Dual quaternion algebra and its derivations

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Kızıl E., Alagöz Y.

TURKISH JOURNAL OF MATHEMATICS, vol.44, pp.2113-2122, 2020 (SCI-Expanded, Scopus, TRDizin)

  • Publication Type: Article / Article
  • Volume: 44
  • Publication Date: 2020
  • Doi Number: 10.3906/mat-1909-73
  • Journal Name: TURKISH JOURNAL OF MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH, TR DİZİN (ULAKBİM)
  • Page Numbers: pp.2113-2122
  • Open Archive Collection: AVESIS Open Access Collection
  • Yıldız Technical University Affiliated: Yes

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.