Deformed solutions of the Yang-Baxter equation associated to dual weak braces

Abstract

A dual weak brace is an algebraic structure $\left(S,+,\circ \right)$ including skew braces and giving rise to a set-theoretic solution of the Yang-Baxter equation. We show that such a map belongs to a family of set-theoretic solutions, called deformed solutions, that are defined on $S$ and depending on certain parameters. We prove these elements are exactly those belonging to the distributor of $S$, i.e., ${\mathcal{D}}_r(S)=\{z\in S\mid \forall a,b\in S(a+b)\circ z=a\circ z-z+b\circ z\}$, that is a full inverse subsemigroup of $\left(S,\circ \right)$. Regarding $S$ as a strong semilattice $[Y,B_{\alpha },{\varphi }_{\alpha ,\beta }]$ of skew braces $B_{\alpha }$, we analyze when ${\mathcal{D}}_r(S)=\underset{\alpha \in Y}{\dot{\bigcup }}{\mathcal{D}}_r(B_{\alpha })$ and in which cases a deformed solution is the strong semilattices of deformed solutions.

arXiv commentIn press

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