Weighted infinitesimal unitary bialgebras on matrix algebras and weighted associative Yang-Baxter equations

TL;DR

This paper develops new algebraic structures on matrix algebras, including weighted infinitesimal unitary bialgebras and solutions to weighted associative Yang-Baxter equations, linking them with Rota-Baxter operators and dendriform algebras.

Contribution

It introduces weighted infinitesimal unitary bialgebras on matrix algebras and explores their connections with Yang-Baxter equations and Rota-Baxter operators, extending prior algebraic frameworks.

Findings

Constructed weighted infinitesimal unitary bialgebras on matrix algebras.

Established a bijection between solutions of weighted AYBEs and Rota-Baxter operators.

Demonstrated that these structures can be endowed with dendriform algebra structures.

Abstract

We equip a matrix algebra with a weighted infinitesimal unitary bialgebraic structure, via a construction of a suitable coproduct. Furthermore, an infinitesimal unitary Hopf algebra, under the view of Aguiar, is constructed on a matrix algebra. By exploring the relationship between weighted infinitesimal bialgebras and pre-Lie algebras, we construct a pre-Lie algebraic structure and then a new Lie algebraic structure on a matrix algebra. We also introduce the weighted associative Yang-Baxter equations (AYBEs) and obtain the relationship between solutions of weighted AYBEs and weighted infinitesimal unitary bialgebras. We give a bijection between the solutions of the associative Yang-Baxter equation of weight $\lambda$ and Rota-Baxter operators of weight $-\lambda$ on matrix algebras. As a consequence, weighted quasitriangular infinitesimal unitary bialgebras are constructed, which generalize the results studied by Aguiar. Finally, We show that any weighted quasitriangular infinitesimal unitary bialgebra can be made into a dendriform algebra.