EIGHT-DIMENSIONAL REAL ABSOLUTE-VALUED ALGEBRAS WITH LEFT UNIT WHOSE AUTOMORPHISM GROUP IS TRIVIAL

We classify, by means of the orthogonal group 7(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.


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 the n-dimensional real vector space R n possesses a bilinear product without zero divisors only in the case where the dimension n = 1, 2, 4, or 8.All eightdimensional real division algebras that occur in the literature contain a fourdimensional 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 containing no two-dimensional subalgebras.On the other hand, any four-dimensional absolute-valued real algebra with left unit contains a two-dimensional subalgebra.Therefore, a natural question to ask is whether an eight-dimensional real absolute-valued algebra with left unit contains a four-dimensional subalgebra.In this note, we give a negative answer and we characterise the eightdimensional absolute-valued real algebras with left unit containing a fourdimensional subalgebra in terms of the automorphism group.

Notation and preliminary results.
For simplicity, we only consider vector spaces over the field R of real numbers.Definition 2.1.Let A be an algebra; A is not assumed to be associative or unital.
(1) An element x ∈ A is called invertible if the linear operators are invertible in the associative unital algebra End(A).The algebra A is called a division algebra if all nonzero elements in A are invertible.
(2) A unital algebra A is called a quadratic algebra if {1,x,x 2 } is linearly dependent for all x ∈ A. If (•/•) is a symmetric bilinear form over A, then a linear operator f on A is called an isometry with respect to (•/•) if (f (x)/f (y)) = (x/y) for all x, y ∈ A. If, moreover, (xy/z) = (x/yz), for all x, y, z ∈ A, then (•/•) is called a trace form over A.
(3) The algebra A is termed normed (resp., absolute-valued) if it is endowed with a space norm • such that xy ≤ x y (resp., xy = x y ) for all x, y ∈ A. A finite-dimensional absolute-valued algebra is obviously a division algebra and has a subjacent Euclidean structure (see [11]). ( We denote by S(E) and vect{x 1 ,...,x n }, respectively, the unit sphere of a normed space E and the vector subspace spanned by x 1 ,...,x n ∈ E.
It is known that a quadratic algebra A is obtained from an anticommutative algebra (V , ∧) and a bilinear form (•, •) over V as follows: A = R⊕V as a vector space, with product ( We have a bilinear form associated to A, namely, (V , ∧) is called the anticommutative algebra associated to A. The elements of V are called vectors, while the elements of R are called scalars.We write A = (V , (•, •), ∧) (see [8]).
We will write (W , (•/•), ×) for the (quadratic) Cayley-Dickson octonions algebra O with its trace form (•/•) and the anticommutative algebra (W , ×).For u ≠ 0 ∈ W , W (u) will be the orthogonal subspace of R•u in W .It is well known that O is an alternative algebra, that is, it satisfies the identities x 2 y = x(xy) and yx 2 = (yx)x.Remark 2.2.Let A be an eight-dimensional absolute-valued algebra with left unit e, and f is an isometry of the Euclidian space A such that f (e) = e.Let A f be equal to A as a vector space, with a new product given by the formula x * y = f (x)y, for all x, y ∈ A. Then A f is also an absolute-valued algebra with left unit e.It is clear that an f -invariant subalgebra of A is a subalgebra of A f .In particular, if we consider the isometry R −1 e , then we obtain an absolute-valued algebra A R −1 e with unit e, which is isomorphic to O (see [12]).

Isometries of O with 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 basis Ꮾ = {1,x 1 ,...,x 7 } of O such that x 1 is an eigenvector of ϕ and , then the basis Ꮾ can be chosen as an extension of an orthonormal basis {1,u,y,z} of B, with u ∈ W an eigenvector of ϕ, and E = vect{y,z} is a ϕ-invariant subspace of B. Thus, B can be written as a direct orthogonal In the following important example, we use the notation introduced above.
Example 3.1.If ϕ fixes x 1 and its restriction to every W k is the rotation with angle kπ /4, then vect{1,x 1 } is the eigenspace E 1 (ϕ) of ϕ associated to the eigenvalue 1.The characteristic polynomial P ϕ (X) of ϕ is then with The characteristic polynomial P ϕ //B (X) of the restriction of ϕ to B is a polynomial of degree 4, a multiple of X − 1, and a divisor of P ϕ (X).Actually, P ϕ //B (X) = (X − 1) 2 P k (X) for k ∈ {1, 2, 3}, and this "forces" B to be of the form E 1 (ϕ) ⊕ W k for a certain k ∈ {1, 2, 3}.In particular, if Ꮾ is obtained from the canonical basis {1,e 1 ,...,e 7 } of O by taking then for each i ≠ j and l, x i × x j and x l are not colinear.This shows that 4. Eight-dimensional real absolute-valued algebras with left unit.First recall 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.
As a consequence we have the following lemma.

