Relative Rota-Baxter operators, modules and projections

TL;DR

This paper introduces modules over relative Rota-Baxter operators in braided monoidal categories, establishes categorical equivalences with Hopf braces, and explores projections that induce modules under certain conditions.

Contribution

It defines modules over relative Rota-Baxter operators, proves an adjunction and equivalence with Hopf brace modules, and introduces strong projections that generate modules in cocommutative contexts.

Findings

Existence of an adjunction between module categories of invertible Rota-Baxter operators and Hopf braces.

Definition of projections between Rota-Baxter operators and conditions for inducing modules.

Establishment of a module structure from strong projections in cocommutative settings.

Abstract

The present article is devoted to introduce, in a braided monoidal setting, the notion of module over a relative Rota-Baxter operator. It is proved that there exists an adjunction between the category of modules associated to an invertible relative Rota-Baxter operator and the category of modules associated to a Hopf brace, which induces an equivalence by assuming certain additional hypothesis. Moreover, the notion of projection between relative Rota-Baxter operators is defined, and it is proved that those which are called ``strong'' give rise to a module according to the previous definition in the cocommutative setting.