Extensions of a family of Linear Cycle Sets

TL;DR

This paper studies the cohomology of linear cycle sets to classify their extensions by abelian groups, providing explicit formulas and conditions for their structure, with applications to different abelian groups.

Contribution

It introduces explicit formulas for the second cohomology group of linear cycle sets and characterizes extensions under specific conditions, advancing understanding of their algebraic structure.

Findings

Derived explicit formulas for second cohomology groups.

Characterized extensions when the abelian group is in the socle and the set is trivial.

Constructed examples of trivial and non-trivial extensions.

Abstract

This paper explores the cohomology of linear cycle sets, focusing on extensions of a specific linear cycle set H by an abelian group I. We derive explicit formulas for the second cohomology group, which classifies these extensions, and establish conditions under which the extensions are fully determined. Key results include a characterization of extensions when I lies in the socle of the extended structure and H is trivial, and the construction of explicit examples for both trivial and non-trivial cases. The paper provides a systematic approach to understanding the structure of these extensions, with applications to various families of abelian groups.