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arXiv · 2003.01983

Primitive set-theoretic solutions of the Yang-Baxter equation

F. Cedo·E. Jespers·J. Okninski
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Abstract

To every involutive non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation on a finite set $X$ there is a naturally associated finite solvable permutation group ${\mathcal G}(X,r)$ acting on $X$. We prove that every primitive permutation group of this type is of prime order $p$. Moreover, $(X,r)$ is then a so called permutation solution determined by a cycle of length $p$. This solves a problem recently asked by A. Ballester-Bolinches. The result opens a new perspective on a possible approach to the classification problem of all involutive non-degenerate set-theoretic solutions.

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F. Cedo, E. Jespers, J. Okninski. 2020-03-04. Primitive set-theoretic solutions of the Yang-Baxter equation. https://arxiv.org/abs/2003.01983

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