Non commutative truncated polynomial extensions

TL;DR

This paper introduces non-commutative truncated polynomial extensions of an algebra A, classifies one family completely, and links obstructions in constructing upper triangular extensions to Hochschild homology.

Contribution

It defines a new concept of non-commutative truncated polynomial extensions and provides a complete classification for one family, connecting obstructions to Hochschild homology.

Findings

Complete classification of the first family of extensions.

Obstructions to constructing upper triangular extensions relate to Hochschild homology.

Introduces a new framework for non-commutative polynomial extensions.

Abstract

We introduce the notion of non commutative truncated polynomial extension of an algebra A. We study two families of these extensions. For the first one we obtain a complete classification and for the second one, which we call upper triangular, we find that the obstructions to inductively construct them, lie in the Hochschild homology of A, with coefficients in a suitable A-bimodule.