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

Abstract

A recent method for acquiring new solutions of the Yang–Baxter equation involves deforming the classical solution associated with a skew brace. In this work, we demonstrate the applicability of this method to a dual weak brace \(\left( S,+,\circ \right) \) and prove that all elements generating deformed solutions belong precisely to the set \(\mathcal {D}_r(S)=\{z \in S \mid \forall a,b \in S \, \, (a+b) \circ z = a\circ z-z+b \circ z\}\), which we term the distributor of S. We show it is a full inverse subsemigroup of \(\left( S, \circ \right) \) and prove it is an ideal for certain classes of braces. Additionally, we express the distributor of a brace S in terms of the associativity of the operation \(\cdot \), with \(\circ \) representing the circle or adjoint operation. In this context, \((\mathcal {D}_r(S),+,\cdot )\) constitutes a Jacobson radical ring contained within S. Furthermore, we explore parameters leading to non-equivalent solutions, emphasizing that even deformed solutions by idempotents may not be equivalent. Lastly, considering S as a strong semilattice \([Y, B_\alpha , \phi _{\alpha ,\beta }]\) of skew braces \(B_\alpha \), we establish that a deformed solution forms a semilattice of solutions on each skew brace \(B_\alpha \) if and only if the semilattice Y is bounded by an element 1 and the deforming element z lies in \(B_1\).