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Ternary Z2 and Z3 Graded Algebras and Generalized Color Dynamics

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Mathematical Structures and Applications

Abstract

We discuss cubic and ternary algebras which are a direct generalization of Grassmann and Clifford algebras, but with Z 3-grading replacing the usual Z 2-grading. Combining Z 2 and Z 3 gradings results in algebras with Z 6 grading, which are also investigated. We introduce a universal constitutive equation combining binary and ternary cases.

Elementary properties and structures of such algebras are discussed, with special interest in low-dimensional ones, with two or three generators.

Invariant anti-symmetric quadratic and cubic forms on such algebras are introduced, and it is shown how the SL(2, C) group arises naturally in the case of lowest dimension, with two generators only, as the symmetry group preserving these forms.

In the case of lowest dimension, with two generators only, it is shown how the cubic combinations of Z 3-graded elements behave like Lorentz spinors, and the binary product of elements of this algebra with an element of the conjugate algebra behave like Lorentz vectors.

Then Pauli’s principle is generalized for the case of the Z 3 graded ternary algebras leading to cubic commutation relations. A generalized Dirac equation is introduced.

The model displays the color SU(3) symmetry of strong interactions, as well as the SU(2) and U(1) symmetries giving rise to the Standard Model gauge fields.

Dedicated to Norbert Hounkonnou for his 60-th birthday

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Acknowledgements

I am greatly indebted to Michel Dubois-Violette, Viktor Abramov, and Karol Penson for many discussions and constructive criticism. I would like to express my sincere thanks to Jan-Willem van Holten, Yuri Dokshitser, Paul Sorba, and Jürg Frölich for important suggestions and remarks.

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Authors and Affiliations

  1. Laboratoire de Physique Théorique de la Matière Condensée, Sorbonne-Universités, CNRS UMR 7600, Paris, France

    Richard Kerner

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  1. Richard Kerner
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Correspondence to Richard Kerner .

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Editors and Affiliations

  1. Mathematical Sciences Department, The University of Alabama in Huntsville, Hunstville, AL, USA

    Toka Diagana

  2. Department of Mathematics, Howard University, Washington, DC, USA

    Bourama Toni

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Kerner, R. (2018). Ternary Z2 and Z3 Graded Algebras and Generalized Color Dynamics. In: Diagana, T., Toni, B. (eds) Mathematical Structures and Applications. STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health. Springer, Cham. https://doi.org/10.1007/978-3-319-97175-9_14

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