Pluriassociative and polydendriform algebras

TL;DR

This paper introduces and studies $ ext{γ}$-pluriassociative and $ ext{γ}$-polydendriform algebras, generalizing classical diassociative and dendriform structures using operad theory, with detailed algebraic and combinatorial properties.

Contribution

It develops a new family of algebraic structures parametrized by $ ext{γ}$, generalizing known operads, and provides their presentations, Koszulity, and free objects.

Findings

The operads are Koszul and have explicit presentations.

Hilbert series of the operads are computed.

Constructs free objects in the categories.

Abstract

We introduce, by adopting the point of view and the tools offered by the theory of operads, a generalization on a nonnegative integer parameter $\gamma$ of diassociative algebras of Loday, called $\gamma$-pluriassociative algebras. By Koszul duality of operads, we obtain a generalization of dendriform algebras, called $\gamma$-polydendriform algebras. In the same manner as dendriform algebras are suitable devices to split associative operations into two parts, $\gamma$-polydendriform algebras seem adapted structures to split associative operations into $2 \gamma$ operations so that some partial sums of these operations are associative. We provide a complete study of the operads governing our generalizations of the diassociative and dendriform operads. Among other, we exhibit several presentations by generators and relations, compute their Hilbert series, show that they are Koszul, and construct free objects in the corresponding categories. We also provide consistent generalizations on a nonnegative integer of the duplicial, triassociative and tridendriform operads, and of some operads of the operadic butterfly.