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Classification of Non-Degenerate Involutive Set-Theoretic Solutions to the Yang-Baxter Equation with Multipermutation Level Two

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Abstract

Set-theoretic solutions to the Yang-Baxter equation of multipermutation level two are classified via transvection orbits which are abelian torsors. The relationship to square-free solutions and rack solutions is determined by using the square map and its relationship to non-degeneracy of the corresponding cycle set. The crucial role of left and right ideal powers of braces is discussed in connection with applications to higher multipermutation level. As an illustration, we give a simple proof of a recent theorem of Smoktunowicz on braces with nilpotent adjoint group, and a classification of braces with multipermutation level two, removing the finiteness in Smoktunowicz’ theorem for such braces.

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Acknowledgements

The author is grateful to an anonymous referee for a thorough inspection of the manuscript and thoughtful remarks, which helped to improve the final version of the paper.

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  1. Institute for Algebra and Number Theory, University of Stuttgart, Pfaffenwaldring 57, D-70550, Stuttgart, Germany

    Wolfgang Rump

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  1. Wolfgang Rump
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Correspondence to Wolfgang Rump.

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Rump, W. Classification of Non-Degenerate Involutive Set-Theoretic Solutions to the Yang-Baxter Equation with Multipermutation Level Two. Algebr Represent Theor 25, 1293–1307 (2022). https://doi.org/10.1007/s10468-021-10067-5

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  • DOI: https://doi.org/10.1007/s10468-021-10067-5

Keywords

Mathematics Subject Classification (2010)

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