Deformations and homotopy theory for Rota-Baxter family algebras

TL;DR

This paper develops a homotopy-theoretic framework for Rota-Baxter family algebras, including cohomology, deformations, and homotopy structures, linking them to dendriform and $A_$-algebras.

Contribution

It constructs an $L_$-algebra for Rota-Baxter family algebras, defines their cohomology and deformations, and introduces homotopy Rota-Baxter structures on $A_$-algebras.

Findings

Established an $L_$-algebra governing Rota-Baxter family algebra structures.

Defined cohomology theory for Rota-Baxter family algebras.

Explored homotopy Rota-Baxter structures and their relation to homotopy dendriform algebras.

Abstract

The concept of Rota-Baxter family algebra is a generalization of Rota-Baxter algebra. It appears naturally in the algebraic aspects of renormalizations in quantum field theory. Rota-Baxter family algebras are closely related to dendriform family algebras. In this paper, we first construct an $L_\infty$-algebra whose Maurer-Cartan elements correspond to Rota-Baxter family algebra structures. Using this characterization, we define the cohomology of a given Rota-Baxter family algebra. As an application of our cohomology, we study formal and infinitesimal deformations of a given Rota-Baxter family algebra. Next, we define the notion of a homotopy Rota-Baxter family algebra structure on a given $A_\infty$-algebra. We end this paper by considering the homotopy version of dendriform family algebras and their relations with homotopy Rota-Baxter family algebras.