Ilaria Colazzo (School of Mathematics, University of Leeds) – Understanding set-theoretic solutions to the pentagon equation through matched product of groups

Motivated by the appearance of the pentagon equation in the algebraic structures underlying integrable systems, and in the spirit of this mathematical physics seminar, my aim in this talk is also to explore what kind of interaction there can be between these very algebraic constructions and questions coming from integrable systems. I will give a complete description of finite bijective set-theoretic solutions to the pentagon equation in terms of matched products of groups, based on joint work with Okninski and Van Antwerpen. The emphasis will be on the underlying group-theoretic structure: I will explain the characterisation via matched products, describe the role of irretractable solutions as building blocks, and show how they can be used to obtain all bijective solutions. Based on ongoing joint work with Janssens, I will also briefly indicate how this description, together with the link between pentagon solutions and Hopf algebras, leads to constructions of Hopf algebras with a positive basis in the sense of Lu–Yan–Zhu.