Local, 2-local derivations and biderivations on 3-parameter generalized quaternion
Hassan Oubba
TL;DR
The paper analyzes the three-parameter generalized quaternion algebra $\mathbb{K}_{\lambda_1,\lambda_2,\lambda_3}$ over $\mathbb{R}$, focusing on local/2-local derivations, biderivations, and commuting maps with the centroid. It proves that for $\lambda_3\neq0$ every local and every 2-local derivation is a derivation, and it provides a complete description of all biderivations, showing they are scalar wedges of vector parts. It also characterizes all commuting linear maps and establishes that the centroid is a field, consisting only of scalar maps; the results connect derivations and centroids in this nonassociative setting. The paper discusses the $\lambda_3=0$ case, indicating possible non-derivation local derivations and detailing symmetric/skew-symmetric biderivations, thereby outlining a fuller structure theory for 3PGQ algebras.
Abstract
This article investigates the recently introduced three-parameter generalized quaternion algebra (3PGQ), denoted here as $\mathbb{K}_{λ_1,λ_2,λ_3}$ . 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.