Classification of Left Octonionic Modules


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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  1. 1.

    Atiyah, M.F., Bott, R., Shapiro, A.: Clifford modules. Topology 3((suppl, suppl. 1)), 3–38 (1964)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Baez, J.C.: The octonions. Bull. Am. Math. Soc. (N.S.) 39(2), 145–205 (2002)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Cartan, H., Eilenberg, S.: Homological Algebra. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1999). With an appendix by David A. Buchsbaum, Reprint of the 1956 original

  4. 4.

    Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative Functional Calculus, Progress in Mathematics, vol. 289. Birkhäuser/Springer Basel AG, Basel (2011). Theory and applications of slice hyperholomorphic functions

  5. 5.

    Eilenberg, S.: Extensions of general algebras. Ann. Soc. Polon. Math. 21, 125–134 (1948)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Ghiloni, R., Moretti, V., Perotti, A.: Continuous slice functional calculus in quaternionic Hilbert spaces. Rev. Math. Phys. 25(4), 1350006 (2013)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Gilbert, J.E., Murray, M.A.M.: Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge Studies in Advanced Mathematics, vol. 26. Cambridge University Press, Cambridge (1991)

  8. 8.

    Goldstine, H.H., Horwitz, L.P.: Hilbert space with non-associative scalars. I. Math. Ann. 154, 1–27 (1964)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Harvey, F.R.: Spinors and Calibrations, Perspectives in Mathematics, vol. 9. Academic Press, Boston (1990)

  10. 10.

    Horwitz, L.P., Razon, A.: Tensor product of quaternion Hilbert modules. In: Classical and Quantum Systems (Goslar, 1991), pp. 266–268. World Sci. Publ., River Edge (1993)

  11. 11.

    Jacobson, N.: Structure of alternative and Jordan bimodules. Osaka Math. J. 6, 1–71 (1954)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Ludkovsky, S.V.: Algebras of operators in Banach spaces over the quaternion skew field and the octonion algebra. Sovrem. Mat. Prilozh. 35, 98–162 (2005)

    Google Scholar 

  13. 13.

    Ludkovsky, S.V., Sprössig, W.: Spectral representations of operators in Hilbert spaces over quaternions and octonions. Complex Var. Elliptic Equ. 57(12), 1301–1324 (2012)

    MathSciNet  Article  Google Scholar 

  14. 14.

    McIntosh, A.: Book Review: Clifford algebra and spinor-valued functions, a function theory for the Dirac operator. Bull. Am. Math. Soc. (N.S.) 32(3), 344–348 (1995)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Ng, C.-K.: On quaternionic functional analysis. Math. Proc. Camb. Philos. Soc. 143(2), 391–406 (2007)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Razon, A., Horwitz, L.P.: Uniqueness of the scalar product in the tensor product of quaternion Hilbert modules. J. Math. Phys. 33(9), 3098–3104 (1992)

    ADS  MathSciNet  Article  Google Scholar 

  17. 17.

    Razon, A., Horwitz, L.P.: Projection operators and states in the tensor product of quaternion Hilbert modules. Acta Appl. Math. 24(2), 179–194 (1991)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Schafer, R.D.: An Introduction to Nonassociative Algebras. Dover Publications, New York (1995). Corrected reprint of the (1966) original

  19. 19.

    Soffer, A., Horwitz, L.P.: \(B^{\ast } \)-algebra representations in a quaternionic Hilbert module. J. Math. Phys. 24(12), 2780–2782 (1983)

    ADS  MathSciNet  Article  Google Scholar 

  20. 20.

    Viswanath, K.: Normal operations on quaternionic Hilbert spaces. Trans. Am. Math. Soc. 162, 337–350 (1971)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Wang, H., Ren, G.: Octonion analysis of several variables. Commun. Math. Stat. 2(2), 163–185 (2014)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Zhevlakov, K.A., Slinko, A.M., Shestakov, I.P., Shirshov, A.I.: Rings that are Nearly Associative, Pure and Applied Mathematics, vol. 104. Academic Press [Harcourt Brace Jovanovich, Publishers], New York (1982). Translated from the Russian by Harry F. Smith

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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).

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Mathematics Subject Classification

  • 17A05


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