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EIGHT-DIMENSIONAL REAL ABSOLUTE-VALUED ALGEBRAS
WITH LEFT UNIT WHOSE AUTOMORPHISM
GROUP IS TRIVIAL
A. ROCHDI
Received 5 May 2002
We classify, by means of the orthogonal groupᏻ7(R), all eight-dimensional real absolute-valued algebras with left unit, and we solve the isomorphism problem. We give an example of those algebras which contain no four-dimensional subalgebras and characterise with the use of the automorphism group those algebras which contain one.
2000 Mathematics Subject Classification: 17A35.
1. Introduction. One of the fundamental results about finite-dimensional real division algebras is due to Kervaire [7] and Bott and Milnor [3], and states that then-dimensional real vector spaceRnpossesses a bilinear product
with-out zero divisors only in the case where the dimensionn=1,2,4, or 8. All eight-dimensional real division algebras that occur in the literature contain a four-dimensional subalgebra (see [1,2,4,5,6]). However, it is still an open problem whether a four-dimensional subalgebra always exists in an eight-dimensional real division algebra, even for quadratic algebras [4]. In [9], Ramírez Álvarez gave an example of a four-dimensional absolute-valued real algebra contain-ing no two-dimensional subalgebras. On the other hand, any four-dimensional absolute-valued real algebra with left unit contains a two-dimensional subal-gebra. Therefore, a natural question to ask is whether an eight-dimensional real absolute-valued algebra with left unit contains a four-dimensional subal-gebra. In this note, we give a negative answer and we characterise the eight-dimensional absolute-valued real algebras with left unit containing a four-dimensional subalgebra in terms of the automorphism group.
2. Notation and preliminary results. For simplicity, we only consider vec-tor spaces over the fieldRof real numbers.
Definition2.1. LetAbe an algebra;Ais not assumed to be associative or unital.
(1) An elementx∈Ais called invertible if the linear operators
are invertible in the associative unital algebra End(A). The algebraAis called a division algebra if all nonzero elements inAare invertible.
(2) A unital algebraAis called a quadratic algebra if{1,x,x2}is linearly
de-pendent for allx∈A. If(·/·)is a symmetric bilinear form overA, then a linear operatorfonAis called an isometry with respect to(·/·)if(f (x)/f (y))= (x/y)for allx,y∈A. If, moreover,(xy/z)=(x/yz), for allx,y,z∈A, then (·/·)is called a trace form overA.
(3) The algebraAis termed normed (resp., absolute-valued) if it is endowed with a space norm · such thatxy ≤ xy(resp., xy = xy) for allx,y ∈A. A finite-dimensional absolute-valued algebra is obviously a division algebra and has a subjacent Euclidean structure (see [11]).
(4) An automorphism f ∈Aut(A)is called a reflexion ofA if f≠IA and
f2=I A.
Write Aut(O)=G2. We denote byS(E) and vect{x1,...,xn}, respectively,
the unit sphere of a normed space E and the vector subspace spanned by x1,...,xn∈E.
It is known that a quadratic algebraAis obtained from an anticommutative algebra(V ,∧)and a bilinear form(·,·)overVas follows:A=R⊕Vas a vector space, with product
(α+x)(β+y)=αβ+(x,y)+(αy+βx+x∧y). (2.2)
We have a bilinear form associated toA, namely,
A×A →R, (α+x,β+y)→αβ+(x,y), (2.3)
(V ,∧)is called the anticommutative algebra associated to A. The elements ofV are called vectors, while the elements of Rare called scalars. We write A=(V ,(·,·),∧)(see [8]).
We will write(W ,(·/·),×)for the (quadratic) Cayley-Dickson octonions al-gebraOwith its trace form(·/·)and the anticommutative algebra(W ,×). For u≠0∈W,W (u)will be the orthogonal subspace ofR·uinW. It is well known thatOis an alternative algebra, that is, it satisfies the identitiesx2y=x(xy)
andyx2=(yx)x.
