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Some Identities in Algebras Obtained by the Cayley-Dickson Process

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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.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Ovidius University, Bd. Mamaia 124, 900527, Constanta, Romania

    Cristina Flaut

  2. Department of Complex Analysis and Potential Theory, Institute of Mathematics of the National Academy of Sciences of Ukraine, 3, Tereshchenkivs’ka st., 01601, Kiev-4, Ukraine

    Vitalii Shpakivskyi

Authors
  1. Cristina Flaut
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  2. Vitalii Shpakivskyi
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Correspondence to Cristina Flaut.

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To my family: Dan and Ana-Theodora.

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Flaut, C., Shpakivskyi, V. Some Identities in Algebras Obtained by the Cayley-Dickson Process. Adv. Appl. Clifford Algebras 23, 63–76 (2013). https://doi.org/10.1007/s00006-012-0344-6

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  • DOI: https://doi.org/10.1007/s00006-012-0344-6

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