Cohomology of Linear Cycle Sets when the adjoint group is finite abelian

TL;DR

This paper studies the second cohomology group of finite abelian linear cycle sets with commutative operations, providing methods to classify extensions via explicit 2-cocycle constructions and illustrative examples.

Contribution

It introduces a systematic approach to compute and classify extensions of finite abelian linear cycle sets using cohomological techniques.

Findings

Explicit construction of 2-cocycles for finite abelian linear cycle sets

Classification of extensions based on cohomology groups

Validation through illustrative examples

Abstract

This paper analyzes the second cohomology group of a linear cycle set with coefficients in an abelian group I, for linear cycle sets with commutative adjoint operation, focusing on the finite abelian case. It aims to classify extensions of such structures through cohomological methods. Techniques are developed to systematically construct explicitly 2-cocycles. Finally, some illustrative examples are explored to validate the theoretical framework.