The Rota-Baxter algebra structures on split semi-quaternion algebra

TL;DR

This paper classifies all Rota-Baxter operators on split semi-quaternion algebra, providing matrix characterizations and linking to known results on Sweedler algebra, thus advancing understanding of algebraic operator structures.

Contribution

It offers a complete classification and matrix representation of Rota-Baxter operators on split semi-quaternion algebra, extending previous work on Sweedler algebra.

Findings

All Rota-Baxter operators characterized by matrices.

Matrix representations explicitly constructed.

Connections to existing results on Sweedler algebra established.

Abstract

In this paper, we shall describe all the Rota-Baxter operators with any weight on split semi-quaternion algebra. Firstly, we give the matrix characterization of the Rota-Baxter operator on split semi-quaternion algebra. Then we give the corresponding matrix representations of all the Rota-Baxter operators with any weight on split semi-quaternion algebra. Finally, we shall prove that the Ma et al. results about the Rota-Baxter operators on Sweedler algebra are just special cases of our results.