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Advances in Applied Clifford Algebras

, Volume 23, Issue 1, pp 63–76 | Cite as

Some Identities in Algebras Obtained by the Cayley-Dickson Process

  • Cristina Flaut
  • Vitalii Shpakivskyi
Article

Abstract

Polynomial identities in algebras are the central objects of Polynomial Identities Theory. They play an important role in learning of algebras properties. In particular, the Hall identity is fulfilled in the quaternion algebra and does not hold in other non-commutative associative algebras. For this reason, the Hall identity is important for the quaternion algebra. The idea of this work is to generalize the Hall identity to algebras obtained by the Cayley-Dickson process.

Starting from the above remarks, in this paper, we prove that the Hall identity is true in all algebras obtained by the Cayley-Dickson process and, in some conditions, the converse of this statement is also true for split quaternion algebras. From Hall identity, we will find some new properties and identities in algebras obtained by the Cayley-Dickson process.

Keywords

Cayley-Dickson process Clifford algebras Hall identity 

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References

  1. Al; 39.
    A. A. Albert, Structure of algebras. Amer. Math. Soc. Colloquium Publications 24, (1939).Google Scholar
  2. Al; 49.
    Albert A. A.: Absolute-valued algebraic algebras. Bull. Amer. Math. Soc. 55, 763–768 (1949)MathSciNetCrossRefGoogle Scholar
  3. Br; 67.
    Brown R. B.: On generalized Cayley-Dickson algebras. Pacific J. of Math. 20(3), 415–422 (1967)CrossRefGoogle Scholar
  4. Fl; 12.
    C. Flaut, Levels and sublevels of algebras obtained by the Cayley- Dickson process. 2011, was accepted in Ann. Mat. Pura Appl. 2012, DOI 10.1007/s10231-012-0260-3.
  5. Fl; 01.
    Flaut C.: Some equations in algebras obtained by the Cayley-Dickson process. An. St. Univ. Ovidius Constanta 9(2), 45–68 (2001)MathSciNetMATHGoogle Scholar
  6. Fl, Şt; 09.
    C. Flaut, M. Ştefănescu, Some equations over generalized quaternion and octonion division algebras. Bull. Math. Soc. Sci. Math. Roumanie, 52 (4), (100) (2009) 427-439.Google Scholar
  7. Ha; 43.
    Hall M.: Projective planes. Trans. Amer. Math. Soc. 54, 229–277 (1943)MathSciNetMATHCrossRefGoogle Scholar
  8. Iv, Za; 05.
    Ivanov S., Zamkovoy S.: Parahermitian and paraquaternionic manifolds. Differential Geometry and its Applications 23, 205–234 (2005)MathSciNetMATHCrossRefGoogle Scholar
  9. Le; 06.
    Lewis D. W.: Quaternion Algebras and the Algebraic Legacy of Hamilton’s Quaternions. Irish Math. Soc. Bulletin 57, 41–64 (2006)Google Scholar
  10. Ki, Ou; 99.
    ElKinani E. H., Ouarab A.: The Embedding of U q(sl (2)) and Sine Algebras in Generalized Clifford Algebras. Adv. Appl. Clifford Algebras 9(1), 103–108 (1999)MathSciNetCrossRefGoogle Scholar
  11. Ko; 10.
    Koç C.: C-lattices and decompositions of generalized Clifford algebras. Adv. Appl. Clifford Algebras 20(2), 313–320 (2010)MATHCrossRefGoogle Scholar
  12. Sc; 66.
    Schafer R. D.: An Introduction to Nonassociative Algebras. Academic Press, New-York (1966)Google Scholar
  13. Sc; 54.
    Schafer R. D.: On the algebras formed by the Cayley-Dickson process. Amer. J. Math. 76, 435–446 (1954)MathSciNetCrossRefGoogle Scholar
  14. Smi; 50.
    Smiley M. F.: A remark on a theorem of Marshall Hall. Proceedings of the American Mathematical Society 1, 342–343 (1950)MathSciNetCrossRefGoogle Scholar
  15. Sm; 91.
    Smith T. L.: Decomposition of Generalized Clifford Algebras. Quart. J. Math. Oxford 42, 105–112 (1991)CrossRefGoogle Scholar
  16. Sz; 09.
    Szpakowski V. S.: Solution of general quadratic quaternionic equations. Bull. Soc. Sci. Lettres Łódź 59, Ser. Rech. Déform 58, 45–58 (2009)Google Scholar
  17. Ti; 99.
    Tian Y.: Similarity and cosimilarity of elements in the real Cayley-Dickson algebras. Adv. Appl. Clifford Algebras 9(1), 61–76 (1999)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceOvidius UniversityConstantaRomania
  2. 2.Department of Complex Analysis and Potential TheoryInstitute of Mathematics of the National Academy of Sciences of UkraineKiev-4Ukraine

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