Twisted algebras and Rota–Baxter type operators

Abstract

We define the concept of weak pseudotwistor for an algebra [Formula: see text] in a monoidal category [Formula: see text], as a morphism [Formula: see text] in [Formula: see text], satisfying some axioms ensuring that [Formula: see text] is also an algebra in [Formula: see text]. This concept generalizes the previous proposal called pseudotwistor and covers a number of examples 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.