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

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.