I am looking for a reference, if it exists, to the study of differential equations defined using Fox derivatives over the group ring, say, of a free group. Is this a topic which has been studied before, and are there any results.
I'll not strive to be as general as possible, but here is an example of the sorts of things I mean.
Fox derivatives
Let $F_n$ be the free group on $\{x_1,\dots,x_n\}$, then $\mathbb{Z}F_n$ is the set of sums $\sum_{g\in F_n}a_gg$ where $a_g\in\mathbb{Z}$, and all but finitely many of the coefficients are $0$. This set can be given the structure of a ring by defining addition linearly, and (non-commutative multiplication) in a the same way multiplication of polynomials is defined. This ring has an identity $1e_{F_n}$ which we can abbreviate to $1$. Given $\alpha=\sum_{g\in F_n}a_gg$, we define $\varepsilon(f)=\sum_{g\in F_n}a_g$ to be the coefficient sum.
A Fox derivative is a linear function $D:\mathbb{Z}F_n\to\mathbb{Z}F_n$ which satisfies a version of the Leibnitz rule: $$D(\alpha\beta)=D(\alpha)\varepsilon(\beta)+\alpha D(\beta).$$ These derivatives turn out to have several surprising parallels to the usual derivative of real functions (and some key differences, of course). There are some simple examples of Fox derivatives are as follows. Fix a generator $x_i$ and define $$\partial_i(x_j)=\delta_{ij}$$ where $\delta_{ij}$ is the Kronecker delta. This extends uniquely to a Fox derivative on $\mathbb{Z}F_n$, and in a sense, all Fox derivatives can be built out of these simple ones.
Differential equations
It is now possible to give equations involving these Fox derivatives, and ask whether they have solutions over $\mathbb{Z}F_n$. Some simple examples might be:
- $\partial_i\alpha=1$ which has $\alpha=x_i$ as a solution.
- $\partial_i\alpha=\alpha$ which has no non-trivial solutions ($\alpha=0$ is a solution), this takes a small argument to see.
- $\partial_i\alpha=-\alpha$ which has $\alpha=\beta x_i^{-1}$ where $\beta$ is any element of $F_n$ whose reduced word representation does not involve $x_i$.
Has anyone studied equations like this in the past?