Remark2.2. LetA be an eight-dimensional absolute-valued algebra with left unite, andf is an isometry of the Euclidian spaceAsuch thatf (e)=e. LetAf be equal toAas a vector space, with a new product given by the formula
x∗y=f (x)y, for allx,y∈A. ThenAfis also an absolute-valued algebra with
left unite. It is clear that anf-invariant subalgebra ofAis a subalgebra ofAf. In
particular, if we consider the isometryR−1
e , then we obtain an absolute-valued
3. Isometries ofOwith no invariant four-dimensional subalgebras. Let ϕ be an isometry of the Euclidian space O= R⊕W, fixing the element 1. Then there exists an orthonormal basisᏮ= {1,x1,...,x7}ofOsuch thatx1
is an eigenvector ofϕandWk=vect{x2k,x2k+1}is aϕ-invariant subspace of O, fork=1,2,3. IfB is a four-dimensionalϕ-invariant subspace ofO con-taining 1, then the basisᏮcan be chosen as an extension of an orthonormal basis{1,u,y,z}ofB, with u∈W an eigenvector ofϕ, and E=vect{y,z}is a ϕ-invariant subspace of B. Thus, B can be written as a direct orthogonal ϕ-invariant sumR⊕R·u⊕E.
In the following important example, we use the notation introduced above.
Example3.1. If ϕfixesx1and its restriction to every Wkis the rotation
with anglekπ/4, then vect{1,x1}is the eigenspaceE1(ϕ)ofϕassociated to
the eigenvalue 1. The characteristic polynomialPϕ(X)ofϕis then
(X−1)2
X2−2Xcos π
4
+1
X2−2Xcos 2π
4
+1
X2−2Xcos 3π 4 +1 =
0≤k≤3
Pk(X)
(3.1)
with
Pk(X)=X2−2Xcos kπ
4
+1. (3.2)
The characteristic polynomialPϕ//B(X)of the restriction ofϕtoBis a
poly-nomial of degree 4, a multiple of X−1, and a divisor of Pϕ(X). Actually,
Pϕ//B(X)=(X−1)2Pk(X) fork∈ {1,2,3}, and this “forces”B to be of the
formE1(ϕ)⊕Wkfor a certaink∈ {1,2,3}. In particular, ifᏮis obtained from
the canonical basis{1,e1,...,e7}ofOby taking
x1=e5, x2=
e1√+e2
2 , x3= e1√−e2
2 , x4= e3√+e4
2 , x5=
e3√−e4
2 , x6= e6√+e7
2 , x7= e6√−e7
2 ,
(3.3)
then for eachi≠j and l, xi×xj and xl are not colinear. This shows that
E1(ϕ)⊕Wkis not a subalgebra ofO, fork=1,2,3. It follows thatOhas no
four-dimensionalϕ-invariant subalgebras.
4. Eight-dimensional real absolute-valued algebras with left unit. First re-call the following result from [11].
Lemma 4.1. Every homomorphism from a normed complete algebra into an absolute-valued algebra is contractive. In particular, every isomorphism of absolute-valued algebras is an isometry.
Lemma4.2. Letψ:A→Bbe an isomorphism of absolute-valuedR-algebras andf:A→Aan isometry. Thenψ◦f◦ψ−1:B→Bis an isometry andψ:A
f →
Bψ◦f◦ψ−1is an isomorphism. In particular,ψ:Af →Ois an isomorphism if and
only ifψ:A→Oψ◦f−1◦ψ−1is an isomorphism.
Proof. The first statement is a consequence ofLemma 4.1. Forx,y∈A, we have
ψf (x)y=ψf (x)ψ(y)=ψ◦f◦ψ−1ψ(x)ψ(y), (4.1)
henceψ:Af→Bψ◦f◦ψ−1is an isomorphism.
Theorem4.3. Every eight-dimensional absolute-valued left unital algebra is isomorphic toOf wherefis an isometry of the Euclidian spaceOwhich fixes 1. Moreover, the following two properties are equivalent:
(1) Of andOg are isomorphic (f,gbeing two isometries ofOfixing1); (2) there existsψ∈G2such thatg=ψ◦f◦ψ−1, that is,f andgare in the
same orbit of conjugations by isometries ofOfixing1.
