On the extraordinary construction of cycle sets by Wolfgang Rump

TL;DR

This paper introduces cycle sets, explores their relation to braces and the Yang-Baxter equation, and provides detailed proofs and new constructions to deepen understanding of their algebraic properties.

Contribution

It offers a comprehensive, accessible introduction to cycle sets, connecting them with brace theory and presenting novel constructions beyond braces.

Findings

Cycle sets are fundamental in studying set-theoretic solutions to the Yang-Baxter equation.

The paper provides detailed proofs of key results on cycle sets.

New constructions of cycle sets that are not directly related to braces are discussed.

Abstract

Cycle sets are algebraic structures introduced by Rump to study set theoretic solutions to the Yang-Baxter equation. While studying cycle sets Rump also introduced braces, which have since overtaken cycle sets as a tool for studying solutions. This survey paper is primarily an introduction to cycle sets, motivating their study and relating them to key results of brace theory and Yang-Baxter theory. It is aimed at anyone from those already very familiar with braces but less familiar with cycle sets, to those with only a basic level of background in ring theory and group theory. We introduce cycle sets following Rump's original results - giving more detailed, easy to follow versions of his proofs - and then relate them back to left braces. We also go on to discuss interesting constructions of cycle sets which do not necessarily correspond directly to braces.