Problems in hypercomplex analysis over some matrix hypercomplex systems

Abstract

In this paper the authors emphasize first on the holomorphy and power series analyticity over the non‐division algebra C(1,j) of double‐complex number z+jw, j4 = −1, (or j2 = i,) z,w∈ C, j∉C, and it real cyclic hypercomplex 4‐dimensional system R(1,j,j2,j3).For general algebra with zero divisors (or hypercomplex number system of finite dimension n+1) we are interested of the stratification of the underline n‐dimensional multiplicative ring system (in fact Rn) consisting of all simply connected open subsets of invertible elements. These subsets are called regular components of the mentioned stratification, contrary to the subset of non‐invertible elements called singular. It is to remark that the classical complex analysis is based on the simplest stratification consisting of one strata only—the subset of regular elements C∗C\ {0}. But the function theory over the algebra of the hyperbolic complex numbers (the double‐numbers) is based on 4 different simply connected subsets (4 regular strata). On each such strata one can develop a kind of complex analysis, but if z is singular element, i.e. non‐invertible, one cannot define near to z0 a Loran series for instance.The problem of finding the concrete stratification for a given non‐division algebra seems to be important. Some other problems, more or less of technical character are formulated too.