LOOP ALGEBRAS OF ZASSENHAUS ALGEBRAS IN CHARACTERISTIC THREE

A. CARANTI

1. Introduction

In [Car97] we studied the graded Lie algebra L associated to the Nottingham group in characteristic p > 3, from the point of view of providing a (finite) presentation of it. L is a loop algebra, with respect to a suitable grading, of the Witt algebra W_{1}. The latter has a one-dimensional Schur multiplier. Therefore the corresponding loop algebra M of the universal covering Ŵ_{1} of W_{1} has an infinitedimensional centre. As M is finitely generated, and M/Z(M) ≅ L, by a result of B. H. Neumann ([Neu37] or [Rob82, 2.2.3]) the algebra L is not finitely presented. We proved in [Car97], however, that M is finitely presented.

When p is 3, the Witt algebra W_{1} is isomorphic to sl(2), and has trivial Schur multiplier. Still, the loop algebra L has a central extension M by an infinitedimensional centre which is finitely presented. In this case, the cocycles needed to construct Z(M) are the products of the Killing form on sl(2) and the cyclic cohomology (see below) on the polynomial part, in a transposition of the theory of [Gar80] to the case of positive characteristic. This is dealt with in [Car].

In [Car97] we studied more generally loop algebras of the Zassenhaus algebras W_{n} in characteristic p > 3. The theory is exactly the same as in the case n = 1.

It is the purpose of this paper to study finite presentations of the analogous loop algebras of the Zassenhaus algebras W_{n} in characteristic 3, for n > 1. As we will see, these algebras combine the features of the two cases described above. The cohomology of these loop algebras could be calculated with the methods developed by Zusmanovich in [Zus92] (see also [Zus94]); however, Zusmanovich considers loop algebras of Zassenhaus algebras only in characteristic greater than 3. We will therefore give a brief self-contained, elementary account of the matter here, stressing in particular the role played by the invariant form on W_{n}.

It is well known (see [Far87] and [Dzh85]) that the Schur multiplier of W_{n} has dimension n − 1 in characteristic 3; as above, we obtain infinitely many cocycles for the loop algebra L. We call these the finite cohomology cocycles (which are nonclassical in the terminology of Zusmanovich).

Date: 30 September 1997.

1991 Mathematics Subject Classification. 17B65 17B56 17B50.

Key words and phrases. Cohomology of Zassenhaus algebras, cyclic cohomology, loop algebras, modular Lie algebras, graded Lie algebras, nondegenerate associative forms.

Partially supported by MURST, Italy. The author is a member of CNR-GNSAGA, Italy. The author is grateful to the Mathematisches Forschungsinstitut Oberwolfach for the kind hospitality while part of this work was being written.

The Killing form of W_{n} is zero, for n > 1; however, because of the characteristic 3, the algebra W_{n} has a nondegenerate associative form ([Far86]). This enables us, as in [Gar80] and [Car], to use the cyclic cohomology of the polynomial part of the loop algebra L to construct infinitely many more cocycles on L. We call these the loop cocycles (these are the classical ones according to Zusmanovich).

We prove that the central extension M of L obtained via the finite cohomology and loop cocycles is finitely presented. Note that by the result of B. H. Neumann mentioned above, this implies that all these cocycles generate a subspace of finite codimension in the second cohomology group of L (and also yields an alternative method for calculating the second cohomology group of W_{n}); one could see with the methods of Zusmanovich that these cocycles really exhaust the whole cohomology group.

Our proofs basically follow the pattern of [Car97], to which we refer. An argument similar to one used in [CMNS96] is employed to determine the occurrence of the loop cocycles.

We are grateful to Mike Newman for reading the manuscript and suggesting several corrections.

2. Preliminaries

We use [Jac79] and [SF88] as general references for Lie algebras. A good reference for the facts about cohomology, central extensions and universal coverings that we will be using is [Gar80].

At one point we will use Lucas’ theorem ([Luc78]), which states that if the two nonnegative integers a and b are written p-adically

a = a_{0} + a_{1}p + a_{2}p^{2} + ⋯ + a_{n}p^{n}, b = b_{0} + b_{1}p + b_{2}p^{2} + ⋯ + b_{n}p^{n},

so that 0 ≤ a_{i}, b_{i} < p, then

\binom{a}{b} ≡ ∏_{i=0}^{n} \binom{a_{i}}{b_{i}} (mod p).

