A note on set theoretical solutions of the Yang-Baxter equation with trivial retraction

TL;DR

This paper investigates finite non-degenerate set-theoretical solutions to the Yang-Baxter equation, revealing their linearization properties and characterizing solutions that retract to a flip solution via their associated Lie algebra.

Contribution

It generalizes previous results by linking retractions to flip solutions with the abelian property of the solution's Lie algebra, introducing a new invariant for analysis.

Findings

Solutions linearize to twists of the flip by roots of unity.

A solution retracts to a flip if and only if its Lie algebra is abelian.

Introduces the Lie algebra as a new invariant for solutions.

Abstract

We show that every finite non-degenerate set theoretical solution to the YBE whose retraction is a flip linearizes to a twist of the flip solution by roots of unity. This generalizes a result of Gateva-Ivanova and Majid. To prove the result we use a new invariant associated to a solution, its Lie algebra. We show also that a solution retracts to a flip solution if and only if its Lie algebra is abelian.