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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 n . Let i 0 ,i 1 ,···,i n-1 be a base of 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 . Every vector x= k=0 n-1 x k i k is invertible with x -1 =|x| -2 x ¯, |x| being the Euclidean norm and ı ¯ h =-i h for h=1,···,n-1, ı ¯ 0 =i 0 . The products of vectors different from zero form the Clifford group Γ 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(x)=(ax+b)(cx+d) -1 induces a bijective mapping of ^ n onto ^ n if and only if g=abcd belongs to GL(Γ n ), which means if (i) a,b,c,dΓ n {0}, (ii) Δ(g)=ad * -bc * {0}, (iii) ab * , cd * , c * a, d * b 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:
30G35Functions of hypercomplex variables and generalized variables
30C35General theory of conformal mappings
15A66Clifford algebras, spinors
16KxxDivision rings and semi-simple Artin rings