Extensions of Linear Cycle Sets

TL;DR

This paper extends the cohomology theory for linear cycle sets to classify more general extensions, broadening the understanding of their algebraic structure beyond central ideals.

Contribution

It generalizes the cohomology theory for linear cycle sets to classify extensions by trivial ideals, not just central ones, providing a cohomological perspective.

Findings

Developed a cohomology theory for extensions by trivial ideals.

Established the equivalence of extensions of linear cycle sets.

Connected the theory to existing work on braces and extensions.

Abstract

We generalize the cohomology theory for linear cycle sets introduced by Lebed and Vendramin. Our cohomology classifies extensions of linear cycle sets by trivial ideals, whereas the cohomology of Lebed and Vendramin only deals with central ideals $I$ (which are automatically trivial). Therefore our theory gives an analog to the theory of extensions of braces by trivial ideals constructed by Bachiller, but from a cohomological point of view. We also study the general notions of extensions of linear cycle sets and the equivalence of extensions.