Proof. The first statement is a consequence of aRemark 2.2andLemma 4.2. The second statement can be proved as follows:ψ:Of→Ogis an isomor-phism if and only ifψ:O→(Og)ψ◦f−1◦ψ−1=Oψ◦f−1◦ψ−1◦gis an isomorphism. This is equivalent to
ψ◦f−1◦ψ−1◦g=I
O, ψ∈G2. (4.2)
5. Subalgebras and automorphisms ofOϕ. The following preliminary re-sult allows us to characterise the subalgebras ofOϕ.
Lemma5.1. IfAis an algebra with left unit and without zero divisors, then every nontrivial finite-dimensional subalgebra ofAcontains the left unit element ofA.
Proof. Such a subalgebraBis a division algebra and for everyx≠0∈B, there existsy∈B such thatyx=x. On the other hand, ifeis the left unit ofA, thenex=x. Then the absence of zero divisors inAshows thaty=e∈B.
What are the subalgebras ofOϕ?
Proposition5.2. Letϕbe an isometry of the Euclidian spaceOthat fixes 1andBis a subspace ofO. Then the following two properties are equivalent:
(1) Bis a subalgebra ofOϕ;
(2) Bis aϕ-invariant subalgebra ofO.
Proof. (1)⇒(2). The subalgebraB contains the left unit element 1 ofOϕ
1∈B, ∀x∈B:ϕ(x)= ϕ(x)1 product inO
=
product inOϕ
x∗1 ∈B. (5.1)
(2)⇒(1). SeeRemark 2.2.
Remark5.3. (1) The algebraOϕhas a two-dimensional subalgebra because ϕhas an eigenvectorx∈Wand the subalgebra vect{1,x}ofOisϕ-invariant. This argument shows thatHϕ has a two-dimensional subalgebra.
(2) Letϕbe the isometry considered inExample 3.1. ThenOϕhas no four-dimensional subalgebras.
The following elementary result is useful for characterising the automor-phisms of the algebraOϕ.
Lemma5.4. LetAbe an algebra with left uniteand without zero divisors. If f∈Aut(A), thenf (e)=e.
Proof. We have(f (e)−e)f (e)=0.
What are the automorphisms of the algebraOϕ?
Proposition5.5. Ifϕis an isometry of the Euclidian spaceOthat fixes1, thenf∈Aut(Oϕ)if and only iff∈G2andfcommutes withϕ.
Proof. For allx,y∈O, we have thatf (ϕ(x)y)=ϕ(f (x))f (y), hence f (ϕ(x))=f (ϕ(x)1)=ϕ(f (x))f (1)=ϕ(f (x)), andf◦ϕ=ϕ◦f andf∈ G2.
Remark5.6. Iff∈Aut(Oϕ)\{IO}is a reflexion, thenB=Ker(f−IO)is a
four-dimensional subalgebra ofOϕ.
6. The relation inOϕbetween four-dimensional subalgebras and nontriv-ial automorphisms. We begin with the following useful preliminary result taken from [10].
Lemma6.1. Every four-dimensional subalgebraBofO=(W ,(·/·),×) coin-cides with the square of its orthogonalB⊥and satisfies the equalityBB⊥=B⊥B= B⊥.
Proof. Letv∈S(B⊥), thenB⊥=vB. Indeed, taking into account the trace property of(·/·), we have for allx,y∈Bthat(vx/y)=(v/xy)=0, hence vB⊂B⊥, and we have equality because the dimensions of both spaces are equal. Using the middle Moufang identity, we compute that
(vx)(vy)= −(vx)(yv)=v(xy)v=xy (6.1)
Proposition6.2. LetBbe aϕ-invariant four-dimensional subalgebra ofO. Then the map
f:O=B⊕B⊥ →O, f (a+b)=a−b, (6.2)
is a reflexion which commutes withϕ.
