Polarization and deformations of generalized dendriform algebras

TL;DR

This paper extends key results on dendriform algebras to a broader class by introducing polarization, weak Rota-Baxter operators, and exploring their deformations, filtrations, and relations to other algebraic structures.

Contribution

It generalizes Aguiar's results to arbitrary dendriform algebras, introduces polarization and weak Rota-Baxter operators, and explores their implications for algebraic deformations and relations.

Findings

Generalized dendriform algebras via polarization

Deformations and filtrations of dendriform algebras

Connection between infinitesimal bialgebras and pre-Lie algebras

Abstract

We generalize three results of M. Aguiar, which are valid for Loday's dendriform algebras, to arbitrary dendriform algebras, i.e., dendriform algebras associated to algebras satisfying any given set of relations. We define these dendriform algebras using a bimodule property and show how the dendriform relations are easily determined. An important concept which we use is the notion of polarization of an algebra, which we generalize here to (arbitrary) dendriform algebras: it leads to a generalization of two of Aguiar's results, dealing with deformations and filtrations of dendriform algebras. We also introduce weak Rota-Baxter operators for arbitrary algebras, which lead to the construction of generalized dendriform algebras and to a generalization of Aguiar's third result, which provides an interpretation of the natural relation between infinitesimal bialgebras and pre-Lie algebras in terms of dendriform algebras. Throughout the text, we give many examples and show how they are related.