Twisted algebras and Rota-Baxter type operators

TL;DR

This paper introduces weak pseudotwistors in monoidal categories, generalizing previous concepts to include Rota-Baxter operators and related twisted algebras, and establishes an equivalence relation among these algebras.

Contribution

It defines weak pseudotwistors, broadening the class of twisted algebras to include Rota-Baxter and similar operators, and introduces twist equivalence among algebras.

Findings

Generalizes pseudotwistors to weak pseudotwistors

Includes Rota-Baxter operators within the framework

Establishes twist equivalence relation for algebras

Abstract

We define the concept of weak pseudotwistor for an algebra $(A, \mu)$ in a monoidal category $\mathcal{C}$, as a morphism $T:A\otimes A\rightarrow A\otimes A$ in $\mathcal{C}$, satisfying some axioms ensuring that $(A, \mu \circ T)$ is also an algebra in $\mathcal{C}$. This concept generalizes the previous proposal called pseudotwistor and covers a number of exemples of twisted algebras that cannot be covered by pseudotwistors, mainly examples provided by Rota-Baxter operators and some of their relatives (such as Leroux's TD-operators and Reynolds operators). By using weak pseudotwistors, we introduce an equivalence relation (called "twist equivalence") for algebras in a given monoidal category.