Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

ON THE CLASSIFICATION OF FOUR-DIMENSIONAL MÖBIUS TRANSFORMATIONS

Wensheng Caoa1a2

a1 School of Mathematical Sciences, Peking University, Beijing 100871, China

a2 Institute of Mathematics, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China (wenscao@yahoo.com.cn)

Abstract

Based on the identification of four-dimensional Möbius transformations

$$ g(x)=(ax+b)(cx+d)^{-1} $$

by the matrix group $\mathrm{PS}_\triangle L(2,\mathbb{H})$ of quaternionic $2\times2$ matrices with Dieudonné determinant equal to $1$, we give an explicit expression for the classification of $g$ in terms of $a$, $b$, $c$ and $d$.

(Online publication February 09 2007)

(Received March 24 2005)

Keywords

  • Primary 30F40;
  • 22E40;
  • 20H10;
  • quaternion;
  • four-dimensional Möbius transformations;
  • classification