[Paper Review] Ennea-algebras

Philippe Leroux|arXiv (Cornell University)|Sep 12, 2003

Advanced Topics in Algebra30 citations

TL;DR

This paper introduces Ennea-algebras, a generalization of Quadri-algebras, and demonstrates that the augmented free Ennea-algebra forms a connected Hopf algebra. Using Baxter operators, the authors construct explicit formal deformations of dendriform dialgebras, quadri-algebras, and Ennea-algebras, extending recent work in nonassociative algebraic structures.

ABSTRACT

We propose a generalisation of a recent work of M. Aguiar and J.-L. Loday on Quadri-algebras, called Ennea-algebras. In this second version, this paper has been extended. We show that the augmented free Ennea-algebra is a connected Hopf algebra and construct explicit formal deformations of dendriform dialgebras, quari-algebras and ennea-algebras via Baxter operators.

Motivation & Objective

• To generalize the concept of Quadri-algebras into a broader algebraic framework, introducing Ennea-algebras.
• To establish that the augmented free Ennea-algebra is a connected Hopf algebra.
• To develop formal deformation theory for dendriform dialgebras, quadri-algebras, and Ennea-algebras using Baxter operators.
• To extend recent results by Aguiar and Loday on algebraic structures with multiple operations.

Proposed method

• Generalize the algebraic structure of Quadri-algebras to define Ennea-algebras through a set of nine operations.
• Construct the augmented free Ennea-algebra and prove its connectedness and Hopf algebra structure.
• Apply Baxter operators to generate formal deformations of dendriform dialgebras, quadri-algebras, and Ennea-algebras.
• Use the properties of Baxter operators to ensure compatibility with the algebraic operations and deformation parameters.
• Leverage the Hopf algebra structure to analyze deformation functors and cohomological properties.
• Establish a hierarchy of algebraic structures by extending known results from smaller algebras to Ennea-algebras.

Experimental results

Research questions

• RQ1 How can the concept of Quadri-algebras be generalized to include more algebraic operations, leading to a new class of algebras called Ennea-algebras?
• RQ2 Is the augmented free Ennea-algebra naturally endowed with a connected Hopf algebra structure?
• RQ3 Can formal deformations of dendriform dialgebras be constructed using Baxter operators?
• RQ4 Can the deformation framework be extended to quadri-algebras and Ennea-algebras?
• RQ5 What is the role of Baxter operators in unifying deformation theories across these generalized algebraic structures?

Key findings

• The augmented free Ennea-algebra is proven to be a connected Hopf algebra, establishing a foundational structural property.
• Explicit formal deformations of dendriform dialgebras are constructed using Baxter operators.
• Formal deformations of quadri-algebras are obtained through the same operator-theoretic framework.
• The deformation mechanism via Baxter operators is extended to Ennea-algebras, unifying the approach across multiple algebraic systems.
• The construction demonstrates compatibility between the Hopf algebra structure and deformation theory in the context of nonassociative algebras.
• The results generalize and extend previous findings by Aguiar and Loday on Quadri-algebras and related structures.