Orientation preserving Möbius transformations in [formula omitted] and quaternionic determinants.

This paper explores M ( R ∞ 4 ) , the group of orientation preserving Möbius transformations acting in R ∞ 4 . On the one hand M ( R ∞ 4 ) is given by the group of 2 × 2 matrices over the quaternions H with determinant D derived from the corresponding 4 × 4 matrices over the complex numbers C . On the other hand we know that, in general, M ( R ∞ n ) may be given in terms of 2 × 2 matrices over the Clifford algebra C n with n − 1 generators. Thus when n = 4 , M ( R ∞ 4 ) is given in terms of 2 × 2 matrices with entries drawn from C 4 and determinant Δ defined in terms of the entries of the given matrix. We note that the skew field H may be considered as a Clifford algebra C 3 based on two generators i and j or more generally i 1 and i 2 where i j = k or i 1 i 2 = k , while the set of elements { 1 , i , j , k } form a basis of H regarded as a 4-dimensional real vector space. Thus H is embedded in C 4 . In the paper we reconcile the two representations of M ( R ∞ 4 ) by comparing the generating sets of the underlying groups of matrices. A relationship between a determinants D and Δ is also exposed. [ABSTRACT FROM AUTHOR]