Solutions of the Yang–Baxter equation associated to skew left braces, with applications to racks

By D. Bachiller. Journal of knot theory and its ramifications, 2016

61 citations recorded by the source corpus.

Given a skew left brace [Formula: see text], a method is given to construct all the non-degenerate set-theoretic solutions [Formula: see text] of the Yang–Baxter equation such that the associated permutation group [Formula: see text] is isomorphic, as a skew left brace, to [Formula: see text]. This method depends entirely on the brace structure of [Formula: see text]. We then adapt this method to show how to construct solutions with additional properties, like square-free, involutive or irretractable solutions. Using this result, it is even possible to recover racks from their permutation group.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/66fffce29f1b17da0f493005811ad8b2f91dea67

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Citations in the corpus

Bi-skew braces and Hopf Galois structures (2019, unknown): “Skew braces have been studied in [4] and in [19].”
Skew left braces of nilpotent type (2018, methodology): “Following [28] we also define the sequence A ⊇ A([2]) ⊇ A([3]) ⊇ · · · , where A([1]) = A and A is the additive subgroup of A generated by all the elements from the sets A ∗A[n+1−i] for 1 i n.”