Wydawnictwa / Banach Center Publications / Wszystkie tomy

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Rump (2007) traced a research line for determining set-theoretic solutions of the Yang–Baxter equation by introducing the algebraic structure of brace. Many authors have intensively studied braces with some interesting generalizations over the years, among these skew braces (Guarnieri and Vendramin, 2017). Any skew brace $(B,+,\circ )$ determines a bijective non-degenerate solution $r: B \times B \to B \times B$ given by $r(a,b)=(-a+a\circ b, (-a+a\circ b)^-\circ a \circ b),$ for all $a,b \in B$. Recently, Doikou and Rybołowicz (2024) showed that a bigger family of bijective solutions can be obtained from any skew brace $B$ by “deforming” the map $r$ above by certain parameters. Namely, given a specific $z \in B$, the map $r_z(a,b)=(-a\circ z+a\circ b \circ z, (-a\circ z+a\circ b \circ z)^-\circ a \circ b),$ for all $a,b \in B$, gives rise to a new solution. In particular, $r_0=r$, where $0$ is the identity of the skew brace $B$. Mazzotta et al. (2024) showed that the parameters that fit well are only those belonging to the set $$\mathcal {D}_r(B) = \{z \in B \, \mid \, \forall \, a,b \in B \quad (a+b) \circ z=a\circ z-z+b \circ z\}$$ called the right distributor of $B$, which is a special subgroup of $(B, \circ )$.