A function $f: \mathbb{C} \rightarrow \mathbb{C}$ is naturally viewed as mapping $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$. Suppose $f = (u,v)$ is continuously differentiable on $D \subseteq \mathbb{C}$, the the following are equivalent:
- the Cauchy Riemann equations $u_x = v_y$, $v_x=-u_y$ hold on $D$
- $\frac{df}{dz}$ exists on $D$; that is, $f$ is complex-differentiable on $D$
- the differential $df_z$ is complex-linear for each $z \in D$
For example, see pages 50-51 of Theory of Complex Functions by Reinhold Remmert.
Suppose $\mathbb{R}^n$ is given some algebraic structure so that we can multiply vectors. For the sake of discussion say $\mathbb{R}^n = \cal{A}$ and for all $v,w \in \cal{A}$ we have $v*w \in \cal{A}$. We should then define $\cal{A}$-linearity of a real-linear mapping $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ by the additional condition $T(v * w) = v*T(w)$ for all $v,w \in \mathbb{R}^n$. When we consider the matrix $B$ inducing $T$ then the condition of $\cal{A}$-linearity will impose certain equations (call these $\star$) on the components of $B$. If there is a natural theory of differentiation on $\cal{A}$ space then we ought to have some intrinsic concept of $\cal{A}$-differentiability. Moreover, it ought to be the case that continuous differentiability paired with the $\star$-equations on the Jacobian matrix provide sufficient conditions to insure $\cal{A}$-differentiability. These equations would naturally be termed generalized CR equations with respect to $\cal{A}$.
I'm curious is the concept I outline above has been studied systematically for a wider class of algebraic examples. Of course $\cal{A}=\mathbb{C}$ is interesting, but is this idea useful beyond complex variables?
I can calculate the so-called CR-equations from the $\cal{A}$-linearity of the differential for algebras other than $\mathbb{C}$, but for what $\cal{A}$ is the concept of $d/dz$ meaningful for $z$ an $\cal{A}$-valued variable?
Thank you in advance for any insights.
