Skip to main content
SpringerLink
Log in

Finite-dimensional absolute-valued algebras

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper we provide some new tools for the study of finite-dimensional absolute-valued algebras. We introduce homotopy notions in this field and develop some of their applications. Next, we parametrize these algebras by spin groups and study their isomorphisms. Finally, we introduce a duplication process for the construction of absolute-valued algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. A. Albert, Absolute-valued real algebras, Annals of Mathematics (2) 48 (1947), 495–501.

    Article  Google Scholar 

  2. T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups, Graduate Texts in Mathematics, 98, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1985.

    MATH  Google Scholar 

  3. A. J. Calderón and C. Martín, Two-graded absolute-valued algebras, Journal of Algebra 292 (2005), 492–515.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. H. Conway and D. A. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, A K Peters, Ltd., Natick, MA, 2003.

    Google Scholar 

  5. J. A. Cuenca and A. Rodriguez-Palacios, Absolute values on H*-algebras, Commuications in Algebra 23 (1995), 1709–1740.

    Article  MATH  Google Scholar 

  6. J. A. Cuenca, E. Darpö and E. Dieterich, Classification of the finite-dimensional absolute-valued algebras having a non-zero central idempotent or a one-sided unity, Bulletin des Sciences Mathématiques 134 (2010), 247–277.

    Article  MATH  Google Scholar 

  7. A. Dold, Lectures on Algebraic Topology, Classic in Mathematics, reprint of the 1980 edition, Springer-Verlag, Berlin Heidelberg New York, 1995.

    MATH  Google Scholar 

  8. H. D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel and R. Remmert, Numbers, Graduate Texts in Mathematics, 123, Readings in Mathematics, Springer-Verlag, New York, 1990.

    MATH  Google Scholar 

  9. A. Elduque and J. M. Pérez, Composition algebras with associative bilinear form, Communications in Algebra 24 (1996), 1091–1116.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Elduque and J. M. Pérez, Composition algebras with large derivation algebras, Journal of Algebra 190 (1997), 372–404.

    Article  MathSciNet  MATH  Google Scholar 

  11. T. H. Kiang, The Theory of Fixed Point Classes, Springer-Verlag, Berlin, Science Press Beijing, 1989.

    MATH  Google Scholar 

  12. M-A. Knus, A. Merkurjev, M. Rost and J-P. Tignol, The Book of Involutions, American Mathematical Society Colloquium Publications, 44, American Mathematical Society, Providence, RI, 1998.

    MATH  Google Scholar 

  13. Y. I. Lyubich, Mathematical Structures in Population Genetics, Biomathematics, 22, Springer-Verlag, Berlin, 1992.

    MATH  Google Scholar 

  14. A. Ostrowsky, Über einige Lösungen der Funktionalgleichung ϕ(x)ϕ(y) = ϕ(xy), Acta Mathematica 41 (1918), 271–284.

    Article  Google Scholar 

  15. M. Postnikov, Leçons de géométrie. Groupes et algèbres de Lie, Editions Mir, Moscow, 1985.

    Google Scholar 

  16. M. I. Ramírez, On four-dimensional absolute-valued algebras, in Proceedings of the International Conference on Jordan Structures (Málaga, 1997), University of Málaga, Málaga, 1999, pp. 169–173.

    Google Scholar 

  17. A. Rochdi, Eight-dimensional real absolute-valued algebras with left unit whose automorphism group is trivial, International Journal of Mathematics and Mathematical sciences 70 (2003), 4447–4454.

    Article  MathSciNet  Google Scholar 

  18. A. Rochdi and A. Rodriguez, Absolute-valued algebras with involution, Communications in Algebra 37 (2009), 1151–1159.

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Rodríguez-Palacios, One-sided division absolute-valued algebras, Publicacions Matemàtiques (2B) 36 (1992), 925–954.

    Article  MATH  Google Scholar 

  20. A. Rodríguez, Absolute-valued algebras of degree two, in Non-associative Algebra and its Applications (Oviedo, 1993), Mathematics and its Applications, 303, Kluwer Acad. Publ., Dordrecht, 1994, pp. 350–356.

    Google Scholar 

  21. A. Rodríguez, Absolute-valued algebras, and absolute-valuable Banach spaces, in Advanced Courses of Mathematical Analysis I, World Sci. Publ., Hackensack, NJ, 2004, pp. 99–155.

    Google Scholar 

  22. B. Segre, La teoria delle algebre ed alcune questione di realtá, Università di Roma Istituto Nazionale Alta Matematica Rediconti di Matematica e Applicazioni (5) 13 (1954), 157–188.

    MathSciNet  MATH  Google Scholar 

  23. E. H. Spanier, Algebraic Topology, McGraw-Hill, New York-Toronto, Ontario-London, 1966.

    MATH  Google Scholar 

  24. T. tom Dieck, Topology, De Gruyter Lehrbuch, Walter de Gruyter, Berlin, New York, 1991.

    MATH  Google Scholar 

  25. K. Urbanik and F. B. Wright, Absolute-valued algebras, Proceedings of the American Mathematical Society 11 (1960), 861–866.

    Article  MathSciNet  Google Scholar 

  26. K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov and A. I. Shirshov, Rings that are Nearly Associative, Pure and Applied Mathematics, 104, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1982.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Cádiz, 11510, Puerto Real, Cádiz, Spain

    A. Calderón

  2. Departamento de Álgebra y Análisis Matemático, Universidad de Almería, 04071, Almería, Spain

    A. Kaidi

  3. Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, Apartado 59, 29080, Málaga, Spain

    C. Martín

  4. Departamento de Álgebra y Análisis Matemático, Universidad de Almería, 04071, Almería, Spain

    A. Morales & M. Ramírez

  5. Département de Mathématiques et Informatique, Faculté des Sciences, Université Hassan II, 7955, Casablanca, Morocco

    A. Rochdi

Authors
  1. A. Calderón
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Kaidi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. Martín
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. Morales
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. M. Ramírez
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. A. Rochdi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Calderón.

Additional information

The authors of this paper are supported in part by the PCI of the UCA ‘Teoría de Lie y teoría de espacios de Banach’, by the PAI with project numbers FQM-194, FQM-298, FQM-336 and FQM-3737, by the projects I+D+I of the Spanish Ministerio de Educación y Ciencia MTM2010-17687 and MTM2004-06580-C02-02, with fondos FEDER and by the project PCI 62/04/R/E of the Spanish AECI.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Calderón, A., Kaidi, A., Martín, C. et al. Finite-dimensional absolute-valued algebras. Isr. J. Math. 184, 193–220 (2011). https://doi.org/10.1007/s11856-011-0065-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-011-0065-x

Keywords

Search

Navigation