Proof. Takea,x∈Bandb,y∈B⊥. UsingLemma 6.1, we compute
f(a+b)(x+y)=f (ax+by+ay+bx) =(ax+by)−(ay+bx) =(a−b)(x−y)
=f (a+b)f (x+y),
(6.3)
B⊥isϕ-invariant sinceBisϕ-invariant, and we have
(f◦ϕ)(a+b)=fϕ(a)+ϕ(b) =ϕ(a)−ϕ(b) =ϕ(a−b) =(ϕ◦f )(a+b).
(6.4)
Theorem6.3. Ifϕ is an isometry of the Euclidian spaceOwhich fixes1, then the following four properties are equivalent:
(1) Oϕ contains a four-dimensional subalgebra;
(2) Ocontains aϕ-invariant four-dimensional subalgebra; (3) Aut(Oϕ)contains a reflexion;
(4) Aut(Oϕ)is not trivial.
Proof. The only thing that remains to be shown is that (4) implies (1). Let g∈Aut(Oϕ)−{IO}. Ifgis a reflexion, then the result follows fromRemark 5.6. By assuming thatgis not a reflexion, we distinguish two cases.
Case1. The automorphismgadmits two linearly independent orthonor-mal eigenvectorsu,y∈W. Theng(uy)=g(u)g(y)=(±u)(±y)= ±uy and vect{1,u,y,uy} =Ker(g2−I
O)is aϕ-invariant four-dimensional subalgebra
ofO.
Case2. The automorphismghas only one eigenvector u∈S(W )except the sign. Thenuis an eigenvector ofϕandgandϕinduce isometries
gu,ϕu:W (u) →W (u). (6.5)
Using the minimal polynomialsP(X)andQ(X)ofguandϕu, we will first show
existence ofEis assured by the fact that the eigenspaces ofϕuaref-invariant,
and their direct sum is of even dimension. So we can assume thatQ(X)is a product of polynomials of degree two with negative discriminant. Now, we have three different cases.
(a)P(X)=X2−αX−βandQ(X)=X2−λX−µ are polynomials of degree
two. Sinceα2+4β <0 andλ2+4µ <0, there existsω∈R∗such thatα2+4β=
ω2(λ2+4µ), and we have
gu−α
2IW (u) 2
= α2
4 +β
IW (u)
=ω2 λ2
4 +µ
IW (u)
=ω2
ϕu−λ
2IW (u) 2
.
(6.6)
Nowguandϕucommute, so
gu−α2IW (u)−ω
ϕu−λ2IW (u)
◦
gu−α2IW (u)+ω
ϕu−λ2
≡0. (6.7)
(i) Ifgu−(α/2)IW (u)= ±ω(ϕu−(λ/2)IW(u)), then everyg-invariant
two-dimensional subspace ofW (u)isϕ-invariant, as well as its orthogonal. (ii) Ifgu−(α/2)IW (u)≠±ω(ϕu−(λ/2)IW (u)), then
H=Ker
gu−α
2IW (u)−ω
ϕu−λ
2IW (u)
(6.8)
andH⊥aregu-invariant andϕu-invariant proper subspaces ofW (u). One of them is necessarily two-dimensional and the other one is four-dimensional.
(b) If deg(P(X)) >2, then we consider an irreducible componentP1(X)of
P(X). The kernel Ker(P1(gu))and its orthogonal are then gu-invariant and
ϕu-invariant proper subspaces ofW (u).
(c) The case deg(Q(X)) >2 is similar to the case deg(P(X)) >2.
The subspace vect{1,u}⊕Eis then a subalgebra ofO. Indeed,E=vect{y,z}, withy,z∈W (u)orthogonal, and there exista,b∈Rwith a2+b2=1 such
that the matrix of the restriction ofgtoEis
a −b
b a
or
a b
b −a
. (6.9)
Thus,g(yz)=g(y)g(z)= ±(a2+b2)yz= ±yz, and consequentlyyz= ±u.
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A. Rochdi: Département de Mathématiques et Informatique, Faculté des Sciences, Université Hassan II, BP 7955, Casablanca, Morocco
