Deformed solutions of the Yang-Baxter equation associated to dual weak braces
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arXiv
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| Format: | Preprint |
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2023
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| author | Mazzotta, Marzia Rybołowicz, Bernard Stefanelli, Paola |
| author_facet | Mazzotta, Marzia Rybołowicz, Bernard Stefanelli, Paola |
| contents | 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., $\mathcal{D}_r(S)=\{z \in S \, \mid \, \forall \, a,b \in S \quad (a+b) \circ z=a\circ z-z+b \circ z\}$, that is a full inverse subsemigroup of $\left(S, \circ\right)$. Regarding $S$ as a strong semilattice $[Y, B_α, ϕ_{α,β}]$ of skew braces $B_α$, we analyze when $\mathcal{D}_r(S)=\mathop{\dot{\bigcup}}\limits_{α\in Y} \mathcal{D}_r(B_α)$ and in which cases a deformed solution is the strong semilattices of deformed solutions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_05235 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Deformed solutions of the Yang-Baxter equation associated to dual weak braces Mazzotta, Marzia Rybołowicz, Bernard Stefanelli, Paola Quantum Algebra Rings and Algebras 16T25, 81R50, 20M18, 16Y99 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., $\mathcal{D}_r(S)=\{z \in S \, \mid \, \forall \, a,b \in S \quad (a+b) \circ z=a\circ z-z+b \circ z\}$, that is a full inverse subsemigroup of $\left(S, \circ\right)$. Regarding $S$ as a strong semilattice $[Y, B_α, ϕ_{α,β}]$ of skew braces $B_α$, we analyze when $\mathcal{D}_r(S)=\mathop{\dot{\bigcup}}\limits_{α\in Y} \mathcal{D}_r(B_α)$ and in which cases a deformed solution is the strong semilattices of deformed solutions. |
| title | Deformed solutions of the Yang-Baxter equation associated to dual weak braces |
| topic | Quantum Algebra Rings and Algebras 16T25, 81R50, 20M18, 16Y99 |
| url | https://arxiv.org/abs/2304.05235 |