Compatible associative bialgebras

ABSTRACT

We introduce a non-symmetric operad 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 𝒩-algebra. The data (As,As²,𝒩) is a good triple of operads, in J.-L. Loday’s sense. Our construction induces another triple of operads (As,As₂,As), where As₂ 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.

KEYWORDS:

• Compatible associative algebras
• infinitesimal bialgebras
• triples of operads

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16T10

Acknowledgment

I would like to express my thanks to Prof. M. Ronco for motivating me to work on this problem and for her constant contributions to it, and to Prof. A. Labra for many useful comments and for encouraging me to continue my research work. My special thanks to the University of Talca for the support provided during this period.