Set-theoretical reflection equation: Classification of reflection maps

The set-theoretical reflection equation and its solutions, the reflection maps, recently introduced by two of the authors, is presented in general and then applied in the context of quadrirational Yang-Baxter maps. We provide a method for constructing reflection maps and we obtain a classification of solutions associated to all the families of quadrirational Yang-Baxter maps that have been classified recently.

💡 Research Summary

The paper introduces the set‑theoretical reflection equation, a natural extension of the set‑theoretical Yang‑Baxter equation that incorporates a boundary (or “reflection”) map. Given a set X, a Yang‑Baxter map R : X×X → X×X and a reflection map B : X → X, the reflection equation reads
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