A CORRECTION OF THE DECOMPOSABILITY RESULT IN A
PAPER BY MEYER-NEUTSCH

arXiv:1801.02339v1 [math.RA] 8 Jan 2018

VLADIMIR G. TKACHEV
Abstract. In this short note, it is shown that there is a gap in the proof of
Theorem 11 in the paper of Meyer and Neutsch (J. of Algebra, 1993). We
prove, nevertheless, that the statement of the theorem is true and fix the proof
by using a certain extremal property of idempotents which has an independent
interest.

1. Introduction
In their 1993 paper [5], Meyer and Neutsch established the existence of a 48dimensional associative subalgebra in the Griess algebra G and conjectured that 48
is the largest possible dimension of an associative subalgebra of G. This conjecture
has been proved by Miyamoto [6].
One of the key results claimed in [5], Theorem 11 asserts that an idempotent c ∈
G is indecomposable if and only if the Peirce eigenspace Gc (1) with the eigenvalue 1
is at most one-dimensional. This result was used by Miyamoto (see Theorem 6.8 in
[6]) and in a later research on associative subalgebras of low-dimensional Majorana
algebras, see for instance p. 576 in [2] and section 3.1 in [4].
Unfortunately, there is a gap in the argument in the proof of Theorem 11 in [5].
Recall the statement of the theorem.
Theorem 1.1 (Theorem 11, in [5]). An idempotent a ∈ G is indecomposable if
and only if the Peirce eigenspace Ga (1) = {x ∈ G : La x = x} is at most onedimensional.
Remark 1.2. Note also that Ga (1) is a subalgebra in G, see [5].
More precisely, in their proof of the implication “dim Gc (1) ≥ 2 ⇒ c is decomposable”, Meyer and Neutsch uses an erroneous argument that a (normalized) cubic
form
hx, x2 i
(1)
ϕ(x) :=
3
hx, xi 2
provides at least two linearly independent idempotents. Let us explain what is a
gap here. It is well known (and easy to see) that x 6= 0 is a stationary point of
ϕ(x) precisely when x2 = λx for some λ ∈ R. If λ 6= 0 then x gives rise to a
(nonzero) idempotent by normalizing: indeed c = x/λ implies c2 = c. In order to
get two distinct (linearly independent) idempotents, Meyer and Neutsch suggest to
consider a minimum and a maximum points of ϕ which would provide us with two
desired idempotents (see p. 15 in [5]). But, since φ is an odd function, this may give
us two anti-collinear, and therefore linearly dependent, elements. In particular, it is
clear from (1) that x is a local maximum of ϕ if and only if −x is a local minimum
of ϕ.
2000 Mathematics Subject Classification. Primary 17A99, 17C27; Secondary 20D08.
Key words and phrases. Commutative nonassociative algebras; Griess algebra; Idempotents;
Associative bilinear form; Metrised algebras .
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VLADIMIR G. TKACHEV

Remark 1.3. We will point out that the variational argument used by the authors
in the proof of Lemma 2 and Theorem 8 in [5] causes no problems because one uses
there an even function (based on a quartic form). This, together with nonzero lower
estimate (76), allows one to guarantee at least two linear independent stationary
points (corresponding to a local maximum and minimum).
The variational argument used in [5] would be easily fixed if one could prove that
a general cubic form always has at least 2 linearly independent stationary points
with nonzero gradient1. However, this property fails in general, as a counterexample in Section 3 below shows.
Thus, the method suggested in [5] does not work in general. Nevertheless, the
claim of Theorem 11 holds true. More precisely, we have the following result which
contains Theorem 11 in [5] as a corollary.
Theorem 1.4. Let V be a commutative nonassociative algebra over R with an
associative inner product h, i. Suppose c 6= 0 is an idempotent of V and the Peirce
eigenspace Vc (1) is an subalgebra of V . Then dim Vc (1) = 1 if and only if c is
indecomposable.
The theorem easily follows from an auxiliary result (Lemma 2.3 below) which
has an independent interest. The proof of our results uses the notion of metrised
algebra. In the next section we recall some standard definitions and introduce the
so-called algebras of cubic forms.
2. Algebras of cubic forms
Let V be a vector space over a field k of characteristic not 2 or 3. A multiplication
on V is a bilinear map (denoted by juxtaposition)
V × V ∋ (x, y) → xy ∈ V.
A vector space with a multiplication is called an algebra over k.
A symmetric k-bilinear form Q(x, y) : V × V → k is called nonsingular if
Q(x, y) = 0 for all y ∈ V implies x = 0 [3]. In what follows, we use the standard inner product notation for Q:
Q(x, y) = hx, yi
and the squared norm notation |x|2 = hx, xi. Note that |x| may be not defined in
general.
If V is an algebra then the bilinear form h·, ·i is called associative [9], [8] if
(2)

hxy, zi = hx, yzi,

∀x, y, z ∈ V,

An algebra carrying a non-singular symmetric bilinear form is called metrised, cf.
[1]. The standard examples are Lie algebras with the Killing form and Jordan
algebras with the generic trace bilinear form. The most striking corollary of the
existence of an associative bilinear form on an commutative algebra is that the
operator of left (=right) multiplication
Lx : y → xy,
is self-adjoint, i.e.
(3)

hLx y, zi = hy, Lx yzi,

∀x, y, z ∈ V.

