Free quadri-algebras and dual quadri-algebras

TL;DR

This paper explores the structure of quadri-algebras and dual quadri-algebras, providing explicit constructions, combinatorial descriptions, and connections to Hopf algebras, thereby advancing the understanding of these algebraic systems.

Contribution

It proves a conjecture relating free quadri-algebras to the Hopf algebra of permutations and describes the operad construction from dendriform algebras.

Findings

Free quadri-algebra on one generator is a subobject of FQSym.

Operad of quadri-algebras derived from dendriform operad via Manin products.

Introduces quadri-bialgebra with applications to FQSym and WQSym.

Abstract

We study quadri-algebras and dual quadri-algebras. We describe the free quadri-algebra on one generator as a subobject of the Hopf algebra of permutations FQSym, proving a conjecture due to Aguiar and Loday, using that the operad of quadri-algebras can be obtained from the operad of dendriform algebras by both black and white Manin products. We also give a combinatorial description of free dual quadri-algebras. A notion of quadri-bialgebra is also introduced, with applications to the Hopf algebras FQSym and WQSym.