For a prime-power q, we write F_{q} for the field with q elements. We will use Kronecker’s delta

δ(a, b) = \begin{cases}1 & if a = b, \\ 0 & otherwise. \end{cases}

We will use several times without explicit mention the generalized Jacobi identity

[v[y \underbrace{x … x}_{n}]] = ∑_{i=0}^{n}(−1)^{i}\binom{n}{i}[v \underbrace{x … x}_{i} y \underbrace{x … x}_{n−i}].

3. The Zassenhaus algebras and their loop algebras

The Zassenhaus algebras are defined as algebras of special derivations of algebras of divided powers in one variable ([SF88]), and can be regarded abstractly as vector spaces

W_{n} = W (1; n) = {y_{i} : −1 ≤ i ≤ q − 2},

of dimension q over the field F_{p} with p elements, p an odd prime, and q = p^{n}, with the product defined by

[y_{i}y_{j}] = (\binom{i + j + 1}{j} − \binom{i + j + 1}{i}) y_{i+j} .

These algebras are graded over the integers, weighting the y_{i} by their indices.

From now on, we will take p = 3, and n > 1. The corresponding theory for p > 3 is dealt with in [Car97], and that for p = 3 and n = 1 in [Car].

The original definition by Zassenhaus regarded W_{n} as an algebra over F_{q}, with basis e_{α}, for α ∈ F_{q}, and multiplication defined by

[e_{α}, e_{β}] = (β − α) · e_{α+β}.

A connection between the two presentations, which we will be handy later, is given by the formulas

\begin{cases}y_{−1} = e_{0} + ∑_{α} e_{α}, \\ y_{i} = − ∑_{α} α^{q−2−i}e_{α} & for i ≠ −1. \end{cases} (1)

We will consider the loop algebra of W_{n}, with respect to a particular grading, as in [Car97]. This is a grading over a cyclic group of order q − 1, and it is derived from the one over the integers by declaring y_{−1} and y_{q−2} to be of weight 1. In other words, we are viewing modulo q − 1 the opposite grading to the above grading over the integers.

Thus y_{q−i} acquires weight i − 1 in this grading, for 2 ≤ i ≤ q, and y_{−1} acquires weight 1. Therefore all homogeneous component have dimension 1, except that the i-th one, for i ≡ 1 (mod q − 1), has dimension 2, and it is spanned by y_{−1} and y_{q−2}.

Now consider the loop algebra of W_{n} with respect to the latter grading. That is, take first the Lie algebra

W_{n} ⊗ F_{p}[t],

where F_{p}[t] is a polynomial algebra, and consider the subalgebra L generated by the two elements

x = y_{−1} ⊗ t, y = y_{q−2} ⊗ t.

One easily sees that the action of the two generators x and y on L is defined as follows. Define formally an element v_{0} with the property

[v_{0}x] = −x, [v_{0}y] = y.

and elements v_{i}, for i > 0

v_{i} = [v_{i−1}xy \underbrace{x … x}_{q−3}].

Then we have, for i ≥ 0,

[v_{i}xyy] = [v_{i}xyxy] = ⋯ = [v_{i}xy \underbrace{x … x}_{q−4} y] = 0,

[v_{i+1}yx] = [v_{i+1}xy], [v_{i+1}xx] = [v_{i+1}yy] = 0.

4. The finite cohomology cocycles

We give here, for the convenience of the reader, a brief treatment of the second cohomology group of W_{n} in characteristic 3, in a form that is convenient for our purposes. This cohomology group is discussed in [Far87] and [Dzh85].

One sees readily that the following maps are 2-cocycles for W_{n}, in characteristic p = 3 and for n > 1, with respect to the presentation of Zassenhaus,

ϕ_{m}(e_{α}, e_{β}) = α^{3^{m}}δ(α + β, 0),

where 1 ≤ m < n. Independence in cohomology follows from the linear independence of the distinct field automorphisms of F_{3^{n}}.

We rewrite these cocycles in terms of the y_{i}, using formulas (1). We get, first for i, j ≠ −1,

ϕ_{m}(y_{i}, y_{j}) = ∑_{α} α^{q−2−i}(−α)^{q−2−j}α^{3^{m}}

= (−1)^{j+1} ∑_{α} α^{2q−i−j+3^{m}−4}

= (−1)^{j+1} ∑_{α} α^{q−i−j+3^{m}−3}

= (−1)^{j+1} \begin{cases}−1 & if q − 1 divides q − i − j + 3^{m} − 3 \\ 0 & otherwise \end{cases} .

