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

TL;DR

This paper develops a comprehensive deformation and homotopy theory for Rota-Baxter algebras of any weight, introducing new algebraic structures and cohomology theories to understand their formal deformations.

Contribution

It defines an $L_$-algebra controlling deformations, develops a cohomology theory, and introduces homotopy Rota-Baxter algebras with a minimal operad model.

Findings

An $L_$-algebra controls simultaneous deformations.

A cohomology theory for Rota-Baxter algebras is established.

Homotopy Rota-Baxter algebras are characterized by a minimal operad model.

Abstract

This paper studies formal deformations and homotopy theory of Rota-Baxter algebras of any weight. We define an $L_\infty$-algebra, which controls simultaneous deformations of associative products and Rota-Baxter operators. As a consequence, we develop a cohomology theory of Rota-Baxter algebras of any weight and justify it by interpreting lower degree cohomology groups as formal deformations and abelian extensions. The notion of homotopy Rota-Baxter algebras is introduced and it is shown that the operad governing homotopy Rota-Baxter algebras is a minimal model of the operad of Rota-Baxter algebras.