On the extraordinary construction of cycle sets by Wolfgang Rump

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.