Now q − 1 | q − i − j + 3^{m} − 3 when i + j = 3^{m} − 2, and when i + j = q + 3^{m} − 3. There are no other possibilities, as i + j ≤ 2q − 4. Therefore

ϕ_{m}(y_{i}, y_{j}) = \begin{cases}(−1)^{j} & if i + j = 3^{m} − 2, \\ (−1)^{j} & if i + j = q + 3^{m} − 3, \\ 0 & otherwise. \end{cases}

Now the map

θ_{m}(y_{i}, y_{j}) = \begin{cases}(−1)^{j} & if i + j = 3^{m} − 2, \\ 0 & otherwise. \end{cases}

is a coboundary, as

θ_{m}(u, v) = h([uv]),

where h is the linear map h : L → F_{3} defined by h(y_{i}) = −δ(i, 3^{m} − 2). We obtain that ψ_{m} = ϕ_{m} − θ_{m} is a cocycle, for 1 ≤ m < n, where

ψ_{m}(y_{i}, y_{j}) = \begin{cases}(−1)^{j} & if i + j = q + 3^{m} − 3, \\ 0 & otherwise. \end{cases}

The same formulas actually hold also when i or j take the value −1, as the extra e_{0} does not play a role. We have thus recovered the cocycles of [Dzh85, Theorem 2].

Take Z to be an abelian Lie algebra with basis

z_{1}, z_{2}, …, z_{n−1},

and construct the universal covering Ŵ_{n} of W_{n} as a central extension

0 → Z → Ŵ_{n} → W_{n} → 0,

with multiplication of (preimages of) y_{i} and y_{j} in Ŵ_{n} defined by

[y_{i}y_{j}] + ∑_{m=1}^{n−1} ψ_{m}(y_{i}, y_{j}) · z_{m}.

The grading over the integers of W_{n} extends to Ŵ_{n}, and z_{m} acquires weight q + 3^{m} − 3 in it. As these weights are distinct, and do not occur in W_{n} itself, we see that Ŵ_{n} is perfect. (This shows once more that the cocycles ψ_{m}, for 1 ≤ m < n, are independent in cohomology.)

In the opposite grading modulo q−1 of the grading over the integers, z_{m} acquires weight q − 3^{m} + 1 = q − (3^{m} − 1), that is, same as y_{3^{m}−2}.

Each of the cocycles ψ will yield infinitely many cocycles of L, the loop algebra of W_{n}. We call these cocycles of L the finite cohomology cocycles, because of the way they arise. The extension M of L via all these cocycles can also be obtained as a loop algebra of Ŵ_{n}, as in the previous section. In the next section we will show that M has infinitely many more cocycles, arising from the loop process.

5. The loop cocycles for the loop algebra

It is shown in [Far86] that the following defines an associative form on W_{n} in characteristic 3:

η(y_{i}, y_{j}) = (−1)^{j+1}δ(i + j, q − 3).

By this we mean that η is bilinear and symmetric, and that for u, v, w ∈ W_{n} we have

η([uv], w) = η(u, [vw]).

Now consider the following bilinear maps, for a ≥ 0,

σ_{ap} : F_{p}[t] × F_{p}[t] → F_{p}
(t^{i}, t^{j}) ↦ j · δ(i + j, ap).

These are easily seen to be elements (actually a basis) of the first cyclic cohomology group of F_{p}[t], as they satisfy the identity

σ_{ap}(uv, w) + σ_{ap}(vw, u) + σ_{ap}(wu, v) = 0

(see for instance [Zus92]). As shown in [Kas84], one can generalize the arguments of [Gar80] by using the first cyclic cohomology group.

In fact, proceed as in [Gar80] and [Car], and combine η with the σ to get the following cocycles on the loop algebras L or M (remember that we are working in characteristic p = 3):

τ_{3a}(y_{i} ⊗ t^{η(q−1)−i}, y_{j} ⊗ t^{μ(q−1)−j}) =
= η(y_{i}, y_{j}) · σ_{3a}(t^{η(q−1)−i}, t^{μ(q−1)−j})
= (−1)^{j+1} · δ(i + j, q − 3) ·
(−j − μ) · δ (λ(q − 1) − i + μ(q − 1) − j, 3a)

= (−1)^{i} · (j + μ) · δ(i + j, q − 3) · δ ((λ + μ)(q − 1) − (i + j), 3a) ,

where we have used the fact that i + j = q − 3 is even. Note that (λ + μ)(q − 1) = i + j + 3a = q − 3 + 3a is divisible by 3, so that λ + μ = 3b for some b. It follows that the values of a in the above for which the τ_{3a} do not obviously vanish on L are given by

3a = 3b(q − 1) − q + 3 = 2q + 3(q − 1)(b − 1).

