ON AN EXTREMAL PROPERTY OF JORDAN ALGEBRAS OF
CLIFFORD TYPE

arXiv:1801.05724v1 [math.RA] 17 Jan 2018

VLADIMIR G. TKACHEV
Abstract. If V is a finite-dimensional unital commutative (maybe nonassociative) algebra carrying an associative positive definite bilinear form h, i then
there exist a nonzero idempotent c 6= e (e being the algebra unit) the shortest
possible length |c|2 := hc, ci. In particular, |c|2 ≤ 21 |e|2 . We prove that the
equality holds exactly when V is a Jordan algebra of Clifford type.

1. Introduction
Throughout this paper, V denotes a finite-dimensional commutative nonassociative algebra over R carrying an associative nonsingular bilinear form:
(1)

hxy, zi = hx, yzi,

∀x, y, z ∈ V,

where (x, y) → xy = yx is the algebra multiplication. Following M. Bordemann
[1], an algebra satisfying (1) is called metrised. We shall always assume that V
is a Euclidean metrised algebra, i.e. the associative bilinear form h·, ·i is positive
definite. The classical example is formal real (Euclidean) Jordan algebras with the
invariant trace form [3], [9]. Another important example is the Griess algebra G
appearing in connection with the Monster sporadic simple group [14] or, in general,
any axial algebra [4], [16], [4], [7]. A more recent example which also appears in
this context is the class of nonassociative algebras decoding the geometric structure
of cubic minimal cones and cubic polynomial solutions to certain elliptic PDEs
[13, Chapter 6], [12], [19], [20]. Some related questions as well as geometry of
idempotents are also discussed in [17], [5], [6], [2].
It is known [20], [10] (see also Proposition 2.1 below) that if V is a Euclidean
metrised algebra then the set of nonzero idempotents Idm(V ) is nonempty and
there exists an idempotent c 6= 0 such that
(2)

|c|2 ≤ |c′ |2 ,

∀c′ ∈ Idm(V ),

where |x|2 = hx, xi. Such an idempotent is called extremal , denoted by c ∈
Idm1 (V ). In other words, Idm1 (V ) denotes the set of shortest nonzero idempotents in V .
If additionally V is a unital algebra with the unit e then the conjugation c →
c̄ := e − c is a natural involution on Idm(V ), indeed,
(3)

c̄2 = e − 2ec + c = e − c = c̄,

and similarly cc̄ = 0. Therefore, using (1) one obtains
(4)

hc, c̄i = hc2 , c̄i = hc, cc̄i = 0,

i.e. c and c̄ are orthogonal. This together with (2) immediately yields
(5)

|e|2 = |c|2 + |c̄|2 ≥ 2|c|2 ,

∀c ∈ Idm1 (V ).

2000 Mathematics Subject Classification. Primary 17A99, 17C27.
Key words and phrases. Commutative nonassociative algebras; Associative bilinear form;
Idempotents; Jordan algebras; Jordan algebras of Clifford type .
1

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VLADIMIR G. TKACHEV

Definition 1.1. A unital Euclidean metrised algebra such that the equality in (5)
attains for some c ∈ Idm1 (V ) is said to be minimal.
It is the main purpose of the present paper to completely characterize the class
of minimal algebras. Let us recall some standard definitions, see [8, p. 13-14], [3],
[11]. Let U be a vector space over a field k of characteristic 6= 2, f (u, v) be a
nonsingular symmetric bilinear form from U to k. Then k ⊕ U together with the
multiplication law
(6)

(a ⊕ u) • (b ⊕ v) = (ab + f (u, v)) ⊕ (av + bu)

is a Jordan algebra, called the Jordan algebra of bilinear form f and denoted by Uf
(also known as Jordan algebra of Clifford type or a spin-factor [11]). Then ǫ = 1 ⊕ 0
is the unit in Uf and any element z = a ⊕ u ∈ Uf satisfies the quadratic relation
z 2 − z tr z + d(z)ǫ = 0,

(7)

where tr z = 2a is the generic trace and d(z) = a2 − f (u, u) is the generic norm
of z. It is well-known that the trace form t(z, w) := tr(z • w) is associative in the
sense of (1). In particular, Uf is a metrised algebra with respect to the trace form.
Our main result states that any minimal algebra is a Jordan algebra of Clifford
type. More precisely, we shall prove
Theorem 1.2. If V is a minimal algebra then
(8)

V = span(Idm1 (V )).

