquely determines the sequence τ ∈ T . Also, it follows from the descriptions of two-step centralizers in an inflated algebra, and from our normalization of sequences τ, that τ ∈ T uniquely determines the sequence of two-step centralizers, and thus the isomorphism type, of ^{τ}L. We have thus

9.1 Proposition. There is a one-to-one correspondence between sequences τ ∈ T and isomorphism types of algebras ^{τ}L.

A presentation for an algebra ^{τ}L, where τ = (t_{i})_{i≥1}, can be read off, as described in Section 3, from the following description of the two-step centralizers C_{i} = C_{^{τ}L_{1}}(^{τ}L_{i}):

C_{i} = Ft_{j} if \{ \begin{matrix}i ≡ 0 (mod p^{j−1}) & and & i ≠ p^{j−1}, \\ and \\ i ≢ 0 (mod p^{j}) & or & i = p^{j}. \end{matrix}

As a first example, if we take τ = (u, u, u, …), we obtain that ^{τ}L is the infinitedimensional metabelian algebra of maximal class.

More interestingly, with this construction we are able to prove

9.2 Theorem. There are covered algebras of maximal class over every field of positive characteristic which is at most countable.

This we obtain simply by taking τ to be a sequence (u, v, u−v, ...) which includes generators for all 1-dimensional subspaces of L_{1}.

We now see which sequences τ ∈ T correspond to periodic algebras. Note first that if τ = (t_{1}, t_{2}, …) ∈ T , then

(^{τ}L)^{↓} = ^{τ^{↓}}L,

where τ^{↓} = (t_{2}, t_{3}, …). Here it is not necessarily true that τ^{↓} ∈ T . Conversely, we also have

t_{1}(^{(t_{2},t_{3},...)}L) = ^{(t_{1},t_{2},...)}L.

Suppose now ^{τ}L is periodic, with (not necessarily minimal) period k. It is easily seen that the deflated algebra (^{τ}L)^{↓} has period k/(k, p). After a finite number m of deflation steps, we will obtain an algebra N with period k′ = k/p^{m} coprime to p. By the above remark, the resulting algebra is still of the form N = ^{σ}L, for some sequence σ, and thus it is obtained by inflation. But we have seen that in an inflated algebra, which is not metabelian, all constituents have lengths congruent to −1 (mod p). Counting also the trailing two-step centralizer of each constituent, we see that the period of an inflated algebra must have length a power of p. This contradicts (k′, p) = 1, unless k′ = 1. If k′ = 1, σ is a constant sequence, and N is the infinite-dimensional metabelian algebra of maximal class. By the above remarks, ^{τ}L is obtained from L by a finite number of inflation steps, and it is therefore soluble. Also, τ is an ultimately constant sequence. Clearly there are max{|F|, ℵ_{0}} such sequences. We have obtained

9.3 Proposition. For τ ∈ T , the following are equivalent (1) ^{τ}L is periodic

(2) ^{τ}L is soluble

(3) τ is ultimately constant.

9.4 Corollary. Over every field F of positive characteristic there are (1) |F|^{ℵ_{0}} (non-periodic) algebras of maximal class.

(2) max{|F|, ℵ_{0}} infinite-dimensional, soluble algebras of maximal class.

As mentioned in the Introduction, the existence of non-periodic algebras of maximal class answers a question of Shalev [Sh3, Problem 28].

A particularly interesting class of examples is provided by the algebras of maximal class, that we are tempted to call fractal, that are transformed back to themselves by a finite number of deflation steps. The algebras of Albert-Frank-Shalev are examples of periodic algebras with this property: this follows easily from Proposition 7.3. The algebras we construct here are non-periodic, and have the additional property that they are transformed back to themselves also under a suitable finite sequence of inflation steps.

The simplest example of this kind is ^{τ}L, with τ = (u, v, u, v, …). Here (^{τ}L)^{↓} is isomorphic to ^{τ}L, and ^{τ}L inflates to itself with respect to the second distinct twostep centralizer Fv. To construct this algebra, one starts by inflating the abelian algebra of maximal class with respect to two independent elements, and then keeps inflating with respect to the second distinct two-step centralizer. For the prime p = 3, this algebra has constituent length sequence

4, 2^{3}, (5, 2)^{3}, (5, 2^{4}, 5, 2)^{3}, (5, 2^{4}, (5, 2)^{4}, 5, 2^{4}, 5, 2)^{3}, …

In fact, we obtain this sequence by starting with an algebra of class 7 with two distinct two-step centralizers with just one constituent of length 4, and applying repeatedly the transformation, described in the section on inflation, that replaces a constituent of length m with the subsequence 2p − 1, (p − 1)^{m−1}. This example can be generalized to the algebras ^{τ_{n}}L, with

τ_{n} = (\underbrace{u, …, u}_{n}, \underbrace{v, …, v}_{n}, \underbrace{u, …, u}_{n}, \underbrace{v, …, u}_{n}, …),

for n ≥ 1. Such an algebra reduces to itself after n deflation steps, and no fewer.

Other examples of algebras that deflate to themselves are obtained by considering sequences like

τ = (u, v, u − v, u, v, u − v, …).

10. Computations

None of the results of this paper rely on computer calculations. Still, a lot of computational work went into the writing of this paper. We report on it briefly here.

Our first computer calculations made use of a simple-minded GAP program [S+] to calculate the constituent lengths in some of the Albert-Frank-Shalev algebras. This lead to the results in Section 4.

We made use of the ANU p-Quotient Program [HNO] to compute with graded Lie algebras over a prime field as explained elsewhere [CMNS].

To simplify the exploration of algebras of maximal class with two distinct twostep centralizers, we built a small C preprocessor, called ppi (for prepare pQ input), which produces the input needed by the ANU p-Quotient Program from a description of the sequence of constituent lengths. This allowed us to explore these algebras to class 127 (the limit under the current version of the program). The theory developed in this paper is able to account for the algebras we have observed in this way. We constructed ppi so that, in producing the presentation to be used as input file by the ANU p-Quotient Program, it exploits our theory. Therefore the final output acted as a confirmation of the correctness of our results, and has actually allowed us to spot a couple of mistakes in earlier versions of this paper.

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(A. Caranti) Dipartimento di Matematica, Università degli Studi di Trento, via Sommarive 14, I-38050 Povo (Trento), Italy

E-mail address: caranti@science.unitn.it

(S. Mattarei) Dipartimento di Matematica ed Applicazioni, Università degli Studi di Padova, via Belzoni 7, I-35131 Padova, Italy

E-mail address: mattarei@pdmat1.math.unipd.it

(M.F. Newman) School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia

E-mail address: newman@maths.anu.edu.au