On set-theoretic solutions of pentagon equation and positive basis Hopf algebras

TL;DR

This paper explores the relationship between set-theoretic solutions of the pentagon equation and Hopf algebras, generalizing classifications and characterizing bases that yield solutions.

Contribution

It establishes a link between finite solutions and positive basis Hopf algebras, extends classification to arbitrary characteristic zero fields, and classifies all (co)commutative solutions.

Findings

Finite solutions correspond to Hopf algebras with positive bases.

Classification of solutions over arbitrary characteristic zero fields.

Description of bases of group algebras yielding set-theoretic solutions.

Abstract

We investigate the connection between bijective, not necessarily finite, set-theoretic solutions of the pentagon equation and Hopf algebras. Firstly, we prove that finite solutions correspond to Hopf algebras with the positive basis property. As a corollary we generalise Lu-Yan-Zhu classification to arbitrary characteristic $0$ fields $k$. Secondly, we study the general problem of when a Hopf algebra has a basis yielding a set-theoretic solution. Finally, we classify all (co)commutative bijective solutions. This result requires to obtain a description of all bases of a group algebra $k[G]$ yielding a set-theoretic solution. We namely show that such bases correspond, through a Fourier transform, to splittings $A \rtimes N$ of $G$ with $A$ a finite abelian group.