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

Ilaria Colazzo, Geoffrey Janssens

TL;DR

The paper investigates bijective set-theoretic solutions of the pentagon equation in relation to Hopf algebras, showing finite solutions correspond precisely to Hopf algebras with a positive basis property and extends the Lu–Yan–Zhu classification to characteristic $0$ fields. It develops the Davydov–Militaru coefficient Hopf algebras framework, connects set-theoretic and vector-space/algebraic PE solutions, and analyzes infinite-dimensional, multiplier, and cocommutative cases, including a full classification of cocommutative (co)set-theoretic solutions via bases of group algebras. A key achievement is the complete description of Φ-set theoretic bases for group algebras, proving that any such basis yields a solution that encodes a split $G o A times N$ with $A$ finite abelian and $A^ ewvar i$ characters, thereby producing explicit RPE solutions from the dual group structure. The results illuminate when Hopf-algebra structures canonically produce set-theoretic PE solutions and highlight obstructions (e.g., restricted to domains like $U( rak g)$) that prevent non-group-like set-theoretic bases. Overall, the work deepens the bridge between combinatorial PE solutions and Hopf-algebra theory, with implications for constructing and classifying RPE and related Yang–Baxter-type 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.

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

TL;DR

The paper investigates bijective set-theoretic solutions of the pentagon equation in relation to Hopf algebras, showing finite solutions correspond precisely to Hopf algebras with a positive basis property and extends the Lu–Yan–Zhu classification to characteristic fields. It develops the Davydov–Militaru coefficient Hopf algebras framework, connects set-theoretic and vector-space/algebraic PE solutions, and analyzes infinite-dimensional, multiplier, and cocommutative cases, including a full classification of cocommutative (co)set-theoretic solutions via bases of group algebras. A key achievement is the complete description of Φ-set theoretic bases for group algebras, proving that any such basis yields a solution that encodes a split with finite abelian and characters, thereby producing explicit RPE solutions from the dual group structure. The results illuminate when Hopf-algebra structures canonically produce set-theoretic PE solutions and highlight obstructions (e.g., restricted to domains like ) that prevent non-group-like set-theoretic bases. Overall, the work deepens the bridge between combinatorial PE solutions and Hopf-algebra theory, with implications for constructing and classifying RPE and related Yang–Baxter-type 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 fields . 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 yielding a set-theoretic solution. We namely show that such bases correspond, through a Fourier transform, to splittings of with a finite abelian group.
Paper Structure (46 sections, 56 theorems, 216 equations)

This paper contains 46 sections, 56 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Connection set-theoretic solutions and positive basis property
  4. Infinite dimensional Hopf algebras yielding set-theoretic solutions
  5. Outline
  6. Overview and connection on the different types of Pentagon Solutions
  7. Background on pentagon equation
  8. Connections between set-theoretic and vector space solutions
  9. Via linearisation.
  10. Via the space of functions
  11. Connection between the two constructions
  12. Relation between vector space and algebra solutions
  13. Hopf algebras from set-theoretic RPE solutions
  14. Combinatorial RPE solutions arising from Hopf algebras
  15. Background on multiplier Hopf algebra
  16. ...and 31 more sections

Key Result

Theorem A

Let $(S,s)$ be a finite bijective solution to RPE. Then $H_{\ell}(s)$ and $H_{r}(s)$ have a basis which is both positive and $\Phi$-set theoretic.

Theorems & Definitions (127)

  • Theorem A: \ref{['pos basis prop theorem']}
  • Corollary B
  • Remark
  • Theorem C: \ref{['classif cocomm sol']}
  • Proposition D: \ref{['thm:dual']}
  • Theorem E: \ref{['Classification theorem basis grp alg']}
  • Theorem F: \ref{['domain no set solution']} & \ref{['no set for Lie']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 117 more