On quadrirational pentagon maps

TL;DR

This paper classifies rational quadrirational solutions to the set-theoretical pentagon equation, introduces entwining solutions with partial inverses, and explores how to derive Yang-Baxter maps from these pentagon solutions.

Contribution

It provides a complete classification of rational quadrirational pentagon maps and introduces methods to generate Yang-Baxter maps from them.

Findings

Classified all quadrirational solutions to the pentagon equation.

Established a link between pentagon maps with partial inverses and entwining solutions.

Demonstrated how to derive Yang-Baxter maps from pentagon solutions.

Abstract

We classify rational solutions of a specific type of the set theoretical version of the pentagon equation. That is, we find all quadrirational maps $R:(x,y)\mapsto (u(x,y),v(x,y)),$ where $u, v$ are two rational functions on two arguments, that serve as solutions of the pentagon equation. Furthermore, provided a pentagon map that admits a partial inverse, we obtain genuine entwining pentagon set theoretical solutions. Finally, we show how to obtain Yang-Baxter maps from entwining pentagon maps.