A family of irretractable square-free solutions of the Yang–Baxter equation
-
David Bachiller
,
Ferran Cedó
Abstract
A new family of non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation is constructed. All these solutions are strong twisted unions of multipermutation solutions of multipermutation level at most two. A large subfamily consists of irretractable and square-free solutions. This subfamily includes a recent example of Vendramin [38, Example 3.9], who first gave a counterexample to Gateva-Ivanova’s Strong Conjecture [19, Strong Conjecture 2.28 (I)]. All the solutions in this subfamily are new counterexamples to Gateva-Ivanova’s Strong Conjecture and also they answer a question of Cameron and Gateva-Ivanova [21, Open Questions 6.13 (II)(4)]. It is proved that the natural left brace structure on the permutation group of the solutions in this family has trivial socle. Properties of the permutation group and of the structure group associated to these solutions are also investigated. In particular, it is proved that the structure groups of finite solutions in this subfamily are not poly-(infinite cyclic) groups.
Funding source: Ministerio de Economía y Competitividad
Award Identifier / Grant number: MTM2011-28992-C02-01
Funding source: Ministerio de Economía y Competitividad
Award Identifier / Grant number: MTM2014-53644-P
Funding source: Narodowe Centrum Nauki
Award Identifier / Grant number: 2013/09/B/ST1/04408
Funding statement: The two first-named authors were partially supported by the grants DGI MICIIN MTM2011-28992-C02-01, and MINECO MTM2014-53644-P. The third author is supported in part by Onderzoeksraad of Vrije Universiteit Brussel and Fonds voor Wetenschappelijk Onderzoek (Belgium). The fourth author is supported by the National Science Centre grant 2013/09/B/ST1/04408 (Poland).
Acknowledgements
The authors thank the referee for his/her comments and suggestions, especially for the information that appears in Remark 3.2.
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Articles in the same Issue
- Frontmatter
- The role of the Rogers–Shephard inequality in the characterization of the difference body
- Test vectors for local periods
- The local Langlands conjecture for p-adic GSpin4, GSpin6, and their inner forms
- A family of irretractable square-free solutions of the Yang–Baxter equation
- On Sylow permutable subgroups of finite groups
- Using Steinberg algebras to study decomposability of Leavitt path algebras
- The ℓ∞-semi-norm on uniformly finite homology
- Minimal potential results for Schrödinger equations with Neumann boundary conditions
- On a unique continuation for Aharonov–Bohm magnetic Schrödinger equation
- Sup-norm bounds for Eisenstein series
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Permutohedral structures on
- On the complete integrability of the periodic quantum Toda lattice
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Fans, decision problems and generators of free abelian
- Standard cocycles: Variations on themes of C. Kassel’s and R. Wilson’s
- Crossed modules as maps between connected components of topological groups
- A local converse theorem for U(2,2)
- Probabilistic trace and Poisson summation formulae on locally compact abelian groups
