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Möbius transformations and Clifford numbers. (English) Zbl 0569.30040
Differential geometry and complex analysis, Vol. dedic. H. E. Rauch, 65-73 (1985).

[For the entire collection see Zbl 0561.00010.]

The author gives an introduction to the use of Clifford numbers for the study of Möbius transformations in ${ℝ}^{n}$. Let ${i}_{0},{i}_{1},···,{i}_{n-1}$ be a base of ${ℝ}^{n}$ and let ${C}_{n-1}$ be the Clifford algebra generated by the rules ${i}_{h}^{2}=-1$, ${i}_{h}{i}_{k}=-{i}_{k}{i}_{h}$, ${i}_{0}$ unit element, over $ℝ$. Every vector $x={\sum }_{k=0}^{n-1}{x}_{k}{i}_{k}$ is invertible with ${x}^{-1}={|x|}^{-2}\overline{x}$, $|x|$ being the Euclidean norm and ${\overline{ı}}_{h}=-{i}_{h}$ for $h=1,···,n-1$, ${\overline{ı}}_{0}={i}_{0}$. The products of vectors different from zero form the Clifford group ${{\Gamma }}_{n}$. Another conjugation ”*” in ${C}_{n-1}$ is generated by reversing the order of the factors in each ${i}_{{h}_{1}}···{i}_{{h}_{p}}·$

The main theorem shows that $g\left(x\right)=\left(ax+b\right){\left(cx+d\right)}^{-1}$ induces a bijective mapping of ${\stackrel{^}{ℝ}}^{n}$ onto ${\stackrel{^}{ℝ}}^{n}$ if and only if $g=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ belongs to $GL\left({{\Gamma }}_{n}\right)$, which means if (i) $a,b,c,d\in {{\Gamma }}_{n}\cup \left\{0\right\}$, (ii) ${\Delta }\left(g\right)=a{d}^{*}-b{c}^{*}\in ℝ\setminus \left\{0\right\}$, (iii) $a{b}^{*}$, $c{d}^{*}$, ${c}^{*}a$, ${d}^{*}b\in {ℝ}^{n}$. Further theorems in this direction are given, a very interesting and clear paper. The author has found first steps in a paper by K. Th. Vahlen [Math. Ann. 55, 585-593 (1902)].

Reviewer: K.Habetha

##### MSC:
 30G35 Functions of hypercomplex variables and generalized variables 30C35 General theory of conformal mappings 15A66 Clifford algebras, spinors 16Kxx Division rings and semi-simple Artin rings