Local, 2-local derivations and biderivations on 3-parameter generalized quaternion

TL;DR

This paper studies the algebraic structure of 3-parameter generalized quaternion algebras, showing that local derivations are derivations, characterizing biderivations, and describing commuting maps and the centroid.

Contribution

It proves that all local and 2-local derivations are derivations and provides a full characterization of biderivations on the algebra.

Findings

Every local and 2-local derivation is a derivation.

Complete characterization of biderivations.

Description of commuting maps and centroid.

Abstract

This article investigates the recently introduced three-parameter generalized quaternion algebra (3PGQ), denoted here as $\mathbb{K}_{\lambda_1,\lambda_2,\lambda_3}$ . Our analysis is structured in three parts. First, we demonstrate that every local and 2-local derivation on this algebra is automatically a derivation. Second, we provide a complete characterization of its biderivations. Finally, we describe its commuting maps and centroid.