Cohomology theory of averaging algebras, $L_\infty$-structures and homotopy averaging algebras

TL;DR

This paper develops a cohomology theory for averaging algebras, linking it to deformations and extensions, and introduces homotopy averaging algebras via $L_ olinebreak_$-structures, advancing the understanding of their algebraic properties.

Contribution

It introduces a cohomology framework for averaging algebras, interprets cohomology groups as deformation and extension tools, and defines homotopy averaging algebras through $L_$-algebra Maurer-Cartan elements.

Findings

Established a cohomology theory for averaging algebras.

Connected cohomology groups to formal deformations and extensions.

Defined homotopy averaging algebras via $L_$-structures.

Abstract

This paper studies averaging algebras, say, associative algebras endowed with averaging operators. We develop a cohomology theory for averaging algebras and justify it by interpreting lower degree cohomology groups as formal deformations and abelian extensions of averaging algebras. We make explicit the $L_\infty$-algebra structure over the cochain complex defining cohomology groups and introduce the notion of homotopy averaging algebras as Maurer-Cartan elements of this $L_\infty$-algebra.