# Weighted infinitesimal unitary bialgebras, pre-Lie, matrix algebras and polynomial algebras

## TL;DR

This paper introduces weighted infinitesimal unitary bialgebras on matrix and polynomial algebras, establishing new algebraic structures such as Newtonian comatrix coalgebras, pre-Lie, and Lie algebras, expanding the understanding of algebraic frameworks.

## Contribution

It constructs new infinitesimal unitary bialgebras and Hopf algebras on matrix and polynomial algebras, and explores their relationships with pre-Lie and Lie algebra structures.

## Key findings

- Established Newtonian comatrix coalgebra.
- Constructed infinitesimal unitary Hopf algebra on matrix algebra.
- Derived pre-Lie and Lie algebra structures on matrix and polynomial algebras.

## Abstract

Motivated by the classical comatrix coalgebra, we introduce the concept of a Newtonian comatrix coalgebra. We construct an infinitesimal unitary bialgebra on a matrix algebra and a weighted infinitesimal unitary bialgebra on a non-commutative polynomial algebra, via two constructions of suitable coproducts. As a consequence, a Newtonian comatrix coalgebra is established. Furthermore, an infinitesimal unitary Hopf algebra, under the view of Aguiar, is constructed on a matrix algebra. By investigating the relationship between weighted infinitesimal bialgebras and pre-Lie algebras, we erect respectively a pre-Lie algebraic structure and further a new Lie algebraic structure on matrix algebras. Finally, a pre-Lie algebraic structure and a Lie algebraic structure on non-commutative polynomial algebras are also given.