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 $$\left( S,+,\circ \right) $$ S , + , ∘ and prove that all elements generating deformed solutions belong precisely to the set $$\mathcal {D}_r(S)=\{z \in S \mid \forall a,b \in S \, \, (a+b) \circ z = a\circ z-z+b \circ z\}$$ D r ( S ) = { z ∈ S ∣ ∀ a , b ∈ S ( a + b ) ∘ z = a ∘ z - z + b ∘ z } , which we term the distributor of S. We show it is a full inverse subsemigroup of $$\left( S, \circ \right) $$ S , ∘ 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, $$(\mathcal {D}_r(S),+,\cdot )$$ ( D r ( S ) , + , · ) 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 $$[Y, B_\alpha , \phi _{\alpha ,\beta }]$$ [ Y , B α , ϕ α , β ] of skew braces $$B_\alpha $$ B α , we establish that a deformed solution forms a semilattice of solutions on each skew brace $$B_\alpha $$ B α if and only if the semilattice Y is bounded by an element 1 and the deforming element z lies in $$B_1$$ B 1 .