Skew braces from Rota--Baxter operators: a cohomological characterisation and some examples

TL;DR

This paper characterizes gamma functions associated with skew braces derived from Rota-Baxter operators on groups using cohomology, and provides examples of skew braces not obtainable from such operators.

Contribution

It offers a cohomological criterion for gamma functions to originate from Rota-Baxter operators, and demonstrates how to reconstruct operators from these functions when possible.

Findings

Cohomological condition characterizes gamma functions from Rota-Baxter operators.

Examples of skew braces with gamma functions not arising from Rota-Baxter operators.

Method to recover Rota-Baxter operators from gamma functions when the cohomological condition is satisfied.

Abstract

Rota-Baxter operators for groups were recently introduced by L. Guo, H. Lang, and Y. Sheng. V. G. Bardakov and V. Gubarev showed that with each Rota-Baxter operator one can associate a skew brace. Skew braces on a group $G$ can be characterised in terms of certain gamma functions from $G$ to its automorphism group $\operatorname{Aut}(G)$, that are defined by a functional equation. For the skew braces obtained from a Rota-Baxter operator the corresponding gamma functions take values in the inner automorphism group $\operatorname{Inn}(G)$ of $G$. In this paper, we give a characterisation of the gamma functions on a group $G$, with values in $\operatorname{Inn}(G)$, that come from a Rota-Baxter operator, in terms of the vanishing of a certain element in a suitable second cohomology group. Exploiting this characterisation, we are able to exhibit examples of skew braces whose corresponding gamma functions take values in the inner automorphism group, but cannot be obtained from a Rota--Baxter operator. For gamma functions that can be obtained from a Rota-Baxter operators, we show how to get the latter from the former, exploiting the knowledge that a suitable central group extension splits.