On Dualities Generated by the Generalised Hurwitz Problem

In a recent paper [8] two of the present authors showed that the pseudo-euclidean Hurwitz pairs of real vector spaces, introduced in [10] as a geometrical realization of solutions of the generalised Hurwitz problem (involving scalar products instead of norms), generate a duality between the pairs described by the generalised Fueter equations of different types. The duality was expressed via one-to-one type-reversing and type-preserving transformations of the pairs. In this paper we give an additional motivation for the duality pairing, characterizing two geometrical realizations of the real Clifford algebra C (r.s) in terms of pseudoeuclidean Hurwitz pairs. The symmetries of both kinds of dualities (type-reversing and type-preserving) are studied in detail and the correspondences between indices are represented by values of explicitly given quasiregular and quasimeromorphic functions. The type-preserving transformation can be chosen as a linear transformation, which is shown to be an isometry if and only if the pseudo-euclidean Hurwitz pairs involved are isomorphic.