Extension realizing affine datum: low-dimensional cohomology

TL;DR

This paper develops a cohomology-based theory for classifying algebraic extensions with affine data across various algebraic varieties, generalizing classical cases like groups with multiple operators.

Contribution

It introduces a unified cohomology framework for extensions in universal algebra varieties, especially those with weak-difference and difference terms, generalizing classical algebraic extension theories.

Findings

Extensions with affine data are characterized by first and second cohomology groups.

In varieties with a weak-difference term, such extensions have abelian kernels.

Central extensions are characterized by properties of their actions in varieties with a difference term.

Abstract

For arbitrary varieties of universal algebras, we develop the theory around the first and second-cohomology groups characterizing extensions realizing affine datum. Restricted to varieties with a weak-difference term, extensions realizing affine datum are exactly extensions with abelian kernels. This recovers many classic examples of extensions with abelian coefficients since varieties with a weak-difference term give a far-reaching generalization of algebras like groups with multiple operators; indeed, any variety of algebras whose congruences form modular lattices. We introduce a notion of action and its model relation with a set of equations. In varieties with a difference term, central extensions are characterized by a property of their actions. Restricting further to a subclass of varieties with a difference term which still includes groups with multiple operators, we recover a special case of the representation of extensions with abelian kernels.