Furthermore, V is isomorphic to the Jordan algebra e⊥
f of the symmetric bilinear
form
1
f (x, y) = 2 hx, yi,
|e|
where e⊥ = {x ∈ V : hx, ei = 0}.
The paper is organized as follows. In Section 2 we give a short overview of
metrised algebras and discuss variational properties of extremal idempotents in
more details. The spectral inequality (11) in Proposition 2.1 and Proposition 2.8
are key ingredients in the classification of minimal algebras. The proof of Theorem 1.2 is given in Section 3.
2. Extremal idempotents
2.1. Euclidean metrised algebras. Let V be a finite dimensional algebra over
R with multiplication denoted by juxtaposition (x, y) → xy ∈ V. A symmetric Rbilinear form hx, yi : V × V → R is called nonsingular if hx, yi = 0 for all y ∈ V
implies x = 0. In what follows, we use the standard squared norm notation
|x|2 = hx, xi.
The bilinear form h·, ·i is called associative [18], [15] if (1) holds. An algebra
carrying a non-singular symmetric bilinear form is called metrised, cf. [1].
The operator of left (=right) multiplication is denoted by Lx : y → xy. If algebra
V is metrised then Lc is self-adjoint:
(9)

hLx y, zi = hy, Lx zi,

∀x, y, z ∈ V.

We recall the following result, see also [20], [21].
Proposition 2.1. Let (V, h·, ·i) be a nonzero Euclidean metrised algebra (i.e. V V 6=
0). Then the set E of constrained stationary points of the variational problem
(10)

hx, x2 i → max

subject to a constraint

hx, xi = 1

ON AN EXTREMAL PROPERTY OF JORDAN ALGEBRAS OF CLIFFORD TYPE

3

is nonempty and the maximum is attained. Denote by E0 ⊂ E the (nonempty) set
of local maxima in (10). 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
(11)

Lc ≤

1
2

on c⊥ := {x ∈ V : hx, ci = 0}.

In particular, the eigenvalue 1 of Lc is simple.
Proof. By the positive definiteness of h·, ·i, the unit sphere S = {x ∈ V : hx, xi = 1}
is compact in the standard Euclidean norm topology on V induced by |x|2 . Since
V V 6= 0, the cubic form f (x) = 61 hx2 , xi 6≡ 0 (this can be easily seen by polarization
of f ). Since f is a continuous odd function on S, f (x) attains its positive maximum
value there. In particular E0 (and, thus, E) is nonempty.
First let x ∈ E be a stationary point. Then Lagranges undetermined multipliers
method yields for any y ∈ V and hx, yi = 0 that the directional derivative ∂y f |x
vanishes:
0 = ∂y f |x = 12 hx2 , yi.
Since the inner product is nonsingular, this implies that x and x2 are parallel,
therefore x2 = λx with λ = hx2 , xi. Note that x2 = 0 if and only if hx2 , xi = 0,
otherwise scaling x appropriately yields that c := x/hx2 , xi is a nonzero idempotent.
Next, to prove (11), let us consider an arbitrary x ∈ E0 , |x| = 1, and define
xt = x cos t + y sin t, where y ∈ V such that hx, yi = 0, |y| = 1 and t ∈ R. Since
|xt | = |x| = 1, we obtain by the extremal property of x0 that
(12)

hx2t , xt i ≤ hx20 , x0 i = hx2 , xi.

