Extensions of associative and Lie algebras via Gr\"obner-Shirshov bases method

TL;DR

This paper develops a method using Gr"obner-Shirshov bases to characterize algebra extensions for associative and Lie algebras, providing a systematic approach to understand their structure when presented by generators and relations.

Contribution

It introduces a complete characterization technique for associative and Lie algebra extensions using Gr"obner-Shirshov bases, applicable to algebras given by generators and relations.

Findings

Provides explicit criteria for algebra extensions

Utilizes Gr"obner-Shirshov bases for systematic analysis

Applicable to algebras presented by generators and relations

Abstract

Let $\mathfrak{a},\mathfrak{b},\mathfrak{e}$ be algebras over a field $k$. Then $\mathfrak{e}$ is an extension of $\mathfrak{a}$ by $\mathfrak{b}$ if $\mathfrak{a}$ is an ideal of $\mathfrak{e}$ and $\mathfrak{b}$ is isomorphic to the quotient algebra $\mathfrak{e}/\mathfrak{a}$. In this paper, by using Gr\"obner-Shirshov bases theory for associative (resp. Lie) algebras, we give complete characterizations of associative (resp. Lie) algebra extensions of $\mathfrak{a}$ by $\mathfrak{b}$, where $\mathfrak{b}$ is presented by generators and relations.