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The classification of nondegenerate uniconnected cycle sets

Wolfgang Rump

Vol. 323 (2023), No. 1, 205–221
DOI: 10.2140/pjm.2023.323.205
Abstract

It is known that the set-theoretic solutions to the Yang–Baxter equation studied by Etingof et al. (1999) are equivalent to a class of sets with a binary operation, called nondegenerate cycle sets. There is a covering theory for cycle sets which associates a universal covering to any indecomposable cycle set. The cycle sets arising as universal covers are said to be uniconnected. In this paper, the category of nondegenerate uniconnected cycle sets is determined, and it is proved that up to isomorphism, a nondegenerate uniconnected cycle set is given by a brace A with a transitive cycle base (an adjoint orbit which generates the additive group of A). The theorem is applied to braces with cyclic additive or adjoint group, where a more explicit classification is obtained.

Dedicated to B. V. M.

Keywords
braces, coverings, cycle sets, Yang–Baxter equation
Mathematical Subject Classification
Primary: 05E18, 16T25, 81R50
Milestones
Received: 23 March 2022
Revised: 8 October 2022
Accepted: 9 October 2022
Published: 29 May 2023
Authors
Wolfgang Rump
Institute for Algebra and Number Theory
University of Stuttgart
Stuttgart
Germany

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