Rota-Baxter systems, dendriform algebras and covariant bialgebras

TL;DR

This paper introduces a generalized Rota-Baxter system with two operators, linking it to dendriform algebras and covariant bialgebras, and explores new algebraic structures and solutions.

Contribution

It generalizes Rota-Baxter operators to systems with two operators, establishing their connection to dendriform algebras and introducing covariant bialgebras.

Findings

Rota-Baxter systems are equivalent to certain dendriform algebra structures.

A Rota-Baxter system induces a weak pseudotwistor leading to a new associative product.

Solutions are constructed from quasitriangular covariant bialgebras.

Abstract

A generalisation of the notion of a Rota-Baxter operator is proposed. This generalisation consists of two operators acting on an associative algebra and satisfying equations similar to the Rota-Baxter equation. Rota-Baxter operators of any weights and twisted Rota-Baxter operators are solutions of the proposed system. It is shown that dendriform algebra structures of a particular kind are equivalent to Rota-Baxter systems. It is shown further that a Rota-Baxter system induces a weak peudotwistor [F. Panaite & F. Van Oystaeyen, Twisted algebras, twisted bialgebras and Rota-Baxter operators, arXiv:1502.05327 (2015)] which can be held responsible for the existence of a new associative product on the underlying algebra. Examples of solutions of Rota-Baxter systems are obtained from quasitriangular covariant bialgebras hereby introduced as a natural extension of infinitesimal bialgebras [M. Aguiar, Infinitesimal Hopf algebras, [in:] New trends in Hopf algebra theory (La Falda, 1999), Contemp. Math., 267, Amer. Math. Soc., Providence, RI, (2000), pp. 1-29].