On the cabling of non-involutive set-theoretic solutions of the Yang--Baxter equation
Ilaria Colazzo, Arne Van Antwerpen
TL;DR
This work generalizes the cabling method for set-theoretic Yang–Baxter solutions to the bijective non-degenerate case by leveraging the Yang–Baxter monoid $M(X,r)$ instead of the group $G(X,r)$. It establishes that cabling is functorial on biquandles, that the diagonal map transforms as $q\mapsto q^k$, and that decomposability is preserved under injectivization and passage to the associated biquandle, enabling robust criteria for (in)decomposability. A key application shows that square-free solutions with nilpotent derived monoid are decomposable, highlighting the method's power in deriving structure results from monoid and biquandle data. Overall, the paper provides new, practical tools for analyzing indecomposability and simplicity in bijective non-degenerate YBE solutions, with clearer pathways to classify and construct such solutions. The results deepen the connections between monoidic/categorical constructions and combinatorial invariants in the Yang–Baxter setting.
Abstract
We extend the cabling method by Lebed, Ramírez and Vendramin from involutive to bijective non-degenerate set-theoretic solutions of the Yang--Baxter equation by working in the Yang--Baxter monoid $M(X,r)$ rather than the group $G(X,r)$. This shift in approach overcomes the obstruction that, for non-involutive solutions, the canonical map from $X$ to the Yang--Baxter group $G(X,r)$ need not be injective and yields a well-defined cabling. We prove that cabling is functorial on biquandles and that the diagonal map transforms as $q\mapsto q^k$. We also show that decomposability is preserved by injectivization and by passing to the associated biquandle, allowing us to work within that class without loss of generality. This leads to criteria for (in)decomposability. As an application, we obtain that square-free solutions with nilpotent derived monoid are decomposable.