Twisted algebras and Rota-Baxter type operators

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.Comment: 15 pages; continues arXiv:math/0605086 and arXiv:0801.2055, some concepts from these papers are recalled; we added a Note and some references. In this final version, accepted for publication in J. Algebra Appl., the title has been slighty modified and few little things have been adde