I have tried searching for something similar to what is described below, but to no avail. It would be great if somebody could show some right references, where this has been done, or explain why such approach is bound to fail.

## Motivation ##

Let's consider algebra $gl_n$ with ''bosonic'' generators $B_{ij}$:
$$[B_{ij},B_{kl}]=\delta_{jk}B_{il}-\delta_{li}B_{jk}$$
They can be realized as bosonic differential operators $B_{ij}=x_i\partial_j$ in commuting variables $x_i x_j = x_j x_i$ 

There is a known way to extend this construction to the so-called supersymmetric ($\mathbb{Z}_2$) case by adding ''fermionic'' generators $F_{rs}$ with grading $\exp(\frac{2\pi i}{2})$, which can be realized with the help of Grassmanian variables $\psi_i$, $\psi_i \psi_j = - \psi_j \psi_i$. These generators satisfy similar relations, but commutation law $[,]$ is turned into anticommutation law $\{,\}$. The pattern is $[B,B]\sim B$, $\{F,F\}\sim B$, $[B,F]\sim F$.

Now I want to generalize it to $\mathbb{Z}_3$ case. 

## My approach ##
consists in introducing another set of generators $C$ and reassigning the gradings: $|B| = 1$, $|F| = \exp(\frac{2\pi i}{3})$, $|C| = \exp(\frac{4\pi i}{3})$ so that the following pattern will take place: $[B,B]\sim B$, $[B,F]\sim F$, $[B,C]\sim C$, $[F,F]\sim C$, $[F,C]\sim B$, $[C,C]\sim F$. Here instead of usual commutators I mean suitably redefined $\mathbb{Z}_3$-graded commutators.

## The problem ##

is that it doesn't work. Indeed, in order to define commutators in this way I need for elements to have commutation relations with cubic roots of $1$. But it isn't possible because if all my variables are similar to each other with the only exception of grading, then, for example, $\psi c = f(\psi,c) c \psi = f(\psi,c) f(c,\psi) \psi c$. Hence, since $f(\psi,c) = f(c,\psi)$ this factor must be a quadratic root of $1$.

The only way out that I could think of was introducing more (I started with two) sets variables which kind of live on a quantum plane: $x_i^{1} x_j^{2} = q x_j^{2} x_i^{1}$. I could write some algebra, but it looked very messy and is probably wrong. 

## The question ##

is whether similar constructions of $\mathbb{Z}_n$ supersymmetry exist in literature or are easy to exclude on some general grounds.

## References where people do something else ##

There's something called ["color Lie algebras"](https://arxiv.org/abs/math/0407165v5), but I haven'r been able to fully digest their ideas.

Richard Kerner et. al. have [considered](https://arxiv.org/abs/hep-th/9607143) a similar problem and solved it by embracing the world of ternary structures. Is there any hope to stay binary in this brave new world?