The first few values of 3a are thus 2q, 5q − 3, 8q − 6, …

We call the cocycles τ_{3a} corresponding to these values of a the loop cocycles of L. To show that they are linearly independent in the second cohomology group of M, note the calculation, for i = (q − 3)/2, any b ≥ 1, and the corresponding value of a as above,

τ_{3a}(y_{i} ⊗ t^{(q−1)−i}, y_{i} ⊗ t^{(3b−1)(q−1)−i}) =
= (−1)^{i}(i − 1)
= (−1)^{i+1}
≠ 0.

Now the elements y_{i} ⊗ t^{(q−1)−i} and y_{i} ⊗ t^{(3b−1)(q−1)−j} commute in M, and there is one value of a for which the cocycle τ_{3a} does not vanish on the corresponding pair. Independence follows.

6. The Presentation

Let now N be the central extension of L via the finite cohomology cocycles and the loop cocycles τ_{3a}. In the rest of the paper we will show that N is finitely presented. As remarked in the Introduction, a result of B. H. Neumann implies that the finite cohomology cocycles and the loop cocycles span a subspace of finite codimension in the second cohomology group of L.

It is not difficult now to write down the action of the generators x and y on N, starting from the presentation of L, and keeping into account the cocycles we have added. Keeping the notation we have already employed for L, and thus slightly abusing it, define formally an element v_{0} with the property

[v_{0}x] = −x, [v_{0}y] = y.

and elements v_{i}, for i > 0

v_{i} = [v_{i−1}xy \underbrace{x … x}_{q−3}].

Write also θ_{i} = [v_{i}yx] − [v_{i}xy].

Then we have, for i ≥ 0,

\begin{cases}[v_{i}xy \underbrace{x … x}_{j} y] = 0, for 0 ≤ j ≤ q − 4, and j ≠ q − 3^{m} − 2, for 1 ≤ m < n \\ [v_{i}xy \underbrace{x … x}_{q−3^{m}−2} yx] = [v_{i}xy \underbrace{x … x}_{q−3^{m}−2} yy] = 0, for 1 ≤ m < n \\ [v_{i+1}xx] = [v_{i+1}yy] = 0 \\ [θ_{i}x] = [θ_{i}y] = 0, \\ θ_{i} = 0 for i ≢ 2 (mod 3). \end{cases} (2)

Now consider the Lie algebra Q given by the presentation

Q = ⟨ x, y : [yx \underbrace{x … x}_{j} y],

for 0 ≤ j ≤ q − 4, and j ≠ q − 3^{m} − 2, for 1 ≤ m < n

[yx \underbrace{x … x}_{q−3^{m}−2} yx], for 1 ≤ m < n

[y \underbrace{x … x}_{q}],

[y \underbrace{x … x}_{q−1} y \underbrace{x … x}_{q−5} yx],

[y \underbrace{x … x}_{q−1} y \underbrace{x … x}_{q−3} yxx] = [y \underbrace{x … x}_{q−1} y \underbrace{x … x}_{q−3} xyx] ⟩.

(3)

Note that the relations in (3) are a subset of those of (2). We want to show that Q is isomorphic to N, by showing that the defining relations (3) of Q imply all of (2).

We follow the pattern of [Car97], omitting the steps that are identical, and dealing only with the peculiarities for p = 3.

We have immediately

0 = [yx \underbrace{x … x}_{q−3^{m}−3}[yxy]] = −[y \underbrace{x … x}_{q−3^{m}−2} yy],

so that [yx \underbrace{x … x}_{q−3^{m}−2} y] is central for all m. The expansion

0 = [[y \underbrace{x … x}_{(q−1)/2}][y \underbrace{x … x}_{(q−1)/2}]]

yields the relation [v_{1}xy] = [v_{1}yx], or θ_{1} = 0.

