Derivations of generalized quaternion algebra

Abstract

The purpose of this paper is to determine derivations of the algebra H α,β of generalized quaternions over the reals and hence to obtain the algebra Der(H α,β ) of derivations of H α,β . Once we know derivations we might decompose Der(H α,β ) in terms of its inner and/or central derivations whenever they exist. Apart from Der(H α,β ) we would also be able to obtain generalized derivations, which have been studied by analysts in the context of algebras of some normed spaces, and of prime and semiprime rings.

De Moivre’s Theorem for the Matrix Representation of Dual Generalized Quaternions

“…In several studies [5][6][7] , dual quaternions and their applications were considered. In other studies [8][9][10][11][12][13][14] , generalized quaternions and their properties were studied, in which De Moivre's theorem and Euler's formula were obtained for generalized quaternions. In 2020, Kong studied commutative quaternions and split semi quaternions, and De Moivre's theorem was obtained for the matrix representation of split (semi) quaternions [15][16][17] .…”

Dual quaternion algebra and its derivations

“…We have considered derivations of Lie algebras and provided for this purpose a simple computational algorithm in [1]. Recently, we have considered quaternions as a class of Lie algebra and given explicitly in [4] derivations of generalized quaternion algebra over the field of real numbers. In this paper, we consider derivations of the algebra H d of dual quaternions since for such quaternions we have no inner derivations in contrast to classical real quaternions.…”