Using the associativity of the inner product we obtain
hx2t , xt i = hx2 cos2 t + 2xy cos t sin t + y 2 cos2 t, x cos t + y sin ti
= cos3 thx2 , xi + 3 cos2 t sin thx2 , yi + 3 cos t sin2 thx, y 2 i + sin3 thy 2 , yi
then using the fact that hx2 , yi = hλx, yi = 0, we find from (12)
3 cos t sin2 thx, y 2 i + sin3 thy 2 , yi ≤ (1 − cos3 t)hx2 , xi,
Dividing the obtained inequality by sin2 t and passing to limit as t → 0 yields
3hx, y 2 i ≤ 23 hx2 , xi.
Now, recall that c = x/hx, x2 i is an extremal idempotent, hence
hy, Lc yi = hy, cyi = hc, y 2 i ≤

1
2

= 21 |y|2

Thus, the latter inequality holds for all y⊥c and |y| = 1, implying by the self
adjointness of Lc that Lc ≤ 21 on c⊥ , as desired.
We shall also need the following generalization.
Corollary 2.2. Let V be as in Proposition 2.1 and let V = U ⊕W be an orthogonal
decomposition, where U, W are nontrivial subspaces of V . If U U 6= 0 (i.e. U is not
zero subalgebra of V ) then there exists u ∈ U , such that u 6= 0 and
u2 = λu + w,

where λ ∈ R, w ∈ W.

Proof. If dim U = 1 then the claim is trivial, therefore assume that dim U ≥ 2.
Consider the variational problem of maximizing of the cubic form f (x) := hx2 , xi
under two conditions: |x|2 = 1 and x ∈ U . If U is a nonzero subalgebra of V then
f (x) 6≡ 0 on U , hence arguing as in Proposition 2.1 we conclude that the (positive)
maximum attains at some u ∈ U , |u| = 1. Therefore for any y ∈ U ∩ u⊥ we have
0 = ∂y f |u = 21 hu2 , yi.

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VLADIMIR G. TKACHEV

The latter implies that
u2 ∈ (U ∩ u⊥ )⊥ = span(W ∪ u),
hence there exists λ ∈ R such that u2 − λu ∈ W , the desired conclusion follows. 
2.2. Variational properties. It follows from Proposition 2.1 that the set of nonzero
idempotents is nonempty for any Euclidean metrised algebra. Moreover, any idempotent arising as in Proposition 2.1 has distinguished spectral properties.
Definition 2.3. An idempotent c ∈ V is called extremal, or c ∈ Idm1 (V ), if the
function hx2 , xi attains its global maximum value at c for all x ∈ V satisfying
|x| = |c|.
It follows from the definition that if c is an extremal idempotent then
hx, x2 i ≤

(13)

1 3
|x| ,
|c|

∀x ∈ V,

and the equality holds for x = c. Furthermore, all extremal idempotent have the
same length. It also follows from (13) that
(14)

|c| ≤ |c′ |,

∀c ∈ Idm1 (V ), ∀c′ ∈ Idm(V ),

i.e. the extremal idempotents have the minimal possible squared length.
Remark 2.4. It is well-known, see for example [3], that if J is an Euclidean Jordan
algebra equipped with the trace form hx, yi = tr(xy) then for any idempotent
c ∈ Idm(J): |c|2 = n ∈ Z+ , i.e. the squared length takes only positive integer
values, and an idempotent c ∈ J is primitive if and only it has minimal possible
squared length. Furthermore, if J is a spin factor (a Jordan algebra associated
with symmetric bilinear form) then all (nonzero) idempotents have the same square
length |c|2 = 1.
Taking into account the previous remark, it is convenient to scale the inner
product such that all extremal idempotents has the unit length. To this end, note
that if h·, ·i is an associative positive definite bilinear form on V then so also is
kh·, ·i for any k > 0. We have the following definition.
Definition 2.5. Let V be a Euclidean metrised algebra with an associative inner
product h·, ·i. The inner product is said to be normalized if
hx, x2 i ≤ |x|3

(15)

holds for all x ∈ V and the equality holds for some x 6= 0. By abuse of terminology,
we call V normalized if its inner product is so.
In other words, the inner product is normalized if and only if any extremal
idempotent has length |c| = 1. Therefore, in an normalized algebra
Idm1 (V ) = {c ∈ V : c2 = c

and |c|2 = 1}.

As a corollary of (14),
(16)

|c| ≥ 1,

∀c ∈ Idm(V ).

If the inner product normalized, Proposition 2.1 can be reformulated as follows.
Proposition 2.6. The equality in (15) is obtained for x 6= 0 if and only if x/|x| ∈
Idm1 (V ). If c ∈ Idm1 (V ) then Lc ≤ 21 on c⊥ .

