On the representation theory of Möbius groups inR ^n*

Abstract

We will solve several fundamental problems of Möbius groupsM(R ⁿ) which have been matters of interest such as the conjugate classification, the establishment of a standard form without finding the fixed points and a simple discrimination method.

Let$g = \left[ {\begin{array}{*{20}c} a & b \\ c & d \\ \end{array} } \right]$ be a Clifford matrix of dimensionn, c ≠ 0. We give a complete conjugate classification and prove the following necessary and sufficient conditions:g is f.p.f. (fixed points free) iff$g \sim \left[ {\begin{array}{*{20}c} \alpha & 0 \\ c & {\alpha '} \\ \end{array} } \right]$, |α|<1 and |E−AE ¹| ≠ 0;g is elliptic iff$g \sim \left[ {\begin{array}{*{20}c} \alpha & \beta \\ c & {\alpha '} \\ \end{array} } \right]$, |α| <1 and |E−AE ¹|=0;g is parabolic iff$g \sim \left[ {\begin{array}{*{20}c} \alpha & 0 \\ c & {\alpha '} \\ \end{array} } \right]$, |α|=1; andg is loxodromic iff$g \sim \left[ {\begin{array}{*{20}c} \alpha & \beta \\ c & {\alpha '} \\ \end{array} } \right]$, |α| >1 or rank (E−AE ¹) ≠ rank (E−AE ¹,ac ⁻¹+c ⁻¹ d), where α is represented by the solutions of certain linear algebraic equations and satisfies

$$\left| {c^{ - 1} \alpha '} \right| = \left| {\left( {E - AE^1 } \right)^{ - 1} \left( {\alpha c^{ - 1} + c^{ - 1} \alpha '} \right)} \right|.$$