Generalized Cauchy–Riemann equations in non-identity bases with application to the algebrizability of vector fields

Abstract

We complete the work done by James A. Ward in the mid-twentieth century on a system of partial differential equations that defines an algebra A for which this system is the generalized Cauchy–Riemann equations for the derivative introduced by Sheffers at the end of the nineteenth century with respect to A, which is also known as the Lorch derivative with respect to A, and recently simply called A -differentiability. We get a characterization of finite-dimensional algebras, which are associative commutative with unity.