A function u : V → k is called a cubic form if its full linearization
u(x, y, z) = u(x + y + z) − u(x + y) − u(x + z) − u(y + z) + u(x) + u(y) + u(z)
1Note that the vanishing of gradient implies rather a 2-nilpotent, i.e. x2 = 0, than idempotent.
Therefore one assumes that λ 6= 0.

A CORRECTION OF THE DECOMPOSABILITY RESULT IN A PAPER BY MEYER-NEUTSCH
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is a trilinear form. Note that thus defined linearization is obviously symmetric and
the cubic form is recovered by
(4)

6u(x) = u(x, x, x).

The vector space of all cubic forms on V will be denoted by C3 (V ).
The following is an immediate corollary of the defintions.
Proposition 2.1. Given a vector space V with a non-singular symmetric bilinear form h·, ·i, there exists a canonical bijection between C3 (V ) and commutative
metrised algebras (V, h·, ·i), where the multiplication (x, y) → xy is uniquely determined by
(5)

xy := the unique element satisfying hxy, zi = u(x, y, z) for all z ∈ V

Given a cubic form u on V , the commutative nonassociative algebra obtained
according to Proposition 2.1 is denoted by V (u) and is called the algebra of the
cubic form u. Conversely, given a commutative metrised algebra A = (V, h·, ·i),
define the cubic form
1
(6)
uA (x) := hxx, xi.
6
1
The factor 6 is chosen to agree with (4). Then A is isomorphic to V (uA ).
We point out that as a corollary of the symmetry of u(x, y, z), the algebra V (u) is
always commutative but may be nonassociative. It is easy to see that the following
is true.
Proposition 2.2. V (u) is a zero algebra if and only if u ≡ 0.
Now we consider the specialization of the above definitions when V is a vector
space over k = R or C. Let h·, ·i be some non-singular symmetric bilinear form on
V and let us fix a cubic form u. Then it readily follows from (6) and (5) that the
gradient of u(x) is essentially the square of the element x (in V (u)):
1
1
(7)
Du(x) = xx = x2 ,
2
2
and, furthermore, the multiplication in V (u) is explicitly recovered by
(D2 u(x))y = xy,

(8)

where D2 u(x) is the Hessian matrix of u at x. This observation is an important
ingredient in the proof of the main results of [5], in particular of Theorem 11. The
relations (7) and (8) are also important tools in recent applications of nonassociative
algebras to global analysis, see [7, Ch. 6], [10]. Note also that if k is an arbitrary
field, then (7) and (8) can be interpreted as formal definitions of the gradient and
the Hessian of u.
The case k = R has a particular interest as the following property shows.
Lemma 2.3. Let (V, h·, ·i) be a (nonzero) commutative metrized algebra over R,
h·, ·i being positive definite. The set E of constrained stationary points of the variational problem
(9)

hx, x2 i → max

subject to a constraint

hx, xi = 1

is nonempty and the maximum is attained. Denote by E0 ⊂ E the (nonempty) set
of local maxima in (9). Then for any x ∈ E, either x2 = 0 or c := x/hx2 , xi is
an idempotent in V . If additionally x ∈ E0 then hx2 , xi > 0 and the corresponding
idempotent c = x/hx2 , xi satisfies the extremal property
(10)

Lc ≤

1
2

on c⊥ := {x ∈ V : hx, ci = 0}.

In particular, the eigenvalue 1 of Lc is simple.

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VLADIMIR G. TKACHEV

Proof. By the positive definiteness of h·, ·i, the unit sphere S = {x ∈ V : hx, xi = 1}
is compact in the standard Euclidean topology on V induced by |x|2 = hx, xi. By
the assumption and Proposition 2.2, the cubic form u(x) = 16 hx2 , xi 6≡ 0. Note
that the algebra V (u) is isomorphic to V .
Since u is continuous as a function on S, it attains its maximum value there. In
particular both E0 and E are nonempty. Let x ∈ E be a stationary point, then
using Lagranges undetermined multipliers and (7) we obtain for the directional
derivative
0 = ∂y u|x = 12 hx2 , yi for any y ∈ V and hx, yi = 0.
which implies that x and x2 are parallel, i.e. x2 = λx where λ = hx2 , xi. Thus,
x2 = 0 if and only if hx2 , xi = 0, otherwise scaling appropriately we have that
c := x/hx2 , xi is a nonzero idempotent in V .
2
To prove the second claim of the lemma, note that the function f (x) = hx|x|, 3xi is
a homogeneous of degree zero and smooth outside the origin. We have
(11)

∂y f |x =

3(hx2 , yi|x|2 − hx2 , xihx, yi)
,
|x|5

implying by duality the expression for the gradient
1
3 Df (x)

=

x2 |x|2 − hx2 , xix
.
|x|5

Arguing similarly, one obtains
1
3 ∂z Df (x)

