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arXiv · 2106.05149

On the extraordinary construction of cycle sets by Wolfgang Rump

Pravin Bhandari·Miguel Córdoba·Jamie Henderson·Scott Warrander
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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.

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Pravin Bhandari, Miguel Córdoba, Jamie Henderson, Scott Warrander. 2021-06-09. On the extraordinary construction of cycle sets by Wolfgang Rump. https://arxiv.org/abs/2106.05149

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