Turkish Journal of Mathematics
Article Title
DOI
10.3906/mat-1909-73
Abstract
It is well known that the automorphism group $Aut(H)$ of the algebra of real quaternions $H$ consists entirely of inner automorphisms $i_{q}:p\rightarrow q\cdot p\cdot q^{-1}$ for invertible $q\in H$ and is isomorphic to the group of rotations $SO(3)$. Hence, $H$ has only inner derivations $D=ad(x),$ $x\in H$. See [4] for derivations of various types of quaternions over the reals. Unlike real quaternions, the algebra $H_{d}$ of dual quaternions has no nontrivial inner derivation. Inspired from almost inner derivations for Lie algebras, which were first introduced in [3] in their study of spectral geometry, we introduce coset invariant derivations for dual quaternion algebra being a derivation that simply keeps every dual quaternion in its coset space. We begin with finding conditions for a linear map on $H_{d}$ become a derivation and show that the dual quaternion algebra $% H_{d}$ consists of only central derivations. We also show how a coset invariant central derivation of $H_{d}$ is closely related with its spectrum.
First Page
2113
Last Page
2122
Recommended Citation
KIZIL, EYÜP and ALAGÖZ, YASEMİN
(2020)
"Dual quaternion algebra and its derivations,"
Turkish Journal of Mathematics: Vol. 44:
No.
6, Article 10.
https://doi.org/10.3906/mat-1909-73
Available at:
https://journals.tubitak.gov.tr/math/vol44/iss6/10
