Extensions of Multilinear Module Expansions

TL;DR

This paper develops a theoretical framework for understanding extensions in algebraic varieties expanded by multilinear operators, connecting derivations, cohomology, and algebraic structures.

Contribution

It introduces a new parametrization of extensions via abelian ideals and establishes a Well's type theorem linking derivations to Lie algebra extensions.

Findings

Characterization of ideal-preserving derivations as Lie algebra extensions

Establishment of a Hochschild-Serre exact sequence for these extensions

Connection between cohomological derivations and algebraic extensions

Abstract

We consider the deconstruction/reconstruction of extensions in varieties of algebras which are modules expanded by multilinear operators. The parametrization of extensions determined by abelian ideals with unary actions agrees with the previous development of extensions realizing affine datum in arbitrary varieties of universal algebras. We establish a Well's type theorem which, for a fixed affine ideal, characterizes those ideal-preserving derivations of a group-trivial extension as a Lie algebra extension of the compatible pairs of derivations of the datum algebras associated to the ideal by the cohomological derivations of the datum. For these varieties, we establish a low-dimensional Hochschild-Serre exact sequence associated to an arbitrary extension equipped with an additional affine action.