On loop extensions and cohomology of loops

TL;DR

This paper develops cohomology-like groups to classify loop extensions satisfying various identities, generalizing previous work, and computes the number of certain metacyclic extensions in specific cases.

Contribution

It introduces a broad framework for cohomology of loops that encompasses multiple identities and extends prior theories, including explicit calculations for metacyclic extensions.

Findings

Defined cohomology-like groups for loop extensions with various identities

Generalized previous cohomology theories for loops

Computed the number of metacyclic extensions in specific cases

Abstract

In this paper are defined cohomology-like groups that classify loop extensions satisfying a given identity in three variables for association identities, and in two variables for the case of commutativity. It is considered a large amount of identities. This groups generalize those defined in works of Nishigori [2] and of Jhonson and Leedham-Green [4]. It is computed the number of metacyclic extensions for trivial action of the quotient on the kernel in one particular case for left Bol loops and in general for commutative loops.