A Note on Standard Composition Algebras of Types II and III

Some identities satisfied by certain standard composition algebras, of types II and III, are studied and become candidates for the characterization of the mentioned types. Composition algebras of arbitrary dimension, over a field F with char(F)≠2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(F) \neq 2}$$\end{document} and satisfying the identity x2y=n(x)y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x^{2}y = n(x)y}$$\end{document} are shown to be standard composition algebras of type II. As a consequence, the identity yx2=n(x)y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${yx^{2} = n(x)y}$$\end{document} characterizes the type III.


Introduction
Among composition algebras, the well known ones are those with identity, that is, Hurwitz algebras. These over a field of characteristic different from 2, by the generalized Hurwitz Theorem in [5], are isomorphic either to the base field, a separable quadratic extension of the base field, a generalized quaternion algebra or a generalized Cayley algebra. In particular, the dimension of any Hurwitz algebra is 1, 2, 4 or 8. This restriction on the dimension was also proved by Chevalley using Clifford algebras (for more details, see [3]).
In general, the classification of all finite dimensional composition algebras is still an open problem. Nevertheless, the imposition of one or more additional conditions on these algebras, in the form of identities, can afford such a classification. For instance, in [8], Okubo proved that, over fields of characteristic different from 2, any finite dimensional composition algebra satisfying the flexible identity (xy)x = x(yx) is either a form of a Hurwitz algebra, a form of a para-Hurwitz algebra or an Okubo algebra.
In [1] we began to approach the problem of characterizing standard composition algebras of type II over a field F with char(F ) = 2. Concretely, we applied the classification of 2-dimensional and of 4-dimensional composition algebras due to, respectively, Petersson, in [9], and Stampfli-Rollier, in *Corresponding author.

Preliminaries
From now on, with exception of Theorems 3.2 and 4.2, F is a field such that char(F ) = 2.
Let A be an algebra over F , with multiplication denoted by juxtaposition.
An identity of A has degree s (i.e., is a s-identity), where s ∈ N and s ≥ 2, if the multiplication appears s − 1 times in each term of the identity.
The opposite algebra A op of A is the algebra with the same underlying vector space as A but with multiplication defined by x • y = yx.
The algebra A is a composition algebra if it is endowed with a nondegenerate quadratic form (the norm) n : A → F which is multiplicative, i.e., for any x, y ∈ A, n(xy) = n(x)n(y).
(1) The form n being nondegenerate means that the associated symmetric bilinear form n(x, y) = 1 2 (n(x + y) − n(x) − n(y)) is nondegenerate. Recall that from (1), by linearization, for all x, y, z ∈ A, n(xy, xz) = n(x)n(y, z) = n(yx, zx) that is, the linear maps of left and right multiplication by x, L x : y → xy and R x : y → yx, are similarities of the norm n(x). As a consequence, L x and R x are injective for all x ∈ A such that n(x) = 0.
A unital composition algebra, that is, a composition algebra with identity is called a Hurwitz algebra.
Hurwitz algebras are the main ingredient in the construction of finite dimensional composition algebras without identity. In fact, given a Hurwitz Vol. 27 (2017) A Note on Standard Composition Algebras 957 algebra with multiplication * , norm n, and φ, ψ two isometries of n, the multiplication defines a new composition algebra with the same norm n but, generally, nonunital [6]. Conversely, given a finite dimensional composition algebra A with product denoted by juxtaposition and quadratic form n, for any element a with n(a) = 0, the left and the right multiplication operators L u and R u , , are isometries of n. Hence, the new multiplication ) gives a Hurwitz algebra with norm n and identity u 2 , [6]. So, the dimension of any finite dimensional composition algebra is equal to 1, 2, 4 or 8.
According to [4], modifying the multiplication * of a Hurwitz algebra (A, * ) over an algebraically closed field as in (2) leads to a new composition algebra, relative to the same quadratic form n, with "fewer degrees of symmetry" except for the multiplications defined by The new algebras are called standard composition algebras of the corresponding type, that is, I, II, III, IV, respectively, associated to (A, * ). If the dimension of A is 1, then all standard composition algebras are the base field F . In higher dimensions, standard composition algebras of different type are not isomorphic.
Kaplansky, in [6], proved that every Hurwitz algebra must be finite dimensional. However, there exist infinite dimensional composition algebras, necessarily non-unital. Namely, as mentioned in [4], Elduque and Pérez-Izquierdo gave examples of such algebras with a one-sided identity element.

