Cocyclic braces and indecomposable cocyclic solutions of the Yang-Baxter equation

TL;DR

This paper investigates indecomposable cocyclic solutions to the Yang-Baxter equation, revealing that such solutions do not correspond directly to cocyclic braces, challenging recent assumptions in the field.

Contribution

It demonstrates that indecomposable cocyclic solutions are not in one-to-one correspondence with cocyclic braces, contradicting previous findings.

Findings

No one-to-one correspondence between cocyclic solutions and cocyclic braces.

Counterexample to recent assumptions in the literature.

Clarification of the structure of cocyclic solutions.

Abstract

We study indecomposable involutive set-theoretic solutions of the Yang-Baxter equation with cyclic permutation groups (cocyclic solutions). In particular, we show that there is no one-to-one correspondence between indecomposable cocyclic solutions and cocyclic braces which contradicts recent results in \cite{Rump21}.