Left braces and skew left braces are excellent tools to study finite bijective non-degenerate set-theoretic solutions of the Yang-Baxter equation. Furthermore, any such solution (X, r) leads to a skew left brace by looking at the structure group G(X, r) associated to the solution. In the non-involutive case, however, the map X → G(X, r) is not necessarily injective. Hence, by looking at the structure group, we might lose some information on the original solution. A possible way to solve this problem is by looking at the structure monoid instead of the structure group, but the structure monoid is no longer a skew left brace.
In this talk, we study left non-degenerate set-theoretic solutions of the Yang-Baxter equation. It was shown by Cedo, Jespers and Verwimp that the structure monoid associated to a left non-degenerate solution is a left semitruss, an associative structure introduced by Brzezinski. However, the structure monoid satisfies more, which leads to the introduction of YB-semitrusses. These are left semitrusses that admit left non-degenerate set-theoretic solutions. Furthermore, any left non-degenerate solution leads to a YB-semitruss by looking at the structure monoid associated to the solution. We will recall all previously mentioned notions, and discuss why YB-semitrusses are the desirable associative structure to study left non-degenerate solutions.
This talk is mainly based on a joint paper with Ilaria Colazzo, Eric Jespers, and Arne Van Antwerpen.
5 jan 2022
Braces in Bracelet Bay: LMS Regional Meeting and Workshop