Pseudocomplex and pseudo(bi)quaternion mappings. (English. Russian original) Zbl 1085.14004

Gorinov, A. G.

Funct. Anal. Appl. 38, No. 2, 149-150 (2004); translation from Funkts. Anal. Prilozh. 38, No. 2, 84-85 (2004).

A pseudobiquaternion homeomorphism of \(\mathbb{H}^n\) is a homeomorphism that takes any left or right affine quaternion lines to a left or right affine quaternion line (a left line can be taken to a right one, and vice versa).
It is proven that for a pseudobiquaternion homeomorphism \(f:\mathbb{H}^n\to \mathbb{H}^n\), \(n\geq 2\), there exist \(q_1,q_2\in \mathbb{H}\), \(v\in\mathbb{H}^n\) and a real nonsingular \(n\times n\) matrix \(A\) such that \(f\) or its quaternion conjugation \(\overline f\) is of the form \[ x\mapsto q_1 Axq_2+ v. \] The main idea of the proof is to consider the continuous mapping \[ F: A^l(\mathbb{H}^n)\cup A^r(\mathbb{H}^n)\to A^l(\mathbb{H}^n)\cup A^r(\mathbb{H}^n) \] induced by \(f\), where \(A^l(\mathbb{H}^n)\) and \(A^r(\mathbb{H}^n)\) are the space of all left and right, respectively, affine quaternion lines in \(\mathbb{H}^n\). By this idea, the author reduces the proof to the known case of \(\mathbb{H}\)-linear isomorphism.

Reviewer: Marek S. Karaś (Kraków)

MSC:

14A25	Elementary questions in algebraic geometry
14R99	Affine geometry

Keywords:

pseudoquaternion homeomorphism; fundamental theorem of projective geometry