Compatible associative bialgebras

TL;DR

This paper introduces a new operad related to compatible associative algebras, proves structural properties of free compatible associative algebras, and connects these concepts to Hopf algebras of paths, expanding algebraic frameworks.

Contribution

It constructs a non-symmetric operad with Catalan number dimensions, establishes a compatible infinitesimal bialgebra structure on free compatible associative algebras, and links these to Hopf algebras of paths via bi-matching dialgebras.

Findings

The operad $ abla$ has dimensions given by Catalan numbers.

Free compatible associative algebras admit compatible infinitesimal bialgebra structures.

The Hopf algebra of paths $P(S)$ is a bi-matching dialgebra.

Abstract

We introduce a non-symmetric operad $\mathcal{N}$, whose dimension in degree $n$ is given by the Catalan number $c_{n-1}$. It arises naturally in the study of coalgebra structures defined on compatible associative algebras. We prove that any free compatible associative algebra admits a compatible infinitesimal bialgebra structure, whose subspace of primitive elements is a $\mathcal{N}$-algebra. The data $({\rm As},{\rm As}^2, \mathcal{N})$ is a good triple of operads, in J.-L. Loday's sense. Our construction induces another triple of operads $({\rm As},{\rm As}_2,{\rm As})$, where ${\rm As}_2$ is the operad of matching dialgebras. Motivated by A. Goncharov's Hopf algebra of paths $P(S)$, we introduce the notion of bi-matching dialgebras and show that the Hopf algebra $P(S)$ is a bi-matching dialgebras.