Bimodules over relative Rota-Baxter algebras and cohomologies

TL;DR

This paper introduces bimodules and cohomology theories for relative Rota-Baxter algebras, connecting them with dendriform algebras and exploring their extensions and homotopy classifications.

Contribution

It develops the theory of bimodules and cohomology for relative Rota-Baxter algebras, extending the understanding of their algebraic and homotopical structures.

Findings

Defined cohomology for relative Rota-Baxter algebras with bimodule coefficients

Studied abelian extensions via second cohomology group

Classified skeletal homotopy relative Rota-Baxter algebras

Abstract

A relative Rota-Baxter algebra is a generalization of a Rota-Baxter algebra. Relative Rota-Baxter algebras are closely related to dendriform algebras. In this paper, we introduce bimodules over a relative Rota-Baxter algebra that fits with the representations of dendriform algebras. We define the cohomology of a relative Rota-Baxter algebra with coefficients in a bimodule and then study abelian extensions of relative Rota-Baxter algebras in terms of the second cohomology group. Finally, we consider homotopy relative Rota-Baxter algebras and classify skeletal homotopy relative Rota-Baxter algebras in terms of the above-defined cohomology.