O-operators on associative algebras and dendriform algebras

TL;DR

This paper introduces a generalization of Rota-Baxter operators called O-operators on associative algebras, establishing their role in constructing and classifying dendriform dialgebras and trialgebras.

Contribution

It extends the construction of dendriform structures from Rota-Baxter algebras to O-operators, providing a complete classification framework.

Findings

All dendriform dialgebras and trialgebras can be constructed from O-operators.

There are bijections between equivalence classes of invertible O-operators and dendriform algebras.

The framework generalizes classical constructions related to the Yang-Baxter equation.

Abstract

An O-operator is a relative version of a Rota-Baxter operator and, in the Lie algebra context, is originated from the operator form of the classical Yang-Baxter equation. We generalize the well-known construction of dendriform dialgebras and trialgebras from Rota-Baxter algebras to a construction from O-operators. We then show that this construction from O-operators gives all dendriform dialgebras and trialgebras. Furthermore there are bijections between certain equivalence classes of invertible O-operators and certain equivalence classes of dendriform dialgebras and trialgebras.