So we can suppose (2) are verified for values less then some i > 2. Let v = v_{i} ∈ Q_{i(q−1)}. In [Car97] we proved by induction on j

[vxy \underbrace{x … x}_{j} y] = 0

for 0 ≤ j ≤ q − 4, and j ≠ q − 3^{m} − 2, for 1 ≤ m < n. For these special values we now show

[vxy \underbrace{x … x}_{q−3^{m}−2} yx] = [vxy \underbrace{x … x}_{q−3^{m}−2} xy] = [vxy \underbrace{x … x}_{q−3^{m}−2} yy] = 0.

Proving [vxy \underbrace{x … x}_{q−3^{m}−2} yy] = 0 is straightforward, as in [Car97]. For the rest, let

j = q − 3^{m} − 2, for 1 ≤ m < n.

We begin, as in [Car97], with

0 = [vx[y \underbrace{x … x}_{q−3^{m}−1} y]]

= [vx[y \underbrace{x … x}_{q−3^{m}−1}]y] − [vxy[y \underbrace{x … x}_{q−3^{m}−1}]]

= [vxy \underbrace{x … x}_{q−3^{m}−1} y]

− (−1)^{q−3^{m}−2}(q − 3^{m} − 1)[vxy \underbrace{x … x}_{q−3^{m}−2} yx] − (−1)^{q−3^{m}−1}[vxy \underbrace{x … x}_{q−3^{m}−1} y]

= −[vxy \underbrace{x … x}_{q−3^{m}−2} xy] + [vxy \underbrace{x … x}_{q−3^{m}−2} yx].

Now we want to prove that the two commutators vanish separately. The argument of [Car97] for m = 1 works also in characteristic 3, so we have to consider the case m > 1, for which there is no counterpart in [Car97]. Here and in the following, write v^{−s}, for 0 ≤ s < q − 1, for a homogeneous element such that

[v^{−s} \underbrace{x … x}_{s}] = v.

Let thus m > 1. Then

0 = [v^{−3^{m}+3^{m−1}+1}[y \underbrace{x … x}_{q−3^{m−1}−1} y]]

= [v^{−3^{m}+3^{m−1}+1}[y \underbrace{x … x}_{q−3^{m−1}−1}]y]

= (−1)^{3^{m}−3^{m−1}−1}\binom{q − 3^{m−1} − 1}{3^{m} − 3^{m−1} − 1}[vyx \underbrace{x … x}_{q−3^{m}−2} xy]

+ (−1)^{3^{m}−3^{m−1}} \binom{q − 3^{m−1} − 1}{3^{m} − 3^{m−1}}[vxy \underbrace{x … x}_{q−3^{m}−2} xy].

Here we apply Lucas’ theorem to get

\binom{q − 3^{m−1} − 1}{3^{m} − 3^{m−1} − 1} = \binom{q − 3^{m} + 3^{m} − 3^{m−1} − 1}{3^{m} − 3^{m−1} − 1}

= \binom{3^{m}(3^{n−m} − 1)}{0} · \binom{3^{m} − 3^{m−1} − 1}{3^{m} − 3^{m−1} − 1}

= 1,

and

\binom{q − 3^{m−1} − 1}{3^{m} − 3^{m−1}} = \binom{3^{n−1}(3 − 1) + ⋯ + 3^{m}(3 − 1) + 3^{m−1}(3 − 2) + ⋯ + (3 − 1)}{3^{m−1}(3 − 1)}

= \binom{1}{2} = 0.

Therefore

0 = ((−1) · 1) · [vxy \underbrace{x … x}_{q−3^{m}−2} y] = −[vxy \underbrace{x … x}_{q−3^{m}−2} xy],

as requested.

We prove easily, as in [Car97]

[v_{i+1}xx] = [v_{i+1}yy] = 0.

We now have to prove that θ_{i+1} = [v_{i+1}yx] − [v_{i+1}xy] is central. We first show that [v_{i+1}xy] and [v_{i+1}yx] are centralized by y. This follows from

0 = −[v_{i+1}[yxy]] = [v_{i+1}xyy] − 2[v_{i+1}yxy] + [v_{i+1}yyx] = [[v_{i+1}xy] + [v_{i+1}yx], y],

and

0 = [v_{i+1}^{−3}[yxxxxy]] = [v_{i+1}^{−3}[yxxxx]y]

= (−1)^{3}\binom{4}{3}[v_{i+1}^{−3}xxxyxy] + (−1)^{4}\binom{4}{4}[v_{i+1}^{−3}xxxxyy]

= [−[v_{i+1}yx] + [v_{i+1}xy], y],

To prove that θ_{i+1} commutes with x, we will expand, in a variation on the corresponding argument in [Car97], and proceeding by induction on i,

0 = [v_{i−1}x, [v_{2}yx] − [v_{2}xy]].

