An associative analogy of Lie H-pseudobialgebra

TL;DR

This paper introduces the concept of infinitesimal H-pseudobialgebras as an associative analogue of Lie H-pseudobialgebras, explores their properties, and links them to the associative Yang-Baxter equation and classical Yang-Baxter equation.

Contribution

It defines infinitesimal H-pseudobialgebras, studies their properties, and establishes connections with the associative and classical Yang-Baxter equations.

Findings

Defined infinitesimal H-pseudobialgebra and studied its properties.

Established the associative Yang-Baxter equation over an H-pseudoalgebra.

Linked infinitesimal H-pseudobialgebras with Lie H-pseudobialgebras and classical Yang-Baxter equation.

Abstract

The purpose of this paper is to study infinitesimal H-pseudobialgebra, which is an associative analogy of Lie H-pseudobialgebra. We first define the infinitesimal H-pseudobialgebra and investigate some properties of this new algebraic structure. Then we consider the coboundary infinitesimal H-pseudobialgebra, which is the subclass of infinitesimal H-pseudobialgebra and we obtain the associative Yang-Baxter equation over an associative H-pseudoalgebra. Finally, we found the connection between the (coboundary) infinitesimal H-pseudobialgebra and the (coboundary) Lie H-pseudobialgebra. Meanwhile, the relationship between the associative Yang-Baxter equation and the classical Yang-Baxter equation (over an H-pseudoalgebra) is established.