Extensions of Multilinear Module Expansions
Alexander Wires
TL;DR
The work develops a cohomological framework for extensions in varieties of $R$-modules expanded by multilinear operations, unifying nonabelian and affine extensions through 2-cocycles and compatible actions. It proves a Wells-type theorem relating kernel-preserving derivations of a group-trivial extension realizing affine datum to a Lie algebra extension of derivations, and constructs a low-dimensional Hochschild-Serre sequence connecting $H^1$ and $H^2$ under a square-compatibility condition. Extensions realizing a fixed affine datum $(Q,I,\ast)$ are classified by the abelian group $H^2_{\mathcal{V}}(Q,I,\ast)$, with a Galois connection linking subvarieties to cohomology classes, and the theory recovers classical classifications in various bilinear/multilinear settings. The results offer a structured path to Schur-type multipliers and nilpotent/central extensions in multilinear-module expansions, and suggest avenues for higher cohomology and category-theoretic reformulations.
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.