ON AN EXTREMAL PROPERTY OF JORDAN ALGEBRAS OF CLIFFORD TYPE

5

2.3. Unital algebras. Recall that an algebra is called unital if there exists e ∈ V
such that ex = xe = x for all x. If a unit exists then it is necessarily unique and
distinct from 0.
Proposition 2.7. If V is a unital Euclidean metrised algebra and dim V ≥ 2 then
there exist at least two different idempotents in V distinct from the unit.
Proof. First note that a unital algebra is obviously nonzero, because ee = e 6=
0, where e ∈ V is the algebra unit. Therefore by Proposition 2.1 there exists a
(nonzero) extremal idempotent c ∈ V . By the extremal property, one also has
Lc ≤ 12 on c⊥ . Note that the subspace c⊥ is nontrivial because by the assumption
dim c⊥ = dim V − 1 ≥ 1. The algebra unit e is an idempotent and e 6= c because
Le ≡ 1 on V . By (3), c̄ = e − c is also an idempotent and c 6= e − c (because
otherwise e = 2c, hence e = e2 = 4c = 2e, implying a contradiction). This proves
the proposition.

In the rest of this section we assume that V is a unital algebra and dim V ≥ 2.
Given an arbitrary idempotent c ∈ Idm(V ), let c̄ = e−c denote its conjugate. Then
cc̄ = c(e − c) = c − c = 0,
and it follows from cc̄ = 0 and c and c̄ are orthogonal, see (4).
Recall that an idempotent distinct from 0 and e is called nontrivial.
By the above, Idm1 (V ) 6= ∅. If c ∈ Idm1 (V ) then c⊥ is nontrivial, hence c 6= e
(since Le = 1 on the whole V ). We have by the orthogonality and (16)
|e|2 = |c|2 + |c̄|2 ≥ 2.

(17)

In the next section we classify all algebras where the equality in (17) is obtained.
2.4. Minimal algebras. For convenience reasons, we shall assume in the rest of
the paper that V is a normalized algebra. Then rephrasing the Definition 1.1 yields
that a unital normalized algebra is minimal if and only if
|e|2 = 2.

(18)

Furthermore, specializing (22) for (18) yields
he, ci = 1,

(19)

∀c ∈ Idm1 (V ).

2

Since |e| = 2 > 1, the unit e is not an extremal idempotent. The next proposition
shows that it is the only (distinct from zero) non-extremal idempotent.
Proposition 2.8. Let V be a minimal algebra. Then all nonzero idempotents in
V , except for the unit, are extremal:
Idm(V ) = Idm1 (V ) ∪ {e}.
Furthermore, if c ∈ Idm1 (V ) then
(20)

Lc =

1
2

on c⊥ ∩ c̄⊥ .

Proof. Let c ∈ Idm(V ), 0 6= c 6= e. Then
2 = |e|2 = |c̄|2 + |c|2 ≥ 2,
therefore |c|2 = 1, which implies that Idm(V ) = Idm1 (V ) ∪ {e}. Next, if c ∈
Idm1 (V ) then c̄ ∈ Idm1 (V ) too. By the extremal property,
Lc ≤

1
2

+ c ⊗ c,

Lc̄ ≤

1
2

+ c̄ ⊗ c̄.

Since Lc̄ = 1 − Lc , this yields
1
2

which implies that Lc =

1
2

− c̄ ⊗ c̄ ≤ Lc ≤
⊥

⊥

1
2

+c⊗c

on c ∩ c̄ , as desired.



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VLADIMIR G. TKACHEV

The identity (20) shows that the multiplication by an extremal idempotent is
essentially 12 . This implies that the multiplication structure on a minimal algebra
is quite special. More precisely we have
Proposition 2.9. If V is a normalized minimal algebra then for any c1 , c2 ∈
Idm1 (V ) there holds
2c1 c2 = c1 + c2 − hc̄1 , c2 ie.

(21)

Proof. First note that for any idempotent c ∈ Idm(V ) there holds
he, ci = he, cci = hec, ci = |c|2 ,

(22)

hence if c1 , c2 ∈ Idm(V ) then
hc1 , c̄2 i = hc1 , e − c2 i = hc1 , ei − hc1 , c2 i = |c1 |2 − hc1 , c2 i.

