Groebner-Shirshov bases for extensions of algebras

TL;DR

This paper uses Groebner-Shirshov bases to fully characterize and provide an algorithm for identifying algebra extensions where a nilpotent ideal is extended by another algebra.

Contribution

It introduces a complete characterization and an explicit algorithm for algebra extensions using Groebner-Shirshov bases.

Findings

Complete characterization of algebra extensions using Groebner-Shirshov bases

Algorithm for determining when an algebra is an extension of a given ideal and quotient

Conditions for algebra extensions derived explicitly

Abstract

An algebra $\cal{R}$ is called an extension of the algebra $M$ by $B$ if $M^2=0$, $M$ is an ideal of $\cal{R}$ and $\cal{R}$$/M\cong B$ as algebras. In this paper, by using the Gr\"{o}bner-Shirshov bases, we characterize completely the extensions of $M$ by $B$. An algorithm to find the conditions of an algebra $A$ to be an extension of $M$ by $B$ is obtained.