Algebres non associatives normees de division. Classification des algebres reelles de Jordan non commutatives de division lineaire de dimension 8

TL;DR

This paper explores the classification of 8-dimensional quadratic flexible real division algebras, introducing new construction methods, analyzing their automorphisms, and solving isomorphism problems within this subclass.

Contribution

It presents a generalized Cayley-Dickson process for constructing new 8-dimensional division algebras and characterizes their automorphism groups and isomorphism classes.

Findings

Introduced a new process generalizing Cayley-Dickson for algebra construction

Identified algebras not obtainable by standard duplication methods

Characterized automorphism groups of these algebras

Abstract

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 eight-dimensional quadratic flexible real division algebras. We give examples of 8-dimensional quadratic flexible real division algebras which cannot be obtained by this first process of duplication and by means of a second process, which consists in making an appropriate deformation of the product of the octonion algebra, we determine these last ones and we resolve the isomorphism problem. Among the eight-dimensional quadratic flexible real division algebras, we study those which possess a nontrivial derivation by mean of the generalized Cayley-Dickson process. We also give examples where the group of automorphisms is trivial, and characterize the algebras whose group of automorphisms is not trivial. This brings to light the unlimitedness of this subclass of algebras.