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 ) 6= 2 and satisfying the identity xy = n(x)y are shown to be standard composition algebras of type II. As a consequence, the identity yx = n(x)y 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 [10]. Unfortunately, since no similar description is known for 8-dimensional composition algebras, the strategy in [1] could not be continued.
In the present work we consider the standard composition algebras of types II and III, over a field F with char(F ) = 2, associated to the Hurwitz algebras H (quaternion algebra) and O (octonion algebra). Both H and O are related to certain Clifford algebras, as can be read, for instance, in [7] and references therein. In Sections 3 and 4, we focus on some identities satisfied by the former algebras and, concretely for the ones of degree 3, we apply the random vectors method. See [2] for more details about the cited process that involves computational linear algebra on matrices.
In Section 5, we analyze if the obtained identities characterize standard composition algebras of types II and III. On the one hand, we see that the degree 3 identity that implies all degree 3 identities of the considered standard composition algebras of type II, associated either to H or to O, can not accomplish that purpose for the mentioned type. The same happens with the corresponding one for type III. On the other hand, we conclude, in arbitrary dimension and over a field of characteristic different from 2, that the identity x 2 y = n(x)y (respectively, yx 2 = n(x)y) characterizes the standard composition algebras of type II (respectively, III).

Preliminaries
From now on, with exception of Theorem 3.2 and Theorem 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 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.
3 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   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 Theorem 3.2. Let H be the standard composition algebra of type II associated to (H, * ) over an algebraically closed field F with char(F ) = 0. The 3-identities satisfied by H are consequences of 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 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 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 2-dimensional 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 −(xz)y + (yx)z + (yz)x − (zx)y + x(yz) − y(xz) − y(zx) + z(yx) = 0, −(yz)x + (xy)z + (xz)y − (zy)x + y(xz) − x(yz) − x(zy) + z(xy) = 0.
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.  Proof. Applying the associativity of (H, * ), we have We now consider H, the standard composition algebra of type III associated to (H, * ), over F . Its multiplication, denoted by juxtaposition, is given by xy := x * y.   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

Characterization
Notice that every standard composition algebra of type I associated to a 2-dimensional Hurwitz algebra (so, commutative and associative) satisfies Hence, the former identity can not characterize all standard composition algebras of type II (respectively, III). We now prove, with char(F ) = 2, that the identities x 2 y = n(x)y and yx 2 = n(x)y allow us to achieve that goal.
Theorem 5.1. Let F be a field with char(F ) = 2. Let A be a composition algebra of arbitrary dimension over F , with multiplication denoted by juxtaposition, that satisfies Then A is a standard composition algebra of type II.
Proof. For any x with n(x) = 0, 1 n(x) x 2 is a left identity of A. In particular, there exists a left identity e: ex = x for any x. But a left identity is unique, because if e 1 and e 2 are left unities, then (e 1 −e 2 )x = 0 for any x. As the right multiplication by any x with n(x) = 0 is injective, we conclude that e 1 = e 2 . Therefore, denote by e the left identity of A. By uniqueness, x 2 = n(x)e for any x with n(x) = 0.
Let us consider the algebra obtained by scalar extension to an algebraic closureF :Â = A ⊗ FF .
For any z ∈ A, the subspace S =F e +F z has finite dimension and is non-isotropic. Notice that the maps defined by x → x 2 and x → n(x)e, from S to S 2 , are polynomial functions. Since x 2 = n(x)e for any x ∈ S with n(x) = 0, by Zariski density, x 2 = n(x)e for any x ∈ S. In particular, z 2 = n(z)e. Linearizing x 2 = n(x)e, we obtain xe + ex = 2n(e, x)e. Hence, xe = 2n(e, x)e − x for any x and so r e , the right multiplication by e, is a linear map such that (r e ) 2 = id. Therefore, it is bijective.
Define a new multiplication on A by x · y = r −1 e (x)y. Then (A, ·) is a Hurwitz algebra (so, finite dimensional) with the same norm and identity e and xy = (xe) · y for any x, y ∈ A.
With x = y this gives x 2 = (xe) · x, but x 2 = n(x)e, so we get n(x)e = (xe) · x for any x. But n(x)e =x · x (wherex denotes the standard involution in the Hurwitz algebra (A, ·)). Hencex · x = (xe) · x for any x. Thenx = xe for any x with n(x) = 0 and, again by Zariski density onÂ, this is valid for any x.
Therefore, xy = (xe) · y =x · y for any x, y, as required.
Corollary 5.2. Let A be a composition algebra of arbitrary dimension over F , with multiplication denoted by juxtaposition, that satisfies yx 2 = n(x)y.
Then A is a standard composition algebra of type III.
Proof. If A satisfies yx 2 = n(x)y, then the opposite algebra A op of A satisfies x 2 y = n(x)y. By Theorem 5.1, A op is a standard composition algebra of type II. Therefore, A is a standard composition algebra of type III.