Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 44 (2003), No. 1, 245-262.

Hopf Mappings for Complex Quaternions Johannes Wallner Institut für Geometrie, Technische Universität Wien Wiedner Hauptstr. 8–10/113, A-1040 Wien e-mail: wallner@geometrie.tuwien.ac.at Abstract. The natural mapping of the right quaternion vector space ℍ^{2} onto he quaternion projective line (identified with the four-sphere) can be defined for omplex quaternions ℍ ⊗_{ℝ} ℂ as well. We discuss its exceptional set, the fiber ubspaces, and how the linear automorphism groups of two-dimensional quaternion ector spaces and modules induce groups of projective automorphisms of the image uadrics. 1. Notation Consider the skew field ℍ of quaternions. We will use the usual notation a = a^{0}+ia^{1}+ja^{2}+ka^{3} with real numbers a_{i} and the ‘quaternion units’ i, j, k. The symbol \overline{a} denotes the quaternion a_{0} − ia_{1} − ja_{2} − ka_{3} conjugate to a, and the quaternion norm is denoted by N(a). It has the properties that N(a) = a\overline{a} = \overline{a}a = a_{0}^{2} + a_{1}^{2} + a_{2}^{2} + a_{3}^{2}, and that N(ab) = N(a)N(b). The group of unit quaternions will be identified with the three-sphere S^{3} = {a ∈ ℍ | N(a) = 1}.

If V is a vector space over the field 𝕂, we write rk_{𝕂}, dim_{𝕂}, etc., to indicate the field, if ambiguity is possible. Likewise we write P_{𝕂}(V) for the projective space defined by V. Nonetheless we use ‘standard’ notation to indicate ‘standard’ objects: The real projective space P_{ℝ}(ℍ^{2}) is of dimension seven and will be denoted simply by P^{7}. The subset {(q_{0}, q_{1}) | N(q_{0}) + N(q_{1}) = 1} of ℍ^{2} is called its unit seven-sphere and is denoted by S^{7}. P^{7} arises from S^{7} by identification of antipodal points.

The projective line P_{ℍ}(ℍ^{2}) is the set of one-dimensional right quaternion linear subspaces of ℍ^{2}, and is denoted by P^{1}(ℍ), i.e.,

P^{1}(ℍ) = {(q_{0}, q_{1})ℍ | (q_{0}, q_{1}) ≠ (0, 0)}.  (1)

0138-4821/93 $ 2.50 \begin{matrix}c \\ ⊗ \end{matrix} 2003 Heldermann Verlag