Totally compatible associative and Lie dialgebras, tridendriform algebras and PostLie algebras

TL;DR

This paper explores the structure of totally compatible dialgebras and Lie dialgebras, establishing their connections to bimodule algebras, Rota-Baxter operators, and PostLie algebras, and constructs free examples.

Contribution

It introduces the concepts of totally compatible dialgebras and Lie dialgebras, linking them to Rota-Baxter operators and PostLie algebras, and constructs free instances.

Findings

Totally compatible dialgebras relate closely to bimodule algebras.

Rota-Baxter operators on these algebras generalize known algebraic structures.

Construction of free totally compatible dialgebras.

Abstract

This paper studies the concepts of a totally compatible dialgebra and a totally compatible Lie dialgebra, defined to be a vector space with two binary operations that satisfy individual and mixed associativity conditions and Lie algebra conditions respectively. We show that totally compatible dialgebras are closely related to bimodule algebras and semi-homomorphisms. More significantly, Rota-Baxter operators on totally compatible dialgebras provide a uniform framework to generalize known results that Rota-Baxter related operators give tridendriform algebras. Free totally compatible dialgebras are constructed. We also show that a Rota-Baxter operator on a totally compatible Lie dialgebra gives rise to a PostLie algebra, generalizing the fact that a Rota-Baxter operator on a Lie algebra gives rise to a PostLie algebra.