Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Full-text PDF

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. E. Goodaire, A brief history of loop rings. 15th Brasilian School of Algebra (Canela, 1998). Mat. Contemp. 16 (1999), 93-109. MR 2001i:17044
  • 2. E. N. Kuzmin and I. P. Shestakov, Nonassociative structures, VINITI, Itogi nauki i tekhniki, seria ``Fundamental Branches'', vol. 57, 179-266, Moscow, 1990; English transl. in ``Encyclopaedia of Math. Sciences, vol. 57, Algebra VI, 199-280'', edited by A. I. Kostrikin and I. R. Shafarevich, Springer-Verlag, 1995. MR 91i:17001
  • 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. Zhevlakov, A. M. Slin'ko, I. P. Shestakov, and A. I. Shirshov, Rings that are nearly associative, Nauka, Moscow, 1978; English transl., Pure and Applied Mathematics, vol. 104, Academic Press, 1982. MR 80h:17002, MR 83i:17001

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 17D05, 20N05

Retrieve articles in all journals with MSC (2000): 17D05, 20N05

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

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

American Mathematical Society