A differential graded Lie algebra approach to non abelian extensions of associative algebras

TL;DR

This paper introduces a differential graded Lie algebra framework to classify non-abelian extensions of associative algebras, linking Maurer-Cartan elements with non-abelian cohomology.

Contribution

It establishes a novel approach connecting non-abelian extensions of associative algebras with Maurer-Cartan elements in a differential graded Lie algebra.

Findings

Maurer-Cartan elements correspond to non-abelian extensions

The Deligne groupoid is in 1-1 correspondence with non-abelian cohomology

Provides a new algebraic framework for classifying extensions

Abstract

In this paper we show that non abelian extensions of an associative algebra $\mathcal{B}$ by an associative algebra $\mathcal{A}$ can be viewed as Maurer-Cartan elements of a suitable differential graded Lie algebra $L$. In particular we show that $\mathcal{MC}(L)$, the Deligne groupoid of $L$, is in 1-1 correspondence with the non-abelian cohomology $H^2_{nab}(\mathcal{B},\mathcal{A})$.