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A classification of polynomial functions satisfying the Jacobi identity over integral domains

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Abstract

The Jacobi identity is one of the properties that are used to define the concept of Lie algebra and in this context is closely related to associativity. In this paper we provide a complete description of all bivariate polynomials that satisfy the Jacobi identity over infinite integral domains. Although this description depends on the characteristic of the domain, it turns out that all these polynomials are of degree at most one in each indeterminate.

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Acknowledgements

This research is partly supported by the Internal Research Project R-AGR-0500 of the University of Luxembourg. The authors thank Michel Rigo of the University of Liège for pointing out Lucas’ theorem. They also thank Jörg Tomaschek of Deloitte Austria for bringing this problem to their attention.

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Authors and Affiliations

  1. Mathematics Research Unit, University of Luxembourg, Maison du Nombre, 6, Avenue de la Fonte, 4364, Esch-sur-Alzette, Luxembourg

    Jean-Luc Marichal

  2. Department of Mathematics, University of Liège, Allée de la Découverte 12 - B37, 4000, Liège, Belgium

    Pierre Mathonet

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  1. Jean-Luc Marichal
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  2. Pierre Mathonet
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Correspondence to Jean-Luc Marichal.

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Marichal, JL., Mathonet, P. A classification of polynomial functions satisfying the Jacobi identity over integral domains. Aequat. Math. 91, 601–618 (2017). https://doi.org/10.1007/s00010-017-0477-8

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  • DOI: https://doi.org/10.1007/s00010-017-0477-8

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