Classification, $\alpha$-Inner Derivations and $\alpha$-Centroids of Finite-Dimensional Complex Hom-Trialgebras

TL;DR

This paper investigates the structure and classification of finite-dimensional Hom-associative trialgebras, focusing on inner derivations and centroids, and provides explicit computations and properties for low-dimensional cases.

Contribution

It introduces a classification of low-dimensional Hom-associative trialgebras and analyzes their inner derivations and centroids, expanding understanding of their algebraic structure.

Findings

Classification of 2- and 3-dimensional Hom-associative trialgebras

Properties of inner derivations and centroids established

Explicit computations of inner derivations and centroids provided

Abstract

In the current research work, our basic objective is to investigate the stucture of Hom-associative trialgebras. Next, we build up one important class of Hom-associative trialgebras and provide properties of right, left and meddle operations in Hom-associative trialgebras. Furthermore, we describe the classification of $n$-dimensional Hom-associative trialgebras for $n\leq 3$. Additionally, the properties of the Inner-derivations and centroids are identified and discussed. Eventually the Inner-derivations and centroids are computed.