Some Standard Composition Algebras Associated to the Quaternion Algebra (H, * )
From now on, we can suppose, extending scalars if necessary, that F is algebraically closed. Let (H, * ) be the quaternion algebra over F with basis {e 0 , e 1 , e 2 , e 3 } and multiplication table given by e 1 * e 1 = e 2 * e 2 = e 3 * e 3 = e 1 * e 2 * e 3 = −e 0 , where e 0 is the identity. We now consider H, the standard composition algebra of type II associated to (H, * ), over F . Its multiplication, denoted by juxtaposition, is given by Proof. The 2-identities satisfied by H are of the form ζxy + ηyx = 0, where ζ and η are scalars. With x = e 0 , we obtain ζy + ηy = 0 for all y. If any of the scalars is non-null, then y and y are linearly dependent, which is a contradiction. Hence ζ = η = 0.
a i e i ∈ H. Let l(a) and r(a) denote, respectively, the coordinate matrices of the left multiplication by a and of the right multiplication by a. Then, calculating ax and xa, we have where (·, ·, ·) denotes the associator in H.
Proof. In degree 3, there are two association types, (· ·) · and · (· ·), i.e., the 3-identities of H are of the form where α σ , β σ ∈ F and S 3 stands for the symmetric group of degree 3. We now apply the random vectors method, using Maple TM , in characteristic 0. From the previous notations, we use l(x)Y for the multiplication given by xy, where Y = [y 0 . . . y 3 ] T . The process begins with the construction of a 16 × 12 matrix M initialized to zero. The first 6 columns and columns 7 to 12 of M are labeled by the 6 monomials (· ·) · and by the 6 monomials · (· ·), respectively, in (4). Furthermore, we think of M as consisting of a 12 × 12 square matrix on top of a 4 × 12 matrix. We generate three pseudo-random vectors with 4 components. We allocate the 4 components of the evaluation of the jth (j ∈ {1, . . . , 6}) monomial (· ·) · and of the jth (j ∈ {7, . . . , 12}) monomial · (· ·) in column j of M , in rows 13 to 16. The computation of the row canonical form of Vol. 27 (2017) A Note on Standard Composition Algebras 959 the obtained matrix completes the first iteration of the algorithm. We repeat this fill and reduce process until the stabilization of the rank of the matrix is reached, arriving at rank 10. The 3-identities satisfied by H lie in the 2dimensional nullspace of this matrix. Expressing the vectors of a basis for this subspace as linear combinations of the original 12 monomials, we obtain the subsequent identities It is clear that the second identity can be obtained by the action of the transposition (xy) over the first one. So, the first identity generates the whole space of 3-identities of H under the action of S 3 .
Although x 2 y = n(x)y is not an identity of a certain degree s, in the sense defined in the Preliminaries, this condition is satisfied by H.

Proposition 3.3. Let H be the standard composition algebra of type II associated to (H, * ) over an algebraically closed field F with char(F ) = 2. The condition x 2 y = n(x)y is an identity of H.
Proof. Notice that x 2 y = (x * x) * y = n(x)y. Proof. Applying the associativity of (H, * ), we have From here and by Proposition 3.3, we conclude that the identity (3) holds in H when char(F ) = 2.
We now consider H, the standard composition algebra of type III associated to (H, * ), over F . Its multiplication, denoted by juxtaposition, is given by

Some Standard Composition Algebras Associated to the Octonion Algebra (O, * )
As in the previous section, we assume that F is algebraically closed. Let (O, * ) be the octonion algebra over F with basis {e 0 , e 1 , e 2 , e 3 , e 4 , e 5 , e 6 , e 7 } and multiplication table given by e i * e i = −e 0 for i ∈ {1, . . . , 7}, being e 0 the identity, and the Fano plane in Fig. 1, where the cyclic ordering of each three elements lying on the same line is shown by the arrows. We now consider O, the standard composition algebra of type II associated to (O, * ), over F . Its multiplication, denoted by juxtaposition, is given by  a i e i ∈ O. Then the coordinate matrices of the left and of the right multiplication by a are given, respectively, by a 0 a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 1 −a 0 −a 3 a 2 −a 5 a 4 −a 7 a 6 a 2 a 3 −a 0 −a 1 −a 6 a 7 a 4 −a 5 a 3 −a 2 a 1 −a 0 a 7 a 6 −a 5 −a 4 a 4 a 5 a 6 −a 7 −a 0 −a 1 −a 2 a 3 a 5 −a 4 −a 7 −a 6 a 1 −a 0 a 3 a 2 a 6 a 7 −a 4 a 5 a 2 −a 3 −a 0 −a 1 a 7 −a 6 Proof. We apply the random vectors method in Maple TM , as in the proof of Theorem 3.2, in characteristic zero. The needed M initialized to zero is a 20 × 12 matrix and the generated pseudo-random vectors have 8 components.
By the previous notations, we write l(w)Z, where w is given by l(x)Y , and l(x)(l(y)Z) instead of (xy)z and x(yz), respectively, where Y = [y 0 . . . y 7 ] T and Z = [z 0 . . . z 7 ] T . The conclusion is similar to that obtained for the degree 3 identities of H, that is, all 3-identities of O are implied by (u, v, v) = (vv)u− v(uv), where (·, ·, ·) denotes the associator in the mentioned algebra.