=

2|x|4 xz − 3|x|2 (x2 hx, zi + hx2 , zix) − hx2 , xi|x|2 z + 5hx, zihx2 , xix
,
|x|7

hence the Hessian is given by
(12)

1 2
3 D f (x)

=

2|x|4 Lx − hx2 , xi|x|2 − 3|x|2 (x ⊗ x2 + x2 ⊗ x) + 5hx2 , xix ⊗ x
.
|x|5

Now let x ∈ E0 . Then x is a local maximum of f and obviously f (x) = hx2 , xi > 0
(because hx2 , xi 6≡ 0 on V ). By the above c = x/hx, x2 i is an idempotent, and by
the homogeneity f (x) = f (c), hence c is a local maximum of f too. Therefore the
maximum condition implies that the Hessian matrix is non-positive definite
0 ≥ 13 D2 f (c) =

(2Lc − 1)|c|2 − c ⊗ c
.
|c|3

Since c ⊗ c = 0 on c⊥ we conclude that 2Lc − 1 ≤ 0 on c⊥ , thus, the desired
conclusion follows.

Definition 2.4. An idempotent c ∈ (V, Q) corresponding to any local maximum
in (9) is said to be extremal.
An important corollary of Lemma 2.3 is the following
Corollary 2.5. Let a nonzero unital commutative algebra V over R possess a symmetric positive definite associative bilinear form and dim V ≥ 2. Then there exists
an idempotent in V distinct from the unit. In particular, the unit is decomposable
and V has at least three distinct nonzero idempotents.
Proof. By Lemma 2.3 there exists a (nonzero) extremal idempotent c ∈ V . By
the extremal property, one also has Lc ≤ 21 on c⊥ . Note that the subspace c⊥ is
nontrivial because dim c⊥ = dim V − 1 ≥ 1. Let e ∈ V be the algebra unit. Then e
is an idempotent and e 6= c because Le ≡ 1 on V . Then (e−c)2 = e−2ec+c = e−c,
hence e − c is also an idempotent. The claim follows.


A CORRECTION OF THE DECOMPOSABILITY RESULT IN A PAPER BY MEYER-NEUTSCH
5

Now we are able to finish the proof of the main result.
Proof of Theorem 1.4. First recall that an idempotent is called decomposable if
it may be expressed as a sum of at least two non-zero idempotents; otherwise,
the idempotent is called indecomposable. Now, if c is decomposable, say c =
c1 +c2 , ci being nonzero idempotents, then c1 c2 = 0 and c1 , c2 are obviously linearly
independent. One has therefore cci = ci , hence ci ∈ Vc (1), implying dim Vc (1) ≥ 2.
In the converse direction, suppose that c 6= 0 be a idempotent in V with
dim Vc (1) ≥ 2 and Vc (1) is a subalgebra of V . Note that c acts on the subalgebra
Vc (1) as a unit. Therefore by Corollary 2.5, c is decomposable.

3. A counterexample
Here we construct an example of a cubic form whose stationary points with
nonzero gradient lies on a line. On the nonassociative algebra level, this shows that
for any n ≥ 2 there exist nontrivial n-dimensional algebras over R with associative
bilinear form having only one nonzero idempotent.
Let n ≥ 2 and consider a cubic form u(x) = 31 x1 (x21 + 3a2 x22 + . . . + 3an x2n ) in
n
R equipped with the standard Euclidean scalar product. Suppose that all ai are
different and 0 < ai < 12 . We claim that any stationary point of ϕ(x) = u(x)/|x|3
with nonzero gradient are parallel to ξ := (1, 0, . . . , 0). Indeed, given a stationary
point x of ϕ(x), the stationary point condition readily yields Du(x) = λx for some
real λ. By the assumption Du(x) 6= 0, hence λ 6= 0, in particular x 6= 0. Since
u(−x) = −u(x), one has Du(x) = Du(−x), thus interchanging x and −x, one may
assume without loss of generality that λ > 0, and, also by the homogeneity of u
that λ = 1. This yields
(13)

x21 + (a2 x22 + . . . + an x2n ) = x1 ,

(14)

2ak xk x1 = xk ,

for k = 2, . . . , n.

If xk = 0 for 2 ≤ k ≤ n then by (13) x21 = x1 , thus x1 = 1, i.e. x = ξ. If at least
two xi and xj are nonzero for 2 ≤ i, j ≤ n then (14) implies x1 = 2a1 i = 2a1 j , a
contradiction with ai 6= aj . Therefore there can exist only one nonzero xi 6= 0 for
some 2 ≤ i ≤ n. But in that case, x1 = 2a1 i and (13) yields by the assumptions on
< 0, a contradiction again. This proves the claim.
ai that x2i = 2a4ai −1
3
i

Remark 3.1. Some final remarks are also in order. In the proof of Lemma 2.3, we
made an essentially use of the condition that the associative form is positive definite.
It is unclear, however, if Theorem 1.4 remains true if one drops the definiteness
condition and consider a weaker assumption, for example, that h·, ·i is merely nonsingular.
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VLADIMIR G. TKACHEV

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Department of Mathematics, Linköping University, Linköping, 58183, Sweden, vladimir.tkatjev@liu.se