Lemma 4.2. Let ψ : A → B be an isomorphism of absolute-valued R-algebras and f :
Proof.The first statement is a consequence of Lemma 4.1.For x, y ∈ A, we have  (1) B is a subalgebra of O ϕ ; (2) B is a ϕ-invariant subalgebra of O.
Remark 5.3.(1) The algebra O ϕ has a two-dimensional subalgebra because ϕ has an eigenvector x ∈ W and the subalgebra vect{1,x} of O is ϕ-invariant.This argument shows that H ϕ has a two-dimensional subalgebra.
The following elementary result is useful for characterising the automorphisms of the algebra O ϕ .Proof.For all x, y ∈ O, we have that f

The relation in O ϕ between four-dimensional subalgebras and nontrivial automorphisms.
We begin with the following useful preliminary result taken from [10].

Lemma 6.1. Every four-dimensional subalgebra B of O = (W , (•/•), ×) coincides with the square of its orthogonal B ⊥ and satisfies the equality
Indeed, taking into account the trace property of (•/•), we have for all x, y ∈ B that (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 for all x, y ∈ B. Taking into account the anticommutativity of the product ×, we find that BB ⊥ = B ⊥ B. Finally, the trace property of (•/•) shows that BB ⊥ is orthogonal to B, hence BB ⊥ ⊂ B ⊥ .Proposition 6.2.Let B be a ϕ-invariant four-dimensional subalgebra of O. Then the map is a reflexion which commutes with ϕ.
Proof.The only thing that remains to be shown is that (4) implies (1).Let g ∈ Aut(O ϕ )−{I O }.If g is a reflexion, then the result follows from Remark 5.6.By assuming that g is not a reflexion, we distinguish two cases.
Case 1.The automorphism g admits two linearly independent orthonormal eigenvectors u, y ∈ W . Then g(uy) = g(u)g(y) = (±u)(±y) = ±uy and vect{1,u,y,uy} = Ker( Case 2. The automorphism g has only one eigenvector u ∈ S(W ) except the sign.Then u is an eigenvector of ϕ and g and ϕ induce isometries Using the minimal polynomials P (X) and Q(X) of g u and ϕ u , we will first show that W (u) contains a two-dimensional g-invariant and ϕ-invariant subspace of E. The irreducible factors of P (X) are polynomials of degree two with negative discriminant.However Q(X) can have a factor of degree one, and then the existence of E is assured by the fact that the eigenspaces of ϕ u are f -invariant, and their direct sum is of even dimension.So we can assume that Q(X) is a product of polynomials of degree two with negative discriminant.Now, we have three different cases.
The subspace vect{1,u}⊕E is then a subalgebra of O. Indeed, E = vect{y,z}, with y,z ∈ W (u) orthogonal, and there exist a, b ∈ R with a 2 + b 2 = 1 such that the matrix of the restriction of g to E is

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
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. 2 ) 5 .Lemma 5 . 1 .Proposition 5 . 2 .
Subalgebras and automorphisms of O ϕ .The following preliminary result allows us to characterise the subalgebras of O ϕ .If A is an algebra with left unit and without zero divisors, then every nontrivial finite-dimensional subalgebra of A contains the left unit element of A.Proof.Such a subalgebra B is a division algebra and for every x ≠ 0 ∈ B, there exists y ∈ B such that yx = x.On the other hand, if e is the left unit of A, then ex = x.Then the absence of zero divisors in A shows that y = e ∈ B.What are the subalgebras of O ϕ ?Let ϕ be an isometry of the Euclidian space O that fixes 1 and B is a subspace of O. Then the following two properties are equivalent:

Lemma 5 . 4 .Proposition 5 . 5 .
Let A be an algebra with left unit e and without zero divisors.If f ∈ Aut (A), then f (e) = e.Proof.We have (f (e) − e)f (e) = 0. What are the automorphisms of the algebra O ϕ ?If ϕ is an isometry of the Euclidian space O that fixes 1, then f ∈ Aut(O ϕ ) if and only if f ∈ G 2 and f commutes with ϕ.
an isomorphism.Every eight-dimensional absolute-valued left unital algebra is isomorphic to O f where f is an isometry of the Euclidian space O which fixes 1.Moreover, the following two properties are equivalent: The first statement is a consequence of a Remark 2.2 and Lemma 4.2.The second statement can be proved as follows: (2)O f and O g are isomorphic (f , g being two isometries of O fixing 1);(2)there exists ψ ∈ G 2 such that g = ψ • f • ψ −1 ,that is, f and g are in the same orbit of conjugations by isometries of O fixing 1.Proof.