Classification of Left Octonionic Modules

Abstract

In this article, we provide a complete classification of left octonionic modules (finite or infinite dimensions) in terms of new notions such as associative elements and conjugate associative elements. We give a simple approach to determine the irreducible left \(\mathbb {O}\)-modules by utilizing the Clifford algebra \(C\ell _{7}\). We find that every left \(\mathbb {O}\)-module has a basis in some sense. This means that every left \(\mathbb {O}\)-module is a “free” module.

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Acknowledgements

The authors would like to thank the referees for the very detailed comments and correct suggestions that really helped to improve this paper.

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Correspondence to Qinghai Huo.

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This work was supported by the NNSF of China (11771412).

Communicated by Jacques Helmstetter.

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Huo, Q., Li, Y. & Ren, G. Classification of Left Octonionic Modules. Adv. Appl. Clifford Algebras 31, 11 (2021). https://doi.org/10.1007/s00006-020-01113-4

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Mathematics Subject Classification

  • 17A05

Keywords

  • Octonionic module
  • Associative element
  • Conjugate associative elements
  • \(C\ell _{7}\)-module