(23)

Next note that (21) trivially holds if c2 = c1 or c2 = c̄1 , therefore we assume
that c2 is distinct from c1 and c̄1 . Let us decompose c2 as
c2 = hc1 , c2 ic1 + hc̄1 , c2 ic̄1 + z,
Since Lc1 =

1
2

⊥
z ∈ c⊥
1 ∩ c̄1 .

⊥
on c⊥
1 ∩ c̄1 we have by virtue of (23)

c1 c2 = hc1 , c2 ic1 + 21 z = hc1 , c2 ic1 + 12 (c2 − hc1 , c2 ic1 − hc̄1 , c2 ic̄1 )
= 21 c2 + 12 hc1 , c2 ic1 − 12 hc̄1 , c2 i(e − c1 ) = 21 c2 + 12 c1 − 21 hc̄1 , c2 ie,
as desired.



Corollary 2.10. Let S ⊂ Idm1 (V ) be such that e ∈ span(S), then span(S) is a
subalgebra of V .
Proof. Follows readily from (21).



3. The proof of the main results
First we establishes (8) in Theorem 1.2.
Proposition 3.1. If V is a minimal algebra then V = span(Idm1 (V )).
Proof. Without loss of generality we may assume that V is a normalized minimal
algebra. Recall that Idm1 (V ) is nonempty by Proposition 2.1. Define
W := span(Idm1 (V )).
Then W is a subalgebra by Corollary 2.10. Since c + c̄ = e for any c ∈ Idm1 (V ),
we have e ∈ W . Assume by contradiction that W 6= V , hence V = W ⊕ W ⊥
with W ⊥ 6= 0. Since W W ⊂ W , the associativity of the inner product implies
that W W ⊥ ⊂ W ⊥ . Given an arbitrary z ∈ W ⊥ , let us consider the orthogonal
decomposition
(24)

z 2 = x + y,

x ∈ W, y ∈ W ⊥ .

Let c ∈ Idm1 (V ) be chosen arbitrarily. By Proposition 2.8,
(25)

Lc =

1
2

on

W ⊥,

hence
hz 2 , ci = hz, zci = 21 |z|2 ,
therefore it follows from (24) that hx, ci = 21 |z|2 . Let x0 = x− 21 |z|2 e. Then x0 ∈ W
and using (19) we obtain
hx0 , ci = hx, ci − 12 |z|2 he, ci = 21 |z|2 − 21 |z|2 = 0,

ON AN EXTREMAL PROPERTY OF JORDAN ALGEBRAS OF CLIFFORD TYPE

7

which by virtue of the arbitrariness of c ∈ Idm(V ) and the definition of W implies
that x0 ∈ W ⊥ . Therefore x0 ∈ W ⊥ ∩ W , i.e. x0 = 0. This proves that for any
z ∈ W⊥
z 2 = 12 |z|2 e + y,

(26)

y ∈ W ⊥.

In particular, z 2 =
6 0 if z 6= 0, hence W ⊥ W ⊥ 6= 0. This implies by Corollary 2.2
that there exists a nonzero vector z ∈ W ⊥ such that
z 2 = 21 |z|2 e + λz,

(27)

λ ∈ R.

With this z in hand, we claim that the idempotent equation p2 = p for p = ae + bz
with a, b being some real numbers has a solution distinct from e. Note that we may
assume that b 6= 0, because otherwise p = e. Then expanding p2 = p by virtue of
(27) yields the system
2
1 2
2 b |z|

+ a2 = a

and 2ab + λb2 = b.

From the second equation we have λb = 1 − 2a, therefore
2
1 2
2 b |z|

= a − a2 =

1
4

− (a − 12 )2 =

1
4

− 41 λ2 b2 .

This yields b2 = 1/(2|z|2 + λ2 ). Since the latter equation is solvable, this proves
that there exists an idempotent p = ae + bz ∈ Idm(V ) with b 6= 0, i.e. p 6= e.
Therefore by Proposition 2.8, p ∈ Idm1 (V ) ⊂ W , which obviously contradicts to
the assumption that 0 6= z ∈ W ⊥ . This proves that assumption W ⊥ 6= 0 is wrong,
thus W = V . The proposition follows.

