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

[For the entire collection see Zbl 0561.00010.] \par The author gives an introduction to the use of Clifford numbers for the study of Möbius transformations in ${\bbfR}\sp n$. Let $i\sb 0,i\sb 1,...,i\sb{n-1}$ be a base of ${\bbfR}\sp n$ and let $C\sb{n-1}$ be the Clifford algebra generated by the rules $i\sp 2\sb h=-1$, $i\sb hi\sb k=- i\sb ki\sb h$, $i\sb 0$ unit element, over ${\bbfR}$. Every vector $x=\sum\sp{n-1}\sb{k=0}x\sb ki\sb k$ is invertible with $x\sp{-1}=\vert x\vert\sp{-2}\bar x$, $\vert x\vert$ being the Euclidean norm and $\bar i\sb h=-i\sb h$ for $h=1,...,n-1$, $\bar i\sb 0=i\sb 0$. The products of vectors different from zero form the Clifford group $\Gamma\sb n$. Another conjugation "*" in $C\sb{n-1}$ is generated by reversing the order of the factors in each $i\sb{h\sb 1}...i\sb{h\sb p}.$ \par The main theorem shows that $g(x)=(ax+b)(cx+d)\sp{-1}$ induces a bijective mapping of ${\hat {\bbfR}}\sp n$ onto ${\hat {\bbfR}}\sp n$ if and only if $g=\pmatrix a&b \\ c&d \endpmatrix$ belongs to $GL(\Gamma\sb n)$, which means if (i) $a,b,c,d\in \Gamma\sb n\cup \{0\}$, (ii) $\Delta (g)=ad\sp*-bc\sp*\in {\bbfR}\setminus \{0\}$, (iii) $ab\sp*$, $cd\sp*$, $c\sp*a$, $d\sp*b\in {\bbfR}\sp n$. Further theorems in this direction are given, a very interesting and clear paper. The author has found first steps in a paper by {\it K. Th. Vahlen} [Math. Ann. 55, 585-593 (1902)].
[K.Habetha]

MSC 2000:: *30G35 Functions of hypercomplex variables and generalized variables
30C35 General theory of conformal mappings
15A66 Clifford algebras
16Kxx Division rings and semi-simple Artin rings

Keywords: Clifford numbers; Möbius transformations; Clifford algebra; Clifford group

Citations: Zbl 0561.00010

Cited in: Zbl 1020.37009 Zbl 0793.15019 Zbl 0673.20022 Zbl 0597.30062

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