Third-power associative absolute valued algebras with a nonzero idempotent commuting with all idempotents
| Autor: | José Antonio Cuenca Mira |
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| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
General Mathematics idempotent Third-power associativity 17A75 Division (mathematics) 17D99 17A80 Division algebra third-power associativity Absolute (philosophy) division algebra Idempotent Idempotence 17A60 Algebra over a field Quaternion Pairwise commuting elements Absolute valued algebra Associative property pairwise commuting elements Mathematics |
| Zdroj: | Recercat: Dipósit de la Recerca de Catalunya Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) Publ. Mat. 58, no. 2 (2014), 469-484 Publicacions Matemàtiques; Vol. 58, Núm. 2 (2014); p. 469-484 Dipòsit Digital de Documents de la UAB Universitat Autònoma de Barcelona Recercat. Dipósit de la Recerca de Catalunya instname |
| Popis: | This paper deals with the determination of the absolute valued algebras with a nonzero idempotent commuting with the remaining idempotents and satisfying $x^2 x = x x^2 $ for every $x$. We prove that, in addition to the absolute valued algebras $\mathbb R $, $\mathbb C $, $\mathbb H $, or $\mathbb O $ of the reals, complexes, division real quaternions or division real octonions, one such absolute valued algebra $A$ can also be isometrically isomorphic to some of the absolute valued algebras $\overset{\star}{\mathbb C}$, $\overset{\star}{\mathbb H}$, or $\overset{\star}{\mathbb O}$, obtained from $\mathbb C $, $\mathbb H$, and $\mathbb O $ by imposing a new product defined by multiplying the conjugates of the elements. In particular, every absolute valued algebra having the above properties is finite-dimensional. This generalizes some well known theorems of Albert, Urbanik and Wright, and El-Mallah. |
| Databáze: | OpenAIRE |
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