Cohomology and extensions of braces

TL;DR

This paper develops new cohomology theories for braces and linear cycle sets, linking algebraic structures to solutions of the Yang-Baxter equation and classifying extensions via second cohomology groups.

Contribution

It introduces two novel (co)homology theories for braces and cycle sets, combining existing theories and classifying extensions through second cohomology.

Findings

Two versions of (co)homology theories are introduced.

Extensions of braces are classified by second cohomology groups.

Theories connect algebraic structures to Yang-Baxter solutions.

Abstract

Braces and linear cycle sets are algebraic structures playing a major role in the classification of involutive set-theoretic solutions to the Yang-Baxter equation. This paper introduces two versions of their (co)homology theories. These theories mix the Harrison (co)homology for the abelian group structure and the (co)homology theory for general cycle sets, developed earlier by the authors. Different classes of brace extensions are completely classified in terms of second cohomology groups.