Now we are ready to finish the proof of Theorem 1.2.
Proposition 3.2. If V is a minimal algebra then V is a Jordan algebra of the
symmetric bilinear form
1
f (x, y) = 2 hx, yi.
|e|
Proof. First let V be a normalized minimal algebra. Note that (8) readily implies
that there exists a basis in V consisting of e and idempotents with the unit norm.
Let {e, c1 , . . . , cn } be such a basis, where n + 1 = dim V ≥ 2. It is easy to see that if
ci is in the basis then c̄i = e − ci is not. This implies that the new set {e, e1 , . . . , en }
with ei = ci − c̄i = 2ci − e, 1 ≤ i ≤ n, is also an basis of V .
Let x ∈ V and let
(28)

x = a0 e +

n
X

ai e i

i=1

be the corresponding basis decomposition. By (19)
he, ei i = he, c − c̄i = 1 − 1 = 0
for all i ≥ 1, hence
a0 =

1
hx, ei
= hx, ei.
he, ei
2

Note also that for all i ≥ 1
e2i = 4c − 4c + e = e,

(29)
and also by virtue of (21)
(30)

ei ej = 4ci cj − 2ci − 2cj + e = (1 − 2hc̄i , cj i)e,

1 ≤ i, j ≤ n.

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VLADIMIR G. TKACHEV

Pn
Rewrite (28) as x = 21 hx, eie + y, where y = i=1 ai ei . Since y is orthogonal to e
we find that |x|2 = 12 hx, ei2 + |y|2 . Also from (29) and (30) follows that y 2 = λe
for some λ, hence
y2 =

hy 2 , ei
|y|2
1
e=
= (|x|2 − 12 hx, ei2 )e.
he, ei
2
2

Since (x − a0 e)2 = y 2 , we obtain
x2 − 2a0 x + a20 e =

1
(|x|2 − 12 hx, ei2 )e.
2

implying
x2 − hx, eix + d(x)e = 0,

(31)
where

d(x) := 12 (hx, ei2 − |x|2 ).
In particular, (31) implies for the commutator
[Lx2 , Lx ] = [Lhx, eix−b(x)e , Lx ] = hx, ei[Lx , Lx ] − b(x)[1, Lx ] = 0
implying that V is a Jordan algebra. Polarizing (31) yields the following explicit
expression for the algebra multiplication:

(32)
xy = 21 hx, eiy + hy, eix − (hx, eihy, ei − hx, yi)e
It remains to establish an isomorphism between V and the Jordan algebra e⊥
f ,
where f (x, y) = 21 hx, yi (recall that in a normalized minimal algebra |c|2 = 2). To
⊥
this end, note that e⊥
with the product given by (6), i.e. the present
f = R ⊕e
notation
(a ⊕ u) • (b ⊕ v) = (ab + 21 hu, vi) ⊕ (av + bu),

u, v ∈ e⊥ , a, b ∈ R.

Let us define a homomorphism of vector spaces
φ(z) = ae + u : e⊥
f → V,

where z = a ⊕ u.

Then for z = a ⊕ u and w = b ⊕ v we have
φ(z • w) = (ab + 21 hu, vi) + (av + bu).
On the other hand, using
hφ(z), ei = 2a,

hφ(w), ei = 2b,

hφ(z), φ(w)i = 2ab + hu, vi,

and applying (32) we find
φ(z)φ(w) =

1
2

2aφ(w) + 2bφ(z) − (2ab − hu, vi)e



= av + bu + (ab + 21 hu, vi)e
= φ(z • w),
hence φ is an algebra isomorphism.
Finally, let V be an arbitrary minimal algebra with inner product h·, ·i. Define
the new inner product by
2hx, yi
hx, yi1 =
,
he, ei
so that he, ei1 = 2. Then (V, h·, ·i1 ) is obviously a normalized minimal algebra.
Thus
h·, ·i
,
V ∼
where f = 12 h·, ·i1 =
= e⊥
f ,
he, ei
as desired. The proposition is proved completely.


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9

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

