Dual quaternion algebra and its derivations

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 -> q . p . q(-1) for invertible q is an element of H and is isomorphic to the group of rotations SO(3). Hence, H has only inner derivations D = ad(x), x is an element of 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.