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Moufang loops and alternative algebras


Author: Ivan P. Shestakov
Journal: Proc. Amer. Math. Soc. 132 (2004), 313-316
MSC (2000): Primary 17D05, 20N05
Published electronically: August 28, 2003
MathSciNet review: 2022350
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $\mathbf{O}$ be the algebra $\mathbf{O}$ of classical real octonions or the (split) algebra of octonions over the finite field $GF(p^2), p>2$. Then the quotient loop $\mathbf{O}^*/Z^*$ of the Moufang loop $\mathbf{O}^*$ of all invertible elements of the algebra $\mathbf{O}$ modulo its center $Z^*$ is not embedded into a loop of invertible elements of any alternative algebra.


References [Enhancements On Off] (What's this?)

  • 1. Edgar G. Goodaire, A brief history of loop rings, Mat. Contemp. 16 (1999), 93–109. 15th School of Algebra (Portuguese) (Canela, 1998). MR 1756829
  • 2. E. N. Kuz′min and I. P. Shestakov, Nonassociative structures, Current problems in mathematics. Fundamental directions, Vol. 57 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990, pp. 179–266 (Russian). MR 1060322
  • 3. R. Moufang, Zur Structur von Alternativekörpern, Math. Ann. 110 (1935), 416-430.
  • 4. G. B. Nagy, Applications of Bol reflections, Ph.D. Dissertation, Szeged (Hungary), 1999.
  • 5. N. Sandu, About the embedding of Moufang loops in alternative algebras, Loops '99 Conference, July 27, 1999-Aug. 1, 1999, Prague, Abstracts, 33-34.
  • 6. K. A. Ževlakov, A. M. Slin′ko, I. P. Šestakov, and A. I. Širšov, Koltsa, blizkie k assotsiativnym, “Nauka”, Moscow, 1978 (Russian). Sovremennaya Algebra. [Monographs in Modern Algebra]. MR 518614
    K. A. Zhevlakov, A. M. Slin′ko, I. P. Shestakov, and A. I. Shirshov, Rings that are nearly associative, Pure and Applied Mathematics, vol. 104, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. Translated from the Russian by Harry F. Smith. MR 668355

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Additional Information

Ivan P. Shestakov
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281 - CEP 05315-970, São Paulo, Brazil and Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia
Email: shestak@ime.usp.br

DOI: https://doi.org/10.1090/S0002-9939-03-07260-5
Keywords: Moufang loop, alternative algebra
Received by editor(s): January 2, 2002
Published electronically: August 28, 2003
Additional Notes: Partially supported by CNPq grant 300528/99-0
Communicated by: Lance W. Small
Article copyright: © Copyright 2003 American Mathematical Society