Holomorphic synthesis of monogenic functions of several quaternionic variables

Abstract

The system of differential equations for polymonogenic functions of several quaternionic variables is an analogue of the\(\bar \partial \)-equation in complex analysis. We give a representation of polymonogenic functions by means of integration of a family of σ-holomorphic functions as σ runs over the variety Σ of all complex structures ℍ ≅ ℂ2 which are consistent with the metric and an orientation in ℍ. The variety Σ is isomorphic to the manifold of all proper right ideals in the complexified quaternionic algebra and has a natural complex analytic structure. We construct a\(\tilde \partial \)-complex on Σ that provides a resolvennt for the sheaf of polymonogenic functions.