Abdellatif Rochdi

• University of Hassan II Casablanca

We study those pre-Hilbert absolute-valued algebras satisfying the identity (x2,y,x2)=0. We prove that such an algebra A is finite-dimensional in each one of the following two cases: (1)A satisfies the additional identity (x,x2,x)=0, (2)A contains a weak left-unit. In the first case A is flexible and isomorphic to either R, C, C⁎, H, H⁎, O, O⁎ or P...

We study algebras $A,$ over a field of characteristic zero, satisfying $(x^p, x^q, x^r)=0$ for $p, q, r$ in ${1, 2}.$ The existence of a unit element in such algebras leads to the third power-associativity. If, in addition, $A$ has degree $\leq 4$ then $A$ is power-commutative. We deduce that any 4-dimensional real division algebra, with unit eleme...

By means of principal isotopes H(a, b) of the algebra H [25], we give an exhaustive and not repetitive description of all 4-dimensional absolute-valued algebras satisfying (x p, x q, x r) = 0 for fixed integers p, q, r ∈ {1, 2}. For such algebras, the number N(p, q, r) of isomorphism classes is either 2 or 3, or is infinite. Concretely: 1. N(1, 1,...

We show that every absolute-valued algebra with left-unit satisfying (x2; x2; x2) = 0 is finite-dimensional of degree at most 4: Next, we determine such an algebras. In addition to the already known algebras R; C; \astC; H; \astH; \astH(i; 1); O; \astO; \astO(i; 1) the list is completed by two new algebras not yet specified in the literature.

In this paper we provide some new tools for the study of finite-dimensional absolute-valued algebras. We introduce homotopy notions in this field and develop some of their applications. Next, we parametrize these algebras by spin groups and study their isomorphisms. Finally, we introduce a duplication process for the construction of absolute-valued...

Let A be an absolute valued algebra containing a nonzero central element a. We prove that A is finite dimensional in the two following cases : 1. A satisfies the identity (x 2, x, x) = 0, 2. A is of left unit e such that (a|e) = 0. In the first case A is isomorphic to \({\mathbb{R}}\), \({\mathbb{C}}\), \({\mathbb{H}}\) or \({\mathbb{O}}\) and in...

Dans ce papier nous montrons que si A est une algèbre complexe, normée, préhilbertienne, algébrique, sans diviseurs de zéro et vérifiant $||a^2|| = ||a||^2$ pour tout $a \in A$ . Alors A est de dimension finie et isomorphe à ${{\mathbb{C}}}$ . Ce dernier nous permet de donner des nouveaux résultats plus généraux que ceux du cas absolument val...

By means of principal isotopes lH(a,b) of the algebra lH [Ra 99] we give an exhaustive and not repetitive description of all 4-dimensional absolute-valued algebras satisfying (x^p, x^q, x^r) = 0 for fixed integers p, q, r \in\{1,2\}. For such an algebras the number N(p,q,r) of isomorphism classes is 2 or 3, or is infinite. Concretely 1. N(1,1,1)=N(...

In this work we are interested in the general problem of the determination of the normed division algebras. Our fundamental results are obtained in the particular subclass of those 8-dimensional quadratic flexible real division algebras. We give a new process which generalizes that of Cayley-Dickson and which allows the obtaining of a new family of...

A classification of all four-dimensional power-commutative real division algebras is given. It is shown that every four-dimensional power-commutative real division algebra is an isotope of a particular kind of a quadratic division algebra. The description of such isotopes in dimension four and eight is reduced to the description of quadratic divisi...

Soit A une algèbre réelle sans diviseurs de zéro. On suppose que l’espace vectoriel A est muni d’une norme ∥.∥ préhilbertienne vérifiant ∥a 2∥ ≤∥a∥2 pour tout a Î Aa \in A. Alors A est de dimension finie dans chacun des quatre cas suivants : 1. A est commutative contenant un élément non nul a tel que ∥ax∥=∥a∥ ∥x∥ pour tout x Î Ax \in A, 2. A es...

Let A be an absolute valued algebra such that there exists a nonzero algebraic element e∈A satisfying some of the following conditions: 1. e(xy)=x(ey) for all x,y∈A. 2. (ex)e=e(xe) for all x∈A. We prove that the norm of A comes from an inner product. This generalizes previously known results in [A. Rodríguez Palacios, Publ. Mat., Barc. 36, 925–954...

Let A be an absolute valued algebra with involution, in the sense of Urbanik [K. Urbanik, Absolute valued algebras with an involution, Fund. Math. 49 (1961) 247–258]. We prove that A is finite-dimensional if and only if the algebra obtained by symmetrizing the product of A is simple, if and only if eAs = As, where e denotes the unique nonzero self-...

We classify, by means of the orthogonal group 𝒪7(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.

In this paper we give a new process called vectorial isotopy, in order to classify the eight-dimensional real quadratic flexible division algebras, and we solve the isomorphism problem.