We will take a more conceptual approach than in [Car97]. One first proves by direct easy expansion, using only the extended Jacobi identity, the following formulas, for appropriate j ≤ i:

[v_{j}xv_{1}] = [v_{j+1}x], [v_{j}yv_{1}] = −[v_{j+1}y]
[v_{j}xyv_{1}] = [v_{j+1}yx] − [v_{j+1}xy] = θ_{j+1}
[v_{j}v_{1}] = 0
[v_{j}xyxv_{1}] = [v_{j}xyx].

We then use these to compute

0 = [v_{i−1}x, −[v_{2}yx] + [v_{2}xy]]

= [v_{i−1}x[v_{1}yv_{1}x]] + [v_{i−1}x[v_{1}xv_{1}y]] = [v_{i−1}x[v_{1}yv_{1}]x] − [v_{i−1}xx[v_{1}yv_{1}]]

+ [v_{i−1}x[v_{1}xv_{1}]y] − [v_{i−1}xy[v_{1}xv_{1}]] = [v_{i−1}x[v_{1}y]v_{1}x] − [v_{i−1}xv_{1}[v_{1}y]x]

+ [v_{i−1}x[v_{1}x]v_{1}y] − [v_{i−1}xv_{1}[v_{1}x]y] − [v_{i−1}xy[v_{1}x]v_{1}] + [v_{i−1}xyv_{1}[v_{1}x]]

= [v_{i−1}x[v_{1}y]v_{1}x] − [v_{i}x[v_{1}y]x] + [v_{i−1}x[v_{1}x]v_{1}y] − [v_{i}x[v_{1}x]y] − [v_{i−1}xy[v_{1}x]v_{1}]

= [v_{i−1}xv_{1}yv_{1}x] − [v_{i−1}xyv_{1}v_{1}x] − [v_{i}xv_{1}yx] + [v_{i}xyv_{1}x] + [v_{i−1}xv_{1}xv_{1}y] − [v_{i−1}xxv_{1}v_{1}y] − [v_{i}xv_{1}xy] + [v_{i}xxv_{1}y] − [v_{i−1}xyv_{1}xv_{1}] + [v_{i−1}xyxv_{1}v_{1}]

= [θ_{i+1}x] − 0 − [v_{i+1}xyx] + [θ_{i+1}x] + 0 − 0 − 0 + 0 − 0 + [v_{i+1}xyx] = −[θ_{i+1}x],

as desired.

We are left with proving that only one in three of the θ_{i} can be nonzero. We first prove, by induction on k, the formula [[v_{h}x][v_{k}y]] = [v_{h+k}xy] − kθ_{h+k}. The basis of the induction is provided by

[[v_{h}x][v_{1}y]] = [v_{h}xv_{1}y] − [v_{h}v_{1}xy] = [v_{h+1}xy] − θ_{h+1}, and then the induction step is

[[v_{h}x][v_{k}y]] = −[[v_{h}x][v_{k−1}yv_{1}]]
= −[[v_{h}x][v_{k−1}y]v_{1}] + [[v_{h}x]v_{1}[v_{k−1}y]]
= −[[v_{h+k−1}xy]v_{1}] + [[v_{h+1}x][v_{k−1}y]]
= −θ_{h+k} + [v_{h+k}xy] − (k − 1)θ_{h+k}
= [v_{h+k}xy] − kθ_{h+k}.

In particular, for h = 1 we have

[[v_{1}x][v_{i}y]] = [v_{i+1}xy] − iθ_{i+1},

and also

[[v_{1}x][v_{i}y]] = −[[v_{i}y][v_{1}x]]
= −[v_{i}yv_{1}x] + [v_{i}yxv_{1}]
= [v_{i+1}yx] + θ_{i+1}.

Comparing the two expressions we obtain

(i + 2)θ_{i+1} = 0,

so that θ_{j} can indeed only be nonzero for

j ≡ 2 (mod 3).

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

  E-mail address: caranti@science.unitn.it
  URL: http://www-math.science.unitn.it/~caranti/