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Advances in Applied Clifford Algebras

, Volume 27, Issue 2, pp 955–964 | Cite as

A Note on Standard Composition Algebras of Types II and III

  • P. D. Beites
  • A. P. Nicolás
Article
  • 56 Downloads

Abstract

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\({(F) \neq 2}\) and satisfying the identity \({x^{2}y = n(x)y}\) are shown to be standard composition algebras of type II. As a consequence, the identity \({yx^{2} = n(x)y}\) characterizes the type III.

Mathematics Subject Classification

17A75 15A21 

Keywords

Standard composition algebra Identity 

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References

  1. 1.
    Beites, P.D., Nicolás, A.P.: Álgebras de composição standard de tipo II. Atas do Encontro Nacional da SPM 2014, Lisboa (2014)Google Scholar
  2. 2.
    Bremner M., Hentzel I.: Identities for algebras of matrices over the octonions. J. Algebra 277, 73–95 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chevalley C.: The Algebraic Theory of Spinors and Clifford Algebras. Springer, New York (1997)zbMATHGoogle Scholar
  4. 4.
    Elduque A., Pérez-Izquierdo J.M.: Composition algebras with large derivation algebras. J. Algebra 190, 372–404 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Jacobson N.: Composition algebras and their automorphisms. Rend. Circ. Mat. Palermo 7, 55–80 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kaplansky I.: Infinite-dimensional quadratic forms admitting composition. Proc. Am. Math. Soc. 4, 956–960 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Lounesto P.: Octonions and triality. Adv. Appl. Clifford Algebras 11, 191–213 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Okubo S.: Classification of flexible composition algebras, I and II. Hadronic J. 5, 1564–1626 (1982)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Petersson H.P.: Quasi composition algebras. Abh. Math. Semin. Univ. Hambg. 35, 215–222 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Stampfli-Rollier C.: 4-dimensionale Quasikompositionsalgebren. Arch. Math. 40, 516–525 (1983)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Departamento de Matemática, Centro de Matemática e Aplicações (CMA-UBI)Universidade da Beira InteriorCovilhãPortugal
  2. 2.Departamento de Matemática Aplicada, Instituto de Investigación en Matemáticas (IMUVa)Universidad de ValladolidValladolidSpain

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