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

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.

Preliminary version. comments are welcome!

Cited by in corpus (1)

• Deformations and homotopy theory of Rota-Baxter algebras of any weight