Tue, 4 November 2025, 1:25 pm MT
The Yang--Baxter equation (YBE) is a fundamental relation in mathematical physics that has its roots in the investigation of many-particle interactions in two-dimensional quantum dynamical systems, where it can be interpreted as governing integrability properties.
When a system presents a boundary that interacts with the objects in motion, the Reflection Equation (RE) - as first studied by Cherednik and Sklyanin in 1984 and 1988, respectively - is a condition that ensures the persistence of integrability.
To determine a uniform procedure that produces special solutions, in 1990 Drinfel'd suggested studying the set-theoretical YBE, thus paving the way for the introduction of new combinatorial tools in the subject.
In recent years, Caudrelier and Zhang initiated the investigation of the set-theoretic formulation of the RE, taking inspiration from the study of soliton collisions on the half-line.
The aim of this talk is to provide a self-contained overview of the set-theoretic YBE with a particular attention to the role played by self-distributive structures.
Moreover, we study the interplay between bijective non-degenerate set-theoretic solutions of the YBE and their reflections, with a focus on solutions of derived type.
Based on joint work with M. Mazzotta and P. Stefanelli.