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
with
a transitive cycle base (an adjoint orbit which generates the additive group of
). The
theorem is applied to braces with cyclic additive or adjoint group, where a more
explicit classification is obtained.
|
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
|
| © 2023 MSP (Mathematical Sciences
Publishers). Distributed under the Creative Commons
Attribution License 4.0 (CC BY). |
Open Access made possible by participating
institutions via Subscribe to Open. 
|
