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

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 $\mathrm{Aut}(G)$, 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 $\mathrm{Inn}(G)$ of $G$. In this paper, we give a characterisation of the gamma functions on a group $G$, with values in $\mathrm{Inn}(G)$, 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.

14 pages; accepted for publication, Annali di Matematica Pura e Applicata

References in corpus (4)

• Integration and geometrization of Rota-Baxter Lie algebras
• Rota-Baxter groups, skew left braces, and the Yang-Baxter equation
• From endomorphisms to bi-skew braces, regular subgroups, the Yang--Baxter equation, and Hopf--Galois structures
• Brace blocks from bilinear maps and liftings of endomorphisms

Cited by in corpus (8)

• Rota-Baxter operators on Clifford semigroups and the Yang-Baxter equation
• Brace blocks from bilinear maps and liftings of endomorphisms
• Extensions and automorphisms of Rota-Baxter groups
• Relative Rota-Baxter groups and skew left braces
• Cohomology and Extensions of Relative Rota-Baxter groups
• Rota-Baxter operators on crossed modules of Lie groups and categorical solutions of the Yang-Baxter equation
• The mutually normalizing regular subgroups of the holomorph of a cyclic group of prime power order
• Relative Rota-Baxter operators of weight 0 on groups, pre-groups, braces, the Yang-Baxter equation and $T$-structures