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On composition and absolute-valued algebras

Published online by Cambridge University Press:  12 July 2007

José Antonio Cuenca Mira
José Antonio Cuenca Mira
Affiliation:
Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, 29080 Málaga, Spain (cuenra@agt.cie.uma.es)
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Abstract

In this note we give a complete description of the composition algebras A over fields of characteristic ≠ 2, 3 in the following cases: if A has an anisotropic norm and x2x = xxx2 for every element; when A has a unitary central idempotent, it satisfies the identity (x2x2)x2 = x2(x2x2), and A is of finite dimension or has anisotropic norm. As a consequence, we obtain the existence, up to an isomorphism, of only seven absolute-valued algebras with a non-zero central idempotent where the last identity holds. This result completes the study of the absolute-valued algebras of this kind that was initiated by El-Mallah and Agawany.

We also introduce the class of e-quadratic algebra, which contains the quadratic algebras, but also includes large classes of composition and absolute-valued algebras. Many results on composition, absolute-valued and e-quadratic algebras are shown, and new proofs of some well-known theorems are given.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 136 , Issue 4 , August 2006 , pp. 717 - 731
Copyright
Copyright © Royal Society of Edinburgh 2006

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