# On quadri-bialgebras

## TL;DR

This paper introduces quadri-bialgebras, a new algebraic structure linked to quadri-algebras, and explores their connections to Manin triples, nondegenerate cocycles, and solutions to the Q-equations, expanding the algebraic framework.

## Contribution

It defines quadri-bialgebras, establishes their equivalence to Manin triples, and connects them to classical Yang-Baxter variations and operator constructions.

## Key findings

- Quadri-bialgebras are equivalent to Manin triples of dendriform and quadri-algebras.
- They relate to nondegenerate 2-cocycles and invariant bilinear forms.
- They originate from a variation of the classical Yang-Baxter equation called Q-equations.

## Abstract

We introduce the notion of a quadri-bialgebra, which gives a bialgebra theory for the quadri-algebra introduced by Aguiar and Loday. We show that a quadri-bialgebra is equivalent to a Manin triple of dendriform algebras associated to a nondegenerate 2-cocycle, and to a Manin triple of quadri-algebras associated to a nondegenerate invariant bilinear form. Quadri-bialgebras also come from a variation of the classical Yang-Baxter equation, called the $Q$-equations. Moreover, quadri-bialgebras fit into the framework of construction of Rota-Baxter operators and Nijenhuis operators on the double spaces of quadri-algebras.