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Turkish Journal of Mathematics

DOI

10.55730/1300-0098.3474

Abstract

Let $Q=(\frac{a,b}{{\Bbb R}})$ denote the quaternion algebra over the reals which is by the Frobenius Theorem either split or the division algebra $H$ of Hamilton's quaternions. We have presented explicitly in \cite{Kizil-Alagoz} the matrix of a typical derivation of $Q$. Given a derivation $d\in Der(H)$, we show that the matrix $D\in M_{3}({\Bbb R})$ that represents $d$ on the linear subspace $% H_{0}\simeq {\Bbb R}^{3}$ of pure quaternions provides a pair of idempotent matrices $AdjD$ and $-D^{2}$ that correspond bijectively to the involutary matrix $\Sigma $ of a quaternion involution $\sigma $ and present several equations involving these matrices. In particular, we deal with commuting derivations of $H$ and introduce some results to guarantee commutativity. We also mention briefly eigenspace decomposition of a derivation.

Keywords

Derivation, quaternion, involution, automorphism

First Page

1944

Last Page

1954

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